Evaluation of Various Probability Distributions for Deriving Design Flood Featuring Right-Tail Events in Pakistan

water Article Evaluation of Various Probability Distributions for Deriving Design Flood Featuring Right-Tail Events in Pakistan Muhammad Rizwan, Shenglian Guo * , Feng Xiong and Jiabo Yin State Key Laboratory of Water Resources and Hydropower Engineering Science, Hubei Provincial Collaborative Innovative Center for Water Resources Security, Wuhan University, Wuhan 430072, China; [email protected] (M.R.); [email protected] (F.X.); [email protected] (J.Y.) * Correspondence: [email protected]; Tel.: +86-27-6877-3568 Received: 25 October 2018; Accepted: 6 November 2018; Published: 8 November 2018 Abstract: Design flood estimation is very important for hydraulic structure design, reservoir operation, and water resources management. During the last few decades, severe flash floods have caused substantial human, agricultural, and economic damages in Pakistan during the Monsoon seasons. However, despite phenomenal losses, the flood characteristics are rarely investigated. In this paper, flood frequency analysis (FFA) on four major rivers over Pakistan is performed to probe probability distributions (PDs)at the right-tail flood events. For this purpose, (i) we employed ten different probability distributions associating with an L-moments method for constructing FFA models across Pakistan; (ii) we evaluated the best-fit distribution by using goodness-of-fit test and statistical criteria; and finally; (iii) we devised a Monte Carlo simulation to systematically evaluate the robustness of a selected distribution’s fitting performance by using a synthetic data series of different sizes. Our results indicated that generalized Pareto and Weibull emerged as the most viable options for quantifying hydrological quantiles for most of the river basins in Pakistan. Our main findings would provide rich information as references for flood risk assessment and water resource management in Pakistan. Keywords: flood frequency analysis; probability distributions; L-moment; Monte Carlo simulation; right-tail behavior 1. Introduction Flooding is among the most threatening natural disasters, and its mitigation and management are pivotal for the design of enormous hydraulic structures, according to regulations administered by flood frequency analysis (FFA) [1,2]. It is projected that flooding phenomena will continue to happen in the future; therefore, FFA is recommended to evaluate the frequency of occurrence of extreme flood events by using several probability distributions (PDs) [3–6]. To achieve this purpose, one key issue is the selection of appropriate PD [6,7]. Cunnane (1989) designated the difficulty of identifying a statistical distribution from a pool of globally used distributions for FFA [8]. During the last few decades, many studies are carried out over the best-fit PD in a certain scenario. Some famous and widely used PDs are log Pearson type III (LP3), Pearson type III (P3), generalized Pareto (GPA), generalized logistic (GLO), generalized extreme value (GEV), exponential (EXP), gamma (GAM), Weibull (WEI), Gumbel (GUM) and generalized normal (GNO) [9]. In addition, many countries use specific standard PD for FFA. For instance, China uses P3 distribution [10,11], whereas the United States has adopted LP3distribution [12–14] and Europe prefers GEV distribution [15,16]. Therefore, a lack of global standard PD has restrained hydrologists to using a generic distribution throughout the Water 2018, 10, 1603; doi:10.3390/w10111603 www.mdpi.com/journal/water Water 2018, 10, 1603 2 of 18 world [17]. Hence, it is a challenging task to identify a best-fit PD for the available record of rivers in a specific region of the world. The process of identifying the most reliable PD requires quantifying various candidate PDs by using a goodness-of-fit test. Moreover, different flood characteristics in different rivers and the availability of a wide range of selection criteria demands the selection of PDs from a wide range of available distributions [18]. Cicioni et al. (1973) engaged P3, GEV, three-parameter log-normal (LN3) and two-parameter log-normal (LN2) for FFA of 108 stations in Italy, with a record length of more than 27 years [19]. Haktanir and Horlacher (1993) compared nine different statistical distributions for 11 unregulated streams in Scotland and Rhine Basin in Germany [20]. Karim and Chowdhury employed four different PDs for selecting the best-fit for basins in Bangladesh with the goodness-of-fit analysis [21]. According to Kumar et al. (2003), GEV is the best-fitted PD for estimation of extreme hydrological events after adopting 12 frequency distributions using linear moments (LMO) for estimation of parameters [22]. Yue and Wang (2004) applied LMO to identify the suitable probability distribution for modeling annual streamflow in different climatic regions of Canada [23]. Saf (2009) observed that the P3 distribution is better suited for modeling hydrological quantiles for Antalya and lower West Mediterranean subregions, whereas GLO has the best fit for the upper region [24]. FFA was performed by Haberlandt and Radtke (2014) for identifying PD for three catchments in different regions of northern Germany [25]. In Iran, the GEV has been identified as the best-fit PD amongst five distributions for modeling annual maximum discharge [26]. Likewise, many such studies have been proposed for selection of PDs, but quite infrequent studies have been conducted regarding selection criteria corresponding to the right-tail behavior. Therefore, the selection of PD should take place by considering right-tail events. The inadequate length of historical data offers a high degree of uncertainty in determining the flood quantiles of a certain magnitude. The right-tail of flood frequency curves also shows the sizeable difference for contending distributions. The Monte Carlo method offers a reduction of uncertainty in the estimation of extreme hydrological variables, and this approach manages the inadequacy by generating longer data series [27–30]. The menace of flooding is projected to continue in developing countries, therefore, it is fundamental to understand the dynamics of floods for urban development and better management of water resources [31]. The prediction of floods has received the considerable attention of hydrologists particularly ungauged mountainous areas that are more prone to flash flooding [32]. Since flash floods possess the potential for instant infrastructural damage in Pakistan, the flood quantiles with higher return periods are significant for the evaluation of PDs. In previous studies, various goodness-of-fit tests had been applied to determine the most reliable PD in Pakistan. Therefore, the primary objective of our study is focusing on the determination of PD at a national scale by probing them at the right-tail segment. This study is intended to determine the best-fit distributions by (i) ascertaining the performance of PD on available records of rivers in Pakistan by hypothesis testing while the selection criteria is useful for evaluating the PD and (ii) establishing results for evaluating the performances of PD featuring their right-tail behavior, and concluding the qualitative evaluation from estimated flood quantiles and flood frequency curves. For this reason, this research is important in planning and managing water resources in the studied rivers, which is significant for agriculture development, construction of hydraulic structures, and conservation of natural resources. The remainder of the paper is organized as follows. The study basin is discussed in Section2. Section3 presents methods applied in the study and procedural advancing of our work. In Section4, we present results obtained by our case study and their evaluation. Section5 manifests the discussion about our results and comparison with similar works to obtain PDs for deriving design flood in Pakistan. Finally, Section5 presents the conclusion of our study. 2. Study Basin and Data Figure1 depicts the study area, and its major fragment is Indus basin system in Pakistan. We selected 11 critical locations situated on the Kabul river, Swat river, Indus river, and Jhelum Water 2018, 10, x FOR PEER REVIEW 3 of 19 provided by the hydrology and irrigation division, and the summary is illustrated in Table 1. We noticed that the length of this historical data from eleven gauging stations varies at every station. The Swat river at Munda Headworks has the longest data length of 56 years, while 3 gauging stations at Adezai, Naguman and Shah Alam have 31 years of historical data. Furthermore, all the sites are highly and positively skewed, which implies that the size of the right-tail is larger than the left tail. As shown in Table 1, the value of skewness ranges from 0.65 (Adezai) to 5.26 (Khiali). The variation in the skewness of different sites owes to floods in Monsoon seasons at different times. In the same manner, the kurtosis values are also indexed in Table 1, which presents positive values of kurtosis except for the two sites. This implies that the distribution is heavy-tailed, except for instances where it has acquired negative values. In addition, we inferred from the unit root stationary test and homogeneity test that the data is stationary and homogeneous for further analysis. It will allow various probability distributions to estimate flood peaks that will not be deviating from the historic trend. Water 2018, 10, 1603 3 of 18 Table 1. Summary statistics of annual maximum (AM)flood series record. 3 river for FFA. AnnualGauging maximum Station (AM) streamflowPeriod Mean data with(m /s) a recordCv lengthCs ofCk more than 30 years Attock 1970–2017 13,028 0.29 1.33 4.73 is provided by the hydrology and irrigation division, and the summary is illustrated in Table1. Jindi 1969–2017 300 0.67 1.85 6.81 We noticed that the length of this historical data from eleven gauging stations varies at every station. Munda Headworks 1962–2017 1748 0.73 4.81 31.33 The Swat river at MundaKhwazakhela Headworks has1983–2016 the longest data1451 length of0.83 56 years,1.56 while1.89 3 gauging stations at Adezai, Naguman andNingolai Shah Alam have1984–2016 31 years of historical242 1.45 data.

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