Probability Distribution and Risk of the First Occurrence of k Extreme Hydrologic Events

Abstract

Statistical techniques have been developed for assessing structures to confront extreme events such as floods. Among them are probability distribution functions (PDFs). For example, the waiting time of the first flood exceeding the design flood is geometric, while the number of floods exceeding the design flood in n years is binomial. The expected waiting time (EWT) and hydrologic risk of structures are commonly used metrics developed assuming stationarity and independence. And newer techniques include PDFs and project evaluation metrics applicable for nonstationary conditions. This article focuses on first arrival time of kth floods exceeding the design flood and associated metrics. The likelihood of first occurrence of 1, 2, 3, etc. floods exceeding the design flood becomes important, as uncertainties of hydrologic regimes increase due to climatic and anthropogenic drivers. We use stationary and nonstationary negative binomial distribution and develop EWT and risk for the kth event. They generalize the traditional return period and risk that refer to the occurrence of the first event, i.e., k=1. The newer metrics consider the probability of k or more events. We tested by simulation the concepts and derived equations and apply them using annual floods of Assunpink Creek. The results show that the return period and risk functions developed can be helpful in assessing flood related projects.