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The Relation between the Mean Areal Rainfall and the Fractional Area Where It Rains above a Given Threshold

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  • 1 Laboraioire d'Etude des Transferts en Hydrologie et Environnement, Domaine Universitaire, Grenoble, France
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Abstract

Experimentally, it appears that, over a fixed domain, the mean areal rain rate and the fractional area covered by rain exceeding a fixed threshold are highly correlated. It leads to methods for estimating mean areal rain rates, hereafter called threshold methods, which will be discussed in this paper. 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. The theoretical values are compared with the experimental ones, computed on a dataset of hourly strong rainfall events in the Cévennes region, a mountainous, Mediterranean area in the southeast of France. The second theoretical expression and its experimental values on the dataset tie in together very well. A validation of a threshold method on this dataset will also be proposed.

Abstract

Experimentally, it appears that, over a fixed domain, the mean areal rain rate and the fractional area covered by rain exceeding a fixed threshold are highly correlated. It leads to methods for estimating mean areal rain rates, hereafter called threshold methods, which will be discussed in this paper. 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. The theoretical values are compared with the experimental ones, computed on a dataset of hourly strong rainfall events in the Cévennes region, a mountainous, Mediterranean area in the southeast of France. The second theoretical expression and its experimental values on the dataset tie in together very well. A validation of a threshold method on this dataset will also be proposed.

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