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

: 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.


Introduction
In flood frequency analysis (FFA), the main goal is to calculate the maximum flow corresponding to the area of low probability, where there are usually no records. In general, this analysis can be performed using the Annual Maximum Series (AMS) or Partial Series (Peak Over Thresholds or the Annual Exceedance Series).
In FFA using AMS, an important aspect is representing the observance of the "separation effect", introduced by Matalas [1], which consists of reducing as much as possible the influence of the small values of the maximum flow, knowing that, in the annual series of maximum flows, they do not always represent flood flows. The advantages and disadvantages of using AMS and PS were presented in previous materials [2].
This research is part of a holistic analysis regarding the probability distributions and estimation methods of their parameters, according to the hydrological and hydrometric characteristics in Romania, taking into account climate change, which amplifies extreme events. This research is a proposal for the future normative for determining the maximum flows, in which there is the applicability of some families of distributions, including the Gamma family (Pearson 3, Log-Pearson, Wilson-Hilferty, Pseudo-Weibull, Chi, Inverse Chi, Pearson V, Krtisky-Menkel) [3,4], the GEV family (Fréchet, Weibull) [5], the generalized Pareto family [1], the generalized beta families [6,7] and the family of quantile functions (Wakeby, Lambda) [4,6].
The "separation effect" of the maximum flow can be achieved by using the distributions introduced in the FFA, especially the LH-moments method or the simultaneous use of multiple distributions, as is the case with this article.
The LH-moments method is a parameter estimation method, introduced by Wang [8], that analyzes the generalized distribution of extreme values and was popularized by Sharwar et al. [9], who formulated the LH-moments for five distributions, namely, Mielke-Johnson's kappa distribution, the three-parameter kappa type II distribution, Beta-k distribution, Beta-p distribution and the generalized Gumbel distribution. Compared to the L-moments method, the LH-moments method has the advantage of assigning a lower importance to small extreme values (right hand, high exceedance probabilities) which do not always represent floods. This also represents the main disadvantage of using the Annual Maximum Series. In general, the LH-moments method can be applied up to level 4, but in this article, only the first two levels are analyzed because the use of the last two levels would constitute a very large deviation from the L-moments method, which is nevertheless a stable and robust method.
Regarding the distributions that naturally achieve a "separation effect", one of the most used probability distributions for achieving this "separation effect" in FFA without sample censoring is the Wakeby distribution [1,4,29]. This was introduced in the analysis of annual maximum flows (without sample censoring) by Houghton [1], precisely to fulfill this principle, having as a particular case the four-parameter Wakeby distribution [29][30][31] and generalized Pareto distribution type II [4,29], the latter being the usual distribution applied in the case of frequency analysis using partial series (with sample censoring) [2,29,32,33]. Since 1978, the Wakeby distribution has been used for flood and regional flood frequency analysis [4,29,32,[34][35][36], low−flow frequency analysis [6,37] and annual maximum rainfall analysis [38][39][40]. Using the LH-moments method, it was applied in various materials [41][42][43].
Other distributions that can achieve this "separation effect" but receive limited attention are the generalized Lambda distribution and the Davis distribution. The five parameters generalized Lambda distribution was generally used alongside the method of weighted moments and the method of linear moments [30]. In [6], it was applied for the first time for FFA using the method of ordinary moments (MOM), and all mathematical relations for parameter estimation using this method are being presented. It is a distribution that has as special cases, the four-parameter Lambda distribution and the three-parameter Tukey distribution [30].
Regarding the Lambda distribution and Davis distribution, they are for the first time applied for flood frequency analysis using the LH-moments method. Moreover, for the first time, the Davis distribution is applied using the L-moments method, and the relations for estimating the distribution parameters using the L-moments method are presented.
The purpose of the article is to present all the mathematical equations to apply the analyzed models, using both the L-moments method and the LH-moments method for parameter estimation. We also want to highlight the particular cases of these distributions, which under certain conditions turn into distributions usually applied in these types of analyses.
The performance of the proposed models and methods is examined, and an FFA is carried out for the Bahna River, Romania. The analysis is based on historical data and assumptions of stationarity. Because of climate change, the assumption that future conditions will be similar to those of the past may lose its validity. Selecting the best model is based on the L-moments and LH-moments values and diagrams.

Methods
Five distributions are analyzed that are part of the family of probability distributions and defined only by the inverse function, with applicability in FFA.
All the elements required to apply these distributions are presented, including the inverse function and the relations for estimating the parameters.
The analysis of the maximum flows for the Bahna River is carried out in stages, according to the flow chart presented in Figure 1. No outliers were detected (Grubb's test). The series is homogeneous and the verification was completed with the von Neumann test [44]. To check the homogeneity, MASH v3.03 software can also be used, which was successfully applied in the area of the Pannonian basin [45].

Quantile Functions
In the section below, the analyzed probability distributions are presented. Distributions are defined only by the quantile function. The characteristic frequency factors of the two methods are highlighted in Appendix A. - The five-parameter Wakeby distribution (WK5) [1,4,29,30]: where  represents the position parameter;  and  represent the scale parameters; and  and  represent the shape parameters.
-The four-parameter Wakeby distribution (WK4) [4,29]: -The five-parameter Lambda distribution (L5) [30]: where  represents the position parameter;  and  represent the scale parameters; and  and  represent the shape parameters. - The four-parameter Lambda distribution (L4): The Davis distribution (DV) [46]: where  represents the scale parameter, and  and  represent the shape parameters.

The L-and LH-Moments Estimations Equations
The methods are the L-and LH-moments. The sample L-moments and LH-moments are determined according to [29][30][31][32], respectively [1]. Only the first and second order LHmoments are analyzed.
The relationship between L-moments and LH-moments are presented in Appendix B.
The first order LH-moments estimations equations:

The Four-Parameter Wakeby Distribution (WK4)
Parameter estimations of the L-moments [29]: The first order LH-moments estimations equations: For linear moments For linear moments

The Five-Parameter Lambda Distribution (L5)
Parameter estimation of the L-moments: The first order LH-moments estimations equations: The second order LH-moments estimations equations:

The Four-Parameter Lambda Distribution (L4)
Parameter estimation of the L-moments: The first order LH-moments estimations equations: For linear moments For linear moments

Davis Distribution (DV)
Parameter estimation of the L-moments: The first order LH-moments estimations equations: The second order LH-moments estimations equations:

Study Area and Main Data
The Bahna River is part of the Danube River basin, being a left tributary. It is located in the south-west of Romania, as shown in Figure 2. The Cadastral code of the Bahna River is XIV.1.21.0.0.0.0 [47]. From a morpho-hydrographic point of view, the Bahna River is part of a sector of the Carpathian Gorges, this sector being the first of five sectors that divide the Danube River on the Romania territory [48]. The river is located on the border of the Pannonian Basin and the Transylvanian Basin. As for its main morphometric characteristics, the Bahna River has a length of 35 km, an average stream slope of 28‰, a sinuosity coefficient of 1.45, an average altitude of 559 m and a watershed area of 137 km 2 [47]. The river is characterized by a very high torrentiality, with maximum recorded flows between 0.55 and 93.5m 3 /s.
The maximum annual series of maximum flows has a length of 30 records, from 1992 to 2021, as presented in Table 1. Year 1992 1993 1994 1995 1996 1997 1998 1999 Table 2 shows the first four linear moments, the L-skewness and the L-kurtosis of the sample, for both methods.

Results and Discussions
The analyzed models were applied for forecasting the rare events, with certain exceedance probabilities, for the Bahna River, Romania, as a case study. Table 3 shows the quantiles results, characteristic of the range of the small exceedance probabilities (0.01-5%) necessary for the design of the hydrotechnical retention construction (dams), as well as the values corresponding to the exceedance probabilities of 40% and 80%, respectively, necessary for the delimitation of the bankfull channel.   Figure 3 shows the quantile graphical results for the Bahna River. The proper empirical probability for L-moments is the Hazen formula [48,49], also chosen for the graphic representation of the data recorded for the Bahna River. The results of interest are those corresponding to probabilities between 0.01% and 1%, these being values of interest in the design of hydrotechnical constructions.

The Results Regarding the Parameter Estimation
Considering that to calibrate the parameters for WK5 and WK4, it is necessary to solve some systems of nonlinear equations, it is recommended to use the approximate relations presented by [29,30] as starting values (initial values), thus facilitating the ease of calculation. Table A1 from Appendix C presents the parameter values for the two methods and for the five analyzed distributions.
As observed in other materials [4,29,30], the Wakeby and Lambda distributions can become, in certain special cases, the generalized Pareto type II distribution and the threeparameter Tukey distribution, respectively, which has the following expressions of the inverse functions: generalized Pareto Type II distribution [2,29,33]: -Tukey distribution [30]: ( )

Selecting the Best Fit Model
The distributions performance and the choice of the best model is based on the natural (theoretical values) L-skewness and L-kurtosis values of the distributions, by comparing them with those of the observed data and choosing the one that has values close to those of the data set.
Additionally, two performance indicators that are usually applied in these types of analysis are presented: relative absolute error (RAE) and relative mean error (RME).
Tables 4-6 present the performance indicators' values for the analyzed methods. For the Bahna River, using the L-moments and the first and second order LH-moments, any of the Wakeby and Lambda distributions can be applied, considering that the quantile results are very close, and these distributions calibrate all linear moments accordingly. The theoretical values ( Regarding the RAE and RME [50] results, they are indicative in nature as they highlight the performance of the distributions only in the area of the probability of exceeding the observed data, losing their relevance outside this domain.

Uncertainties
All the quantiles that exceed the probabilities of the recorded values are characterized by a high degree of uncertainty [51,52], due to the relatively short length of the observed data, a general disadvantage encountered in hydrometry in Romania. In the case of the Davis distribution, the high level of uncertainty is also due to the fact that, being a threeparameter distribution, it does not properly calibrate the fourth order linear moment. As could be seen in previous materials [2,3,8,9,29,32], although the L-moments method is more stable and robust than other methods, in terms of the variation of statistical indicators depending on the length of the observed data, this method also requires a minor correction, and the advantage of the stability of the indicators can be lost due to the nonlinear functions for obtaining the parameters, an aspect highlighted very well by Gaume [53].
In general, the relative errors generated by the variability of the length of the recorded data are determined both in the estimation of the statistical indicators specific to the method used, as well as for the values of the parameters and quantiles, thus resulting in three levels of uncertainty. Thus, it is necessary to determine the confidence interval for each distribution and the method of estimating the parameters.
In Figure 4, the confidence interval for the KW5, KW4 and DV distributions is presented, for both methods, using Chow's relation [54][55][56][57] for a 95% confidence level, based on a Gaussian assumption.

Conclusions
The article highlights the main elements necessary to apply the Wakeby, Lambda and Davis distributions in FFA using L-moments and the first and second LH-moments.
The estimation relations, as well as the frequency factors necessary to determine the confidence interval, are presented.
As a case study, the stages and rare events results on the Bahna river, Romania, using the first two levels of linear moments for parameter estimation are presented. The results are presented in comparison with the L-moments method, this being the proposed method to be implemented as the future norm in Romania.
The LH-moments method is a method usually applied in FFA without censored samples, which achieves the "separation effect" by giving less importance to the lower recorded values, which are, not in every case, floods.
Among the analyzed distributions, the Wakeby distribution is a general distribution applied in FFA. It was introduced by Hougthon precisely to fulfill this "separation effect", and it is a distribution that has, as a special case, the generalized Pareto distribution of type II.
The purpose of this article is to present the main elements of the family of distributions, defined only by the quantile function, in FFA, using the L-moments method and the first and second level LH-moments method.
Regarding the results obtained for the Bahna river, the best model has the distributions WK5, WK4, L5 and L4, and the natural values of the statistical indicators Lskewness and L-kurtosis, best modeling the corresponding values of the observed data. This is due to the fact that the distributions are at least 4, thus succeeding in calibrating the L-moments tours. For the Davis distribution, it fails to properly calibrate the fourth order linear moment because it is a three-parameter distribution. This, like all distributions with a maximum of 3 parameters, is recommended to be applied only for data series with values of statistical indicators close to the natural ones of the distribution. This is also the main selection criterion of the best fit distribution, specific to these methods.
This result does not exclude other distributions, in a broader and more complex analysis, in determining the maximum flow on this river.
The research is part of a larger research within the Faculty of Hydrotechnics, in which all families of distributions used in FFA were analyzed, including the results presented in previous materials. The final goal of the research is the identification of the distribution probabilities and methods of estimating their parameters with applicability in FFA to the hydrological specifics of Romania.
The results will be concretized in open-source computer applications, which will be included in the new regulations.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A. The Frequency Factors
The frequency factor of the WK5 distribution and WK4 distribution is the same because the position parameter is reduced. The same is for the L5 and L4 distributions.

Appendix B. The Relationship between L-Moments and LH-Moments
Relations for LH-moments [8]:  The relations between the high-order linear moments and the normalized weighted moments are the following [  represent the natural estimators expressed by the L-moments method. For 0  = , the method become the L-moments method. Table A1 shows the values of the estimated parameters with the two analyzed estimation methods. This is an important aspect regarding the transparency of the analysis and the possibility that the results can be reproduced