Extreme Events Analysis Using LH-Moments Method and Quantile Function Family

Abstract

A direct way to estimate the likelihood and magnitude of extreme events is frequency analysis. This analysis is based on historical data and assumptions of stationarity, and is carried out with the help of probability distributions and different methods of estimating their parameters. Thus, this article presents all the relations necessary to estimate the parameters with the LH-moments method for the family of distributions defined only by the quantile function, namely, the Wakeby distribution of 4 and 5 parameters, the Lambda distribution of 4 and 5 parameters, and the Davis distribution. The LH-moments method is a method commonly used in flood frequency analysis, and it uses the annual series of maximum flows. The frequency characteristics of the two analyzed methods, which are both involved in expressing the distributions used in the first two linear moments, as well as in determining the confidence interval, are presented. The performances of the analyzed distributions and the two presented methods are verified in the following maximum flows, with the Bahna river used as a case study. The results are presented in comparison with the L-moments method. Following the results obtained, the Wakeby and Lambda distributions have the best performances, and the LH-skewness and LH-kurtosis statistical indicators best model the indicators’ values of the sample (0.5769, 0.3781, 0.548 and 0.3451). Similar to the L-moments method, this represents the main selection criterion of the best fit distribution.

Revisiting the use of the Gumbel distribution: A Frequency Analysis Perspective

“…This section presents in detail the statistical and mathematical elements necessary to apply the Gumbel distribution using the 9 methods of estimating the analyzed parameters. The variation graphs of the inverse function related to the variability of the available data lengths and the data variability (the theoretical choice of the usual Cv in FFA) are also presented, highlighting the systematic biases of the distribution for the annual maximum values (with MOM and L-moments) for the annual exceedance probabilities of interest: 1%, 0.1% and 0.01% [1,2,[71][72][73].…”

Revisiting the use of the Gumbel distribution: A Frequency Analysis Perspective

“…This section presents in detail the statistical and mathematical elements necessary to apply the Gumbel distribution using the 9 methods of estimating the analyzed parameters. The variation graphs of the inverse function related to the variability of the available data lengths and the data variability (the theoretical choice of the usual Cv in FFA) are also presented, highlighting the systematic biases of the distribution for the annual maximum values (with MOM and L-moments) for the annual exceedance probabilities of interest: 1%, 0.1% and 0.01% [1,2,[71][72][73].…”

“…In the case of the L-moments method, there are clear criteria for selecting the best model, namely the calibration of the indicator values of L-skewness ( 3 ) and L-kurtozis ( 4 ) of the observed data [2,[65][66][67]69,[71][72][73][87][88][89][90][91][92][93][94]. Unfortunately, the Gumbel distribution is not defined, like the twoparameter Log-normal or Gamma distribution, by a variation curve of these indicators, but has constant values regardless of the observed data analyzed [91].…”