Mean hydrograph event from exact ensemble averages of stochastic trajectories

The design of hydraulic structures, flood protection systems, and river management strategies often relies on the peak-flow discharge associated with a specific return period. However, many river processes are related to the shape and duration of the entire flow event. Nevertheless, only a few, mostly empirical, formulations are available in the literature to describe the whole duration of a hydrograph. In this work, a novel framework based on the stochastic Compound Poisson Process approximation of the hydrological regime is proposed, and accordingly, mathematically exact solutions for the duration of the rising and falling limbs of the hydrograph are provided in terms of the initial and peak flow discharges. Then, the proposed relationships are specifically calculated in the case of Nash's hydrograph, and tested against data from rivers with different hydro-morphological characteristics. The framework is readily applicable to model the flow hydrograph in a broad range of time-dependent fluvial processes, including bank erosion, pier scouring, and uprooting of in-channel vegetation, for which an application example is also presented.