On the probability of droughts: The compound renewal model

Abstract

Droughts influence the planning and design of water supply infrastructure. Hydrologists ascertain drought duration, severity, and pattern of recurrence from instrumental and reconstructed records (e.g., using tree rings) of streamflow and precipitation. This work introduces a compound renewal model for the probabilistic analysis of multiyear drought recurrence. The compound renewal process generalizes the Poisson process. In the former the interarrival time between two consecutive events is the duration of nondrought conditions, and the events (i.e., droughts) have a probabilistic duration of at least θ years. The sum of the interarrival time and its subsequent drought duration is called the renewal time, which regenerates over time according to probabilistic laws derived in this work. Drought severity is incorporated in the analysis by means of a threshold quantile (e.g., the median or the average), so that low-streamflow conditions become a drought whenever they last over θ years. A case study dealing with a river basin that has multiyear storage capacity, and in which droughts recurred frequently in the twentieth century, demonstrates the analytical power of the compound renewal model. Copyright 2005 by the American Geophysical Union.