📝 SummarXiv

Abstract

Since its publication in 1949, Langbein's formula has been applied ubiquitously in both research documents and national guidelines concerning frequency analyses (FAs) of hydrologic extremes. Given a time series of independent peak-over-threshold (POT) events and the corresponding annual maxima (AM) series-defined as the subset of extremes representing the largest event in each year-the formula provides a theoretical relationship between the return period T derived from the AM series and the average recurrence interval ARI from the POT series, for any fixed event magnitude. Despite the minimal assumptions required-specifically, that exceedance counts follow a homogeneous Poisson process-there are real-world situations where the validity of the formula may be compromised. Typical cases include non-stationary processes, mixed-event populations, and over- or under-dispersion in exceedance counts. In this work, we extend Langbein's formula to account for these three cases. We demonstrate that, with appropriate adaptations to the definitions of T and ARI, the traditional functional form of Langbein's relationship remains valid for non-stationary processes and mixed populations. However, accounting for dispersion effects in exceedance counts requires a generalization of Langbein's relationship, of which the traditional version represents a limiting case.