Analytical derivation of the Budyko curve based on rainfall characteristics and a simple evaporation model

The Budyko curve is often used to estimate the actual evaporation as a function of the aridity index in a catchment. Different empirical equations exist to describe this relationship; however, these equations have very limited physical background. The model concept presented in this paper is physically based and uses only measurable parameters. It makes use of two types of evaporation: interception and transpiration. It assumes that interception can be modeled as a threshold process on a daily time scale. If multiplied with the rainfall distribution function, integrated, and multiplied with the expected number of rain days per month, the monthly interception is obtained. In a similar way, the monthly interception can be upscaled to annual interception. Analogous to the interception process, transpiration can be modeled as a threshold process at a monthly time scale and can be upscaled by integration and multiplication with the expected number of rain months. The expected rain days per month are modeled in two ways: as a fixed proportion of the monthly rainfall and as a power function based on Markov properties of rainfall. The latter is solved numerically. It appears that on an annual basis the analytical model does not differ much from the numerical solution. Hence, the analytical model is used and applied on 10 locations in different climates. This paper shows that the empirical Budyko curve can be constructed on the basis of measurable parameters representing evaporation threshold values and the expected number of rain days and rain months and, in addition, a monthly moisture carryover amount for semiarid zones.