Extreme Precipitation and Floods: Monitoring, Modelling, and Forecasting
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Said Arab Khan, Ijaz Hussain, Tajammal Hussain, Muhammad Faisal, Yousaf Shad Muhammad, Alaa Mohamd Shoukry, "Regional Frequency Analysis of Extremes Precipitation Using LMoments and Partial LMoments", Advances in Meteorology, vol. 2017, Article ID 6954902, 20 pages, 2017. https://doi.org/10.1155/2017/6954902
Regional Frequency Analysis of Extremes Precipitation Using LMoments and Partial LMoments
Abstract
Extremes precipitation may cause a series of social, environmental, and ecological problems. Estimation of frequency of extreme precipitations and its magnitude is vital for making decisions about hydraulic structures such as dams, spillways, and dikes. In this study, we focus on regional frequency analysis of extreme precipitation based on monthly precipitation records (1999–2012) at 17 stations of Northern areas and Khyber Pakhtunkhwa, Pakistan. We develop regional frequency methods based on Lmoment and partial Lmoments (L and PLmoments). The L and PLmoments are derived for generalized extreme value (GEV), generalized logistic (GLO), generalized normal (GNO), and generalized Pareto (GPA) distributions. The statistics and L and PLmoments ratio diagrams of GNO, GEV, and GPA distributions were identified to represent the statistical properties of extreme precipitation in Northern areas and Khyber Pakhtunkhwa, Pakistan. We also perform a Monte Carlo simulation study to examine the sampling properties of L and PLmoments. The results show that PLmoments perform better than Lmoments for estimating large return period events.
1. Introduction
Hydraulic and hydrologic designs are key steps in planning of any water project. Any problem pitched at designing stage will result in the failure of design irrespective of the fact how correctly the other steps are taken. Hydrologists deal with waterrelated issues, problems of quantity, quality, and availability, in the society that known as hydrologic events. Stochastic methods are often used to understand sources of uncertainties in physical processes that give rise to observed hydrologic events, as precipitation and stream flow estimates depend on the past or future events. Several statistical methods offered to minimize and summarize the uncertainties of observed data and frequency analysis is one of them. It determines that how often a particular event will occur by estimating the quantile for return period of , where is the magnitude of the event that occurs at a given time and location.
Dalrymple [1] proposed regional frequency analysis (RFA) method for pooling various data samples. It is indexflood procedure in hydrology. Hosking et al. [2] studied the properties of probabilityweighted moments (PWMs) method based on RFA method. This method is first used by Greis and Wood [3] and Wallis [4]. Cunnane [5] reviewed twelve methods of RFA and related regional PWMs algorithm.
Initially, PWMs are considered as an alternative parameter estimation method; however, it was difficult to interpret directly as measures of the shape and scale parameters of distribution. RFA can forecast the flood flow using empirical formula and unithydrograph procedure Subramanya [6], and it can also estimate the quantiles of extreme precipitation.
Hosking and Wallis [7] showed that RFA method based on Lmoments is used to detect homogeneous regions, to select suitable regional frequency distribution, and to predict extreme precipitation quantiles at region of interest.
Whilst Lmoment methodology is effective in estimating parameters, it may not valid for predicting high return period events. Wang [8] suggested that relatively small floods might create disturbance in the analysis. To overcome such situation, a censored sample can be used as by Cunnane [9].
Wang [8] proposed partial probabilityweighted moments (PPWMs) for fitting the probability distribution function to the censored sample. Partial Lmoments (PLmoments) are variants of the Lmoments and similar to PPWMs. PLmoments method has used in fitting generalized extreme value (GEV) distribution for censored flood samples (see Wang [8, 10, 11] and Bhattarai [12]). Bhattarai [12] found that censoring flood samples are nearly thirty percent of basic Lmoments.
Shabri et al. [13] used Trim Lmoments (TLmoments) for the RFA and compared its performance with Lmoments. Saf [14] determined hydrologically homogeneous region and regional flood frequency estimates by using indexflood technique along with Lmoments for the West Mediterranean River, Turkey. Lmoments method is also used to assign a suitable regional distribution for the individual subregions and to assess their homogeneity; see Abolverdi and Khalili [15]. Hussain and Pasha [16] suggested the regional flood frequency analysis based on Lmoments. They used discordancy measure for data screening and used the fourparameter Kappa distribution with 500 simulations for the heterogeneity analysis. Zakaria et al. [17] used the PLmoments technique and found another link for the homogeneity analysis. Shahzadi et al. [18] showed that the generalized normal (GNO) distribution is suitable for the regional quantile estimation at maximum return period and the GEV distribution for the overall regions at low return period based on relative RMSE and relative absolute bias. Most commonly used statistical distributions for high climate modeling are as follows: the logistic distribution with three parameters, lognormal distribution with three parameters, Log Pearson type III, GEV, and generalized Pareto (GPA) (Coles [19]; Katz et al. [20]; Abida and Ellouze [21]; Feng et al. [22]; Yang et al. [23]; Villarini et al. [24]; Zakaria et al. [17]; She et al. [25]). Moreover, GEV and GPA distributions are suitable if data contains extremes values. Over the years, the GNO, GEV, generalized logistic (GLO), and GPA distributions have been widely employed in the extreme value estimation of annual flood peaks. In this study, we aim to develop RFA method based on L and PLmoments approach. Our proposed method can be used at all levels of regional analysis such as identification of homogeneous regions, identification, and also testing of the suitable probability regional frequency distribution based on statistic and the L and PLmoment ratio diagram and estimation of the flood quantile at site of interest. We use 17 sites of Northern areas and Khyber Pakhtunkhwa as a case study to perform the analysis.
We explain the methodology of the Lmoments and PLmoments in Section 2.2. Section 2.3 shows the application of the RFA where we choose the appropriate distribution for the regional analysis. Section 3.3 provides the estimation of quantile for both, the small and large return period. The results of our simulations study will be presented in Section 4.
2. Material and Methods
2.1. Study Area and Data Sources
The data was collected for this study from Karachi Data Processing Center through Pakistan Meteorological (PMD) Islamabad. Monthly precipitation data has been recorded from 1999 to 2012. There are 17 meteorological stations of Northern areas and Khyber Pakhtunkhwa, Pakistan. These stations are full precipitation regions that affect areas in Pakistan, where water is essential for hydropower and flood plains. Figure 1 shows the location of the study area and geographic distribution of precipitation stations. There is no missing value in this data set.
2.2. Statistical Methods
2.2.1. The LMoments
Conventional moments method may be used for estimating the parameters of probability distribution. However, this approach has some serious drawbacks. Ratio of moment estimators is biased and often assumption of being normally distributed is violated, Wallis et al. [26]. Furthermore, it is sensitive to outliers, Pearson [27]. Therefore, it is unreliable for skewed distributions.
Hosking [28] proposed Lmoments approach to overcome the above problems. The Lmoments may precisely describe the statistical properties of hydrological information, and it can write as a linear function of the PWMs. The PWMs of order were properly described by Greenwood et al. [29] as where , and are the real numbers and is the cumulative distribution function of . A functional case is Therefore, where is the cumulative distribution function of a random variable and is a nonnegative integer of the real number that is . Therefore the first four Lmoments, which are the linear combinations of the PWMs, are The Lmoments have no units of measurement, which are called the Lmoments ratios. The Lmoments ratios, proposed by Hosking [28], are computed as where represents the Lcoefficient of variation , represents the Lcoefficient of skewness , and represents the Lcoefficient of kurtosis .
The arranged sample is given as . Wang [8] stated that the statistic is an unbiased estimator of .
So where , and are the first four Lmoments of the sample. And similarly are the sample Lmoments ratios.
2.2.2. The Partial LMoments
Wang [8, 10, 11] introduced a concept of partial probabilityweighted moments (PPWMs) that will estimate the higher quantiles of flood flows. Data can be censored to the right tail or left tail.
Initially, PPWMs were to take out the smaller values from the process of distribution fitting because such values have slight influence on the frequency analysis and are nuisance to the fitting process.
The left tail PPWMs are defined by Wang [8, 11] as where which is the lower limit of the censored observations and is the censoring threshold value.
The PPWMs elongated form described by Wang [10] are to be given a censored sample as If the value of is starting from the zero, then the result of PPWMs will be the same as the usual PWMs. As is the arranged sample, Wang [10] describes the unbiased estimator of as where The censoring level, , is the prior selection of the number of censored sample data. The procedure that determines the number of sample data points are to be censored: where and are the lengths of sample which are to be censored and uncensored, respectively. Similarly, is the highest value of the censored sample. The first four PLmoments for the PPWMs are Similarly, the PLmoments ratios can be written as where , and denote the partial Lcoefficient of variation , partial Lcoefficient of skewness , and partial Lcoefficient of kurtosis , respectively. Therefore, the first four sample PLmoments can be computed as And the first four sample PLmoments ratios can be computed as where , , and represent the sample partial Lmoments ratios of the , , and , respectively. The derivation is Lmoment and PLmoments are given in Appendix. In the present study, different level of censoring threshold is selected.
2.3. Regional Frequency Analysis
Hosking and Wallis [7, 30] identified the following four steps to explain the procedure of the RFA:(1)Data screening(2)Designing of the homogeneous region(3)Selection of an appropriate probability distribution(4)Parameters estimation of the appropriate probability distribution
2.3.1. Data Screening
We screened data anomalies before applying any statistical analysis.
2.3.2. Discordance Test
Hosking and Wallis [7] suggested a discordancy measure test that recognizes the locations where sample Lmoments are marked contrarily from the most other locations. Locations with the large flaws in the data will be flagged as discordant.
The discordancy test for a region containing locations, for site , is proposed by Hosking and Wallis [7] as follows:where is the vector containing the three sample Lmoments ratios for the site expressed as is the average vector of for the overall region that is and is the covariance matrix for the sample that can be expressed as Broadly speaking, a location or a site is considered to be discordant from the whole region or group if the value of is larger than the critical value.
2.3.3. Heterogeneity Test
The homogeneity measure identifies homogenous regions, Hosking and Wallis [7]. It is also useful to tag the locations if they are plausible to handle as a homogeneous region. It estimates the amount of heterogeneity in the overall region. The heterogeneity test is computed as where and are representing the mean and standard deviation of the simulated values. Also, Here, , and are region average Lmoments or PLmoments ratios. We assessed the heterogeneity of a region as suggested by Hosking and Wallis [7]: Region is acceptably homogeneous if . Region is possibly heterogeneous if . Region is definitely heterogeneous if .
2.4. Selection of the Appropriate Probability Distribution
Hosking and Wallis [7] proposed two approaches to select the distribution that fitted best the data: the Lmoment ratios diagram and the test. The Lmoment ratios diagram is using the unbiased estimators, Hosking [28], Stedinger et al. [31], Vogel and Fennessey [32], and Hosking [33]. The Lmoments ratio diagram is a plot of the computed values and the observed values of the distribution function. The curves indicate the hypothetical connections between and of the candidate distribution. The Lmoment ratio diagrams have been proposed for discriminating between the candidate probability distributions in describing the regional information (Hosking [28]; Stedinger et al. [31]; Hosking and Wallis [7]). The Lmoments ratio diagrams have been used as a component of probability distribution process for regional information (Schaefer [34]; Pearson [35]; Vogel and Fennessey [32], Vogel et al. [36]; Chow and Watt [37]; ÖnÖz and Bayazit [38]; Vogel and Wilson [39]; Peel et al. [40]).
Hosking and Wallis [7] suggested a measure to see how well the and of the fitted probability distribution match the regional average and of the observed information.
The measure goodness of fit for every single selected probability distribution is computed as follows: where represents the value of the of the fitted distribution, represents the weighted regional average , and represents standard deviation of the , which is obtained from the simulation of the Kappa probability distribution.
If the computed value of is equal to zero, the probability distribution will be the most suitable fit. If the computed value of statistic is less than 1.64 at 90% confidence level (i.e., ), it will indicate that the distribution qualifies the goodness of fit criteria. If there are more than one distribution that qualify the criteria, the most suitable distribution has the minimum value.
3. Estimation
The sites’ information and statistic by using Lmoments for the present study are presented in Table 1. In Table 1, mean represents the first sample L/PLmoments and , and are the sample L/PLmoments ratios of the , , and , respectively. The lower level censoring threshold is selected from 10 to 23%. Table 2 expresses the feasible threshold values according to the percentile technique along with Average Annual Occurrence Number (AAON). Jiang et al. [41] and Yuguo [42] suggested that the optimal threshold can be obtained if the values of AAON lie between and . Table 2 shows that 90th percentile observations are suitable for the optimum threshold selection of most of areas in the present study. We have 168 values in each station; according to the above table, Astore station has 3.2 threshold values due to which 16 values are being censored in 168. According to censored level, 10.2% censored level was selected. Similarly, 10.5% was selected for Balakot and Muzaffarabad. According to this process, maximum threshold level 22.3% was selected for Gupis. By using the above process for each station, 17 different censoring levels were selected. So we decide that the range from 10 to 23% of censoring level should be kept for selecting threshold values.


3.1. Regional Frequency Analysis
The following four steps are considered as prerequisite for frequency analysis, Hosking and Wallis [7, 30]:(1)Data screening(2)Designing of the homogeneous region(3)Selection of an appropriate probability distribution(4)Parameters estimation of the appropriate probability distribution
3.1.1. Data Screening
In this study, we use secondary data after carefully examining all locations for abnormalities and missing observations. Therefore, we use 14 years of data for RFA that were obtained from seventeen locations.
3.1.2. Discordance Test
Table 3 shows result of (18) for 17 locations of this study region. It can be observed from the results of L/PLmoments in Table 3 that the value of statistic varies from 0.07 to 2.44. If is greater than 3, the location is considered to be discordant from the rest of the regional data, Hosking and Wallis [7]. In this study region, no location is diagnosed as discordant . Therefore, we use all data for the development of the RFA based on Lmoments and PLmoments.

3.1.3. Regional Heterogeneity Measure
The next step is the formation of the homogeneous region, which is conventionally tougher and needs the higher number of subjective judgments. The homogeneity conditions are defined as the locations that have the same frequency distributions.
For the present study area, realization of the Kappa probability distribution is used to conduct the heterogeneity test based on the Lmoments and PLmoments.
Number of simulations are 10,000 for computing the heterogeneity. We computed the regional average Lmoments ratios, the regional PLmoments ratios, and the corresponding parameter values of the fitted Kappa probability distribution (see Table 4). Table 4 shows the results of the heterogeneity measure using Lmoments and PLmoments methods. It can be observed from Table 4 that the different values for the statistic are −0.41, −1.60, and −2.89 based on Lmoments and −0.06, −1.8, and −3.17 based on PLmoments. Therefore, we concluded that, by comparing these results and the heterogeneity conditions, study region is acceptably homogeneous for Lmoments and PLmoments. No further subdivisions of the present study are necessary.

3.2. Fitting Appropriate Probability Distribution
After homogeneity analysis of the study area, a suitable probability distribution is required for the RFA. The objective is not only to recognize a suitable probability distribution for RFA but also to observe a probability distribution that will provide robust quantile estimate for each location and for the regional growth cure. List of candidate probability distributions for RFA is GLO, GEV, GPA, and GNO.
We plotted Lmoments and PLmoments diagrams for preliminary evaluation of the probability distribution for the study area.
Figure 2 illustrates an analogy of the observed and hypothetical relationships of the probability distribution. Figure 2 shows that GLO distribution is not a suitable candidate for the Lmoments and PLmoments.
Interestingly, both analyses of the Lmoments and the PLmoments diagram show that the sample average values are appropriately distinguished by the hypothetical Lmoments and PLmoments for GPA and GNO distributions.
However, it is hard to find a suitable probability distribution that fits most of the regional observed data. Table 5 shows the goodness of fit test results for candidate probability distributions.

Table 5 shows that GLO distribution failed the goodness of fit test for both Lmoments and for PLmoments methods as the calculated value of the test for the GLO distribution is larger than the critical value of 1.64 (at 90% confidence level).
It has been observed that the computed values of are less than 1.64 (at 90% confidence level), namely, GEV, GNO, and GPA distributions. However, GEV, GNO, and GPA distributions are suitable for regional distribution based on Lmoments and PLmoments methods and for obtaining the future estimates of the quantile.
Further, it can be noted that GPA distribution is suitable for Lmoments method (lowest critical value). Similarly, GNO distribution is suitable for PLmoments method (lowest critical value). Table 6 shows the estimates of the regional parameters for Lmoments and PLmoments for the suitable probability distribution.

3.3. Estimation of the Quantiles
The regional quantile estimates , with varying nonexceedance probability for the GNO, GEV, and GPA distributions, are presented in Table 7 based on Lmoments and PLmoments. Quantile function is normally represented as for fitted regional frequency distribution. The quantile estimate at location is established by joining the estimate of and .

Mathematical form of the quantile estimate with nonexceedance probability isThe regional growth curves for the GEV, GNO, and GPA distributions are shown in Figure 3.
Figure 3 shows the regional growth curves of each candidate distribution for Lmoments and PLmoments. GEV, GNO, and GPA distributions are approximately identical up until year return period ( for both Lmoments and PLmoments. However, afterward the growth curves of the GPA distribution lie below the GEV and GNO distributions.
Therefore, it is necessary to assess the performance of regional quantile estimates.
4. Accuracy of the Estimated Quantiles and the Regional Growth Curve
A Monte Carlo simulation is designed to assess accuracy of the regional quantile estimates that are obtained by the RFA. We use logical Lmoments algorithm that has been reported by Hosking and Wallis [7] in Section . This algorithm takes samples from a region that has comparable characteristics as of the actual region, such as having the same record length, same number of locations, and the regional Lmoments ratios. The area used for simulation should report the plausible heterogeneity in the area and intersite dependency if exist (Hosking and Wallis [7]). In the repeated sampling procedure, the quantile estimates are computed for the different nonexceedance probabilities. Suppose that, at th repetition and location , quantile estimate can be written as for the nonexceedance probability . The relative error for this estimator is The bias and the RMSE of the above quantity over all repetition are Also, for the estimated quantile, the regional average bias and the relative RMSE are We use empirical quantities of quantile distribution for the assessment analysis that can be computed by taking the ratio of estimated to true values, for the quantile, and for the regional growth curves. Therefore, 90% of the regional growth curve lie in between the interval: Inverting the expression for , we have The 90% confidence interval limits show the measure of variation between the estimated and the true quantiles. These limits provide the expected magnitude of errors in the estimated quantiles and the regional growth curves.
We computed Lmoments ratios to find the most suitable distribution and the precision of original growth curves. The correlation between the study region sites varies from −0.05 to 0.86 with an average of 0.40. Therefore, we use algorithm from Table of Hosking and Wallis [7].
We held out the analysis for recurrence of different years. We run 10,000 simulations with sample size of 30, 60, and 90 in each case. The whole process is repeated for GEV, GNO, and GPA distributions. From these repetitions, we computed several performance measures, such as the regional average relative bias, regional average RMSE, regional average relative RMSE, and the error bounds for the estimated regional growth curves for the selected nonexceedance probability . Overall results for the suitable probability distribution for both methods are presented in Tables 8, 9, and 10 for sample size of 30, 60, and 90, respectively.



Figures 4, 5, and 6 show estimated regional growth curves for sample sizes 30, 60, and 90, respectively, and also GEV, GNO, and GPA distributions with the 90% error bounds.
Tables 8, 9, and 10 show that increase in the sample size such as 30 to 90 improved the performance particularly in the prediction of the large nonexceedance probability . Lmoments method provides similar performance for the GNO and GPA distributions in terms of relative bias that is presented in Tables 8, 9, and 10. We found that the GPA distribution produced the lowest relative bias compared to GEV and GNO distribution for the PLmoment for the various values of the nonexceedance probability . However, the GPA distribution performs better in terms of RMSE than the GEV and GNO distribution for both methods (Lmoments and PLmoments). Furthermore, RMSE is lowest for PLmoments compared to Lmoments. In addition, the error bounds for the GPA distribution of regional quantiles are narrow compared to GEV and GNO distributions. It shows that the estimation of censored sample improves the prediction of extreme precipitation explicitly at large nonexceedance probability .
5. Discussion and Conclusion
This study provides a comprehensive evaluation of the Lmoments and the PLmoments. First, revisiting RFA on Lmoments by Hosking and Wallis [7], we aimed to develop similar connections of regional homogeneity for PLmoments. The Lmoments and the PLmoments for candidate probability distributions (GLO, GEV, GNO, and GPA) are also developed for presenting the corresponding Lratio and PLratio diagrams with the goodness of fit test results. The regional growth curves for the selected distribution have been shown in Figures 4, 5, and 6. At the lower tail, GEV, GPA, and GNO distributions are approximately the same, but, at the upper tail, there is variation between the regional quantiles. The regional homogeneity analysis starts by assuming 17 locations of Northern areas and Khyber Pakhtunkhwa, Pakistan, as one homogeneous region, based on Lmoments and PLmoments at censoring level ranging from 10 to 23%. This assumption is statistically accepted after applying the heterogeneity and discordancy tests. The statistic provides appropriate distribution for modeling the monthly extreme precipitation in Northern areas and Khyber Pakhtunkhwa, Pakistan. We found that GPA distribution is suitable for the Lmoments and GNO distribution for the PLmoments.
Finally, Monte Carlo simulation used for performance evaluation by commonly used error functions. Several accuracy measures such as relative bias, RMSE, relative RMSE, and error function bounds for the regional quantiles are computed with 10,000 runs of Monte Carlo simulations. We found that GPA distribution produced robust quantile estimates for both return periods and methods (Lmoments and PLmoments). Our results support the finding of previous study (e.g., Cunnane [9]; Bhattarai [12]) for censored sample analysis where PLmoments method outperformed the Lmoments method for the estimation of large return periods events.
Appendix
The partial Lmoments (PLmoments) for generalized logistic (GLO), generalized Pareto (GPA), generalized normal (GNO), and generalized extreme value (GEV) distributions were derived based on the formula defined by Wang (Water Resour Res 32:1767Ö 1771, 1996 (In references lines from 442 to 444 mentioned that in study)). The summary of the derived distributions and parameters estimation for these distributions is as follows.
PLmoments for the GEV Distribution [10] are as follows.
The CDF and quantile function of the GEV are given byand quantile function
Wang [10] developed the partial probabilityweighted moments (PWMs) of the GEV aswhere The first four PLmoments of the GEV are defined as
In previous equation, is an Incomplete Gamma function:Then the first four PLmoments are computed to develop the PLmoment ratios (PLCv, PLCs, and PLCk) for the GEV distribution.
The PLmoments for the GLO Distribution are as follows.
The CDF and quantile function of the GLO are given byand quantile function The partial PWMs of the GLO are developed as follows: where is an Incomplete Beta function
The first four PLmoments of the GLO are defined as
Then the first four PLmoments are computed to develop the PLmoment ratios (PLCv, PLCs, and PLCk) for the GLO distribution.
The PLmoments for the GPA Distribution are as follows.
The CDF and quantile function of the GPA are given byand quantile function
The partial PWMs of the GPA are developed as follows:
The first four PLmoments of the GPA are defined aswhere
Then the first four PLmoments are computed to develop the PLmoment ratios (PLCv, PLCs, and PLCk) for the GPA distribution.
Ethical Approval
The manuscript is prepared in accordance with the ethical standards of the responsible committee on human experimentation and with the latest (2008) version of Helsinki Declaration of 1975.
Conflicts of Interest
The manuscript is prepared by using secondary data and authors declared that there are no conflicts of interest.
Acknowledgments
The authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through Research Group no. RG1437027.
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