Scaling of the network instantaneous response function from basin geomorphology and hydraulic geometry

A challenging problem in basin-scale hydrology is the ability to represent and quantify the nonlinearity of the output hydrograph or response. An important aspect of this challenge is to clearly distinguish among different sources of nonlinearity. To address the previous, a scale-dependent network instantaneous response function (IRF) is obtained by using the theory of transport by travel times, relationships of basin geomorphology, hydraulic geometry, and the inverse Gaussian (IG) distribution. Inverse Gaussian is used to represent the network response because of its connection with diffusion processes. It is shown that both geometric and hydraulic regularity in the stream network, together with the scaling property of IG, can result in the network response being approximately scaling, implying in turn that the peak flow and time to peak of the response are scaling as well. The derived IRF has the advantage over previous formulations of explicitly distinguishing between nonlinearity and scaling. In addition, it provides an explanation for the scaling in terms of well-known hydrogeomorphic relationships. The scaling is found to arise from the dependence of the network length (i.e., the mean path length) and hydraulic geometry, specifically the mean velocity and the cross-sectional area, on the basin size. Moreover, after dividing the network travel times by the mean travel time, a response that is independent of scale is obtained. This response can be seen as analogous to the instantaneous unit hydrograph but representing instead the output associated with a unit travel time. Furthermore, the derived IRF may be useful in design applications as it provides a way for estimating the response across basin sizes by using readily available data sets. For the latter, additional comparison against observed data is necessary to better understand the applicability of the proposed scaling under a variety of climatic and terrain scenarios.