Improved Extreme Rainfall Frequency Analysis Using a Two-Step Kappa Approach

Accurate estimation of annual exceedance probabilities (AEPs) of extreme rainfalls through rainfall frequency analysis is a critical step in engineering design for flood mitigation and disaster response. Here we show how the estimation of rainfall frequency curves can be improved by fitting a four-parameter Kappa distribution to a peaks-over-threshold (POT) series. To fit the Kappa distribution to POT data we present a two-step fitting approach based on maximum likelihood estimation which separately models storm intensity and the arrival frequency. First, a Generalized Pareto distribution (GPA) describing storm intensity is fitted, followed by a Binomial distribution for storm arrivals. We compare the performance of this two-step Kappa approach to an analogous two-step Generalized Extreme Value (GEV) approach and to Kappa and GEV distributions fitted to annual maxima series (AMS), using both synthetic and real-world data representative of Australian climatic conditions. The experiments show that leveraging additional information from the POT series in the two-step Kappa approach dramatically improves quantile estimation and reduces uncertainty compared to fitting either the GEV or the Kappa distributions to AMS, particularly for rare quantiles. When skew-kurtosis properties of extreme rainfalls are well represented by the Kappa but not the GEV, the two-step Kappa approach yields unbiased quantiles under extrapolation, while the use of GEV can lead to highly biased estimates. From these results, we believe the two-step Kappa approach is suitable for both at-site and regional rainfall frequency analyses as it can accommodate distributions ranging from GPA to GEV without encountering over-fitting problems generally associated with four-parameter distributions.Plain Language Summary Estimating the annual probability of extreme storms is a key step in estimating the risk of flooding for infrastructure design. The most common approach for estimating rainfall probabilities involves fitting a model to a series of data of the single largest storm in each year (called an annual maxima series [AMS]). This approach does not use all the information in the rainfall record and generally limits hydrologists to adopting three-parameter models. These models are often not flexible enough to represent the behavior of extreme rainfalls. A more information-rich alternative is to fit a probability model to all storm events larger than a specified threshold (called a peaks-over-threshold [POT] series). Here, we assess the advantages of using the additional information contained in the POT series to fit a four-parameter model (the Kappa distribution) to capture the behavior of extreme rainfalls. We introduce a two-step fitting approach that separates the parameters relevant to storm intensity from those governing storm arrival frequency. Our results show that the Kappa model performs as well or better than a three-parameter model fitted to either POT or AMS in a range of scenarios and conclude that the proposed approach is a practical alternative to traditional methods.

Automated summary

Accurate estimation of annual exceedance probabilities (AEPs) of extreme rainfalls through rainfall frequency analysis is essential for flood mitigation and disaster response in engineering design. This study demonstrates that fitting a four-parameter Kappa distribution to a peaks-over-threshold (POT) series can improve the estimation of rainfall frequency curves. The two-step Kappa approach, which separately models storm intensity and arrival frequency, significantly enhances quantile estimation and reduces uncertainty compared to traditional methods.