A distribution function approach to rainfall runoff modeling

This paper begins with a critique of existing rainfall runoff models and proceeds to a largely new formulation in which the single store (representing, for example, interception of rainfall by vegetation, or retention of water in upper soil layers, or possibly both) is replaced by a statistical population of stores. The consequences of such an assumption are illustrated for the simplest, one‐parameter case in which the distribution of store depths is exponential. It is demonstrated that the use of a population of stores, even with but one parameter, can (1) afford a plausible description of the relation between actual evaporation and soil moisture deficit and (2) remove discontinuities of gradient in the objective function, optimization of which gives estimates of model parameters. The new formulation also permits observed runoff to be written down as a relatively simple function of past rainfall, potential evaporation, and the parameters in the statistical distribution of storages, with the consequence that gradient methods can be used to optimize the objective function in place of more time‐consuming direct search methods. An extension of the model to account for the translation of runoff to the basin outfall is accomplished by using a bivariate distribution of translation times and store depths. A simple recursive equation relating current flow to a proportion of the previous flow and an additive function of rainfall is obtained under the assumption that translation times and store depths are independent and exponentially distributed. More complex models are derived by relaxing the assumption of independence and by considering distributions other than exponential; expressions for two positively skewed density functions, the Weibull and gamma, are obtained. Series and parallel configurations of distribution function models are considered, and the relation of the models' elemental structure to different types of store commonly employed in conceptual modeling is discussed. The new formulation includes, as particular cases, all models based on linear systems theory. Application of the modeling approach to hourly values of flow, rainfall, and evapotranspiration from a number of the Institute of Hydrology's experimental basins results in very good model predictions of flows over the calibration period, with R 2 values above 0.9. However, this level of performance as measured by the R 2 statistic is not maintained over the test period, although quite reasonable predictions of the flood peaks are still obtained. The drop in performance is partly ascribed to the nature of the calibration period during which the basins were ‘wetting up’ after two years of relatively extreme drought. Model performance over the test period is improved by using a more realistic initial condition for the store contents but only at the expense of reduced R 2 values in the calibration period. The need to assess the new model approach in a range of hydrological environments is recognised, especially where evapotranspiration forms an important component of the basin water balance and where the effects of soil moisture deficits on the generation of flood runoff can be expected to be greatest.