Conceptual rainfall-runoff model with a two-parameter, infinite characteristic time transfer function

A two-parameter transfer function with an infinite characteristic time is proposed for conceptual rainfall-runoff models. The large time behaviour of the unit response is an inverse power function of time. The infinite characteristic time allows long term memory effects to be accounted for. Such effects are observed in mountainous and karst catchments. The governing equation of the model is a fractional differential equation in the limit of long times. Although linear, the proposed transfer function yields discharge signals that can usually be obtained only using non-linear models. The model is applied successfully to two catchments, the Dud Koshi mountainous catchment in the Himalayas and the Durzon karst catchment in France. It compares favourably to the linear, non-linear single reservoir models and to the GR4J model. With a single reservoir and a single transfer function, the model is capable of reproducing hysteretic behaviours identied as typical of long term memory effects. Computational efficiency is enhanced by approximating the infinite characteristic time transfer function with a sum of simpler, exponential transfer functions. This amounts to partitioning the reservoir into several linear subreservoirs, the output discharges of which are easy to compute. An efficient partitioning strategy is presented to facilitate the practical implementation of the model.