Time-Scale Invariance As an Emergent Property in Water Balance

Abstract:

The Darwinian modeling approach seeks to explain the behavior of a hydrologic system as a whole by identifying simple and robust temporal or spatial patterns that capture the relevant processes. Darwinian-based hydrologic models include the Soil Conservation Service (SCS) curve number model, the “abcd” model, and the Budyko-type models. However, these models were developed based on widely differing principles and assumptions and applied to distinct time scales. Here, we derive a one-parameter Budyko-type model for mean annual water balance which is based on a generalization of the proportionality hypothesis of the SCS model and therefore is independent of temporal scale. Furthermore, we show that the new model is equivalent to the key equation of the “abcd” model. Theoretical lower and upper bounds of the new model are identified and validated based on previous observations. Thus, we illustrate a time-scale invariance property in water balance, which allows for synthesis with the Newtonian approach and offers opportunities for progress in hydrologic modeling.

In the derivation of Budyko equation, total evaporation is divided into initial evaporation (E0) and continuing evaporation. Runoff does not compete with initial evaporation for water storage by interception and top soils. The assumption behind the derived equation is that the ratio of continuing evaporation to its potential value is equal to the ratio of runoff to the maximum possible value of runoff. The derived equation satisfies the boundary conditions of Budyko hypothesis, and includes one parameter (ε). From the perspective of evaporation, is the ratio between initial evaporation and total evaporation; from the soil wetting (W) perspective, e can be interpreted as the ratio between initial evaporation ratio (λ =E0/W) and Horton index (H=E/W), i.e. ε =λ/H. H is dominantly controlled by vegetation represented by NDVI; l is found to increase with decreasing product between NDVI and the fraction of rainy days.