Enhanced profile likelihood method for the nonstationary hydrological frequency analysis

Abstract

In the nonstationary hydrological frequency analysis (NS-HFA), the profile likelihood (PL) method has shown superiority in uncertainty estimation due to its higher accuracy and realistic asymmetric estimates. However, its wide application has been hindered by two issues, namely the high computational burden and the numerical instability problems arising when dealing with short datasets. This paper aimed to solve or mitigate these two issues. To reduce its computational burden, the classical regula-falsi numerical method was incorporated into the PL method (namely, the RF-PL method) for estimating the bounds of the confidence intervals through an explicit analytical expression of the reparametrized log-likelihood function rather than computing the full profile. The generalized maximum likelihood principle, which constrains the distribution shape parameter to a physically/statistically reasonable range, was extended to the proposed RF-PL method (namely, the RF-GPL method) to handle short datasets. The proposed methods were applied to eight annual maximum series of flow and precipitation from North America, which present temporal trends in the mean and/or the standard deviation, to demonstrate their efficiency. The results showed that the RF-PL method substantially reduced the computational time of the PL method by 94–96% without degrading its estimation accuracy. Moreover, the RF-GPL was proven to be effective in avoiding or substantially mitigating the numerical instability issue of the RF-PL method for small sample sizes in addition to reducing the uncertainty in the estimates. As expected, the superiority of the RF-GPL method decreases with the increase of the sample size, which is beneficial in enhancing the estimation of the shape parameter in the RF-PL method. Therefore, the RF-GPL method is advantageous over the RF-PL method for short datasets, whereas its outperformance would be case-dependent for long datasets. These advancements overcome the common hurdles of the PL method and consequently enable its practical implementation more widely in the NS-HFA.