Generalizations of Langbein's Formula under Non-Stationarity, Mixed Populations, and Over- or Under-Dispersion in the Number of Exceedances

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.