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

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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. Furthermore,2.30 5.61 all the sites are highly and positively skewed,Khiali which implies1969–2017 that the size1648 of the0.82 right-tail 5.26 is34.87 larger than the left tail. As shown in Table1, theShah value Alam of skewness 1987–2017 ranges from207 0.65 (Adezai)0.71 to0.86 5.26 (Khiali).−0.13 The variation in the skewness of differentNaguman sites owes to1987–2017 ﬂoods in Monsoon533 seasons0.84 at1.74 different 4.39 times. In the same manner, the kurtosis valuesAdezai are also indexed1987–2017 in Table 1,868 which presents0.67 0.65 positive −0.44 values of kurtosis except for the two sites. ThisTimergara implies that1984–2017 the distribution 736 is heavy-tailed,0.85 3.09 except 14.77 for instances where it Karot 1969–2010 3725 0.67 2.82 9.59 has acquired negative values.

Figure 1. Location map of ﬂow gauging stations and rivers and digital elevation of the study area. Figure 1. Location map of flow gauging stations and rivers and digital elevation of the study area. Table 1. Summary statistics of annual maximum (AM) ﬂood series record.

3 Gauging Station Period Mean (m /s) Cv Cs Ck Attock 1970–2017 13,028 0.29 1.33 4.73 Jindi 1969–2017 300 0.67 1.85 6.81 Munda Headworks 1962–2017 1748 0.73 4.81 31.33 Khwazakhela 1983–2016 1451 0.83 1.56 1.89 Ningolai 1984–2016 242 1.45 2.30 5.61 Khiali 1969–2017 1648 0.82 5.26 34.87 Shah Alam 1987–2017 207 0.71 0.86 −0.13 Naguman 1987–2017 533 0.84 1.74 4.39 Adezai 1987–2017 868 0.67 0.65 −0.44 Timergara 1984–2017 736 0.85 3.09 14.77 Karot 1969–2010 3725 0.67 2.82 9.59 Water 2018, 10, 1603 4 of 18

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 Waterestimate 2018, ﬂood10, x FOR peaks PEER that REVIEW will not be deviating from the historic trend. 4 of 19

33.. Methods Methods InIn order to find find the best-fitbest-fit PD for river basins in Pakistan,Pakistan, we employed an AM floodflood series method. Considering Considering the the right right-tail-tail events events scenario scenario of of our our study, study, we followed the procedur proceduree in three stepssteps, , suchsuch asas (i)(i) selectionselection ofof PDs;PDs; (ii)(ii) selectionselection ofof parameterparameter estimationestimation method;method; (iii)(iii) carrying out hypothesis test test for evaluating goodness goodness-of-fit-of-fit of of PDs to the historical record and applying selection criteria for choic choicee of statistical distribution distribution;; (iv (iv)) evaluation of of the selected PDs with respect to ri rightght tailtail behavior behavior by by employi employingng Monte Carlo simulation simulation;; and (v (v)) further analyzing the selected PDs with floodflood frequency curves and estimation of flood flood quantiles. The fl flowchartowchart in Figure 22 showsshows thethe mainmain proceduprocedurere for evaluating floodflood quantiles over Pakistan.

Evaluation of PDs emphasizing right- tail

Flood frequency Monte-carlo method At-site flood analysis model frequency analysis Different sized Four candidate Eleven locations Synthetic data PDs and five in Pakistan generation locations

Performance Ten different PDs Return periods indicators

LM estimation Selecting best-fit Determination of method PDs flood quantiles

Adopting results for deriving design flood

FigureFigure 2. 2. FlowchartFlowchart of of the the procedure procedure for evaluating flood ﬂood quantiles over Pakistan Pakistan..

3.1. Probability Probability Distributions We considered various familiar familiar PDs PDs in in this this study study,, namely namely EXP, EXP, GAM, GAM, GEV, GEV, GLO, GLO, GPA, GPA, GNO, GNO, Gumbel extreme extreme-value-value type type I (GUM), normal (NOR), P3 P3andand WEI for carrying out probabilistic modeling of of extreme extreme events. events. Six Six PDs PDs (GEV, (GEV, GLO, GLO, GNO, GNO, GPA, GPA, P3 and WEI) have three parameters parameters,, and four PDs (EXP, GAM, NOR,NOR, and GUM) have two parameters.parameters. The probability density functions (PDF(PDFs)s) and parameters of PDs are illustrated in Table2 2..

Water 2018, 10, 1603 5 of 18

Table 2. The parameters and probability density functions (PDFs) of contending distributions.

Number Distribution Probability Density Function Parameters scale parameter = α 1 EXP F(x) = 1 − exp{−(x − ξ)/α} location parameter = ξ

α−1 −x x exp( β ) F(x) = α shape parameter = α 2 GAM β Γ(α) Γ is the gamma function scale parameter = β shape parameter = κ F(x) = exp{− exp(−y)} 3 GEV scale parameter = α y = κ−1 log{1 − κ(x − ξ)/α} location parameter = ξ = κ F(x) = 1 shape parameter 4 GLO {1+exp(−y)} scale parameter = α −1 y = −κ log{1 − k(x − ξ)/α} location parameter = ξ shape parameter = κ F(x) = φ(y) 5 GNO scale parameter = α y = −κ−1 log{1 − k(x − ξ)/α} location parameter = ξ

F(x) = 1 − exp(−y) shape parameter = κ 6 GPA n κ( −ξ) o scale parameter = α y = −k−1 log 1 − x α location parameter = ξ scale parameter = α 7 GUM F(x) = exp[− exp{−(x − ξ)/α}] location parameter = ξ ( −µ)2 scale parameter = σ 8 NOR F(x) = √1 exp − x σ 2π (2σ2) location paramete = µ

α− |x−ε| 1 exp(−|x−ε|/β) scale parameter = σ F(x) = α 9 P3 β Γ(α) shape parameter = γ 4 1 If γ 6= 0, then let α = γ2 ,β = 2 σ|γ|, and ξ = µ − 2σ/γ location parameter = µ = β h i scale parameter 10 WEI F(x) = 1 − exp −{(x − ζ)/β}δ shape parameter = δ location parameter = ζ

3.2. Parameter Estimation Method Linear moment (LM)method offers small bias, which is an advantage over all other parameter estimation methods [12,33]. Let X1,X2, ... ,Xr be a conceptual random sample of size r, from a −1 continuous function which is quantiﬁed by the function QX(u) = F X(u); let X1r ≤ X2r ≤ ... ≤ Xrr denote the corresponding ordered statistics. Hosking (1990) deﬁned the rth L-moment as follows [34]:

r−1 ! −1 k r − 1 λr = r ∑ (−1) E(Xr−kr), (1) k=0 k where the order of moment is denoted by r (r = 1, 2 . . . ,) and

1 r! Z − E(X ) = Q(u)ut−1(1 − u)r tdu (2) tr (t − 1)!(r − t)! 0

L-moment ratios, the L-coefﬁcient of variation (τ2), L-skewness (τ3), and L-kurtosis (τ4) are deﬁned as λ2 λ3 λ4 τ2 = , τ3 = , τ4 = (3) λ1 λ2 λ2 Water 2018, 10, 1603 6 of 18

3.3. The Goodness-of-Fit and Selection Criteria The Kolmogorov-Smirnov (KS) is a goodness-of-fit statistics which is used to decide if a sample comes from a population with a specific distribution [33]. At the selected level of significance, the p-value is scrutinized for testing the hypothesis. The KS test is defined by the following equation:

i − 1 i D = max F(Xi) − , − F(Xi) , (4) 1≤i≤n n n where sample size is denoted by n, and the cumulative distribution function is expressed by F(Xi). We conducted the KS test for 11 locations shown in Table3 to determine whether the data obeys the distribution. We computed the p-values for each distribution to provide a testimony of ﬁtness of PDs for obeying the empirical data. When the p-value is greater than 0.05, it implies that the data does not obey the distribution at 5% signiﬁcance level [34,35].

Table 3. Computed p-values of contending distributions for the selected locations.

Gauging Stations EXP GUM GPA GNO GLO GEV GAM P3 WEI NOR Attock 0.032 0.015 0.048 0.015 0.015 0.031 0.016 0.015 0.032 0.038 Jindi 0.046 0.004 0.034 0.010 0.009 0.014 0.024 0.032 0.016 0.019 Munda Headworks 0.038 0.085 0.038 0.027 0.022 0.025 0.965 0.049 0.051 0.989 Khwazakhela 0.014 0.051 0.002 0.002 0.014 0.014 0.014 0.002 0.001 0.061 Ningolai 0.071 0.049 0.035 0.035 0.035 0.035 0.035 0.070 0.015 0.075 Khiali 0.072 0.072 0.041 0.045 0.042 0.049 0.045 0.045 0.046 0.037 Shah Alamriver 0.018 0.073 0.018 0.039 0.071 0.050 0.018 0.018 0.018 0.054 Naguman 0.002 0.018 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.051 Adezai 0.080 0.039 0.018 0.018 0.039 0.039 0.018 0.018 0.018 0.052 Timergara 0.013 0.014 0.053 0.014 0.014 0.019 0.016 0.016 0.014 0.061 Karot 0.029 0.051 0.029 0.007 0.0009 0.0009 0.050 0.050 0.041 0.082

An appropriate PD must be selected by making use of the statistical criteria [18]. In this regard, the PDs are evaluated by two criterions, the Akaike information criteria (AIC) and the root mean square error (RMSE). AIC refers to an information-based criterion which encourages the selection of PDs under definite conditions [35,36]. We employed RMSE, which is another criterion offering selection in this study, and is commonly directed for measuring the fit of the PDs to the available record. We persisted with these criterions to measure the descriptive strength of PDS. These are defined by the following Equations [33,37]: r ∑n (P(i) − P(i))2 RMSE = i=0 , (5) n " " n ## 1 2 AIC = ln ∑ Pthe(i) − Pemp(i) + 2K (6) n i=0 where n is the size of the sample, K is the number of parameters of the distribution, Pthe is the theoretical probability of the distribution, and Pemp is empirical probability measured by the Gringorten plotting position formula [38]:

P i − 0.44 . (7) emp(i)= n + 0.12 The comparative performances of PD are measured by AIC, while RMSE measures the ﬁtting of candidate distribution to the AM series of a gauging station. We used Equations (5) and (6) to compute the values of AIC and RMSE corresponding to each PD. The lower AIC and RMSE values, as illustrated in Table4, indicates a better distribution so the best-ﬁt PD will have the minimum value of AIC and RMSE. Water 2018, 10, 1603 7 of 18

Table 4. Calculated values of Akaike information criteria (AIC) and root mean square error (RMSE) of probability distributions for different locations.

Distributions Criteria Attock Jindi Adezai Naguman Timergara Khiali Khwazakhela Munda Ningolai Shah Alam Karot NOR RMSE 0.0457 0.0566 0.0823 0.0989 0.0854 0.1187 0.1385 0.1177 0.179 0.986 0.1471 AIC −290.16 −275.33 −148.77 −137.43 −161.29 −202.77 −128.39 −223.58 −107.52 137.61 −161.13 WEI RMSE 0.0404 0.0404 0.0601 0.0319 0.0494 0.092 0.0393 0.0671 0.0732 0.056 0.0755 AIC −301.98 −308.33 −168.28 −207.58 −198.4 −227.73 −214 −296.47 −166.47 −172.7 −201.55 P3 RMSE 0.0385 0.0394 0.0644 0.0318 0.0486 0.0909 0.0407 0.068 0.1045 0.0584 0.0893 AIC −306.65 −310.77 −163.97 −207.7 −199.51 −228.98 −211.55 −294.99 −143.01 −170.1 −192.08 GAM RMSE 0.0385 0.0422 0.062 0.0357 0.0487 0.0921 0.0792 0.082 0.0741 0.0606 0.1053 AIC −306.48 −304.15 −166.3 −200.46 −199.49 −227.65 −166.38 −274.04 −165.69 −167.76 −178.55 GEV RMSE 0.0387 0.0392 0.0712 0.0301 0.0477 0.0813 0.0489 0.0472 0.0587 0.0711 0.0446 AIC −306.18 −311.37 −157.74 −211.13 −200.81 −239.85 −199.15 −335.73 −181.03 −157.87 −248.84 GLO RMSE 0.0374 0.0393 0.0804 0.0337 0.0486 0.0759 0.0535 0.0424 0.0596 0.0801 0.0414 AIC −309.45 −310.95 −150.27 −204.02 −199.63 −246.59 −193.08 −347.81 −180.07 −150.5 −255.08 GNO RMSE 0.0382 0.039 0.0689 0.0281 0.0474 0.0848 0.0414 0.054 0.0572 0.0663 0.0603 AIC −307.44 −311.77 −159.79 −215.33 −201.24 −235.72 −210.45 −320.7 −182.75 −162.18 −224.25 GPA RMSE 0.0569 0.0486 0.0511 0.032 0.0514 0.0949 0.0408 0.0655 0.0566 0.0512 0.0655 AIC −269.03 −290.11 −178.26 −207.37 −195.22 −224.73 −211.39 −299.16 −183.42 −178.26 −217.39 GUM RMSE 0.0417 0.0391 0.0711 0.0597 0.0556 0.0888 0.1018 0.0783 0.1425 0.0772 0.1049 AIC −298.95 −311.44 −157.9 −168.67 −190.48 −231.25 −149.33 −279.22 −122.58 −152.78 −178.84 EXP RMSE 0.0786 0.066 0.064 0.0317 0.053 0.096 0.0638 0.67 0.104 0.0515 0.0779 AIC −238.13 −260.41 −163.87 −207.96 −193.66 −223.64 −181.12 −296.73 −143.07 −177.89 −203.25

3.4. Monte Carlo Simulation The goodness-of-fit indicates the goodness within the reach of available record; therefore, it is unable to indicate the performance of PDs at the right-tail segment. The performance of PDs regarding right-tail events is significant for the design of critical hydraulic structures for selection of design flood against a small exceedance probability [27]. Moreover, to reduce the lifetime risk of hydraulic structures, the flood quantile with a very small exceedance probability has to be chosen. Therefore, in order to develop the convincing idea of the behavior of contending PD at the extreme tail, we probed PD by carrying out the Monte Carlo simulation. Accordingly, a base distribution should be chosen other than the contending distributions. Cunnane (1989) proposed the five-parameter Wakeby distribution as a parent distribution for its versatility [8]. Considering its credibility, we adopted it as a base distribution for the generation of long synthetic data series to examine the performance of PDs corresponding to rare events. We employed a historical record of four major rivers to obtain parameters of parent distribution, and later estimated random number samples of different sizes n (n = 20, 50, 100). The parameters of the Wakeby distribution are summarized in Table5. Houghton (1978) defined the inverse function of Wakeby distribution as follows [39]:

α β γ −δ x(F) = ξ + 1 − (1 − F) − 1 − (1 − F) . (8) β δ

Table 5. Parameters values of Wakeby distribution for ﬁve stations in Pakistan.

Parameter ξ α β Υ δ Panjkora river 130.97 677.34 0.12 0.00 0.00 Khiali 293.02 4122.62 2.87 52.32 0.82 Shah Alamriver 30.11 206.28 0.164 0.00 0.00 Ningolai 0.83 0.00 0.00 135.22 0.44 Adezai 97.43 1032.69 0.34 0.00 0.00

We obtained parameters of base distribution from the observed data and generated the random samples of different lengths n (n = 20, 50, 100). Next, we calculated flood quantiles of contending distributions from the random samples against a higher return periods T (T = 100, 200). The higher return periods of 100 and 200 years induce right-tail for each PD, so the simulation setup was repeated 1000 times to obtain flood quantiles at the extreme tail for different random samples. Similarly, to quantify the degree of prediction of extreme events, the composition proceeded for the selected Water 2018, 10, 1603 8 of 18 four rivers for each of the candidate distributions. Later, we employed three performance indexes, namely, relative bias (RB), absolute relative bias (ARB), and relative root mean square error (RRMSE), for the flood quantiles secured by PDs at the extreme events. These are defined by the following equations [40,41]: S ! 1 Qˆ T − QT RB = ∑ i 2, (9) S i=1 QT

s Qˆ − Q 1 T i T ARB = ∑ , (10) S i=1 QT v u ! u 1 n Qˆ − Q RRMSE = t T T 2, (11) − ∑ S 1 i=0 QT ˆ ˆ where QT is given parent quantile, QT 1 ... QT s are estimators for the samples generated by parent distribution, and S is the number of Monte Carlo trials. The lower RB, ARB and RRMSE values endorse the most reliable distribution. We used Equations (9)–(11) to compute the values of RB, ARB, and RRMSE. We considered the ﬂood quantiles having return periods of 100 and 200 years, to probe the PDs at the right-tail. Additionally, we engaged boxplots with quantiles estimates for higher return periods of T = 100 and T = 200, and a small return period of T = 10. Boxplots for each contending PD at every station was plotted to assess their performance. We analyzed narrowness of 90% percentile of boxplots and parent quantiles to evaluate the performance of PDs. The 90% conﬁdence interval is represented by each boxplot in each case. The 50% interquartile range is represented by the box with 75% and 25% as upper and lower bounds. Moreover, the upper and lower bounds of whiskers represent 95% and 5% percentile.

4. Results

4.1. The Goodness-of-Fit and Selection of PDs The p-value manifested in Table3 indicates the fitting of sample data to a particular distribution at 5% significance level. To this extent, the p-values lower than 0.05 indicates that the historical record of the AM series belongs to the particular distribution [18]. It can be inferred from the results illustrated in Table5 that NOR distribution is not in consonance with the historical record of six locations at 5% significance level. Next, the GUM distribution does not obey the available record of Munda Headworks, Khwazakhela, Karot, Khiali and Shah Alam. In addition to that, EXP distribution does not follow the empirical data of Ningolai, Khialiand Adezai, with higher p-value. Meanwhile, P3, GAM and GLO failed to follow AM series at one location each. For other locations, the assumption that the historical data belongs to the candidate distribution cannot be rejected at the 5% significance level. Later, we employed RMSE and AIC to compare the results of contending distributions as manifested in Table4. We computed values of RMSE and AIC for each PD at all stations. RMSE and AIC values for GLO are smallest at Attock, Khiali and Munda Headworks. Moreover, GNO adhered to the lowest values for Jindi, Naguman and Panjkora. GPA and WEI fit the AM peak flow series of Adezai and Khwazakhela, respectively possessing lowest RMSE and AIC values at these gauging stations. It can be inferred from the above that different PDs fits the historical AM flood series of different gauging stations. As indicated in Table4, GLO persisted with better performance at four locations, while GNO and GPA advanced with smaller magnitudes of goodness criteria on three locations each. WEI was chosen as the best distribution for Khwazakhela gauging station. Table4 demonstrates the fitted PDs corresponding to each location in the Indus basin system in Pakistan. The plot in Figure3a–d illustrates the fitted distributions for observed data at Karot, Khwazakhela, Timergara and Ningolai. Figure3 yields the fitted distributions to the empirical frequencies of the respective stations. The empirical frequencies are shown by lines, while the points represent the fitted distribution. The plots demonstrated that GLO, WEI, GNO and GPA are the best-fit for the AM Water 2018, 10, 1603 9 of 18 series of the respective gauging stations. It can be inferred from the empirical histogram plotted in Figure2a–d that the distributions that describe the datasets of the respective gauging stations also represent the AM flood series very well. The PDFs of the distributions are plotted to fit the empirical histograms of available records. GLO generally fit the data series with very high skewness, while GPA fit with moderate skewness, except for Ningolai. WEI and GNO also fit the AM flood series with high skewness.

4.2. Performance at Right-Tail of PDs We engaged Wakeby distribution for generation of synthetic data series of different sizes n (n = 20, 50, 100) from the historical data series of Indus basin system of Pakistan to assess the quantile estimates of four PDs with return periods T (T = 100 and 200). The computed values of RB, ARB, and RRMSE are illustrated in Table6. The smallness in the magnitude of RB, ARB, and RRMSE indicates better performance of the candidate distribution. Moreover, we also used the boxplots to determine the goodness of prediction.

Table 6. Calculated relative bias (RB), absolute relative bias (ARB) and relative root mean square error (RRMSE) values for different probability distributions.

Title Sample Size Return Period GLO GNO GPA WEI RB ARB RRMSERB ARB RRMSERB ARB RRMSERB ARB RRMSE T = 100 18.24 27.07 35.31 11.75 24.30 31.43 33.38 36.73 42.78 4.46 21.05 26.52 n = 20 T = 200 33.69 40.16 52.11 18.36 31.63 41.03 3.90 28.14 36.20 7.17 25.08 32.30 Panjkora T = 100 17.76 20.98 25.82 9.57 16.10 20.55 1.18 14.93 18.84 3.61 13.23 17.03 n = 50 T = 200 31.92 33.43 40.26 18.65 22.72 28.38 3.59 16.88 21.70 7.09 15.53 19.85 T = 100 17.58 18.69 22.31 11.16 13.61 16.74 1.36 10.03 12.68 2.95 9.56 12.31 n = 100 T = 200 30.72 31.03 35.18 16.79 18.58 23.18 0.43 12.37 15.47 5.69 10.92 14.08 T = 100 22.74 27.00 33.19 17.47 22.80 28.46 5.09 15.21 19.55 11.69 18.31 23.19 n = 20 T = 200 42.36 43.84 52.62 25.90 30.09 37.36 16.44 17.59 23.54 17.91 22.70 28.57 Adezai T = 100 24.13 24.78 28.56 14.83 16.47 19.82 6.19 8.70 11.49 9.03 12.25 15.19 n = 50 T = 200 39.41 39.59 44.00 24.22 25.02 29.22 9.63 10.76 13.28 15.16 16.81 20.74 T = 100 22.67 22.72 24.76 14.89 15.25 17.44 0.47 7.53 9.33 9.03 10.43 12.54 n = 100 T = 200 38.87 38.88 41.09 23.49 23.55 25.98 4.69 6.58 8.70 14.02 14.49 16.82 T = 100 21.21 27.75 35.79 12.13 23.50 30.79 7.55 16.54 23.97 6.29 20.94 26.77 n = 20 T = 200 35.70 40.98 52.12 21.77 30.43 40.00 6.09 26.19 34.67 9.85 22.94 30.35 Shah Alam T = 100 18.41 20.67 25.47 12.23 17.14 21.39 6.43 11.99 15.54 4.30 12.63 15.94 n = 50 T = 200 34.92 35.68 42.24 18.94 22.13 28.02 3.87 14.57 19.21 7.62 15.26 19.26 T = 100 17.98 18.45 21.62 10.92 12.89 16.00 4.12 9.32 11.52 4.91 9.80 12.26 n = 100 T = 200 32.74 32.87 36.43 19.28 20.38 24.31 2.62 9.81 12.86 8.00 11.40 14.10 T = 100 −8.75 44.99 58.65 3.37 53.79 84.19 −1.14 42.05 55.51 5.13 60.97 117.79 n = 20 T = 200 −2.94 51.54 70.27 −1.16 58.74 97.14 14.45 55.43 77.57 −8.95 60.81 92.93 Ningolai T = 100 −7.46 30.21 37.94 1.41 40.17 62.79 −7.45 32.66 43.03 0.83 43.18 79.70 n = 50 T = 200 −3.60 36.65 49.91 1.80 45.10 70.49 −3.33 42.80 57.98 −4.84 47.76 79.09 T = 100 −4.49 24.39 31.87 −0.24 29.81 41.78 −4.43 26.49 34.33 −2.26 34.32 58.51 n = 100 T = 200 1.18 28.16 39.73 −0.17 35.33 61.11 −5.77 32.47 43.63 −9.64 37.68 63.03

For Panjkora, the RB of GLO is high, and its performance descends when the return period is increased for all sample sizes. By contrast, GPA performed better than other contending distributions, with respect to RB, especially with the growth of the return period for all sample sizes. In terms of ARB and RRMSE, WEI and GPA perform better for all return periods and sample sizes, particularly, GPA behaved well for all sample sizes. To this extent, the robustness of GPA is observed, which is closely followed by WEI. From Figure4, it can be concluded that GPA and WEI maintained narrowest 90% confidence interval for right-tail events, and the parent quantile is also observed within a 50% interquartile range. GLO and GNO failed to maintain the narrowest 90% confidence interval, although the true population was observed within a 90% confidence interval. The results for a smaller return period of T=10 are comparable, however, GPA and WEI performed slightly better here as well. Water 2018, 10, 1603 10 of 18 Water 2018, 10, x FOR PEER REVIEW 10 of 19

2000 4000 6000 8000 10,000 14,000 0 5000 10,000 15,000 Flow (m3/s) Flow (m3/s) (a) Karot

1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000

Flow (m3/s) Flow (m3/s) (b) Khwazakhela

0 1000 2000 3000 0 1000 2000 3000 4000

Flow (m3/s) Flow (m3/s) (c) Timergara

0 500 1000 1500 0 500 1000 1500

Flow (m3/s) Flow (m3/s) (d) Ningolai

FigureFigure 3. Cumulative 3. Cumulative distribution distribution and theand empirical the empirical histograms histogram of ﬁtteds of distributionsfitted distributions for (a) for Karot, (a) (Karotb) , Khwazakhela,(b) Khwazak (c)hela Timergaraand, (c) Timergaraand (d) Ningolai. (d) Ningolai. Water 2018, 10, x FOR PEER REVIEW 12 of 19

illustrates that all the contending distributions contained the true population within the 50% interquartile range, and the boxplots for smaller return periods are comparable. However, for right-tail events, WEI exhibited the smallest 90% confidence interval for all sample sizes. GPA also performed better, except for a single event of smaller sample size. On the other hand, GNO and GLO performed quite poorly in this case. From these observations, it can be inferred that GPA and Weibull persistently performed better than other distributions, whereas the performance of GNO and GLO is poor for all cases. Hence, GPA and WEI predicted rare floods of 100 years and 200 years, better than the other contending distributions and these can be trusted by engaging in FFA of Waterselected2018, 10 ,rivers 1603 in Pakistan. 11 of 18 Water 2018, 10, x FOR PEER REVIEW 12 of 19

FigureFigure 4. Boxplot4. Boxplot for for GLO, GLO, GNO, GNO, GPA GPA and and WEI WEI for for Panjkora Panjkora river. river.

For Adezai, again, the RB of GLO endured higher magnitudes among all PDs at the right-tail portion for all sample sizes. The performance of GPA is significant in contrast to other PDs in terms of RB with high return periods T = 100 and200 for all sample sizes. The performance of GPA is significant when evaluated in terms of ARB and RRMSE, while the performances of GNO and GLO are poor. WEI also advanced with better performance in contrast to an inferior showing of GNO. Figure5 exhibits the robustness of GPA when compared to other available options, especially GNO and GLO. The true population is not falling within the 90% confidence interval for GLO, particularly for larger sample sizes in contrast to other available options. Additionally, GPA observed the narrowest 90% confidence interval for right-tail events. WEI also performed better particularly when compared to Figure 4. Boxplot for GLO, GNO, GPA and WEI for Panjkora river. GLO and GNO.

Figure 5. Boxplot for GLO, GNO, GPA and WEI for Adezai river.

Figure 5. Boxplot for GLO, GNO, GPA and WEI for Adezai river. Figure 5. Boxplot for GLO, GNO, GPA and WEI for Adezai river. For Shah Alam river, the trend of poor performance of GLO continued as the performance indicator shows a downward trend. GPA behaved much better in terms of RB when the return period increased, regardless of sample size. Again, GPA adhered to smaller values of ARB and RRMSE for different sample sizes and return periods, which is closely followed by the WEI distribution. The values of ARB and RRMSE areFigure considerably 6. Boxplot for smaller GLO, forGNO, GNO GPAand with WEI a larger for Shah sample Alam size river. of 100. In this way, the behavior of GPA is superior to other distributions when evaluated in terms of these indicators. According to Figure6, the narrowest conﬁdence interval of 90% is observed by WEI, which is followed by GPA. Additionally, the parent quantile for GPA and WEI is also falling within the 50% interquartile range except for a single instance for both GPA and WEI. Furthermore, GLO performed poorly by failing to contain the true population within a 90% conﬁdence interval for large sample size. Hence, the behavior of GLO and GNO is quite poor.

Figure 6. Boxplot for GLO, GNO, GPAand WEI for Shah Alam river.

Water 2018, 10, x FOR PEER REVIEW 12 of 19

Figure 4. Boxplot for GLO, GNO, GPA and WEI for Panjkora river.

Water 2018, 10, 1603 Figure 5. Boxplot for GLO, GNO, GPA and WEI for Adezai river. 12 of 18

Figure 6. Boxplot for GLO, GNO, GPAand WEI for Shah Alam river. Figure 6. Boxplot for GLO, GNO, GPAand WEI for Shah Alam river. Whereas for Ningolai, GLO performed substantially well in terms of RB, whereby the lowest RB was observed for all sample sizes by GLO. RB value of WEI was observed as lowest for T = 200 for larger sample sizes. Additionally, in terms of ARB and RRMSE, the behavior of GLO is superior to other distributions, whereas GPA observed lower ARB and RRMSE values. Moreover, GPA also persisted with good performance by obtaining lower values of RB, ARB and RRMSE. Figure7 illustrates that all the contending distributions contained the true population within the 50% interquartile range, and the boxplots for smaller return periods are comparable. However, for right-tail events, WEI exhibited the smallest 90% conﬁdence interval for all sample sizes. GPA also performed better, except for a single event of smaller sample size. On the other hand, GNO and GLO performed quite poorly in this case. From these observations, it can be inferred that GPA and Weibull persistently performed better than other distributions, whereas the performance of GNO and GLO is poor for all cases. Hence, GPA and WEIWater predicted 2018, 10, x rare FORﬂoods PEER REVIEW of 100 years and 200 years, better than the other contending distributions14 of 19 and these can be trusted by engaging in FFA of selected rivers in Pakistan.

Figure 7. Boxplot for GLO, GNO, GPA and WEI for Ningolai river. Figure 7. Boxplot for GLO, GNO, GPA and WEI for Ningolai river. 4.3. Comparison of Flood Quantiles 4.3.To Comparison address any of Flood uncertainty, Quantiles we carried out at-site frequency analysis for the available record of five locationsTo address within any the uncertainty, Indus basin we of Pakistancarried out for at-site adopting frequency the best-fit analysis distribution. for the available The magnitudes record of of floodfive locations quantiles correspondingwithin the Indus to return basin period of Pakistan are illustrated for adopting in Table 7the. The best-fit quantiles distribution. estimated byThe differentmagnitudes PDs against of flood smaller quantiles return corresponding periods are comparable to return to period each other. are illustrated However, thein differenceTable 7. The in magnitudequantiles is estimated sizeable forby higherdifferent return PDs periods.against smalle In thisr case,return GLO periods hasthe are largest comparable quantiles to each against other. a higherHowever, return the period difference of 200 years,in magnitude whereas is the sizeable lower for return higher periods return almost periods. induce In this similar case, magnitudesGLO has the of thelargest hydrological quantiles variable.against a Figurehigher8 a–ereturn demonstrates period of 200 the years, flood frequencywhereas the plots lower of empirical return periods and estimatedalmost floodinduce quantiles similar ofmagnitudes the four distributions. of the hydrologic Fromal the variable. flood frequency Figure 8a–e curves, demonstrates it can be concluded the flood thatfrequency higher estimates plots of empirical of quantiles and areestimated consummated flood quantiles by GLO of and the GNO,four distributions. but the frequency From the curves flood frequency curves, it can be concluded that higher estimates of quantiles are consummated by GLO and GNO, but the frequency curves for Karot, Khwazakhela and Attock are linear for WEI. Furthermore, frequency curves at Khiali and Adezai are linear for GPA. Considering the frequency curves, it can be inferred that GPA and WEI are unfolded as more viable options for estimation of flood quantiles.

Table 7. Flood quantiles estimated by different PDs in Pakistan.

Location Distribution Return Period 20 50 80 100 200 GLO 19,284 21,813 23,177 23,843 25,995 GNO 19,322 21,324 22,298 22,750 24,122 Attock GPA 19,128 19,163 20,232 20,333 20,568 WEI 19,284 21,042 21,858 22,229 23,322 GLO 3306 4337 4975 5308 6490 GNO 3402 4289 4776 5016 5794 Khiali GPA 3447 4142 4475 4628 5081 WEI 3431 4181 4558 4735 5280 GLO 1995 2570 2904 3073 3646 GNO 2028 2503 2748 2866 3235 Adezai GPA 2037 2331 2425 2500 2633 WEI 2031 2435 2630 2720 2991 GLO 7567 11,071 13,545 14,926 20,287 GNO 8156 11,579 13,711 15,823 18,721 Karot GPA 8099 11,401 13,480 14,575 18,479 WEI 8484 11,490 13,160 13,980 16,638 GLO 3647 5462 6701 7381 9956 GNO 3915 5619 6645 7171 8979 Khwazakhela GPA 3934 5542 6491 6975 8627 WEI 4050 5521 6314 6700 7928

Water 2018, 10, 1603 13 of 18

for Karot, Khwazakhela and Attock are linear for WEI. Furthermore, frequency curves at Khiali and Adezai are linear for GPA. Considering the frequency curves, it can be inferred that GPA and WEI are Waterunfolded 2018, 10 as, x more FOR PEER viable REVIEW options for estimation of ﬂood quantiles. 15 of 19

(a) (b)

(e)

FigureFigure 8 8.. FloodFlood fre frequencyquency curves ofof GLO,GLO, GNO,GNO, GPA GPA and and WEI WEI for for (a ()a Attock,) Attock (b, )(b Khiali,) Khiali (c,) ( Adezai,c) Adezai (d,) (Karotd) Karot and and (e) ( Khwazakhela.e) Khwazakhela.

Water 2018, 10, 1603 14 of 18

Table 7. Flood quantiles estimated by different PDs in Pakistan.

5. Discussions This study gives us a convincing lead after probing PDs with more consideration being given to the right-tail quantiles to minimize risk to important hydraulic structures in Pakistan. We used the KS method to evaluate the fitting of PDs to the historical data of the selected stations. From our results, the goodness-of-fit of NOR distribution could not concede to six stations. In addition, we observed that GUM could not follow the empirical data of Shah Alam, Munda Headworks, Khwazakhela, Karot and Khiali, while the rest of the PDs fit to the historical data series, except for a few instances. The performance indicators for evaluating descriptive ability were applied for selection of PDs for various locations. In this case, the AIC and RMSE values of all the PDs were estimated, and divergent results were received in return. GLO advanced with the lowest AIC and RMSE values for four stations, while GPA and GNO were found fitting the AM series of three stations each. Additionally, WEI fitted the historical data series of Khwazakhela. The Cumulative Distribution Function (CDF) plots for GLO, GNO, GPA and WEI indicate the fitting of theoretical frequencies of these distributions to the empirical frequencies of respective gauging stations. The CDF plots of PDs supported the results presented by goodness-of-fit test by fitting to the empirical frequencies. The empirical histograms are also plotted for GLO, GNO, GPA and WEI against the respective fitted observed data. The empirical histograms indicated the fitting of these distributions to the observed data. Next, we evaluated four PDs, namely GLO, GNO, GPA and WEI, for their predictive behaviors by using Monte Carlo (MC) simulations. The results of the Monte Carlo simulation were assessed through performance indicators, such as RB, ARB and RRMSE. The PDs were also evaluated with the help of boxplots. As a result, we identified GPA and WEI as the best choice for estimating right-tail quantiles in all cases. The flood quantiles for different return periods were estimated by GLO, GNO, GPA and WEI for five stations. The largest flood quantiles are exhibited by GLO for higher return periods, whereas, for lower return periods, the flood quantiles estimated by all distributions are almost identical. The plots for flood frequency curves were developed for the observed and estimated flood quantiles of contending distributions. The curves demonstrate the linearity of frequency curves of GPA and WEI Water 2018, 10, 1603 15 of 18 for the selected five locations. Since our study focuses on proposing PDs for Pakistan to counter the menace of flash floods, therefore, the performance against higher return periods is highly endorsed. Different PDs have been identified by researchers in Pakistan during the last decade employing goodness-of-fit tests. Zamir (2009) identified GNO as the best-fit PD followed by GPA from estimated flood quantiles and regional growth curve using the Monte Carlo method [42]. Batool (2017) carried out FFA on 17 locations of Chenab, Ravi, Sutlej, Indus, and Jhelum river, and found that GPA suits most of the locations after applying the goodness-of-fit test [43]. Zakaullah (2012) advocated GUM distribution after carrying out FFA of Jhelum river in Pakistan by applying chi-square test [44]. Ishfaq (2015) recommended GPA by conducting at-site flood frequency analysis using L-moments, TL-moments, and maximum likelihood estimation, using annual maximum stream flows in Pakistan after employing goodness-of-fit tests [45]. Another study carried out by Ishfaq (2015) suggested the use of GPA for different locations in Pakistan using annual maximum rainfall data, by adopting L-moments and TL-moments [46]. The PDs identified in these studies are obtained usually by the goodness-of-fit tests. Therefore, we performed a Monte Carlo simulation to thoroughly analyze the performance of PDs at the right-tail section, followed by estimation of flood quantile against return periods for several locations in Pakistan. Overall, even though a few studies used GNO or GUM distribution functions to characterize the dynamics of the flood, most previous studies are still consistent with our finding that the GPA usually performs best over Pakistan. The classical goodness-of-fit test in this study also unveiled GNO and GLO as the most reliable PD for a few locations, but they consistently exhibited poor performance when subjected to estimating flood quantiles against small exceedance probabilities. On the contrary, when we employed WEI distribution, our results confirmed its robustness, particularly when the performance is evaluated at the extreme-tail. WEI distribution also proved an excellent fit to the frequency curves of historical records of several locations, making it a practical option for deriving design flood in Pakistan. The flood frequency analysis is very important for measuring the flood risk and hydrologic structure design, so our finding of best-fit distribution functions may provide rich information as a reference for water resource utilization and management in Pakistan.

6. Conclusions In this study, we characterized the selection of appropriate PD for quantile estimation reflecting right-tail events for different return periods. It is apparent that peak flood with small exceedance probability has to be considered for the safety of the hydraulic structures. Beginning with quantifying the degree of fit, we carried out hypothesis test KS, and engaged goodness criteria of selection. The goodness criteria of AIC and RMSE evaluated the performance of all PDs, which signified four distributions, namely GLO, GNO, GPA and WEI for FFA of rivers. We analyzed the right-tail events with the assistance of RB, ARB, RRMSE and boxplots. The lower values of these criterions and the smallness of boxplots, along with containing parent quantile within 90% confidence interval of the boxplots, suggested goodness of predictive behavior. We used the selected four PDs for four locations which embraced GPA and WEI as the most viable options. To address any uncertainty, we quantified the flood quantiles and plotted frequency curves. The results of frequency curves suggest GPA and WEI as practical options for carrying out FFA of rivers in Pakistan. Based on the above analysis of our results, we conclude that GPA and WEI distributions provide excellent fitness criteria for all cases. Therefore, we can choose these two PDs for estimation of flood quantiles within various locations at river basins in Pakistan. These results can be consumed for construction of hydraulic structures, development of agriculture, and conservation of natural resources. Our findings can help the national government for planning and managing water resources accordingly.

Author Contributions: Conceptualization and software, M.R. and S.G.; Formal analysis, X.F. and J.Y.; Writing-Original Draft, M.R.; Writing-Review and Editing, S.G. Water 2018, 10, 1603 16 of 18

Funding: We are grateful for the funding from National Key R&D Plan of China (Grant No. 2016YFC0402206) and the National Natural Science Foundation of China (Grant No. 51879192). We also thank the Hydrology and irrigation Division, Peshawar in Pakistan for providing the hydrological data. Acknowledgments: We are very grateful to the editor and two anonymous reviewers for their valuable comments and constructive suggestions that helped us to greatly improve the manuscript. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

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