Budyko-Type Models and the Proportionality Hypothesis in Long-Term Water and Energy Balances

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In the perspective of Darwinian hydrology, Budyko hypotheses can be the foundation of approaches for developing models. Numerous Budyko-type models meeting established boundary conditions (water and energy limits) have been developed based on the Budyko hypothesis on the long-term-average annual mass and energy balance. Some of these models are grounded on empirical bases, while others have been formulated on sophisticated mathematical developments. We analyze the basic hypotheses underlying some Budyko-type models; we first describe some published models and then examine their underlying hypotheses in a hydrologically intuitive space (precipitation versus runoff). The analyses show that the models studied are a consequence of assuming that two parallel straight lines (of unit slope) of different intercepts are indeed equal (proportionality hypothesis). This hypothesis gives rise to different Budyko-type models that, although mathematically correct and meeting the limits (partially) related to the Budyko hypotheses, do not yield any information about what happens between those limits. To overcome the extreme energy limit, an expolinear model is introduced.