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.