Technical Note: Analytical Inversion of the Parametric Budyko Equations

Abstract. The non-parametric Budyko framework provides empirical relationships between a catchment's long-term mean evapotranspiration (E) and the aridity index, defined as the ratio of mean rainfall depth (P) to mean potential evapotranspiration (E₀). The parametric Budyko equations attempt to generalize this framework by introducing a catchment-specific parameter (n or w), intended to represent differences in catchment climate and landscape features. Many studies have developed complex regression relationships for the catchment-specific parameter in terms of biophysical features, all of which use known values of P, E₀, and E to numerically invert the parametric Budyko equations to obtain values of n or w. In this study, we analytically invert both forms of the parametric Budyko equations, producing expressions for n and w only in terms of P, E₀, and E. These expressions allow for n and w to be explicitly expressed in terms of biophysical features through the dependence of P, E₀, and E on those same features.