Weakly Non-Stationary Return Period Analysis of Rainfall Events Modeled by a Poisson Process with Trends

Climate change is acknowledged to affect the occurrence of extreme events and the time-series characteristics of hydrological variables. Consequently, the average statistical properties may vary through time and, as such, they are considered to be non-stationary. For non-stationary processes, the classical theory of extreme values analysis based on the Peak-Over-Threshold (POT) method cannot be applied. In this work, we extend the POT method to weakly non-stationary processes, whose change of statistical properties occurs on a timescale longer than the accounted future timeframe. Accordingly, we derive a mathematical framework to calculate the return period of extreme events, and apply the proposed methodology to the stochastic process of rainfall modelled by a Poisson Process with constant average frequency. We consider a time-varying mean magnitude due to the presence of trends in the time-series. A closed-form solution for the maximum allowed value of the mean magnitude is derived and graphically discussed for several values of the initial conditions, in terms of mean frequency and return period. The results are readily applicable to various applications, particularly to the design of hydraulic structures based on the return period of overthreshold events.