The complementary relationship and generation of the Budyko functions

The Budyko hypothesis states that the ratio of the actual evapotranspiration over precipitation (E/P) is fundamentally related to the ratio of the potential evapotranspiration over precipitation (E-0/P). A number of Budyko functions have been proposed to describe such a relationship between E-0/P and E/P. There is, however, no simple method to generate Budyko functions that meet the water and energy constraints. This study showed analytically that for any Budyko function, the sum of elasticity of evapotranspiration with respect to potential evapotranspiration and that with respect to precipitation is equal to unity. This complementary relationship for sensitivity of evapotranspiration has important implications for evaluating hydrologic impact of change in climate and/or catchment characteristics. More importantly, this study found a function that is monotonically increasing with simple limiting properties. This function can be used to generate numerous valid Budyko functions and can also be used to test the validity of the existing Budyko functions.