Stochastic description of mean flow hydrograph shape for flood modeling and river engineering design purposes

The engineering design of hydraulic infrastructures in either urban or rural environments requires flow hydrograph scenarios to be tested in numerical models. Whilst peak flow discharge is calculated according to a return period analysis of historical data,  the definition of the corresponding hydrograph duration and shape is not unique. The first one is typically considered either equal to the concentration time of the catchment or dependent on rainfall event intensity and duration. The shape of the hydrograph is then defined as a rectangle (e.g., constant flow discharge) or other simple shapes (e.g., triangle) according to empirical rules. In this work, we propose the definition of an average hydrograph shape based on the stochastic analysis of the Compound Poisson Process, which is usually considered as a proxy model for hydrological data in several applications. Once the peak flow discharge value is derived by means of the Peak Over Threshold analysis and a baseflow value is set, we calculate the duration of both the raising and the falling limbs of the hydrograph based on the concept of mean first passage time across thresholds for non-Markovian processes. As this technique considers the ensemble of the infinite possible stochastic trajectories reaching the threshold, it then returns a more comprehensive description of the possible mean shape of the hydrograph. Such a shape can also be approximated by using relationships already available in the literature (e.g., the Yevdjevich function). In definitive, the proposed approach provides more reliable results when the hydraulic processes being modelled (e.g., flow erosion) require that not only the peak but also the shape and the hydrograph duration are important for verification and design purposes.