Modeling the probability distribution of rainfall intensities at different aggregation scales, say from sub-hourly to weekly, has always played a key role in most hydrological risk analysis, in particular in the computation of Intensity-Duration-Frequency (IDF) curves. Since any aggregation procedure involves accumulating rainfall over a prescribed time window, it naturally induces simple mathematical constraints related to summation. In particular, return levels inferred from a statistical model should be ordered across time scales, reflecting for example the fact that observed daily accumulations necessarily exceed those at sub-daily scales. From a statistical modeling perspective, each aggregation step combines information from shorter time scales without introducing additional data. Consequently, the number of model parameters should remain limited. Still, parsimonious aggregation models that describe the full distribution of rainfall intensities are sparse in the hydrological literature. In particular, most studies focus on extremes, e.g. by taking seasonal block maxima at different aggregation scales. In this study, we propose a statistical framework that allows to model all rainfall intensities (low, medium and large) at different aggregation scales, while being parsimonious. To reach this goal, we use the extended generalized Pareto distribution (EGPD), which complies with extreme value theory for both low and high extremes and is flexible enough to capture the bulk of the distribution. We show a general result that explains how EGPD random variables behave under different types of aggregation procedures. Direct likelihood inference is difficult in our setting. However, by linking the EGPD class to Poisson compound sums, we can use the Panjer algorithm to quickly and efficiently evaluate the composite likelihood of our proposed model. As a result, return levels can be obtained for any return period, particularly those below the annual and seasonal scales. In addition, our approach insures that return levels do not cross with aggregation. To demonstrate the applicability of our method, we analyze sub-hourly time series from six gauging stations in France that have different climatological features. For each station, we only need a total of eight parameters to capture aggregation scales from six minutes to three days. IDF curves above and below the annual scale are provided. • A parsimonious statistical model for rainfall across multiple aggregation scales is proposed. • Extended Generalized Pareto and compound Poisson distributions serve as core building blocks. • Lower and upper tail properties are shown to remain largely invariant under aggregation. • The framework jointly captures bulk behavior and extreme rainfall in a parsimonious way. • The model is computationally efficient and can be used to compute IDF curves.
Ailliot et al. (Wed,) studied this question.