Return Periods for Continuous-Time Processes in Hydrology: A new Heuristic

The return period is a fundamental concept in hydrological engineering and risk analysis. In this work, we propose a novel formulation of the return period in the context of time-continuous stochastic processes and introduce a new expression to estimate it under the assumptions of a conceptual stochastic rainfall-runoff model. In classical hydrology literature, an extreme event occurs when a random variable is greater or equal than a certain threshold. If a random variable representing a hydrological process is measured every fixed time interval, and assuming that those observations are independent and identically distributed, then the return period of an extreme event is defined as the inverse of its probability of occurrence. This traditional interpretation of the concept relies on a binomial heuristic since the return period is the expected value of a binomial random variable representing the time to the first success in a sequence of Bernoulli trials. However, this approach is not adequate for hydrological modeling because observations of hydrological variables do not necessarily represent independent Bernoulli trials. On the other hand, there are available modern mathematical techniques to formulate questions about large deviations in random processes in terms of sparse random sets. Here, we use the Poisson clumping heuristic to give an interpretation of the return period in the framework of continuous-time Markov processes, extending this concept to events separated by continuous inter-arrival times. We show that the binomial heuristic is a discrete approximation to the Poisson heuristic, and we provide a new expression to estimate the return period of hydrological extreme events through the Rice formula for level crossing counting of a continuous stochastic process. Finally, we replace this formula in the context of a conceptual stochastic rainfall-runoff model to find an expression for the return period of floods.