Fractional Linear Reservoir Model as Elementary Hydrologic Response Function

This paper presents a fractional linear reservoir model as the elementary response function of hydrologic systems corresponding to the classical linear reservoir model and tests its applicability to rainfall-runoff modeling. To this end, we formulate a fractional linear reservoir model in terms of fractional calculus following the same procedure as the classical linear reservoir model and, at the simplest level, compare its performance of rainfall-runoff modeling with the linear and nonlinear reservoir models. The impulse response function of a fractional linear reservoir model, a probability density function (PDF) following the Mittag-Leffler distribution, shows nonlinearity due to its time-variant behavior compared to that of a linear reservoir model. In traditional linear hydrologic system theory, the lag and route version of a fractional linear reservoir model produces the fast-rising and slow-recession of runoff hydrographs, implying the mixed response of linear and nonlinear reservoir models to rainfall. So, a fractional linear reservoir model could be considered a fundamental tool to effectively reflect the nonlinearity of rainfall-runoff phenomena within the framework of the linear hydrologic system theory. In this respect, the fractional order of the storage relationship specifying a fractional linear reservoir model can be viewed as a kind of parameter to quantify the heterogeneity of runoff generation within a river basin.