regional frequency analysis of extremes precipitation...
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Research ArticleRegional Frequency Analysis of Extremes Precipitation UsingL-Moments and Partial L-Moments
Said Arab Khan12 Ijaz Hussain1 Tajammal Hussain3 Muhammad Faisal45
Yousaf ShadMuhammad1 and Alaa Mohamd Shoukry67
1Department of Statistics Quaid-i-Azam University Islamabad Pakistan2Government Degree College Lahore Swabi Khyber Pakhtunkhwa Pakistan3Department of Statistics COMSATS Institute of Information Technology Lahore Pakistan4Faculty of Health Studies University of Bradford Bradford UK5Bradford Institute for Health Research Bradford Teaching Hospitals NHS Foundation Trust Bradford UK6Arriyadh Community College King Saud University Riyadh Saudi Arabia7Workers University Cairo Egypt
Correspondence should be addressed to Ijaz Hussain ijazqauedupk
Received 21 December 2016 Revised 26 January 2017 Accepted 7 February 2017 Published 7 March 2017
Academic Editor Mouleong Tan
Copyright copy 2017 Said Arab Khan et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Extremes precipitation may cause a series of social environmental and ecological problems Estimation of frequency of extremeprecipitations and its magnitude is vital for making decisions about hydraulic structures such as dams spillways and dikes In thisstudy we focus on regional frequency analysis of extreme precipitation based on monthly precipitation records (1999ndash2012) at 17stations of Northern areas and Khyber Pakhtunkhwa Pakistan We develop regional frequency methods based on L-moment andpartial L-moments (L- and PL-moments) The L- and PL-moments are derived for generalized extreme value (GEV) generalizedlogistic (GLO) generalized normal (GNO) and generalized Pareto (GPA) distributions The 119885-statistics and L- and PL-momentsratio diagrams of GNO GEV and GPA distributions were identified to represent the statistical properties of extreme precipitationin Northern areas and Khyber Pakhtunkhwa Pakistan We also perform a Monte Carlo simulation study to examine the samplingproperties of L- and PL-moments The results show that PL-moments perform better than L-moments for estimating large returnperiod events
1 Introduction
Hydraulic and hydrologic designs are key steps in planningof any water project Any problem pitched at designingstage will result in the failure of design irrespective of thefact how correctly the other steps are taken Hydrologistsdeal with water-related issues problems of quantity qualityand availability in the society that known as hydrologicevents Stochastic methods are often used to understandsources of uncertainties in physical processes that give riseto observed hydrologic events as precipitation and streamflow estimates depend on the past or future events Severalstatistical methods offered to minimize and summarize theuncertainties of observed data and frequency analysis is oneof them It determines that how often a particular event will
occur by estimating the quantile 119876119879 for return period of 119879where 119876 is the magnitude of the event that occurs at a giventime and location
Dalrymple [1] proposed regional frequency analysis(RFA) method for pooling various data samples It is index-flood procedure in hydrology Hosking et al [2] studiedthe properties of probability-weighted moments (PWMs)method based on RFA method This method is first used byGreis and Wood [3] and Wallis [4] Cunnane [5] reviewedtwelve methods of RFA and related regional PWMs algo-rithm
Initially PWMs are considered as an alternative parame-ter estimation method however it was difficult to interpretdirectly as measures of the shape and scale parameters ofdistribution RFA can forecast the flood flow using empirical
HindawiAdvances in MeteorologyVolume 2017 Article ID 6954902 20 pageshttpsdoiorg10115520176954902
2 Advances in Meteorology
formula and unit-hydrograph procedure Subramanya [6]and it can also estimate the quantiles of extreme precipitation
Hosking and Wallis [7] showed that RFA method basedon L-moments is used to detect homogeneous regions toselect suitable regional frequency distribution and to predictextreme precipitation quantiles at region of interest
Whilst L-moment methodology is effective in estimatingparameters it may not valid for predicting high returnperiod events Wang [8] suggested that relatively small floodsmight create disturbance in the analysis To overcome suchsituation a censored sample can be used as by Cunnane [9]
Wang [8] proposed partial probability-weightedmoments (PPWMs) for fitting the probability distributionfunction to the censored sample Partial L-moments(PL-moments) are variants of the L-moments and similar toPPWMs PL-momentsmethod has used in fitting generalizedextreme value (GEV) distribution for censored flood samples(see Wang [8 10 11] and Bhattarai [12]) Bhattarai [12] foundthat censoring flood samples are nearly thirty percent ofbasic L-moments
Shabri et al [13] used Trim L-moments (TL-moments)for the RFA and compared its performance with L-momentsSaf [14] determined hydrologically homogeneous region andregional flood frequency estimates by using index-floodtechnique along with L-moments for the West Mediter-ranean River Turkey L-moments method is also used toassign a suitable regional distribution for the individualsubregions and to assess their homogeneity see Abolverdiand Khalili [15] Hussain and Pasha [16] suggested theregional flood frequency analysis based on L-momentsTheyused discordancy measure for data screening and used thefour-parameter Kappa distribution with 500 simulationsfor the heterogeneity analysis Zakaria et al [17] used thePL-moments technique and found another link for thehomogeneity analysis Shahzadi et al [18] showed that thegeneralized normal (GNO) distribution is suitable for theregional quantile estimation at maximum return period andthe GEV distribution for the overall regions at low returnperiod based on relative RMSE and relative absolute biasMost commonly used statistical distributions for high climatemodeling are as follows the logistic distribution with threeparameters lognormal distribution with three parametersLog 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] Sheet al [25])Moreover GEV andGPAdistributions are suitableif data contains extremes values Over the years the GNOGEV generalized logistic (GLO) and GPA distributionshave been widely employed in the extreme value estimationof annual flood peaks In this study we aim to developRFA method based on L- and PL-moments approach Ourproposed method can be used at all levels of regional analysissuch as identification of homogeneous regions identificationand also testing of the suitable probability regional frequencydistribution based on 119885-statistic and the L- and PL-momentratio diagram and estimation of the flood quantile at siteof interest We use 17 sites of Northern areas and KhyberPakhtunkhwa as a case study to perform the analysis
We explain the methodology of the L-moments and PL-moments in Section 22 Section 23 shows the applicationof the RFA where we choose the appropriate distributionfor the regional analysis Section 33 provides the estimationof quantile for both the small and large return period Theresults of our simulations studywill be presented in Section 4
2 Material and Methods
21 Study Area and Data Sources The data was collectedfor this study from Karachi Data Processing Center throughPakistan Meteorological (PMD) Islamabad Monthly precip-itation data has been recorded from 1999 to 2012 There are17 meteorological stations of Northern areas and KhyberPakhtunkhwa Pakistan These stations are full precipitationregions that affect areas in Pakistan where water is essentialfor hydropower and flood plains Figure 1 shows the locationof the study area and geographic distribution of precipitationstations There is no missing value in this data set
22 Statistical Methods
221 The L-Moments Conventional moments method maybe used for estimating the parameters of probability distri-bution However this approach has some serious drawbacksRatio ofmoment estimators is biased and often assumption ofbeing normally distributed is violated Wallis et al [26] Fur-thermore it is sensitive to outliers Pearson [27]Therefore itis unreliable for skewed distributions
Hosking [28] proposed L-moments approach to over-come the above problems The L-moments may preciselydescribe the statistical properties of hydrological informa-tion and it can write as a linear function of the PWMs ThePWMs of order 119903 were properly described by Greenwood etal [29] as
119872119901119903119904 = 119864 [119909119901 119865 (119909)119903 1 minus 119865 (119909)119904] (1)
where 119903 119904 and 119901 are the real numbers and 119865(119909) is thecumulative distribution function of 119909 A functional case is
120573119903 = 11987211199030 (2)
Therefore
120573119903 = int10119909 (119865) 119865119903119889119865 (3)
where 119865 = 119865(119909) is the cumulative distribution function of arandom variable 119909 and 119903 is a nonnegative integer of the realnumber that is 119903 = 0 1 2 3 Therefore the first four L-moments which are the linear combinations of the PWMsare
1205821 = 12057301205822 = 21205731 minus 12057301205823 = 1205732 minus 61205731 + 12057301205824 = 201205733 minus 301205732 + 121205731 minus 1205730
(4)
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37∘0㰀0㰀㰀N
36∘0㰀0㰀㰀N
35∘0㰀0㰀㰀N
34∘0㰀0㰀㰀N
33∘0㰀0㰀㰀N
32∘0㰀0㰀㰀N
37∘0㰀0㰀㰀N
36∘0㰀0㰀㰀N
35∘0㰀0㰀㰀N
34∘0㰀0㰀㰀N
33∘0㰀0㰀㰀N
32∘0㰀0㰀㰀N
71∘0㰀0㰀㰀E 72∘0㰀0㰀㰀E 73∘0㰀0㰀㰀E 74∘0㰀0㰀㰀E 75∘0㰀0㰀㰀E 76∘0㰀0㰀㰀E
71∘0㰀0㰀㰀E 72∘0㰀0㰀㰀E 73∘0㰀0㰀㰀E 74∘0㰀0㰀㰀E 75∘0㰀0㰀㰀E 76∘0㰀0㰀㰀E
W
N
E
S
0 30 60 120 180 240(km)
StationsWater bodiesStudy area
Figure 1 Locations of Northern areas and Khyber Pakhtunkhwa and the meteorological stations (119909 axis indicates Longitude [E] 119910 axisindicates Latitude [N])
The L-moments have no units of measurement which arecalled the L-moments ratios The L-moments ratios pro-posed by Hosking [28] are computed as
120591 = 12058221205821
1205913 = 12058231205822
1205914 = 12058241205822 (5)
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where 120591 represents the L-coefficient of variation (L-Cv)1205913 represents the L-coefficient of skewness (L-Cs) and 1205914represents the L-coefficient of kurtosis (L-Ck)
The arranged sample is given as 119909(1) le 119909(2) le 119909(3) sdot sdot sdot le119909(119899) Wang [8] stated that the statistic
119887119903 = 1119899119899sum119894=1
(119894 minus 1) (119894 minus 2) sdot sdot sdot (119894 minus 119903)(119899 minus 1) (119899 minus 2) sdot sdot sdot (119899 minus 119903)119909(119894) (6)
is an unbiased estimator of 120573119903So
1198971 = 11988701198972 = 21198871 minus 11988701198973 = 61198872 minus 61198871 + 11988701198974 = 201198873 minus 301198872 + 121198871 minus 1198870
(7)
where 1198971 1198972 1198973 and 1198974 are the first four L-moments of thesample And similarly
119905 = 11989721198971 1199053 = 11989731198972 1199054 = 11989741198972
(8)
are the sample L-moments ratios
222 The Partial L-Moments Wang [8 10 11] introduced aconcept of partial probability-weighted moments (PPWMs)that will estimate the higher quantiles of flood flows Data canbe censored to the right tail or left tail
Initially PPWMswere to take out the smaller values fromthe process of distribution fitting because such values haveslight influence on the frequency analysis and are nuisance tothe fitting process
The left tail PPWMs are defined by Wang [8 11] as
120573lowast119903 = int11198650
119909 (119865) 119865119903119889119865 (9)
where 1198650 = 119865(1199090) which is the lower limit of the censoredobservations and 1199090 is the censoring threshold value
The PPWMs elongated form described by Wang [10] areto be given a censored sample as
120573119903 = 11 minus 119865119903+10 int11198650
119909 (119865) 119865119903119889119865 (10)
If the value of 1198650 is starting from the zero then the result ofPPWMswill be the same as the usual PWMs As 119909(1) le 119909(2) le119909(3) sdot sdot sdot le 119909(119899) is the arranged sample Wang [10] describes theunbiased estimator of 120573119903 as
119903 = 1(1 minus 119865119903+10 ) 119899119899sum119894=1
(119894 minus 1) (119894 minus 2) sdot sdot sdot (119894 minus 119903)(119899 minus 1) (119899 minus 2) sdot sdot sdot (119899 minus 119903)119909lowast(119894) (11)
where
119909lowast(119894) = 0 for 119909(119894) le 119909(0)119909lowast(119894) = 119909(119894) for 119909(119894) gt 119909(0) (12)
The censoring level 1198650 is the prior selection of the numberof censored sample data The procedure that determines thenumber of sample data points are to be censored
1198650 = 1198990119899 (13)
where 1198990 and 119899 are the lengths of sample which are to becensored and uncensored respectively Similarly 1199090 is thehighest value of the censored sample The first four PL-moments for the PPWMs are
1 = 12057302 = 2 1205731 minus 12057303 = 1205732 minus 6 1205731 + 12057304 = 20 1205733 minus 30 1205732 + 12 1205731 minus 1205730
(14)
Similarly the PL-moments ratios can be written as
120591 = 21
1205913 = 32
1205914 = 42
(15)
where 120591 1205913 and 1205914 denote the partial L-coefficient of variation(PL-Cv) partial L-coefficient of skewness (PL-Cs) and par-tial L-coefficient of kurtosis (PL-Ck) respectively Thereforethe first four sample PL-moments can be computed as
1198971 = 01198972 = 21 minus 01198973 = 62 minus 61 + 01198974 = 203 minus 302 + 121 minus 0
(16)
And the first four sample PL-moments ratios can be com-puted as
119905 = 11989721198971 1199053 = 11989731198972 1199054 = 11989741198972
(17)
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where 119905 1199053 and 1199054 represent the sample partial L-momentsratios of the PL-Cv PL-Cs and PL-Ck respectively Thederivation is L-moment and PL-moments are given inAppendix In the present study different level of censoringthreshold is selected
23 Regional Frequency Analysis Hosking and Wallis [7 30]identified the following four steps to explain the procedure ofthe 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
231 Data Screening We screened data anomalies beforeapplying any statistical analysis
232 Discordance Test Hosking and Wallis [7] suggested adiscordancy measure (119863119894) test that recognizes the locationswhere sample L-moments are marked contrarily from themost other locations Locations with the large flaws in thedata will be flagged as discordant
The discordancy test for a region containing119873 locationsfor site 119894 is proposed by Hosking and Wallis [7] as follows
119863119894 = 13119873 (119906119894 minus 119906)119879 119878minus1 (119906119894 minus 119906) 119894 = 1 2 119873 (18)
where 119906119894 is the vector containing the three sample L-momentsratios for the site 119894 expressed as
119906119894 = [119905(119894)2 119905(119894)3 119905(119894)4 ]119879 (19)
119906 is the average vector of 119906119894 for the overall region that is
119906 = 1119873119873sum119894=1
119906119894 (20)
and 119878 is the covariance matrix for the sample that can beexpressed as
119878 = 119873sum119894=1
(119906119894 minus 119906) (119906119894 minus 119906)119879 (21)
Broadly speaking a location or a site is considered to bediscordant from the whole region or group if the value of 119863119894is larger than the critical value
233 Heterogeneity Test The homogeneity measure (119867119895)identifies homogenous regions Hosking and Wallis [7] Itis also useful to tag the locations if they are plausible tohandle as a homogeneous region It estimates the amountof heterogeneity in the overall region The heterogeneity test(119867119895) is computed as
119867119895 = 119881119895 minus 120583V119895120590V119895 119895 = 1 2 3 (22)
where 120583V119895 and 120590V119895 are representing the mean and standarddeviation of the simulated 119881119895 values Also
1198811 = 119873sum119894=1
119899119894 (119905(119894)2 minus 1199051198772 )sum119873119894=1 119899119894
12
1198812 =sum119873119894=1 [119899119894 (119905(119894)2 minus 1199051198772 )2 + (119905(119894)3 minus 1199051198773 )212]
sum119873119894=1 119899119894
1198813 =sum119873119894=1 [119899119894 (119905(119894)3 minus 1199051198773 )2 + (119905(119894)4 minus 1199051198774 )212]
sum119873119894=1 119899119894
(23)
Here 1199051198772 1199051198773 and 1199051198774 are region average L-moments or PL-moments ratios We assessed the heterogeneity of a region assuggested by Hosking and Wallis [7]
Region is acceptably homogeneous if119867 lt 1Region is possibly heterogeneous if 1 le 119867 lt 2Region is definitely heterogeneous if119867 ge 2
24 Selection of the Appropriate Probability DistributionHosking andWallis [7] proposed two approaches to select thedistribution that fitted best the data the L-moment ratios dia-gram and the 119885-test The L-moment ratios diagram is usingthe unbiased estimators Hosking [28] Stedinger et al [31]Vogel and Fennessey [32] and Hosking [33]The L-momentsratio diagram is a plot of the computed values L-Cs andthe observed values L-Ck of the distribution function Thecurves indicate the hypothetical connections between L-Csand L-Ck of the candidate distribution The L-moment ratiodiagrams have been proposed for discriminating between thecandidate probability distributions in describing the regionalinformation (Hosking [28] Stedinger et al [31] Hoskingand Wallis [7]) The L-moments ratio diagrams have beenused as a component of probability distribution process forregional information (Schaefer [34] Pearson [35] Vogel andFennessey [32] Vogel et al [36] Chow and Watt [37] OnOzand Bayazit [38] Vogel and Wilson [39] Peel et al [40])
Hosking and Wallis [7] suggested a measure to see howwell the L-Cs and L-Ck of the fitted probability distributionmatch the regional average L-Cs and L-Ck of the observedinformation
The measure goodness of fit for every single selectedprobability distribution is computed as follows
119885Dis = (120591Dis4 minus 1199051198774 )1205904 (24)
where 120591Dis4 represents the value of the L-Ck of the fitteddistribution 1199051198774 represents the weighted regional averageL-Ck and 1205904 represents standard deviation of the 1199051198774 whichis obtained from the simulation of the Kappa probabilitydistribution
If the computed value of 119885Dis is equal to zero the proba-bility distributionwill be themost suitable fit If the computed
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Table 1 Statistics of annual extreme monthly precipitation for study region based on L-moments and PL-moments
Sites L-moments PL-momentsMean 119905 1199053 1199054 Mean 119905 1199053 1199054
Astore 37624 0518 0366 0193 41752 0467 0377 0183Balakot 120450 0478 0322 0192 133259 0428 0335 0198Bunji 14297 0609 0456 0259 16339 0553 0450 0257Chilas 15091 0585 0392 0194 17603 0516 0375 0200Cherat 49038 0580 0375 0162 55289 0526 0355 0161DI khan 28885 0689 0516 0262 32351 0651 0487 0250Dir 105694 0432 0216 0098 116741 0378 0222 0097Drosh 45284 0525 0317 0136 50316 0474 0307 0137Garhi Dupatta 111352 0454 0227 0111 123374 0398 0229 0120Gilgit 12677 0591 0412 0228 14290 0539 0399 0239Gupis 23446 0678 0529 0331 30299 0583 0506 0350Kakul 99431 0453 0247 0102 110068 0399 0254 0093Kotli 95070 0542 0332 0136 106903 0487 0314 0138Muzaffarabad 118439 0459 0284 0182 131122 0405 0301 0194Peshawar 46216 0572 0398 0216 51761 0520 0392 0222Saidu Sharif 85667 0455 0265 0181 94978 0399 0283 0200Skardu 20438 0616 04564 0238 22885 0570 0444 0231
value of 119885-statistic is less than 164 at 90 confidence level(ie |119885Dis| le 164) it will indicate that the distributionqualifies the goodness of fit criteria If there are more thanone distribution that qualify the criteria the most suitabledistribution has the minimum |119885Dis| value3 Estimation
The sitesrsquo information and statistic by using L-moments forthe present study are presented in Table 1 In Table 1 meanrepresents the first sample LPL-moments and 119905 1199053 and 1199054are the sample LPL-moments ratios of the L-CvPL-CvL-CsPL-Cs and L-CkPL-Ck respectively The lower levelcensoring threshold is selected from 10 to 23 Table 2expresses the feasible threshold values according to thepercentile technique along with Average Annual OccurrenceNumber (AAON) Jiang et al [41] and Yuguo [42] suggestedthat the optimal threshold can be obtained if the values ofAAON lie between 1 and 2 Table 2 shows that 90th percentileobservations are suitable for the optimum threshold selectionof most of areas in the present study We have 168 valuesin each station according to the above table Astore stationhas 32 threshold values due to which 16 values are beingcensored in 168 According to censored level 102 censoredlevel was selected Similarly 105 was selected for Balakotand Muzaffarabad According to this process maximumthreshold level 223 was selected for Gupis By using theabove process for each station 17 different censoring levelswere selected So we decide that the range from 10 to 23of censoring level should be kept for selecting thresholdvalues
31 Regional Frequency Analysis The following four steps areconsidered as prerequisite for frequency analysis Hoskingand 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
311 Data Screening In this study we use secondary dataafter carefully examining all locations for abnormalities andmissing observations Therefore we use 14 years of data forRFA that were obtained from seventeen locations
312 Discordance Test Table 3 shows 119863119894 result of (18) for17 locations of this study region It can be observed fromthe results of LPL-moments in Table 3 that the value of 119863-statistic varies from 007 to 244 If 119863119894 is greater than 3 thelocation is considered to be discordant from the rest of theregional data Hosking and Wallis [7] In this study regionno location is diagnosed as discordant (119863119894 ge 3) Thereforewe use all data for the development of the RFA based on L-moments and PL-moments
313 Regional Heterogeneity Measure The next step is theformation of the homogeneous region which is conven-tionally tougher and needs the higher number of subjectivejudgments The homogeneity conditions are defined as thelocations that have the same frequency distributions
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Table 2 Precipitation threshold selection in GPA distribution for 17 stations
Station 80th 90th 95th 975th 99thAstore
Threshold 98 32 02 0 0AAON 243 121 064 043 043
BalakotThreshold 325 148 51 3 0AAON 243 129 057 029 014
MuzaffarabadThreshold 304 118 5 1 0AAON 25 121 057 029 021
Garhi DupattaThreshold 288 10 38 0 0AAON 243 121 057 029 029
DirThreshold 31 20 5 005 0AAON 243 121 057 029 021
KakulThreshold 289 134 21 05 0AAON 164 129 057 029 021
KotliThreshold 13 6 005 0 0AAON 243 135 057 036 036
Saidu SharifThreshold 256 83 2 0 0AAON 243 121 057 036 036
cheratThreshold 5 005 0 0 0AAON 243 143 093 093 093
PeshawarThreshold 6 005 0 0 0AAON 243 129 057 057 057
DroshThreshold 76 15 02 005 0AAON 243 121 057 05 021
D I KhanThreshold 03 0 0 0 0AAON 257 129 129 129 129
GupisThreshold 0 0 0 0 0AAON 271 271 271 271 271
SkarduThreshold 2 03 0 0 0AAON 243 071 071 071 071
ChilasThreshold 1 05 0 0 0AAON 171 107 107 107 107
BunjiThreshold 13 0 0 0 0AAON 15 15 15 15 15
GilgitThreshold 12 005 0 0 0AAON 243 135 071 071 071
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04
03
02
01
00
04
03
02
01
00
00 01 02 03 04 05 06 00 01 02 03 04 05 06
GLOGEVGPA
GNOAverageData
GLOGEVGPA
GNOAverageData
L-ku
rtos
is
L㰀-k
urto
sis
L-skewness L-skewness
Figure 2 L-diagram and PL-diagram of the GLO GEV GPA and GNO distributions
Table 3 Discordance test result based on L-moments and PL-moments
Name of site L-moments (119863119894) PL-moments (119863119894)Astore 205 244Balakot 142 085Bunji 055 060Chilas 029 007Cherat 079 064DI khan 129 190Dir 082 086Drosh 051 043Garhi Dupatta 143 133Gilgit 026 059Gupis 237 231Kakul 082 122Kotli 077 059Muzaffarabad 099 085Peshawar 007 008Saidu Sharif 191 157Skardu 067 065
For the present study area realization of the Kappaprobability distribution is used to conduct the heterogeneitytest based on the L-moments and PL-moments
Number of simulations are 10000 for computing theheterogeneityWe computed the regional average L-momentsratios the regional PL-moments ratios and the correspond-ing parameter values of the fitted Kappa probability dis-tribution (see Table 4) Table 4 shows the results of theheterogeneity measure using L-moments and PL-momentsmethods It can be observed from Table 4 that the different
values for the119867-statistic are minus041 minus160 and minus289 based onL-moments andminus006minus18 andminus317 based onPL-momentsTherefore we concluded that by comparing these resultsand the heterogeneity conditions study region is acceptablyhomogeneous for L-moments and PL-moments No furthersubdivisions of the present study are necessary
32 Fitting Appropriate Probability Distribution After homo-geneity analysis of the study area a suitable probabilitydistribution is required for the RFAThe objective is not onlyto recognize a suitable probability distribution for RFA butalso to observe a probability distribution that will providerobust quantile estimate for each location and for the regionalgrowth cure List of candidate probability distributions forRFA is GLO GEV GPA and GNO
We plotted L-moments and PL-moments diagrams forpreliminary evaluation of the probability distribution for thestudy area
Figure 2 illustrates an analogy of the observed and hypo-thetical relationships of the probability distribution Figure 2shows that GLO distribution is not a suitable candidate forthe L-moments and PL-moments
Interestingly both analyses of the L-moments and the PL-moments diagram show that the sample average values areappropriately distinguished by the hypothetical L-momentsand PL-moments for GPA and GNO distributions
However it is hard to find a suitable probability distri-bution that fits most of the regional observed data Table 5shows the goodness of fit test results for candidate probabilitydistributions
Table 5 shows that GLO distribution failed the goodnessof fit test for both L-moments and for PL-moments methodsas the calculated value of the 119885-test for the GLO distribution
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Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments
Heterogeneity measures L-moment PL-momentHeterogeneity measure1198671
Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060
Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800
Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170
Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments
Method Distribution Parameters120585 120572 119870L-moments
GEV 04660 05644 minus02751GNO 06671 07565 minus07594GPA minus00551 09944 minus00575
PL-momentsGEV 05220 05116 minus02686GNO 07046 06837 minus07487GPA 00478 09073 minus00472
is larger than the critical value of 164 (at 90 confidencelevel)
It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile
Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution
33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7
based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)
Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is
119876 (119865) = 1198971198941119902 (119865) (25)
The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3
Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions
Therefore it is necessary to assess the performance ofregional quantile estimates
4 Accuracy of the Estimated Quantiles and theRegional Growth Curve
A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten
10 Advances in Meteorology
Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865
01000 05000 08000 09000 09500 09800 09900 09975 099875 09990
L-momentsGEV 00454 06836 15139 22247 30591 44161 56871 90748 113165 121338GNO 00473 06671 15585 23072 31448 44098 54997 80676 95666 100823GPA 00500 06481 16217 23931 31958 43073 51878 70578 80502 83782
PL-momentsGEV 01397 07190 14670 21034 28469 40497 51703 81363 10086 107950GNO 01412 07046 15062 21752 29203 40410 50032 72608 85739 90249GPA 01436 06871 15649 22547 29674 39460 47149 63304 71786 74577
Qua
ntile
Regional growth curve (L-moments)
GPAGNOGEV
GPAGNOGEV
Qua
ntile
Return period
Regional growth gurve (PL-moments)
5
4
3
2
1
0
5
4
3
2
1
0
minus2 minus1 0 1 2 3 4 5
2 5 10 20 50 100
Return period2 5 10 20 50 100
Reduced variate minuslog (minus log (F))
minus2 minus1 0 1 2 3 4 5
Reduced variate minuslog (minus log (F))
Figure 3 L-moments and PL-moments regional growth curves
as 119876(119898)119894 (119865) for the nonexceedance probability 119865 The relativeerror for this estimator is
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) (26)
The bias and the RMSE of the above quantity over all 119872repetition are
Bias = 119861119894 (119865) = 1119872119872sum119898=1
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)
= 1119872 [[119872sum119898=1
(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12
(27)
Also for the estimated quantile the regional average bias andthe relative RMSE are
119861119877 (119865) = 1119873119873sum119894=1
119861119894 (119865)
119877119877 (119865) = 1119873119873sum119894=1
119877119894 (119865) (28)
We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval
119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)
Advances in Meteorology 11
Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le
119902 (119865)119871005 (119865) (30)
The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves
We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]
We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively
Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds
Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion
This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail
GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments
Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents
Appendix
The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows
PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by
119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (minus ln119865)119896 119896 = 0 (A2)
Wang [10] developed the partial probability-weightedmoments (PWMs) of the GEV as
(119903 + 1) 1205731015840119903 = 120577 + 120572119867 (120574 1198650 119896) (A3)
where
119867(120574 1198650 119896) = 1119896 [1 minus119875 1 + 119896 minus (119903 + 1) ln1198650(1 minus 119865119903+10 ) (119903 + 1)119896 ] (A4)
12 Advances in Meteorology
Table8Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f30
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus0540
0minus19
7674
minus195328
minus193602
minus191579
minus188659
minus186376
minus1840
97minus18
1165
minus179056
119877(119865)
00900
00695
00963
01952
03334
06038
09057
13261
21337
300
48119877119877 (119865)
19873
01019
00635
00876
01090
01369
01595
01839
02191
02478
119871119861119864lowast
minus20937
08829
09296
08842
08411
07862
07460
07065
06557
06177
119880119861119864lowast
18714
11949
11208
11580
11884
12244
12516
12818
13285
13683
L-mom
ents
GNO
119861119877 (119865)
08124
minus196279
minus193846
minus192837
minus191816
minus190506
minus189573
minus188699
minus187636
minus186899
119877(119865)
00986
00696
00917
02051
03520
06027
08421
11312
16008
20312
119877119877 (119865)
20766
01045
00587
00889
01120
01368
01533
01686
01878
02016
119871119861119864lowast
minus21331
08683
09271
08780
08379
07958
07691
07453
07172
06978
119880119861119864lowast
16569
11974
11077
11569
11954
12355
12615
12851
13149
13369
L-mom
ents
GPA
119861119877 (119865)
38654
minus194351
minus193564
minus193226
minus192616
minus191706
minus191060
minus190506
minus189966
minus189727
119877(119865)
01242
006
6900998
02136
03552
05968
08313
11188
15941
20368
119877119877 (119865)
24785
01033
00614
00892
01111
01386
01602
01833
02162
02431
119871119861119864lowast
minus25192
08591
09215
08817
08444
07979
07646
07327
06919
06623
119880119861119864lowast
14272
11911
11115
11588
11976
12459
12841
13257
13859
14367
PL-m
oments
GEV
119861119877 (119865)
minus02662
minus197059
minus194770
minus193340
minus191600
minus188996
minus186909
minus184797
minus182052
minus1800
66119877(119865)
05379
006
8700947
02023
03467
06171
09087
13058
20551
2854
119877119877 (119865)
38117
00948
006
4900960
01210
01508
01736
01977
02324
02609
119871119861119864lowast
minus77814
08892
09252
08725
08259
07697
07296
06911
064
1706050
119880119861119864lowast
45516
11827
11207
11716
12117
12554
12881
13211
13693
14104
PL-m
oments
GNO
119861119877 (119865)
35142
minus195922
minus193485
minus192704
minus191894
minus190806
minus1900
03minus18
9235
minus188284
minus187617
119877(119865)
06203
006
8900939
02141
03644
06143
08483
11272
15747
19809
119877119877 (119865)
43522
00971
00626
00980
01238
01504
01675
01831
02024
02162
119871119861119864lowast
minus10618
08772
09204
08655
08233
07800
07530
07290
07012
06822
119880119861119864lowast
46353
11849
11122
11734
12189
12653
12949
13218
13546
13780
PL-m
oments
GPA
119861119877 (119865)
1212
3minus19
4300
minus193238
minus193113
minus192716
minus192046
minus191555
minus19114
1minus19
0768
minus1906
49119877(119865)
08513
006
6601041
02250
03684
06021
08215
10853
15144
19094
119877119877 (119865)
58811
00962
006
6900996
01233
01509
01720
01941
02258
02518
119871119861119864lowast
minus17507
08698
09130
08680
08299
07846
07524
07220
06831
06552
119880119861119864lowast
47307
11791
11189
11774
12215
12739
1314
13567
14195
14722
Advances in Meteorology 13
Table9Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f60
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus13271
minus194230
minus193551
minus193241
minus192649
minus191612
minus190736
minus189839
minus188690
minus187888
119877(119865)
00846
00624
00851
01873
03250
05841
08652
12505
19839
27721
119877119877 (119865)
19155
00833
00612
00855
01024
01213
01360
01523
01770
01983
119871119861119864lowast
minus19593
08756
09254
08851
08491
08074
07796
07501
07118
06834
119880119861119864lowast
16242
11675
11015
11472
11879
12309
12678
13038
13659
14141
L-mom
ents
GNO
119861119877 (119865)
08417
minus193335
minus192795
minus192984
minus192999
minus192879
minus192742
minus192593
minus192399
minus192264
119877(119865)
00968
00634
00858
02000
03439
05850
08123
10845
15235
19239
119877119877 (119865)
21055
00881
00582
00861
01050
01235
01354
01463
01599
01699
119871119861119864lowast
minus22887
08647
09244
08794
08466
08141
07957
07798
07597
07443
119880119861119864lowast
15863
11714
1096
11511
11934
1240
512
666
12939
13253
13505
L-mom
ents
GPA
119861119877 (119865)
46371
minus192512
minus192631
minus193160
minus193417
minus193606
minus193745
minus193933
minus194306
minus194708
119877(119865)
01280
00624
00950
02094
03468
05740
07903
10536
14878
18928
119877119877 (119865)
264
4200915
00614
00882
01064
01270
01429
01599
01849
02059
119871119861119864lowast
minus38088
08628
09186
08815
08510
08167
07924
07703
07422
07178
119880119861119864lowast
14699
11721
11020
11534
11925
12454
12791
13236
13869
1440
5
PL-m
oments
GEV
119861119877 (119865)
minus16968
minus193984
minus193601
minus193623
minus193319
minus192576
minus191861
minus191081
minus190026
minus189255
119877(119865)
05431
00596
00888
01978
03366
05844
08417
11838
18179
24868
119877119877 (119865)
38656
00783
00634
0095
01166
01387
01541
01700
01928
02120
119871119861119864lowast
minus92921
08907
09201
08750
08403
07984
07714
07434
07043
06783
119880119861119864lowast
53553
11521
11099
1166
412
098
12572
12869
13216
13697
14105
PL-m
oments
GNO
119861119877 (119865)
23785
minus193306
minus192979
minus193417
minus193628
minus193678
minus193617
minus193509
minus193331
minus193185
119877(119865)
06278
006
0100911
02100
03523
05804
0788
10303
14113
17516
119877119877 (119865)
43743
00812
00623
00965
01186
01391
01513
01618
01742
01830
119871119861119864lowast
minus12083
08856
09188
08709
08389
08061
07870
07706
07517
07380
119880119861119864lowast
52707
11561
11059
11704
12152
12613
12898
13138
13361
1356
PL-m
oments
GPA
119861119877 (119865)
1118
7minus19
2590
minus192910
minus193681
minus1940
84minus19
4322
minus1944
04minus19
4463
minus194560
minus1946
73119877(119865)
08596
00597
01009
02229
03581
05615
07391
09418
12558
15341
119877119877 (119865)
58310
00840
006
6000991
0119
501391
01522
01650
01828
01971
119871119861119864lowast
minus18977
08800
09127
08699
08413
08098
07912
07682
07447
07255
119880119861119864lowast
51281
11551
11117
11769
12173
12615
12887
13227
1360
913
899
14 Advances in Meteorology
Table10A
ccuracymeasure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f90
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus00117
minus194226
minus193109
minus192926
minus192542
minus191809
minus191151
minus1904
48minus18
9497
minus188790
119877(119865)
00787
00556
00823
01850
03151
05459
07840
10989
16793
22883
119877119877 (119865)
17329
00814
00544
00832
01031
01238
01381
01525
01727
01892
119871119861119864lowast
minus19303
08935
09287
08885
08587
08246
08006
07772
07463
07229
119880119861119864lowast
15717
11499
10973
11461
11804
12156
12393
12635
12966
13242
L-mom
ents
GNO
119861119877 (119865)
15851
minus193606
minus192517
minus192695
minus192747
minus192678
minus192569
minus192437
minus192245
minus192098
119877(119865)
00917
00567
00846
01967
03305
05446
07392
09660
13223
1640
0119877119877 (119865)
19328
00850
00543
00853
01052
01237
01347
01442
01554
01632
119871119861119864lowast
minus23268
08848
09261
08844
08568
08301
08139
08000
07833
07722
119880119861119864lowast
15168
11542
10942
11489
11851
12197
1240
112
575
12779
12925
L-mom
ents
GPA
119861119877 (119865)
49081
minus192995
minus192417
minus192865
minus193078
minus193175
minus193195
minus193209
minus193256
minus193330
119877(119865)
01234
00571
00935
02081
03365
05322
07054
09053
12180
14975
119877119877 (119865)
246
8800882
00576
00870
01054
01239
01364
01489
01660
01798
119871119861119864lowast
minus45353
08783
09210
08838
08594
08326
08141
07961
07731
07557
119880119861119864lowast
14833
11574
10996
11532
11870
12223
1246
712
711
1304
413
312
PL-m
oments
GEV
119861119877 (119865)
minus00877
minus194145
minus193060
minus193062
minus192847
minus192278
minus191698
minus191038
minus190100
minus189374
119877(119865)
05458
00565
00855
01963
03338
05712
08090
11156
16650
22276
119877119877 (119865)
39079
00784
00585
00934
0117
201406
01558
01703
01897
02050
119871119861119864lowast
minus10190
08983
09236
08765
08431
08067
07818
07579
07271
07041
119880119861119864lowast
58730
1144
811036
11652
12081
12504
12771
13026
13364
13623
PL-m
oments
GNO
119861119877 (119865)
38414
minus193665
minus192536
minus192859
minus193044
minus193102
minus193057
minus192966
minus192804
minus192665
119877(119865)
06308
00578
00896
02088
03488
05690
07659
09926
13434
16524
119877119877 (119865)
44651
00819
00596
00961
0119
301405
01526
01628
01744
01822
119871119861119864lowast
minus13011
08910
09192
08716
08411
08122
07949
07801
07634
07519
119880119861119864lowast
56756
11491
11030
11693
12128
12539
12775
12974
13197
13347
PL-m
oments
GPA
119861119877 (119865)
126398
minus193142
minus192484
minus193077
minus193404
minus193589
minus193637
minus193655
minus193680
minus193720
119877(119865)
08641
00588
00999
02220
03558
05535
07227
09125
1200
914
521
119877119877 (119865)
60116
00855
006
4000985
0119
901400
01529
01651
01812
01937
119871119861119864lowast
minus19889
08842
09130
08700
08435
08156
07970
07789
07566
07399
119880119861119864lowast
53743
11528
11103
11755
12158
12552
12807
13055
13385
13636
Advances in Meteorology 15
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
0 2 4 6
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
Regional growth curve for GPA
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F))
0
5
10
15
20
Qua
ntile
2 4 60Reduced variate minuslog (minus log (F))
0
5
10
15
Qua
ntile
(based on L-moments) Regional growth curve for GPA
(based on PL-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30
16 Advances in Meteorology
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60
Advances in Meteorology 17
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90
18 Advances in Meteorology
The first four PL-moments of the GEV are defined as
12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)
(A5)
In previous equation 119875(sdot sdot) is an Incomplete Gammafunction
119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500
120579119896expminus120579119889Θ (A6)
Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution
The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by
119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)
The partial PWMs of the GLO are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)
(A9)
where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function
1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Geological ResearchJournal of
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Geology Advances in
2 Advances in Meteorology
formula and unit-hydrograph procedure Subramanya [6]and it can also estimate the quantiles of extreme precipitation
Hosking and Wallis [7] showed that RFA method basedon L-moments is used to detect homogeneous regions toselect suitable regional frequency distribution and to predictextreme precipitation quantiles at region of interest
Whilst L-moment methodology is effective in estimatingparameters it may not valid for predicting high returnperiod events Wang [8] suggested that relatively small floodsmight create disturbance in the analysis To overcome suchsituation a censored sample can be used as by Cunnane [9]
Wang [8] proposed partial probability-weightedmoments (PPWMs) for fitting the probability distributionfunction to the censored sample Partial L-moments(PL-moments) are variants of the L-moments and similar toPPWMs PL-momentsmethod has used in fitting generalizedextreme value (GEV) distribution for censored flood samples(see Wang [8 10 11] and Bhattarai [12]) Bhattarai [12] foundthat censoring flood samples are nearly thirty percent ofbasic L-moments
Shabri et al [13] used Trim L-moments (TL-moments)for the RFA and compared its performance with L-momentsSaf [14] determined hydrologically homogeneous region andregional flood frequency estimates by using index-floodtechnique along with L-moments for the West Mediter-ranean River Turkey L-moments method is also used toassign a suitable regional distribution for the individualsubregions and to assess their homogeneity see Abolverdiand Khalili [15] Hussain and Pasha [16] suggested theregional flood frequency analysis based on L-momentsTheyused discordancy measure for data screening and used thefour-parameter Kappa distribution with 500 simulationsfor the heterogeneity analysis Zakaria et al [17] used thePL-moments technique and found another link for thehomogeneity analysis Shahzadi et al [18] showed that thegeneralized normal (GNO) distribution is suitable for theregional quantile estimation at maximum return period andthe GEV distribution for the overall regions at low returnperiod based on relative RMSE and relative absolute biasMost commonly used statistical distributions for high climatemodeling are as follows the logistic distribution with threeparameters lognormal distribution with three parametersLog 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] Sheet al [25])Moreover GEV andGPAdistributions are suitableif data contains extremes values Over the years the GNOGEV generalized logistic (GLO) and GPA distributionshave been widely employed in the extreme value estimationof annual flood peaks In this study we aim to developRFA method based on L- and PL-moments approach Ourproposed method can be used at all levels of regional analysissuch as identification of homogeneous regions identificationand also testing of the suitable probability regional frequencydistribution based on 119885-statistic and the L- and PL-momentratio diagram and estimation of the flood quantile at siteof interest We use 17 sites of Northern areas and KhyberPakhtunkhwa as a case study to perform the analysis
We explain the methodology of the L-moments and PL-moments in Section 22 Section 23 shows the applicationof the RFA where we choose the appropriate distributionfor the regional analysis Section 33 provides the estimationof quantile for both the small and large return period Theresults of our simulations studywill be presented in Section 4
2 Material and Methods
21 Study Area and Data Sources The data was collectedfor this study from Karachi Data Processing Center throughPakistan Meteorological (PMD) Islamabad Monthly precip-itation data has been recorded from 1999 to 2012 There are17 meteorological stations of Northern areas and KhyberPakhtunkhwa Pakistan These stations are full precipitationregions that affect areas in Pakistan where water is essentialfor hydropower and flood plains Figure 1 shows the locationof the study area and geographic distribution of precipitationstations There is no missing value in this data set
22 Statistical Methods
221 The L-Moments Conventional moments method maybe used for estimating the parameters of probability distri-bution However this approach has some serious drawbacksRatio ofmoment estimators is biased and often assumption ofbeing normally distributed is violated Wallis et al [26] Fur-thermore it is sensitive to outliers Pearson [27]Therefore itis unreliable for skewed distributions
Hosking [28] proposed L-moments approach to over-come the above problems The L-moments may preciselydescribe the statistical properties of hydrological informa-tion and it can write as a linear function of the PWMs ThePWMs of order 119903 were properly described by Greenwood etal [29] as
119872119901119903119904 = 119864 [119909119901 119865 (119909)119903 1 minus 119865 (119909)119904] (1)
where 119903 119904 and 119901 are the real numbers and 119865(119909) is thecumulative distribution function of 119909 A functional case is
120573119903 = 11987211199030 (2)
Therefore
120573119903 = int10119909 (119865) 119865119903119889119865 (3)
where 119865 = 119865(119909) is the cumulative distribution function of arandom variable 119909 and 119903 is a nonnegative integer of the realnumber that is 119903 = 0 1 2 3 Therefore the first four L-moments which are the linear combinations of the PWMsare
1205821 = 12057301205822 = 21205731 minus 12057301205823 = 1205732 minus 61205731 + 12057301205824 = 201205733 minus 301205732 + 121205731 minus 1205730
(4)
Advances in Meteorology 3
37∘0㰀0㰀㰀N
36∘0㰀0㰀㰀N
35∘0㰀0㰀㰀N
34∘0㰀0㰀㰀N
33∘0㰀0㰀㰀N
32∘0㰀0㰀㰀N
37∘0㰀0㰀㰀N
36∘0㰀0㰀㰀N
35∘0㰀0㰀㰀N
34∘0㰀0㰀㰀N
33∘0㰀0㰀㰀N
32∘0㰀0㰀㰀N
71∘0㰀0㰀㰀E 72∘0㰀0㰀㰀E 73∘0㰀0㰀㰀E 74∘0㰀0㰀㰀E 75∘0㰀0㰀㰀E 76∘0㰀0㰀㰀E
71∘0㰀0㰀㰀E 72∘0㰀0㰀㰀E 73∘0㰀0㰀㰀E 74∘0㰀0㰀㰀E 75∘0㰀0㰀㰀E 76∘0㰀0㰀㰀E
W
N
E
S
0 30 60 120 180 240(km)
StationsWater bodiesStudy area
Figure 1 Locations of Northern areas and Khyber Pakhtunkhwa and the meteorological stations (119909 axis indicates Longitude [E] 119910 axisindicates Latitude [N])
The L-moments have no units of measurement which arecalled the L-moments ratios The L-moments ratios pro-posed by Hosking [28] are computed as
120591 = 12058221205821
1205913 = 12058231205822
1205914 = 12058241205822 (5)
4 Advances in Meteorology
where 120591 represents the L-coefficient of variation (L-Cv)1205913 represents the L-coefficient of skewness (L-Cs) and 1205914represents the L-coefficient of kurtosis (L-Ck)
The arranged sample is given as 119909(1) le 119909(2) le 119909(3) sdot sdot sdot le119909(119899) Wang [8] stated that the statistic
119887119903 = 1119899119899sum119894=1
(119894 minus 1) (119894 minus 2) sdot sdot sdot (119894 minus 119903)(119899 minus 1) (119899 minus 2) sdot sdot sdot (119899 minus 119903)119909(119894) (6)
is an unbiased estimator of 120573119903So
1198971 = 11988701198972 = 21198871 minus 11988701198973 = 61198872 minus 61198871 + 11988701198974 = 201198873 minus 301198872 + 121198871 minus 1198870
(7)
where 1198971 1198972 1198973 and 1198974 are the first four L-moments of thesample And similarly
119905 = 11989721198971 1199053 = 11989731198972 1199054 = 11989741198972
(8)
are the sample L-moments ratios
222 The Partial L-Moments Wang [8 10 11] introduced aconcept of partial probability-weighted moments (PPWMs)that will estimate the higher quantiles of flood flows Data canbe censored to the right tail or left tail
Initially PPWMswere to take out the smaller values fromthe process of distribution fitting because such values haveslight influence on the frequency analysis and are nuisance tothe fitting process
The left tail PPWMs are defined by Wang [8 11] as
120573lowast119903 = int11198650
119909 (119865) 119865119903119889119865 (9)
where 1198650 = 119865(1199090) which is the lower limit of the censoredobservations and 1199090 is the censoring threshold value
The PPWMs elongated form described by Wang [10] areto be given a censored sample as
120573119903 = 11 minus 119865119903+10 int11198650
119909 (119865) 119865119903119889119865 (10)
If the value of 1198650 is starting from the zero then the result ofPPWMswill be the same as the usual PWMs As 119909(1) le 119909(2) le119909(3) sdot sdot sdot le 119909(119899) is the arranged sample Wang [10] describes theunbiased estimator of 120573119903 as
119903 = 1(1 minus 119865119903+10 ) 119899119899sum119894=1
(119894 minus 1) (119894 minus 2) sdot sdot sdot (119894 minus 119903)(119899 minus 1) (119899 minus 2) sdot sdot sdot (119899 minus 119903)119909lowast(119894) (11)
where
119909lowast(119894) = 0 for 119909(119894) le 119909(0)119909lowast(119894) = 119909(119894) for 119909(119894) gt 119909(0) (12)
The censoring level 1198650 is the prior selection of the numberof censored sample data The procedure that determines thenumber of sample data points are to be censored
1198650 = 1198990119899 (13)
where 1198990 and 119899 are the lengths of sample which are to becensored and uncensored respectively Similarly 1199090 is thehighest value of the censored sample The first four PL-moments for the PPWMs are
1 = 12057302 = 2 1205731 minus 12057303 = 1205732 minus 6 1205731 + 12057304 = 20 1205733 minus 30 1205732 + 12 1205731 minus 1205730
(14)
Similarly the PL-moments ratios can be written as
120591 = 21
1205913 = 32
1205914 = 42
(15)
where 120591 1205913 and 1205914 denote the partial L-coefficient of variation(PL-Cv) partial L-coefficient of skewness (PL-Cs) and par-tial L-coefficient of kurtosis (PL-Ck) respectively Thereforethe first four sample PL-moments can be computed as
1198971 = 01198972 = 21 minus 01198973 = 62 minus 61 + 01198974 = 203 minus 302 + 121 minus 0
(16)
And the first four sample PL-moments ratios can be com-puted as
119905 = 11989721198971 1199053 = 11989731198972 1199054 = 11989741198972
(17)
Advances in Meteorology 5
where 119905 1199053 and 1199054 represent the sample partial L-momentsratios of the PL-Cv PL-Cs and PL-Ck respectively Thederivation is L-moment and PL-moments are given inAppendix In the present study different level of censoringthreshold is selected
23 Regional Frequency Analysis Hosking and Wallis [7 30]identified the following four steps to explain the procedure ofthe 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
231 Data Screening We screened data anomalies beforeapplying any statistical analysis
232 Discordance Test Hosking and Wallis [7] suggested adiscordancy measure (119863119894) test that recognizes the locationswhere sample L-moments are marked contrarily from themost other locations Locations with the large flaws in thedata will be flagged as discordant
The discordancy test for a region containing119873 locationsfor site 119894 is proposed by Hosking and Wallis [7] as follows
119863119894 = 13119873 (119906119894 minus 119906)119879 119878minus1 (119906119894 minus 119906) 119894 = 1 2 119873 (18)
where 119906119894 is the vector containing the three sample L-momentsratios for the site 119894 expressed as
119906119894 = [119905(119894)2 119905(119894)3 119905(119894)4 ]119879 (19)
119906 is the average vector of 119906119894 for the overall region that is
119906 = 1119873119873sum119894=1
119906119894 (20)
and 119878 is the covariance matrix for the sample that can beexpressed as
119878 = 119873sum119894=1
(119906119894 minus 119906) (119906119894 minus 119906)119879 (21)
Broadly speaking a location or a site is considered to bediscordant from the whole region or group if the value of 119863119894is larger than the critical value
233 Heterogeneity Test The homogeneity measure (119867119895)identifies homogenous regions Hosking and Wallis [7] Itis also useful to tag the locations if they are plausible tohandle as a homogeneous region It estimates the amountof heterogeneity in the overall region The heterogeneity test(119867119895) is computed as
119867119895 = 119881119895 minus 120583V119895120590V119895 119895 = 1 2 3 (22)
where 120583V119895 and 120590V119895 are representing the mean and standarddeviation of the simulated 119881119895 values Also
1198811 = 119873sum119894=1
119899119894 (119905(119894)2 minus 1199051198772 )sum119873119894=1 119899119894
12
1198812 =sum119873119894=1 [119899119894 (119905(119894)2 minus 1199051198772 )2 + (119905(119894)3 minus 1199051198773 )212]
sum119873119894=1 119899119894
1198813 =sum119873119894=1 [119899119894 (119905(119894)3 minus 1199051198773 )2 + (119905(119894)4 minus 1199051198774 )212]
sum119873119894=1 119899119894
(23)
Here 1199051198772 1199051198773 and 1199051198774 are region average L-moments or PL-moments ratios We assessed the heterogeneity of a region assuggested by Hosking and Wallis [7]
Region is acceptably homogeneous if119867 lt 1Region is possibly heterogeneous if 1 le 119867 lt 2Region is definitely heterogeneous if119867 ge 2
24 Selection of the Appropriate Probability DistributionHosking andWallis [7] proposed two approaches to select thedistribution that fitted best the data the L-moment ratios dia-gram and the 119885-test The L-moment ratios diagram is usingthe unbiased estimators Hosking [28] Stedinger et al [31]Vogel and Fennessey [32] and Hosking [33]The L-momentsratio diagram is a plot of the computed values L-Cs andthe observed values L-Ck of the distribution function Thecurves indicate the hypothetical connections between L-Csand L-Ck of the candidate distribution The L-moment ratiodiagrams have been proposed for discriminating between thecandidate probability distributions in describing the regionalinformation (Hosking [28] Stedinger et al [31] Hoskingand Wallis [7]) The L-moments ratio diagrams have beenused as a component of probability distribution process forregional information (Schaefer [34] Pearson [35] Vogel andFennessey [32] Vogel et al [36] Chow and Watt [37] OnOzand Bayazit [38] Vogel and Wilson [39] Peel et al [40])
Hosking and Wallis [7] suggested a measure to see howwell the L-Cs and L-Ck of the fitted probability distributionmatch the regional average L-Cs and L-Ck of the observedinformation
The measure goodness of fit for every single selectedprobability distribution is computed as follows
119885Dis = (120591Dis4 minus 1199051198774 )1205904 (24)
where 120591Dis4 represents the value of the L-Ck of the fitteddistribution 1199051198774 represents the weighted regional averageL-Ck and 1205904 represents standard deviation of the 1199051198774 whichis obtained from the simulation of the Kappa probabilitydistribution
If the computed value of 119885Dis is equal to zero the proba-bility distributionwill be themost suitable fit If the computed
6 Advances in Meteorology
Table 1 Statistics of annual extreme monthly precipitation for study region based on L-moments and PL-moments
Sites L-moments PL-momentsMean 119905 1199053 1199054 Mean 119905 1199053 1199054
Astore 37624 0518 0366 0193 41752 0467 0377 0183Balakot 120450 0478 0322 0192 133259 0428 0335 0198Bunji 14297 0609 0456 0259 16339 0553 0450 0257Chilas 15091 0585 0392 0194 17603 0516 0375 0200Cherat 49038 0580 0375 0162 55289 0526 0355 0161DI khan 28885 0689 0516 0262 32351 0651 0487 0250Dir 105694 0432 0216 0098 116741 0378 0222 0097Drosh 45284 0525 0317 0136 50316 0474 0307 0137Garhi Dupatta 111352 0454 0227 0111 123374 0398 0229 0120Gilgit 12677 0591 0412 0228 14290 0539 0399 0239Gupis 23446 0678 0529 0331 30299 0583 0506 0350Kakul 99431 0453 0247 0102 110068 0399 0254 0093Kotli 95070 0542 0332 0136 106903 0487 0314 0138Muzaffarabad 118439 0459 0284 0182 131122 0405 0301 0194Peshawar 46216 0572 0398 0216 51761 0520 0392 0222Saidu Sharif 85667 0455 0265 0181 94978 0399 0283 0200Skardu 20438 0616 04564 0238 22885 0570 0444 0231
value of 119885-statistic is less than 164 at 90 confidence level(ie |119885Dis| le 164) it will indicate that the distributionqualifies the goodness of fit criteria If there are more thanone distribution that qualify the criteria the most suitabledistribution has the minimum |119885Dis| value3 Estimation
The sitesrsquo information and statistic by using L-moments forthe present study are presented in Table 1 In Table 1 meanrepresents the first sample LPL-moments and 119905 1199053 and 1199054are the sample LPL-moments ratios of the L-CvPL-CvL-CsPL-Cs and L-CkPL-Ck respectively The lower levelcensoring threshold is selected from 10 to 23 Table 2expresses the feasible threshold values according to thepercentile technique along with Average Annual OccurrenceNumber (AAON) Jiang et al [41] and Yuguo [42] suggestedthat the optimal threshold can be obtained if the values ofAAON lie between 1 and 2 Table 2 shows that 90th percentileobservations are suitable for the optimum threshold selectionof most of areas in the present study We have 168 valuesin each station according to the above table Astore stationhas 32 threshold values due to which 16 values are beingcensored in 168 According to censored level 102 censoredlevel was selected Similarly 105 was selected for Balakotand Muzaffarabad According to this process maximumthreshold level 223 was selected for Gupis By using theabove process for each station 17 different censoring levelswere selected So we decide that the range from 10 to 23of censoring level should be kept for selecting thresholdvalues
31 Regional Frequency Analysis The following four steps areconsidered as prerequisite for frequency analysis Hoskingand 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
311 Data Screening In this study we use secondary dataafter carefully examining all locations for abnormalities andmissing observations Therefore we use 14 years of data forRFA that were obtained from seventeen locations
312 Discordance Test Table 3 shows 119863119894 result of (18) for17 locations of this study region It can be observed fromthe results of LPL-moments in Table 3 that the value of 119863-statistic varies from 007 to 244 If 119863119894 is greater than 3 thelocation is considered to be discordant from the rest of theregional data Hosking and Wallis [7] In this study regionno location is diagnosed as discordant (119863119894 ge 3) Thereforewe use all data for the development of the RFA based on L-moments and PL-moments
313 Regional Heterogeneity Measure The next step is theformation of the homogeneous region which is conven-tionally tougher and needs the higher number of subjectivejudgments The homogeneity conditions are defined as thelocations that have the same frequency distributions
Advances in Meteorology 7
Table 2 Precipitation threshold selection in GPA distribution for 17 stations
Station 80th 90th 95th 975th 99thAstore
Threshold 98 32 02 0 0AAON 243 121 064 043 043
BalakotThreshold 325 148 51 3 0AAON 243 129 057 029 014
MuzaffarabadThreshold 304 118 5 1 0AAON 25 121 057 029 021
Garhi DupattaThreshold 288 10 38 0 0AAON 243 121 057 029 029
DirThreshold 31 20 5 005 0AAON 243 121 057 029 021
KakulThreshold 289 134 21 05 0AAON 164 129 057 029 021
KotliThreshold 13 6 005 0 0AAON 243 135 057 036 036
Saidu SharifThreshold 256 83 2 0 0AAON 243 121 057 036 036
cheratThreshold 5 005 0 0 0AAON 243 143 093 093 093
PeshawarThreshold 6 005 0 0 0AAON 243 129 057 057 057
DroshThreshold 76 15 02 005 0AAON 243 121 057 05 021
D I KhanThreshold 03 0 0 0 0AAON 257 129 129 129 129
GupisThreshold 0 0 0 0 0AAON 271 271 271 271 271
SkarduThreshold 2 03 0 0 0AAON 243 071 071 071 071
ChilasThreshold 1 05 0 0 0AAON 171 107 107 107 107
BunjiThreshold 13 0 0 0 0AAON 15 15 15 15 15
GilgitThreshold 12 005 0 0 0AAON 243 135 071 071 071
8 Advances in Meteorology
04
03
02
01
00
04
03
02
01
00
00 01 02 03 04 05 06 00 01 02 03 04 05 06
GLOGEVGPA
GNOAverageData
GLOGEVGPA
GNOAverageData
L-ku
rtos
is
L㰀-k
urto
sis
L-skewness L-skewness
Figure 2 L-diagram and PL-diagram of the GLO GEV GPA and GNO distributions
Table 3 Discordance test result based on L-moments and PL-moments
Name of site L-moments (119863119894) PL-moments (119863119894)Astore 205 244Balakot 142 085Bunji 055 060Chilas 029 007Cherat 079 064DI khan 129 190Dir 082 086Drosh 051 043Garhi Dupatta 143 133Gilgit 026 059Gupis 237 231Kakul 082 122Kotli 077 059Muzaffarabad 099 085Peshawar 007 008Saidu Sharif 191 157Skardu 067 065
For the present study area realization of the Kappaprobability distribution is used to conduct the heterogeneitytest based on the L-moments and PL-moments
Number of simulations are 10000 for computing theheterogeneityWe computed the regional average L-momentsratios the regional PL-moments ratios and the correspond-ing parameter values of the fitted Kappa probability dis-tribution (see Table 4) Table 4 shows the results of theheterogeneity measure using L-moments and PL-momentsmethods It can be observed from Table 4 that the different
values for the119867-statistic are minus041 minus160 and minus289 based onL-moments andminus006minus18 andminus317 based onPL-momentsTherefore we concluded that by comparing these resultsand the heterogeneity conditions study region is acceptablyhomogeneous for L-moments and PL-moments No furthersubdivisions of the present study are necessary
32 Fitting Appropriate Probability Distribution After homo-geneity analysis of the study area a suitable probabilitydistribution is required for the RFAThe objective is not onlyto recognize a suitable probability distribution for RFA butalso to observe a probability distribution that will providerobust quantile estimate for each location and for the regionalgrowth cure List of candidate probability distributions forRFA is GLO GEV GPA and GNO
We plotted L-moments and PL-moments diagrams forpreliminary evaluation of the probability distribution for thestudy area
Figure 2 illustrates an analogy of the observed and hypo-thetical relationships of the probability distribution Figure 2shows that GLO distribution is not a suitable candidate forthe L-moments and PL-moments
Interestingly both analyses of the L-moments and the PL-moments diagram show that the sample average values areappropriately distinguished by the hypothetical L-momentsand PL-moments for GPA and GNO distributions
However it is hard to find a suitable probability distri-bution that fits most of the regional observed data Table 5shows the goodness of fit test results for candidate probabilitydistributions
Table 5 shows that GLO distribution failed the goodnessof fit test for both L-moments and for PL-moments methodsas the calculated value of the 119885-test for the GLO distribution
Advances in Meteorology 9
Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments
Heterogeneity measures L-moment PL-momentHeterogeneity measure1198671
Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060
Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800
Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170
Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments
Method Distribution Parameters120585 120572 119870L-moments
GEV 04660 05644 minus02751GNO 06671 07565 minus07594GPA minus00551 09944 minus00575
PL-momentsGEV 05220 05116 minus02686GNO 07046 06837 minus07487GPA 00478 09073 minus00472
is larger than the critical value of 164 (at 90 confidencelevel)
It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile
Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution
33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7
based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)
Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is
119876 (119865) = 1198971198941119902 (119865) (25)
The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3
Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions
Therefore it is necessary to assess the performance ofregional quantile estimates
4 Accuracy of the Estimated Quantiles and theRegional Growth Curve
A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten
10 Advances in Meteorology
Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865
01000 05000 08000 09000 09500 09800 09900 09975 099875 09990
L-momentsGEV 00454 06836 15139 22247 30591 44161 56871 90748 113165 121338GNO 00473 06671 15585 23072 31448 44098 54997 80676 95666 100823GPA 00500 06481 16217 23931 31958 43073 51878 70578 80502 83782
PL-momentsGEV 01397 07190 14670 21034 28469 40497 51703 81363 10086 107950GNO 01412 07046 15062 21752 29203 40410 50032 72608 85739 90249GPA 01436 06871 15649 22547 29674 39460 47149 63304 71786 74577
Qua
ntile
Regional growth curve (L-moments)
GPAGNOGEV
GPAGNOGEV
Qua
ntile
Return period
Regional growth gurve (PL-moments)
5
4
3
2
1
0
5
4
3
2
1
0
minus2 minus1 0 1 2 3 4 5
2 5 10 20 50 100
Return period2 5 10 20 50 100
Reduced variate minuslog (minus log (F))
minus2 minus1 0 1 2 3 4 5
Reduced variate minuslog (minus log (F))
Figure 3 L-moments and PL-moments regional growth curves
as 119876(119898)119894 (119865) for the nonexceedance probability 119865 The relativeerror for this estimator is
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) (26)
The bias and the RMSE of the above quantity over all 119872repetition are
Bias = 119861119894 (119865) = 1119872119872sum119898=1
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)
= 1119872 [[119872sum119898=1
(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12
(27)
Also for the estimated quantile the regional average bias andthe relative RMSE are
119861119877 (119865) = 1119873119873sum119894=1
119861119894 (119865)
119877119877 (119865) = 1119873119873sum119894=1
119877119894 (119865) (28)
We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval
119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)
Advances in Meteorology 11
Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le
119902 (119865)119871005 (119865) (30)
The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves
We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]
We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively
Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds
Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion
This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail
GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments
Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents
Appendix
The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows
PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by
119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (minus ln119865)119896 119896 = 0 (A2)
Wang [10] developed the partial probability-weightedmoments (PWMs) of the GEV as
(119903 + 1) 1205731015840119903 = 120577 + 120572119867 (120574 1198650 119896) (A3)
where
119867(120574 1198650 119896) = 1119896 [1 minus119875 1 + 119896 minus (119903 + 1) ln1198650(1 minus 119865119903+10 ) (119903 + 1)119896 ] (A4)
12 Advances in Meteorology
Table8Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f30
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus0540
0minus19
7674
minus195328
minus193602
minus191579
minus188659
minus186376
minus1840
97minus18
1165
minus179056
119877(119865)
00900
00695
00963
01952
03334
06038
09057
13261
21337
300
48119877119877 (119865)
19873
01019
00635
00876
01090
01369
01595
01839
02191
02478
119871119861119864lowast
minus20937
08829
09296
08842
08411
07862
07460
07065
06557
06177
119880119861119864lowast
18714
11949
11208
11580
11884
12244
12516
12818
13285
13683
L-mom
ents
GNO
119861119877 (119865)
08124
minus196279
minus193846
minus192837
minus191816
minus190506
minus189573
minus188699
minus187636
minus186899
119877(119865)
00986
00696
00917
02051
03520
06027
08421
11312
16008
20312
119877119877 (119865)
20766
01045
00587
00889
01120
01368
01533
01686
01878
02016
119871119861119864lowast
minus21331
08683
09271
08780
08379
07958
07691
07453
07172
06978
119880119861119864lowast
16569
11974
11077
11569
11954
12355
12615
12851
13149
13369
L-mom
ents
GPA
119861119877 (119865)
38654
minus194351
minus193564
minus193226
minus192616
minus191706
minus191060
minus190506
minus189966
minus189727
119877(119865)
01242
006
6900998
02136
03552
05968
08313
11188
15941
20368
119877119877 (119865)
24785
01033
00614
00892
01111
01386
01602
01833
02162
02431
119871119861119864lowast
minus25192
08591
09215
08817
08444
07979
07646
07327
06919
06623
119880119861119864lowast
14272
11911
11115
11588
11976
12459
12841
13257
13859
14367
PL-m
oments
GEV
119861119877 (119865)
minus02662
minus197059
minus194770
minus193340
minus191600
minus188996
minus186909
minus184797
minus182052
minus1800
66119877(119865)
05379
006
8700947
02023
03467
06171
09087
13058
20551
2854
119877119877 (119865)
38117
00948
006
4900960
01210
01508
01736
01977
02324
02609
119871119861119864lowast
minus77814
08892
09252
08725
08259
07697
07296
06911
064
1706050
119880119861119864lowast
45516
11827
11207
11716
12117
12554
12881
13211
13693
14104
PL-m
oments
GNO
119861119877 (119865)
35142
minus195922
minus193485
minus192704
minus191894
minus190806
minus1900
03minus18
9235
minus188284
minus187617
119877(119865)
06203
006
8900939
02141
03644
06143
08483
11272
15747
19809
119877119877 (119865)
43522
00971
00626
00980
01238
01504
01675
01831
02024
02162
119871119861119864lowast
minus10618
08772
09204
08655
08233
07800
07530
07290
07012
06822
119880119861119864lowast
46353
11849
11122
11734
12189
12653
12949
13218
13546
13780
PL-m
oments
GPA
119861119877 (119865)
1212
3minus19
4300
minus193238
minus193113
minus192716
minus192046
minus191555
minus19114
1minus19
0768
minus1906
49119877(119865)
08513
006
6601041
02250
03684
06021
08215
10853
15144
19094
119877119877 (119865)
58811
00962
006
6900996
01233
01509
01720
01941
02258
02518
119871119861119864lowast
minus17507
08698
09130
08680
08299
07846
07524
07220
06831
06552
119880119861119864lowast
47307
11791
11189
11774
12215
12739
1314
13567
14195
14722
Advances in Meteorology 13
Table9Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f60
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus13271
minus194230
minus193551
minus193241
minus192649
minus191612
minus190736
minus189839
minus188690
minus187888
119877(119865)
00846
00624
00851
01873
03250
05841
08652
12505
19839
27721
119877119877 (119865)
19155
00833
00612
00855
01024
01213
01360
01523
01770
01983
119871119861119864lowast
minus19593
08756
09254
08851
08491
08074
07796
07501
07118
06834
119880119861119864lowast
16242
11675
11015
11472
11879
12309
12678
13038
13659
14141
L-mom
ents
GNO
119861119877 (119865)
08417
minus193335
minus192795
minus192984
minus192999
minus192879
minus192742
minus192593
minus192399
minus192264
119877(119865)
00968
00634
00858
02000
03439
05850
08123
10845
15235
19239
119877119877 (119865)
21055
00881
00582
00861
01050
01235
01354
01463
01599
01699
119871119861119864lowast
minus22887
08647
09244
08794
08466
08141
07957
07798
07597
07443
119880119861119864lowast
15863
11714
1096
11511
11934
1240
512
666
12939
13253
13505
L-mom
ents
GPA
119861119877 (119865)
46371
minus192512
minus192631
minus193160
minus193417
minus193606
minus193745
minus193933
minus194306
minus194708
119877(119865)
01280
00624
00950
02094
03468
05740
07903
10536
14878
18928
119877119877 (119865)
264
4200915
00614
00882
01064
01270
01429
01599
01849
02059
119871119861119864lowast
minus38088
08628
09186
08815
08510
08167
07924
07703
07422
07178
119880119861119864lowast
14699
11721
11020
11534
11925
12454
12791
13236
13869
1440
5
PL-m
oments
GEV
119861119877 (119865)
minus16968
minus193984
minus193601
minus193623
minus193319
minus192576
minus191861
minus191081
minus190026
minus189255
119877(119865)
05431
00596
00888
01978
03366
05844
08417
11838
18179
24868
119877119877 (119865)
38656
00783
00634
0095
01166
01387
01541
01700
01928
02120
119871119861119864lowast
minus92921
08907
09201
08750
08403
07984
07714
07434
07043
06783
119880119861119864lowast
53553
11521
11099
1166
412
098
12572
12869
13216
13697
14105
PL-m
oments
GNO
119861119877 (119865)
23785
minus193306
minus192979
minus193417
minus193628
minus193678
minus193617
minus193509
minus193331
minus193185
119877(119865)
06278
006
0100911
02100
03523
05804
0788
10303
14113
17516
119877119877 (119865)
43743
00812
00623
00965
01186
01391
01513
01618
01742
01830
119871119861119864lowast
minus12083
08856
09188
08709
08389
08061
07870
07706
07517
07380
119880119861119864lowast
52707
11561
11059
11704
12152
12613
12898
13138
13361
1356
PL-m
oments
GPA
119861119877 (119865)
1118
7minus19
2590
minus192910
minus193681
minus1940
84minus19
4322
minus1944
04minus19
4463
minus194560
minus1946
73119877(119865)
08596
00597
01009
02229
03581
05615
07391
09418
12558
15341
119877119877 (119865)
58310
00840
006
6000991
0119
501391
01522
01650
01828
01971
119871119861119864lowast
minus18977
08800
09127
08699
08413
08098
07912
07682
07447
07255
119880119861119864lowast
51281
11551
11117
11769
12173
12615
12887
13227
1360
913
899
14 Advances in Meteorology
Table10A
ccuracymeasure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f90
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus00117
minus194226
minus193109
minus192926
minus192542
minus191809
minus191151
minus1904
48minus18
9497
minus188790
119877(119865)
00787
00556
00823
01850
03151
05459
07840
10989
16793
22883
119877119877 (119865)
17329
00814
00544
00832
01031
01238
01381
01525
01727
01892
119871119861119864lowast
minus19303
08935
09287
08885
08587
08246
08006
07772
07463
07229
119880119861119864lowast
15717
11499
10973
11461
11804
12156
12393
12635
12966
13242
L-mom
ents
GNO
119861119877 (119865)
15851
minus193606
minus192517
minus192695
minus192747
minus192678
minus192569
minus192437
minus192245
minus192098
119877(119865)
00917
00567
00846
01967
03305
05446
07392
09660
13223
1640
0119877119877 (119865)
19328
00850
00543
00853
01052
01237
01347
01442
01554
01632
119871119861119864lowast
minus23268
08848
09261
08844
08568
08301
08139
08000
07833
07722
119880119861119864lowast
15168
11542
10942
11489
11851
12197
1240
112
575
12779
12925
L-mom
ents
GPA
119861119877 (119865)
49081
minus192995
minus192417
minus192865
minus193078
minus193175
minus193195
minus193209
minus193256
minus193330
119877(119865)
01234
00571
00935
02081
03365
05322
07054
09053
12180
14975
119877119877 (119865)
246
8800882
00576
00870
01054
01239
01364
01489
01660
01798
119871119861119864lowast
minus45353
08783
09210
08838
08594
08326
08141
07961
07731
07557
119880119861119864lowast
14833
11574
10996
11532
11870
12223
1246
712
711
1304
413
312
PL-m
oments
GEV
119861119877 (119865)
minus00877
minus194145
minus193060
minus193062
minus192847
minus192278
minus191698
minus191038
minus190100
minus189374
119877(119865)
05458
00565
00855
01963
03338
05712
08090
11156
16650
22276
119877119877 (119865)
39079
00784
00585
00934
0117
201406
01558
01703
01897
02050
119871119861119864lowast
minus10190
08983
09236
08765
08431
08067
07818
07579
07271
07041
119880119861119864lowast
58730
1144
811036
11652
12081
12504
12771
13026
13364
13623
PL-m
oments
GNO
119861119877 (119865)
38414
minus193665
minus192536
minus192859
minus193044
minus193102
minus193057
minus192966
minus192804
minus192665
119877(119865)
06308
00578
00896
02088
03488
05690
07659
09926
13434
16524
119877119877 (119865)
44651
00819
00596
00961
0119
301405
01526
01628
01744
01822
119871119861119864lowast
minus13011
08910
09192
08716
08411
08122
07949
07801
07634
07519
119880119861119864lowast
56756
11491
11030
11693
12128
12539
12775
12974
13197
13347
PL-m
oments
GPA
119861119877 (119865)
126398
minus193142
minus192484
minus193077
minus193404
minus193589
minus193637
minus193655
minus193680
minus193720
119877(119865)
08641
00588
00999
02220
03558
05535
07227
09125
1200
914
521
119877119877 (119865)
60116
00855
006
4000985
0119
901400
01529
01651
01812
01937
119871119861119864lowast
minus19889
08842
09130
08700
08435
08156
07970
07789
07566
07399
119880119861119864lowast
53743
11528
11103
11755
12158
12552
12807
13055
13385
13636
Advances in Meteorology 15
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
0 2 4 6
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
Regional growth curve for GPA
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F))
0
5
10
15
20
Qua
ntile
2 4 60Reduced variate minuslog (minus log (F))
0
5
10
15
Qua
ntile
(based on L-moments) Regional growth curve for GPA
(based on PL-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30
16 Advances in Meteorology
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60
Advances in Meteorology 17
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90
18 Advances in Meteorology
The first four PL-moments of the GEV are defined as
12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)
(A5)
In previous equation 119875(sdot sdot) is an Incomplete Gammafunction
119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500
120579119896expminus120579119889Θ (A6)
Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution
The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by
119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)
The partial PWMs of the GLO are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)
(A9)
where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function
1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Geology Advances in
Advances in Meteorology 3
37∘0㰀0㰀㰀N
36∘0㰀0㰀㰀N
35∘0㰀0㰀㰀N
34∘0㰀0㰀㰀N
33∘0㰀0㰀㰀N
32∘0㰀0㰀㰀N
37∘0㰀0㰀㰀N
36∘0㰀0㰀㰀N
35∘0㰀0㰀㰀N
34∘0㰀0㰀㰀N
33∘0㰀0㰀㰀N
32∘0㰀0㰀㰀N
71∘0㰀0㰀㰀E 72∘0㰀0㰀㰀E 73∘0㰀0㰀㰀E 74∘0㰀0㰀㰀E 75∘0㰀0㰀㰀E 76∘0㰀0㰀㰀E
71∘0㰀0㰀㰀E 72∘0㰀0㰀㰀E 73∘0㰀0㰀㰀E 74∘0㰀0㰀㰀E 75∘0㰀0㰀㰀E 76∘0㰀0㰀㰀E
W
N
E
S
0 30 60 120 180 240(km)
StationsWater bodiesStudy area
Figure 1 Locations of Northern areas and Khyber Pakhtunkhwa and the meteorological stations (119909 axis indicates Longitude [E] 119910 axisindicates Latitude [N])
The L-moments have no units of measurement which arecalled the L-moments ratios The L-moments ratios pro-posed by Hosking [28] are computed as
120591 = 12058221205821
1205913 = 12058231205822
1205914 = 12058241205822 (5)
4 Advances in Meteorology
where 120591 represents the L-coefficient of variation (L-Cv)1205913 represents the L-coefficient of skewness (L-Cs) and 1205914represents the L-coefficient of kurtosis (L-Ck)
The arranged sample is given as 119909(1) le 119909(2) le 119909(3) sdot sdot sdot le119909(119899) Wang [8] stated that the statistic
119887119903 = 1119899119899sum119894=1
(119894 minus 1) (119894 minus 2) sdot sdot sdot (119894 minus 119903)(119899 minus 1) (119899 minus 2) sdot sdot sdot (119899 minus 119903)119909(119894) (6)
is an unbiased estimator of 120573119903So
1198971 = 11988701198972 = 21198871 minus 11988701198973 = 61198872 minus 61198871 + 11988701198974 = 201198873 minus 301198872 + 121198871 minus 1198870
(7)
where 1198971 1198972 1198973 and 1198974 are the first four L-moments of thesample And similarly
119905 = 11989721198971 1199053 = 11989731198972 1199054 = 11989741198972
(8)
are the sample L-moments ratios
222 The Partial L-Moments Wang [8 10 11] introduced aconcept of partial probability-weighted moments (PPWMs)that will estimate the higher quantiles of flood flows Data canbe censored to the right tail or left tail
Initially PPWMswere to take out the smaller values fromthe process of distribution fitting because such values haveslight influence on the frequency analysis and are nuisance tothe fitting process
The left tail PPWMs are defined by Wang [8 11] as
120573lowast119903 = int11198650
119909 (119865) 119865119903119889119865 (9)
where 1198650 = 119865(1199090) which is the lower limit of the censoredobservations and 1199090 is the censoring threshold value
The PPWMs elongated form described by Wang [10] areto be given a censored sample as
120573119903 = 11 minus 119865119903+10 int11198650
119909 (119865) 119865119903119889119865 (10)
If the value of 1198650 is starting from the zero then the result ofPPWMswill be the same as the usual PWMs As 119909(1) le 119909(2) le119909(3) sdot sdot sdot le 119909(119899) is the arranged sample Wang [10] describes theunbiased estimator of 120573119903 as
119903 = 1(1 minus 119865119903+10 ) 119899119899sum119894=1
(119894 minus 1) (119894 minus 2) sdot sdot sdot (119894 minus 119903)(119899 minus 1) (119899 minus 2) sdot sdot sdot (119899 minus 119903)119909lowast(119894) (11)
where
119909lowast(119894) = 0 for 119909(119894) le 119909(0)119909lowast(119894) = 119909(119894) for 119909(119894) gt 119909(0) (12)
The censoring level 1198650 is the prior selection of the numberof censored sample data The procedure that determines thenumber of sample data points are to be censored
1198650 = 1198990119899 (13)
where 1198990 and 119899 are the lengths of sample which are to becensored and uncensored respectively Similarly 1199090 is thehighest value of the censored sample The first four PL-moments for the PPWMs are
1 = 12057302 = 2 1205731 minus 12057303 = 1205732 minus 6 1205731 + 12057304 = 20 1205733 minus 30 1205732 + 12 1205731 minus 1205730
(14)
Similarly the PL-moments ratios can be written as
120591 = 21
1205913 = 32
1205914 = 42
(15)
where 120591 1205913 and 1205914 denote the partial L-coefficient of variation(PL-Cv) partial L-coefficient of skewness (PL-Cs) and par-tial L-coefficient of kurtosis (PL-Ck) respectively Thereforethe first four sample PL-moments can be computed as
1198971 = 01198972 = 21 minus 01198973 = 62 minus 61 + 01198974 = 203 minus 302 + 121 minus 0
(16)
And the first four sample PL-moments ratios can be com-puted as
119905 = 11989721198971 1199053 = 11989731198972 1199054 = 11989741198972
(17)
Advances in Meteorology 5
where 119905 1199053 and 1199054 represent the sample partial L-momentsratios of the PL-Cv PL-Cs and PL-Ck respectively Thederivation is L-moment and PL-moments are given inAppendix In the present study different level of censoringthreshold is selected
23 Regional Frequency Analysis Hosking and Wallis [7 30]identified the following four steps to explain the procedure ofthe 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
231 Data Screening We screened data anomalies beforeapplying any statistical analysis
232 Discordance Test Hosking and Wallis [7] suggested adiscordancy measure (119863119894) test that recognizes the locationswhere sample L-moments are marked contrarily from themost other locations Locations with the large flaws in thedata will be flagged as discordant
The discordancy test for a region containing119873 locationsfor site 119894 is proposed by Hosking and Wallis [7] as follows
119863119894 = 13119873 (119906119894 minus 119906)119879 119878minus1 (119906119894 minus 119906) 119894 = 1 2 119873 (18)
where 119906119894 is the vector containing the three sample L-momentsratios for the site 119894 expressed as
119906119894 = [119905(119894)2 119905(119894)3 119905(119894)4 ]119879 (19)
119906 is the average vector of 119906119894 for the overall region that is
119906 = 1119873119873sum119894=1
119906119894 (20)
and 119878 is the covariance matrix for the sample that can beexpressed as
119878 = 119873sum119894=1
(119906119894 minus 119906) (119906119894 minus 119906)119879 (21)
Broadly speaking a location or a site is considered to bediscordant from the whole region or group if the value of 119863119894is larger than the critical value
233 Heterogeneity Test The homogeneity measure (119867119895)identifies homogenous regions Hosking and Wallis [7] Itis also useful to tag the locations if they are plausible tohandle as a homogeneous region It estimates the amountof heterogeneity in the overall region The heterogeneity test(119867119895) is computed as
119867119895 = 119881119895 minus 120583V119895120590V119895 119895 = 1 2 3 (22)
where 120583V119895 and 120590V119895 are representing the mean and standarddeviation of the simulated 119881119895 values Also
1198811 = 119873sum119894=1
119899119894 (119905(119894)2 minus 1199051198772 )sum119873119894=1 119899119894
12
1198812 =sum119873119894=1 [119899119894 (119905(119894)2 minus 1199051198772 )2 + (119905(119894)3 minus 1199051198773 )212]
sum119873119894=1 119899119894
1198813 =sum119873119894=1 [119899119894 (119905(119894)3 minus 1199051198773 )2 + (119905(119894)4 minus 1199051198774 )212]
sum119873119894=1 119899119894
(23)
Here 1199051198772 1199051198773 and 1199051198774 are region average L-moments or PL-moments ratios We assessed the heterogeneity of a region assuggested by Hosking and Wallis [7]
Region is acceptably homogeneous if119867 lt 1Region is possibly heterogeneous if 1 le 119867 lt 2Region is definitely heterogeneous if119867 ge 2
24 Selection of the Appropriate Probability DistributionHosking andWallis [7] proposed two approaches to select thedistribution that fitted best the data the L-moment ratios dia-gram and the 119885-test The L-moment ratios diagram is usingthe unbiased estimators Hosking [28] Stedinger et al [31]Vogel and Fennessey [32] and Hosking [33]The L-momentsratio diagram is a plot of the computed values L-Cs andthe observed values L-Ck of the distribution function Thecurves indicate the hypothetical connections between L-Csand L-Ck of the candidate distribution The L-moment ratiodiagrams have been proposed for discriminating between thecandidate probability distributions in describing the regionalinformation (Hosking [28] Stedinger et al [31] Hoskingand Wallis [7]) The L-moments ratio diagrams have beenused as a component of probability distribution process forregional information (Schaefer [34] Pearson [35] Vogel andFennessey [32] Vogel et al [36] Chow and Watt [37] OnOzand Bayazit [38] Vogel and Wilson [39] Peel et al [40])
Hosking and Wallis [7] suggested a measure to see howwell the L-Cs and L-Ck of the fitted probability distributionmatch the regional average L-Cs and L-Ck of the observedinformation
The measure goodness of fit for every single selectedprobability distribution is computed as follows
119885Dis = (120591Dis4 minus 1199051198774 )1205904 (24)
where 120591Dis4 represents the value of the L-Ck of the fitteddistribution 1199051198774 represents the weighted regional averageL-Ck and 1205904 represents standard deviation of the 1199051198774 whichis obtained from the simulation of the Kappa probabilitydistribution
If the computed value of 119885Dis is equal to zero the proba-bility distributionwill be themost suitable fit If the computed
6 Advances in Meteorology
Table 1 Statistics of annual extreme monthly precipitation for study region based on L-moments and PL-moments
Sites L-moments PL-momentsMean 119905 1199053 1199054 Mean 119905 1199053 1199054
Astore 37624 0518 0366 0193 41752 0467 0377 0183Balakot 120450 0478 0322 0192 133259 0428 0335 0198Bunji 14297 0609 0456 0259 16339 0553 0450 0257Chilas 15091 0585 0392 0194 17603 0516 0375 0200Cherat 49038 0580 0375 0162 55289 0526 0355 0161DI khan 28885 0689 0516 0262 32351 0651 0487 0250Dir 105694 0432 0216 0098 116741 0378 0222 0097Drosh 45284 0525 0317 0136 50316 0474 0307 0137Garhi Dupatta 111352 0454 0227 0111 123374 0398 0229 0120Gilgit 12677 0591 0412 0228 14290 0539 0399 0239Gupis 23446 0678 0529 0331 30299 0583 0506 0350Kakul 99431 0453 0247 0102 110068 0399 0254 0093Kotli 95070 0542 0332 0136 106903 0487 0314 0138Muzaffarabad 118439 0459 0284 0182 131122 0405 0301 0194Peshawar 46216 0572 0398 0216 51761 0520 0392 0222Saidu Sharif 85667 0455 0265 0181 94978 0399 0283 0200Skardu 20438 0616 04564 0238 22885 0570 0444 0231
value of 119885-statistic is less than 164 at 90 confidence level(ie |119885Dis| le 164) it will indicate that the distributionqualifies the goodness of fit criteria If there are more thanone distribution that qualify the criteria the most suitabledistribution has the minimum |119885Dis| value3 Estimation
The sitesrsquo information and statistic by using L-moments forthe present study are presented in Table 1 In Table 1 meanrepresents the first sample LPL-moments and 119905 1199053 and 1199054are the sample LPL-moments ratios of the L-CvPL-CvL-CsPL-Cs and L-CkPL-Ck respectively The lower levelcensoring threshold is selected from 10 to 23 Table 2expresses the feasible threshold values according to thepercentile technique along with Average Annual OccurrenceNumber (AAON) Jiang et al [41] and Yuguo [42] suggestedthat the optimal threshold can be obtained if the values ofAAON lie between 1 and 2 Table 2 shows that 90th percentileobservations are suitable for the optimum threshold selectionof most of areas in the present study We have 168 valuesin each station according to the above table Astore stationhas 32 threshold values due to which 16 values are beingcensored in 168 According to censored level 102 censoredlevel was selected Similarly 105 was selected for Balakotand Muzaffarabad According to this process maximumthreshold level 223 was selected for Gupis By using theabove process for each station 17 different censoring levelswere selected So we decide that the range from 10 to 23of censoring level should be kept for selecting thresholdvalues
31 Regional Frequency Analysis The following four steps areconsidered as prerequisite for frequency analysis Hoskingand 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
311 Data Screening In this study we use secondary dataafter carefully examining all locations for abnormalities andmissing observations Therefore we use 14 years of data forRFA that were obtained from seventeen locations
312 Discordance Test Table 3 shows 119863119894 result of (18) for17 locations of this study region It can be observed fromthe results of LPL-moments in Table 3 that the value of 119863-statistic varies from 007 to 244 If 119863119894 is greater than 3 thelocation is considered to be discordant from the rest of theregional data Hosking and Wallis [7] In this study regionno location is diagnosed as discordant (119863119894 ge 3) Thereforewe use all data for the development of the RFA based on L-moments and PL-moments
313 Regional Heterogeneity Measure The next step is theformation of the homogeneous region which is conven-tionally tougher and needs the higher number of subjectivejudgments The homogeneity conditions are defined as thelocations that have the same frequency distributions
Advances in Meteorology 7
Table 2 Precipitation threshold selection in GPA distribution for 17 stations
Station 80th 90th 95th 975th 99thAstore
Threshold 98 32 02 0 0AAON 243 121 064 043 043
BalakotThreshold 325 148 51 3 0AAON 243 129 057 029 014
MuzaffarabadThreshold 304 118 5 1 0AAON 25 121 057 029 021
Garhi DupattaThreshold 288 10 38 0 0AAON 243 121 057 029 029
DirThreshold 31 20 5 005 0AAON 243 121 057 029 021
KakulThreshold 289 134 21 05 0AAON 164 129 057 029 021
KotliThreshold 13 6 005 0 0AAON 243 135 057 036 036
Saidu SharifThreshold 256 83 2 0 0AAON 243 121 057 036 036
cheratThreshold 5 005 0 0 0AAON 243 143 093 093 093
PeshawarThreshold 6 005 0 0 0AAON 243 129 057 057 057
DroshThreshold 76 15 02 005 0AAON 243 121 057 05 021
D I KhanThreshold 03 0 0 0 0AAON 257 129 129 129 129
GupisThreshold 0 0 0 0 0AAON 271 271 271 271 271
SkarduThreshold 2 03 0 0 0AAON 243 071 071 071 071
ChilasThreshold 1 05 0 0 0AAON 171 107 107 107 107
BunjiThreshold 13 0 0 0 0AAON 15 15 15 15 15
GilgitThreshold 12 005 0 0 0AAON 243 135 071 071 071
8 Advances in Meteorology
04
03
02
01
00
04
03
02
01
00
00 01 02 03 04 05 06 00 01 02 03 04 05 06
GLOGEVGPA
GNOAverageData
GLOGEVGPA
GNOAverageData
L-ku
rtos
is
L㰀-k
urto
sis
L-skewness L-skewness
Figure 2 L-diagram and PL-diagram of the GLO GEV GPA and GNO distributions
Table 3 Discordance test result based on L-moments and PL-moments
Name of site L-moments (119863119894) PL-moments (119863119894)Astore 205 244Balakot 142 085Bunji 055 060Chilas 029 007Cherat 079 064DI khan 129 190Dir 082 086Drosh 051 043Garhi Dupatta 143 133Gilgit 026 059Gupis 237 231Kakul 082 122Kotli 077 059Muzaffarabad 099 085Peshawar 007 008Saidu Sharif 191 157Skardu 067 065
For the present study area realization of the Kappaprobability distribution is used to conduct the heterogeneitytest based on the L-moments and PL-moments
Number of simulations are 10000 for computing theheterogeneityWe computed the regional average L-momentsratios the regional PL-moments ratios and the correspond-ing parameter values of the fitted Kappa probability dis-tribution (see Table 4) Table 4 shows the results of theheterogeneity measure using L-moments and PL-momentsmethods It can be observed from Table 4 that the different
values for the119867-statistic are minus041 minus160 and minus289 based onL-moments andminus006minus18 andminus317 based onPL-momentsTherefore we concluded that by comparing these resultsand the heterogeneity conditions study region is acceptablyhomogeneous for L-moments and PL-moments No furthersubdivisions of the present study are necessary
32 Fitting Appropriate Probability Distribution After homo-geneity analysis of the study area a suitable probabilitydistribution is required for the RFAThe objective is not onlyto recognize a suitable probability distribution for RFA butalso to observe a probability distribution that will providerobust quantile estimate for each location and for the regionalgrowth cure List of candidate probability distributions forRFA is GLO GEV GPA and GNO
We plotted L-moments and PL-moments diagrams forpreliminary evaluation of the probability distribution for thestudy area
Figure 2 illustrates an analogy of the observed and hypo-thetical relationships of the probability distribution Figure 2shows that GLO distribution is not a suitable candidate forthe L-moments and PL-moments
Interestingly both analyses of the L-moments and the PL-moments diagram show that the sample average values areappropriately distinguished by the hypothetical L-momentsand PL-moments for GPA and GNO distributions
However it is hard to find a suitable probability distri-bution that fits most of the regional observed data Table 5shows the goodness of fit test results for candidate probabilitydistributions
Table 5 shows that GLO distribution failed the goodnessof fit test for both L-moments and for PL-moments methodsas the calculated value of the 119885-test for the GLO distribution
Advances in Meteorology 9
Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments
Heterogeneity measures L-moment PL-momentHeterogeneity measure1198671
Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060
Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800
Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170
Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments
Method Distribution Parameters120585 120572 119870L-moments
GEV 04660 05644 minus02751GNO 06671 07565 minus07594GPA minus00551 09944 minus00575
PL-momentsGEV 05220 05116 minus02686GNO 07046 06837 minus07487GPA 00478 09073 minus00472
is larger than the critical value of 164 (at 90 confidencelevel)
It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile
Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution
33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7
based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)
Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is
119876 (119865) = 1198971198941119902 (119865) (25)
The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3
Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions
Therefore it is necessary to assess the performance ofregional quantile estimates
4 Accuracy of the Estimated Quantiles and theRegional Growth Curve
A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten
10 Advances in Meteorology
Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865
01000 05000 08000 09000 09500 09800 09900 09975 099875 09990
L-momentsGEV 00454 06836 15139 22247 30591 44161 56871 90748 113165 121338GNO 00473 06671 15585 23072 31448 44098 54997 80676 95666 100823GPA 00500 06481 16217 23931 31958 43073 51878 70578 80502 83782
PL-momentsGEV 01397 07190 14670 21034 28469 40497 51703 81363 10086 107950GNO 01412 07046 15062 21752 29203 40410 50032 72608 85739 90249GPA 01436 06871 15649 22547 29674 39460 47149 63304 71786 74577
Qua
ntile
Regional growth curve (L-moments)
GPAGNOGEV
GPAGNOGEV
Qua
ntile
Return period
Regional growth gurve (PL-moments)
5
4
3
2
1
0
5
4
3
2
1
0
minus2 minus1 0 1 2 3 4 5
2 5 10 20 50 100
Return period2 5 10 20 50 100
Reduced variate minuslog (minus log (F))
minus2 minus1 0 1 2 3 4 5
Reduced variate minuslog (minus log (F))
Figure 3 L-moments and PL-moments regional growth curves
as 119876(119898)119894 (119865) for the nonexceedance probability 119865 The relativeerror for this estimator is
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) (26)
The bias and the RMSE of the above quantity over all 119872repetition are
Bias = 119861119894 (119865) = 1119872119872sum119898=1
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)
= 1119872 [[119872sum119898=1
(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12
(27)
Also for the estimated quantile the regional average bias andthe relative RMSE are
119861119877 (119865) = 1119873119873sum119894=1
119861119894 (119865)
119877119877 (119865) = 1119873119873sum119894=1
119877119894 (119865) (28)
We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval
119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)
Advances in Meteorology 11
Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le
119902 (119865)119871005 (119865) (30)
The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves
We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]
We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively
Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds
Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion
This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail
GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments
Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents
Appendix
The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows
PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by
119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (minus ln119865)119896 119896 = 0 (A2)
Wang [10] developed the partial probability-weightedmoments (PWMs) of the GEV as
(119903 + 1) 1205731015840119903 = 120577 + 120572119867 (120574 1198650 119896) (A3)
where
119867(120574 1198650 119896) = 1119896 [1 minus119875 1 + 119896 minus (119903 + 1) ln1198650(1 minus 119865119903+10 ) (119903 + 1)119896 ] (A4)
12 Advances in Meteorology
Table8Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f30
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus0540
0minus19
7674
minus195328
minus193602
minus191579
minus188659
minus186376
minus1840
97minus18
1165
minus179056
119877(119865)
00900
00695
00963
01952
03334
06038
09057
13261
21337
300
48119877119877 (119865)
19873
01019
00635
00876
01090
01369
01595
01839
02191
02478
119871119861119864lowast
minus20937
08829
09296
08842
08411
07862
07460
07065
06557
06177
119880119861119864lowast
18714
11949
11208
11580
11884
12244
12516
12818
13285
13683
L-mom
ents
GNO
119861119877 (119865)
08124
minus196279
minus193846
minus192837
minus191816
minus190506
minus189573
minus188699
minus187636
minus186899
119877(119865)
00986
00696
00917
02051
03520
06027
08421
11312
16008
20312
119877119877 (119865)
20766
01045
00587
00889
01120
01368
01533
01686
01878
02016
119871119861119864lowast
minus21331
08683
09271
08780
08379
07958
07691
07453
07172
06978
119880119861119864lowast
16569
11974
11077
11569
11954
12355
12615
12851
13149
13369
L-mom
ents
GPA
119861119877 (119865)
38654
minus194351
minus193564
minus193226
minus192616
minus191706
minus191060
minus190506
minus189966
minus189727
119877(119865)
01242
006
6900998
02136
03552
05968
08313
11188
15941
20368
119877119877 (119865)
24785
01033
00614
00892
01111
01386
01602
01833
02162
02431
119871119861119864lowast
minus25192
08591
09215
08817
08444
07979
07646
07327
06919
06623
119880119861119864lowast
14272
11911
11115
11588
11976
12459
12841
13257
13859
14367
PL-m
oments
GEV
119861119877 (119865)
minus02662
minus197059
minus194770
minus193340
minus191600
minus188996
minus186909
minus184797
minus182052
minus1800
66119877(119865)
05379
006
8700947
02023
03467
06171
09087
13058
20551
2854
119877119877 (119865)
38117
00948
006
4900960
01210
01508
01736
01977
02324
02609
119871119861119864lowast
minus77814
08892
09252
08725
08259
07697
07296
06911
064
1706050
119880119861119864lowast
45516
11827
11207
11716
12117
12554
12881
13211
13693
14104
PL-m
oments
GNO
119861119877 (119865)
35142
minus195922
minus193485
minus192704
minus191894
minus190806
minus1900
03minus18
9235
minus188284
minus187617
119877(119865)
06203
006
8900939
02141
03644
06143
08483
11272
15747
19809
119877119877 (119865)
43522
00971
00626
00980
01238
01504
01675
01831
02024
02162
119871119861119864lowast
minus10618
08772
09204
08655
08233
07800
07530
07290
07012
06822
119880119861119864lowast
46353
11849
11122
11734
12189
12653
12949
13218
13546
13780
PL-m
oments
GPA
119861119877 (119865)
1212
3minus19
4300
minus193238
minus193113
minus192716
minus192046
minus191555
minus19114
1minus19
0768
minus1906
49119877(119865)
08513
006
6601041
02250
03684
06021
08215
10853
15144
19094
119877119877 (119865)
58811
00962
006
6900996
01233
01509
01720
01941
02258
02518
119871119861119864lowast
minus17507
08698
09130
08680
08299
07846
07524
07220
06831
06552
119880119861119864lowast
47307
11791
11189
11774
12215
12739
1314
13567
14195
14722
Advances in Meteorology 13
Table9Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f60
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus13271
minus194230
minus193551
minus193241
minus192649
minus191612
minus190736
minus189839
minus188690
minus187888
119877(119865)
00846
00624
00851
01873
03250
05841
08652
12505
19839
27721
119877119877 (119865)
19155
00833
00612
00855
01024
01213
01360
01523
01770
01983
119871119861119864lowast
minus19593
08756
09254
08851
08491
08074
07796
07501
07118
06834
119880119861119864lowast
16242
11675
11015
11472
11879
12309
12678
13038
13659
14141
L-mom
ents
GNO
119861119877 (119865)
08417
minus193335
minus192795
minus192984
minus192999
minus192879
minus192742
minus192593
minus192399
minus192264
119877(119865)
00968
00634
00858
02000
03439
05850
08123
10845
15235
19239
119877119877 (119865)
21055
00881
00582
00861
01050
01235
01354
01463
01599
01699
119871119861119864lowast
minus22887
08647
09244
08794
08466
08141
07957
07798
07597
07443
119880119861119864lowast
15863
11714
1096
11511
11934
1240
512
666
12939
13253
13505
L-mom
ents
GPA
119861119877 (119865)
46371
minus192512
minus192631
minus193160
minus193417
minus193606
minus193745
minus193933
minus194306
minus194708
119877(119865)
01280
00624
00950
02094
03468
05740
07903
10536
14878
18928
119877119877 (119865)
264
4200915
00614
00882
01064
01270
01429
01599
01849
02059
119871119861119864lowast
minus38088
08628
09186
08815
08510
08167
07924
07703
07422
07178
119880119861119864lowast
14699
11721
11020
11534
11925
12454
12791
13236
13869
1440
5
PL-m
oments
GEV
119861119877 (119865)
minus16968
minus193984
minus193601
minus193623
minus193319
minus192576
minus191861
minus191081
minus190026
minus189255
119877(119865)
05431
00596
00888
01978
03366
05844
08417
11838
18179
24868
119877119877 (119865)
38656
00783
00634
0095
01166
01387
01541
01700
01928
02120
119871119861119864lowast
minus92921
08907
09201
08750
08403
07984
07714
07434
07043
06783
119880119861119864lowast
53553
11521
11099
1166
412
098
12572
12869
13216
13697
14105
PL-m
oments
GNO
119861119877 (119865)
23785
minus193306
minus192979
minus193417
minus193628
minus193678
minus193617
minus193509
minus193331
minus193185
119877(119865)
06278
006
0100911
02100
03523
05804
0788
10303
14113
17516
119877119877 (119865)
43743
00812
00623
00965
01186
01391
01513
01618
01742
01830
119871119861119864lowast
minus12083
08856
09188
08709
08389
08061
07870
07706
07517
07380
119880119861119864lowast
52707
11561
11059
11704
12152
12613
12898
13138
13361
1356
PL-m
oments
GPA
119861119877 (119865)
1118
7minus19
2590
minus192910
minus193681
minus1940
84minus19
4322
minus1944
04minus19
4463
minus194560
minus1946
73119877(119865)
08596
00597
01009
02229
03581
05615
07391
09418
12558
15341
119877119877 (119865)
58310
00840
006
6000991
0119
501391
01522
01650
01828
01971
119871119861119864lowast
minus18977
08800
09127
08699
08413
08098
07912
07682
07447
07255
119880119861119864lowast
51281
11551
11117
11769
12173
12615
12887
13227
1360
913
899
14 Advances in Meteorology
Table10A
ccuracymeasure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f90
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus00117
minus194226
minus193109
minus192926
minus192542
minus191809
minus191151
minus1904
48minus18
9497
minus188790
119877(119865)
00787
00556
00823
01850
03151
05459
07840
10989
16793
22883
119877119877 (119865)
17329
00814
00544
00832
01031
01238
01381
01525
01727
01892
119871119861119864lowast
minus19303
08935
09287
08885
08587
08246
08006
07772
07463
07229
119880119861119864lowast
15717
11499
10973
11461
11804
12156
12393
12635
12966
13242
L-mom
ents
GNO
119861119877 (119865)
15851
minus193606
minus192517
minus192695
minus192747
minus192678
minus192569
minus192437
minus192245
minus192098
119877(119865)
00917
00567
00846
01967
03305
05446
07392
09660
13223
1640
0119877119877 (119865)
19328
00850
00543
00853
01052
01237
01347
01442
01554
01632
119871119861119864lowast
minus23268
08848
09261
08844
08568
08301
08139
08000
07833
07722
119880119861119864lowast
15168
11542
10942
11489
11851
12197
1240
112
575
12779
12925
L-mom
ents
GPA
119861119877 (119865)
49081
minus192995
minus192417
minus192865
minus193078
minus193175
minus193195
minus193209
minus193256
minus193330
119877(119865)
01234
00571
00935
02081
03365
05322
07054
09053
12180
14975
119877119877 (119865)
246
8800882
00576
00870
01054
01239
01364
01489
01660
01798
119871119861119864lowast
minus45353
08783
09210
08838
08594
08326
08141
07961
07731
07557
119880119861119864lowast
14833
11574
10996
11532
11870
12223
1246
712
711
1304
413
312
PL-m
oments
GEV
119861119877 (119865)
minus00877
minus194145
minus193060
minus193062
minus192847
minus192278
minus191698
minus191038
minus190100
minus189374
119877(119865)
05458
00565
00855
01963
03338
05712
08090
11156
16650
22276
119877119877 (119865)
39079
00784
00585
00934
0117
201406
01558
01703
01897
02050
119871119861119864lowast
minus10190
08983
09236
08765
08431
08067
07818
07579
07271
07041
119880119861119864lowast
58730
1144
811036
11652
12081
12504
12771
13026
13364
13623
PL-m
oments
GNO
119861119877 (119865)
38414
minus193665
minus192536
minus192859
minus193044
minus193102
minus193057
minus192966
minus192804
minus192665
119877(119865)
06308
00578
00896
02088
03488
05690
07659
09926
13434
16524
119877119877 (119865)
44651
00819
00596
00961
0119
301405
01526
01628
01744
01822
119871119861119864lowast
minus13011
08910
09192
08716
08411
08122
07949
07801
07634
07519
119880119861119864lowast
56756
11491
11030
11693
12128
12539
12775
12974
13197
13347
PL-m
oments
GPA
119861119877 (119865)
126398
minus193142
minus192484
minus193077
minus193404
minus193589
minus193637
minus193655
minus193680
minus193720
119877(119865)
08641
00588
00999
02220
03558
05535
07227
09125
1200
914
521
119877119877 (119865)
60116
00855
006
4000985
0119
901400
01529
01651
01812
01937
119871119861119864lowast
minus19889
08842
09130
08700
08435
08156
07970
07789
07566
07399
119880119861119864lowast
53743
11528
11103
11755
12158
12552
12807
13055
13385
13636
Advances in Meteorology 15
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
0 2 4 6
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
Regional growth curve for GPA
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F))
0
5
10
15
20
Qua
ntile
2 4 60Reduced variate minuslog (minus log (F))
0
5
10
15
Qua
ntile
(based on L-moments) Regional growth curve for GPA
(based on PL-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30
16 Advances in Meteorology
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60
Advances in Meteorology 17
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90
18 Advances in Meteorology
The first four PL-moments of the GEV are defined as
12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)
(A5)
In previous equation 119875(sdot sdot) is an Incomplete Gammafunction
119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500
120579119896expminus120579119889Θ (A6)
Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution
The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by
119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)
The partial PWMs of the GLO are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)
(A9)
where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function
1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Geology Advances in
4 Advances in Meteorology
where 120591 represents the L-coefficient of variation (L-Cv)1205913 represents the L-coefficient of skewness (L-Cs) and 1205914represents the L-coefficient of kurtosis (L-Ck)
The arranged sample is given as 119909(1) le 119909(2) le 119909(3) sdot sdot sdot le119909(119899) Wang [8] stated that the statistic
119887119903 = 1119899119899sum119894=1
(119894 minus 1) (119894 minus 2) sdot sdot sdot (119894 minus 119903)(119899 minus 1) (119899 minus 2) sdot sdot sdot (119899 minus 119903)119909(119894) (6)
is an unbiased estimator of 120573119903So
1198971 = 11988701198972 = 21198871 minus 11988701198973 = 61198872 minus 61198871 + 11988701198974 = 201198873 minus 301198872 + 121198871 minus 1198870
(7)
where 1198971 1198972 1198973 and 1198974 are the first four L-moments of thesample And similarly
119905 = 11989721198971 1199053 = 11989731198972 1199054 = 11989741198972
(8)
are the sample L-moments ratios
222 The Partial L-Moments Wang [8 10 11] introduced aconcept of partial probability-weighted moments (PPWMs)that will estimate the higher quantiles of flood flows Data canbe censored to the right tail or left tail
Initially PPWMswere to take out the smaller values fromthe process of distribution fitting because such values haveslight influence on the frequency analysis and are nuisance tothe fitting process
The left tail PPWMs are defined by Wang [8 11] as
120573lowast119903 = int11198650
119909 (119865) 119865119903119889119865 (9)
where 1198650 = 119865(1199090) which is the lower limit of the censoredobservations and 1199090 is the censoring threshold value
The PPWMs elongated form described by Wang [10] areto be given a censored sample as
120573119903 = 11 minus 119865119903+10 int11198650
119909 (119865) 119865119903119889119865 (10)
If the value of 1198650 is starting from the zero then the result ofPPWMswill be the same as the usual PWMs As 119909(1) le 119909(2) le119909(3) sdot sdot sdot le 119909(119899) is the arranged sample Wang [10] describes theunbiased estimator of 120573119903 as
119903 = 1(1 minus 119865119903+10 ) 119899119899sum119894=1
(119894 minus 1) (119894 minus 2) sdot sdot sdot (119894 minus 119903)(119899 minus 1) (119899 minus 2) sdot sdot sdot (119899 minus 119903)119909lowast(119894) (11)
where
119909lowast(119894) = 0 for 119909(119894) le 119909(0)119909lowast(119894) = 119909(119894) for 119909(119894) gt 119909(0) (12)
The censoring level 1198650 is the prior selection of the numberof censored sample data The procedure that determines thenumber of sample data points are to be censored
1198650 = 1198990119899 (13)
where 1198990 and 119899 are the lengths of sample which are to becensored and uncensored respectively Similarly 1199090 is thehighest value of the censored sample The first four PL-moments for the PPWMs are
1 = 12057302 = 2 1205731 minus 12057303 = 1205732 minus 6 1205731 + 12057304 = 20 1205733 minus 30 1205732 + 12 1205731 minus 1205730
(14)
Similarly the PL-moments ratios can be written as
120591 = 21
1205913 = 32
1205914 = 42
(15)
where 120591 1205913 and 1205914 denote the partial L-coefficient of variation(PL-Cv) partial L-coefficient of skewness (PL-Cs) and par-tial L-coefficient of kurtosis (PL-Ck) respectively Thereforethe first four sample PL-moments can be computed as
1198971 = 01198972 = 21 minus 01198973 = 62 minus 61 + 01198974 = 203 minus 302 + 121 minus 0
(16)
And the first four sample PL-moments ratios can be com-puted as
119905 = 11989721198971 1199053 = 11989731198972 1199054 = 11989741198972
(17)
Advances in Meteorology 5
where 119905 1199053 and 1199054 represent the sample partial L-momentsratios of the PL-Cv PL-Cs and PL-Ck respectively Thederivation is L-moment and PL-moments are given inAppendix In the present study different level of censoringthreshold is selected
23 Regional Frequency Analysis Hosking and Wallis [7 30]identified the following four steps to explain the procedure ofthe 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
231 Data Screening We screened data anomalies beforeapplying any statistical analysis
232 Discordance Test Hosking and Wallis [7] suggested adiscordancy measure (119863119894) test that recognizes the locationswhere sample L-moments are marked contrarily from themost other locations Locations with the large flaws in thedata will be flagged as discordant
The discordancy test for a region containing119873 locationsfor site 119894 is proposed by Hosking and Wallis [7] as follows
119863119894 = 13119873 (119906119894 minus 119906)119879 119878minus1 (119906119894 minus 119906) 119894 = 1 2 119873 (18)
where 119906119894 is the vector containing the three sample L-momentsratios for the site 119894 expressed as
119906119894 = [119905(119894)2 119905(119894)3 119905(119894)4 ]119879 (19)
119906 is the average vector of 119906119894 for the overall region that is
119906 = 1119873119873sum119894=1
119906119894 (20)
and 119878 is the covariance matrix for the sample that can beexpressed as
119878 = 119873sum119894=1
(119906119894 minus 119906) (119906119894 minus 119906)119879 (21)
Broadly speaking a location or a site is considered to bediscordant from the whole region or group if the value of 119863119894is larger than the critical value
233 Heterogeneity Test The homogeneity measure (119867119895)identifies homogenous regions Hosking and Wallis [7] Itis also useful to tag the locations if they are plausible tohandle as a homogeneous region It estimates the amountof heterogeneity in the overall region The heterogeneity test(119867119895) is computed as
119867119895 = 119881119895 minus 120583V119895120590V119895 119895 = 1 2 3 (22)
where 120583V119895 and 120590V119895 are representing the mean and standarddeviation of the simulated 119881119895 values Also
1198811 = 119873sum119894=1
119899119894 (119905(119894)2 minus 1199051198772 )sum119873119894=1 119899119894
12
1198812 =sum119873119894=1 [119899119894 (119905(119894)2 minus 1199051198772 )2 + (119905(119894)3 minus 1199051198773 )212]
sum119873119894=1 119899119894
1198813 =sum119873119894=1 [119899119894 (119905(119894)3 minus 1199051198773 )2 + (119905(119894)4 minus 1199051198774 )212]
sum119873119894=1 119899119894
(23)
Here 1199051198772 1199051198773 and 1199051198774 are region average L-moments or PL-moments ratios We assessed the heterogeneity of a region assuggested by Hosking and Wallis [7]
Region is acceptably homogeneous if119867 lt 1Region is possibly heterogeneous if 1 le 119867 lt 2Region is definitely heterogeneous if119867 ge 2
24 Selection of the Appropriate Probability DistributionHosking andWallis [7] proposed two approaches to select thedistribution that fitted best the data the L-moment ratios dia-gram and the 119885-test The L-moment ratios diagram is usingthe unbiased estimators Hosking [28] Stedinger et al [31]Vogel and Fennessey [32] and Hosking [33]The L-momentsratio diagram is a plot of the computed values L-Cs andthe observed values L-Ck of the distribution function Thecurves indicate the hypothetical connections between L-Csand L-Ck of the candidate distribution The L-moment ratiodiagrams have been proposed for discriminating between thecandidate probability distributions in describing the regionalinformation (Hosking [28] Stedinger et al [31] Hoskingand Wallis [7]) The L-moments ratio diagrams have beenused as a component of probability distribution process forregional information (Schaefer [34] Pearson [35] Vogel andFennessey [32] Vogel et al [36] Chow and Watt [37] OnOzand Bayazit [38] Vogel and Wilson [39] Peel et al [40])
Hosking and Wallis [7] suggested a measure to see howwell the L-Cs and L-Ck of the fitted probability distributionmatch the regional average L-Cs and L-Ck of the observedinformation
The measure goodness of fit for every single selectedprobability distribution is computed as follows
119885Dis = (120591Dis4 minus 1199051198774 )1205904 (24)
where 120591Dis4 represents the value of the L-Ck of the fitteddistribution 1199051198774 represents the weighted regional averageL-Ck and 1205904 represents standard deviation of the 1199051198774 whichis obtained from the simulation of the Kappa probabilitydistribution
If the computed value of 119885Dis is equal to zero the proba-bility distributionwill be themost suitable fit If the computed
6 Advances in Meteorology
Table 1 Statistics of annual extreme monthly precipitation for study region based on L-moments and PL-moments
Sites L-moments PL-momentsMean 119905 1199053 1199054 Mean 119905 1199053 1199054
Astore 37624 0518 0366 0193 41752 0467 0377 0183Balakot 120450 0478 0322 0192 133259 0428 0335 0198Bunji 14297 0609 0456 0259 16339 0553 0450 0257Chilas 15091 0585 0392 0194 17603 0516 0375 0200Cherat 49038 0580 0375 0162 55289 0526 0355 0161DI khan 28885 0689 0516 0262 32351 0651 0487 0250Dir 105694 0432 0216 0098 116741 0378 0222 0097Drosh 45284 0525 0317 0136 50316 0474 0307 0137Garhi Dupatta 111352 0454 0227 0111 123374 0398 0229 0120Gilgit 12677 0591 0412 0228 14290 0539 0399 0239Gupis 23446 0678 0529 0331 30299 0583 0506 0350Kakul 99431 0453 0247 0102 110068 0399 0254 0093Kotli 95070 0542 0332 0136 106903 0487 0314 0138Muzaffarabad 118439 0459 0284 0182 131122 0405 0301 0194Peshawar 46216 0572 0398 0216 51761 0520 0392 0222Saidu Sharif 85667 0455 0265 0181 94978 0399 0283 0200Skardu 20438 0616 04564 0238 22885 0570 0444 0231
value of 119885-statistic is less than 164 at 90 confidence level(ie |119885Dis| le 164) it will indicate that the distributionqualifies the goodness of fit criteria If there are more thanone distribution that qualify the criteria the most suitabledistribution has the minimum |119885Dis| value3 Estimation
The sitesrsquo information and statistic by using L-moments forthe present study are presented in Table 1 In Table 1 meanrepresents the first sample LPL-moments and 119905 1199053 and 1199054are the sample LPL-moments ratios of the L-CvPL-CvL-CsPL-Cs and L-CkPL-Ck respectively The lower levelcensoring threshold is selected from 10 to 23 Table 2expresses the feasible threshold values according to thepercentile technique along with Average Annual OccurrenceNumber (AAON) Jiang et al [41] and Yuguo [42] suggestedthat the optimal threshold can be obtained if the values ofAAON lie between 1 and 2 Table 2 shows that 90th percentileobservations are suitable for the optimum threshold selectionof most of areas in the present study We have 168 valuesin each station according to the above table Astore stationhas 32 threshold values due to which 16 values are beingcensored in 168 According to censored level 102 censoredlevel was selected Similarly 105 was selected for Balakotand Muzaffarabad According to this process maximumthreshold level 223 was selected for Gupis By using theabove process for each station 17 different censoring levelswere selected So we decide that the range from 10 to 23of censoring level should be kept for selecting thresholdvalues
31 Regional Frequency Analysis The following four steps areconsidered as prerequisite for frequency analysis Hoskingand 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
311 Data Screening In this study we use secondary dataafter carefully examining all locations for abnormalities andmissing observations Therefore we use 14 years of data forRFA that were obtained from seventeen locations
312 Discordance Test Table 3 shows 119863119894 result of (18) for17 locations of this study region It can be observed fromthe results of LPL-moments in Table 3 that the value of 119863-statistic varies from 007 to 244 If 119863119894 is greater than 3 thelocation is considered to be discordant from the rest of theregional data Hosking and Wallis [7] In this study regionno location is diagnosed as discordant (119863119894 ge 3) Thereforewe use all data for the development of the RFA based on L-moments and PL-moments
313 Regional Heterogeneity Measure The next step is theformation of the homogeneous region which is conven-tionally tougher and needs the higher number of subjectivejudgments The homogeneity conditions are defined as thelocations that have the same frequency distributions
Advances in Meteorology 7
Table 2 Precipitation threshold selection in GPA distribution for 17 stations
Station 80th 90th 95th 975th 99thAstore
Threshold 98 32 02 0 0AAON 243 121 064 043 043
BalakotThreshold 325 148 51 3 0AAON 243 129 057 029 014
MuzaffarabadThreshold 304 118 5 1 0AAON 25 121 057 029 021
Garhi DupattaThreshold 288 10 38 0 0AAON 243 121 057 029 029
DirThreshold 31 20 5 005 0AAON 243 121 057 029 021
KakulThreshold 289 134 21 05 0AAON 164 129 057 029 021
KotliThreshold 13 6 005 0 0AAON 243 135 057 036 036
Saidu SharifThreshold 256 83 2 0 0AAON 243 121 057 036 036
cheratThreshold 5 005 0 0 0AAON 243 143 093 093 093
PeshawarThreshold 6 005 0 0 0AAON 243 129 057 057 057
DroshThreshold 76 15 02 005 0AAON 243 121 057 05 021
D I KhanThreshold 03 0 0 0 0AAON 257 129 129 129 129
GupisThreshold 0 0 0 0 0AAON 271 271 271 271 271
SkarduThreshold 2 03 0 0 0AAON 243 071 071 071 071
ChilasThreshold 1 05 0 0 0AAON 171 107 107 107 107
BunjiThreshold 13 0 0 0 0AAON 15 15 15 15 15
GilgitThreshold 12 005 0 0 0AAON 243 135 071 071 071
8 Advances in Meteorology
04
03
02
01
00
04
03
02
01
00
00 01 02 03 04 05 06 00 01 02 03 04 05 06
GLOGEVGPA
GNOAverageData
GLOGEVGPA
GNOAverageData
L-ku
rtos
is
L㰀-k
urto
sis
L-skewness L-skewness
Figure 2 L-diagram and PL-diagram of the GLO GEV GPA and GNO distributions
Table 3 Discordance test result based on L-moments and PL-moments
Name of site L-moments (119863119894) PL-moments (119863119894)Astore 205 244Balakot 142 085Bunji 055 060Chilas 029 007Cherat 079 064DI khan 129 190Dir 082 086Drosh 051 043Garhi Dupatta 143 133Gilgit 026 059Gupis 237 231Kakul 082 122Kotli 077 059Muzaffarabad 099 085Peshawar 007 008Saidu Sharif 191 157Skardu 067 065
For the present study area realization of the Kappaprobability distribution is used to conduct the heterogeneitytest based on the L-moments and PL-moments
Number of simulations are 10000 for computing theheterogeneityWe computed the regional average L-momentsratios the regional PL-moments ratios and the correspond-ing parameter values of the fitted Kappa probability dis-tribution (see Table 4) Table 4 shows the results of theheterogeneity measure using L-moments and PL-momentsmethods It can be observed from Table 4 that the different
values for the119867-statistic are minus041 minus160 and minus289 based onL-moments andminus006minus18 andminus317 based onPL-momentsTherefore we concluded that by comparing these resultsand the heterogeneity conditions study region is acceptablyhomogeneous for L-moments and PL-moments No furthersubdivisions of the present study are necessary
32 Fitting Appropriate Probability Distribution After homo-geneity analysis of the study area a suitable probabilitydistribution is required for the RFAThe objective is not onlyto recognize a suitable probability distribution for RFA butalso to observe a probability distribution that will providerobust quantile estimate for each location and for the regionalgrowth cure List of candidate probability distributions forRFA is GLO GEV GPA and GNO
We plotted L-moments and PL-moments diagrams forpreliminary evaluation of the probability distribution for thestudy area
Figure 2 illustrates an analogy of the observed and hypo-thetical relationships of the probability distribution Figure 2shows that GLO distribution is not a suitable candidate forthe L-moments and PL-moments
Interestingly both analyses of the L-moments and the PL-moments diagram show that the sample average values areappropriately distinguished by the hypothetical L-momentsand PL-moments for GPA and GNO distributions
However it is hard to find a suitable probability distri-bution that fits most of the regional observed data Table 5shows the goodness of fit test results for candidate probabilitydistributions
Table 5 shows that GLO distribution failed the goodnessof fit test for both L-moments and for PL-moments methodsas the calculated value of the 119885-test for the GLO distribution
Advances in Meteorology 9
Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments
Heterogeneity measures L-moment PL-momentHeterogeneity measure1198671
Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060
Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800
Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170
Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments
Method Distribution Parameters120585 120572 119870L-moments
GEV 04660 05644 minus02751GNO 06671 07565 minus07594GPA minus00551 09944 minus00575
PL-momentsGEV 05220 05116 minus02686GNO 07046 06837 minus07487GPA 00478 09073 minus00472
is larger than the critical value of 164 (at 90 confidencelevel)
It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile
Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution
33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7
based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)
Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is
119876 (119865) = 1198971198941119902 (119865) (25)
The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3
Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions
Therefore it is necessary to assess the performance ofregional quantile estimates
4 Accuracy of the Estimated Quantiles and theRegional Growth Curve
A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten
10 Advances in Meteorology
Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865
01000 05000 08000 09000 09500 09800 09900 09975 099875 09990
L-momentsGEV 00454 06836 15139 22247 30591 44161 56871 90748 113165 121338GNO 00473 06671 15585 23072 31448 44098 54997 80676 95666 100823GPA 00500 06481 16217 23931 31958 43073 51878 70578 80502 83782
PL-momentsGEV 01397 07190 14670 21034 28469 40497 51703 81363 10086 107950GNO 01412 07046 15062 21752 29203 40410 50032 72608 85739 90249GPA 01436 06871 15649 22547 29674 39460 47149 63304 71786 74577
Qua
ntile
Regional growth curve (L-moments)
GPAGNOGEV
GPAGNOGEV
Qua
ntile
Return period
Regional growth gurve (PL-moments)
5
4
3
2
1
0
5
4
3
2
1
0
minus2 minus1 0 1 2 3 4 5
2 5 10 20 50 100
Return period2 5 10 20 50 100
Reduced variate minuslog (minus log (F))
minus2 minus1 0 1 2 3 4 5
Reduced variate minuslog (minus log (F))
Figure 3 L-moments and PL-moments regional growth curves
as 119876(119898)119894 (119865) for the nonexceedance probability 119865 The relativeerror for this estimator is
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) (26)
The bias and the RMSE of the above quantity over all 119872repetition are
Bias = 119861119894 (119865) = 1119872119872sum119898=1
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)
= 1119872 [[119872sum119898=1
(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12
(27)
Also for the estimated quantile the regional average bias andthe relative RMSE are
119861119877 (119865) = 1119873119873sum119894=1
119861119894 (119865)
119877119877 (119865) = 1119873119873sum119894=1
119877119894 (119865) (28)
We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval
119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)
Advances in Meteorology 11
Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le
119902 (119865)119871005 (119865) (30)
The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves
We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]
We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively
Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds
Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion
This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail
GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments
Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents
Appendix
The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows
PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by
119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (minus ln119865)119896 119896 = 0 (A2)
Wang [10] developed the partial probability-weightedmoments (PWMs) of the GEV as
(119903 + 1) 1205731015840119903 = 120577 + 120572119867 (120574 1198650 119896) (A3)
where
119867(120574 1198650 119896) = 1119896 [1 minus119875 1 + 119896 minus (119903 + 1) ln1198650(1 minus 119865119903+10 ) (119903 + 1)119896 ] (A4)
12 Advances in Meteorology
Table8Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f30
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus0540
0minus19
7674
minus195328
minus193602
minus191579
minus188659
minus186376
minus1840
97minus18
1165
minus179056
119877(119865)
00900
00695
00963
01952
03334
06038
09057
13261
21337
300
48119877119877 (119865)
19873
01019
00635
00876
01090
01369
01595
01839
02191
02478
119871119861119864lowast
minus20937
08829
09296
08842
08411
07862
07460
07065
06557
06177
119880119861119864lowast
18714
11949
11208
11580
11884
12244
12516
12818
13285
13683
L-mom
ents
GNO
119861119877 (119865)
08124
minus196279
minus193846
minus192837
minus191816
minus190506
minus189573
minus188699
minus187636
minus186899
119877(119865)
00986
00696
00917
02051
03520
06027
08421
11312
16008
20312
119877119877 (119865)
20766
01045
00587
00889
01120
01368
01533
01686
01878
02016
119871119861119864lowast
minus21331
08683
09271
08780
08379
07958
07691
07453
07172
06978
119880119861119864lowast
16569
11974
11077
11569
11954
12355
12615
12851
13149
13369
L-mom
ents
GPA
119861119877 (119865)
38654
minus194351
minus193564
minus193226
minus192616
minus191706
minus191060
minus190506
minus189966
minus189727
119877(119865)
01242
006
6900998
02136
03552
05968
08313
11188
15941
20368
119877119877 (119865)
24785
01033
00614
00892
01111
01386
01602
01833
02162
02431
119871119861119864lowast
minus25192
08591
09215
08817
08444
07979
07646
07327
06919
06623
119880119861119864lowast
14272
11911
11115
11588
11976
12459
12841
13257
13859
14367
PL-m
oments
GEV
119861119877 (119865)
minus02662
minus197059
minus194770
minus193340
minus191600
minus188996
minus186909
minus184797
minus182052
minus1800
66119877(119865)
05379
006
8700947
02023
03467
06171
09087
13058
20551
2854
119877119877 (119865)
38117
00948
006
4900960
01210
01508
01736
01977
02324
02609
119871119861119864lowast
minus77814
08892
09252
08725
08259
07697
07296
06911
064
1706050
119880119861119864lowast
45516
11827
11207
11716
12117
12554
12881
13211
13693
14104
PL-m
oments
GNO
119861119877 (119865)
35142
minus195922
minus193485
minus192704
minus191894
minus190806
minus1900
03minus18
9235
minus188284
minus187617
119877(119865)
06203
006
8900939
02141
03644
06143
08483
11272
15747
19809
119877119877 (119865)
43522
00971
00626
00980
01238
01504
01675
01831
02024
02162
119871119861119864lowast
minus10618
08772
09204
08655
08233
07800
07530
07290
07012
06822
119880119861119864lowast
46353
11849
11122
11734
12189
12653
12949
13218
13546
13780
PL-m
oments
GPA
119861119877 (119865)
1212
3minus19
4300
minus193238
minus193113
minus192716
minus192046
minus191555
minus19114
1minus19
0768
minus1906
49119877(119865)
08513
006
6601041
02250
03684
06021
08215
10853
15144
19094
119877119877 (119865)
58811
00962
006
6900996
01233
01509
01720
01941
02258
02518
119871119861119864lowast
minus17507
08698
09130
08680
08299
07846
07524
07220
06831
06552
119880119861119864lowast
47307
11791
11189
11774
12215
12739
1314
13567
14195
14722
Advances in Meteorology 13
Table9Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f60
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus13271
minus194230
minus193551
minus193241
minus192649
minus191612
minus190736
minus189839
minus188690
minus187888
119877(119865)
00846
00624
00851
01873
03250
05841
08652
12505
19839
27721
119877119877 (119865)
19155
00833
00612
00855
01024
01213
01360
01523
01770
01983
119871119861119864lowast
minus19593
08756
09254
08851
08491
08074
07796
07501
07118
06834
119880119861119864lowast
16242
11675
11015
11472
11879
12309
12678
13038
13659
14141
L-mom
ents
GNO
119861119877 (119865)
08417
minus193335
minus192795
minus192984
minus192999
minus192879
minus192742
minus192593
minus192399
minus192264
119877(119865)
00968
00634
00858
02000
03439
05850
08123
10845
15235
19239
119877119877 (119865)
21055
00881
00582
00861
01050
01235
01354
01463
01599
01699
119871119861119864lowast
minus22887
08647
09244
08794
08466
08141
07957
07798
07597
07443
119880119861119864lowast
15863
11714
1096
11511
11934
1240
512
666
12939
13253
13505
L-mom
ents
GPA
119861119877 (119865)
46371
minus192512
minus192631
minus193160
minus193417
minus193606
minus193745
minus193933
minus194306
minus194708
119877(119865)
01280
00624
00950
02094
03468
05740
07903
10536
14878
18928
119877119877 (119865)
264
4200915
00614
00882
01064
01270
01429
01599
01849
02059
119871119861119864lowast
minus38088
08628
09186
08815
08510
08167
07924
07703
07422
07178
119880119861119864lowast
14699
11721
11020
11534
11925
12454
12791
13236
13869
1440
5
PL-m
oments
GEV
119861119877 (119865)
minus16968
minus193984
minus193601
minus193623
minus193319
minus192576
minus191861
minus191081
minus190026
minus189255
119877(119865)
05431
00596
00888
01978
03366
05844
08417
11838
18179
24868
119877119877 (119865)
38656
00783
00634
0095
01166
01387
01541
01700
01928
02120
119871119861119864lowast
minus92921
08907
09201
08750
08403
07984
07714
07434
07043
06783
119880119861119864lowast
53553
11521
11099
1166
412
098
12572
12869
13216
13697
14105
PL-m
oments
GNO
119861119877 (119865)
23785
minus193306
minus192979
minus193417
minus193628
minus193678
minus193617
minus193509
minus193331
minus193185
119877(119865)
06278
006
0100911
02100
03523
05804
0788
10303
14113
17516
119877119877 (119865)
43743
00812
00623
00965
01186
01391
01513
01618
01742
01830
119871119861119864lowast
minus12083
08856
09188
08709
08389
08061
07870
07706
07517
07380
119880119861119864lowast
52707
11561
11059
11704
12152
12613
12898
13138
13361
1356
PL-m
oments
GPA
119861119877 (119865)
1118
7minus19
2590
minus192910
minus193681
minus1940
84minus19
4322
minus1944
04minus19
4463
minus194560
minus1946
73119877(119865)
08596
00597
01009
02229
03581
05615
07391
09418
12558
15341
119877119877 (119865)
58310
00840
006
6000991
0119
501391
01522
01650
01828
01971
119871119861119864lowast
minus18977
08800
09127
08699
08413
08098
07912
07682
07447
07255
119880119861119864lowast
51281
11551
11117
11769
12173
12615
12887
13227
1360
913
899
14 Advances in Meteorology
Table10A
ccuracymeasure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f90
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus00117
minus194226
minus193109
minus192926
minus192542
minus191809
minus191151
minus1904
48minus18
9497
minus188790
119877(119865)
00787
00556
00823
01850
03151
05459
07840
10989
16793
22883
119877119877 (119865)
17329
00814
00544
00832
01031
01238
01381
01525
01727
01892
119871119861119864lowast
minus19303
08935
09287
08885
08587
08246
08006
07772
07463
07229
119880119861119864lowast
15717
11499
10973
11461
11804
12156
12393
12635
12966
13242
L-mom
ents
GNO
119861119877 (119865)
15851
minus193606
minus192517
minus192695
minus192747
minus192678
minus192569
minus192437
minus192245
minus192098
119877(119865)
00917
00567
00846
01967
03305
05446
07392
09660
13223
1640
0119877119877 (119865)
19328
00850
00543
00853
01052
01237
01347
01442
01554
01632
119871119861119864lowast
minus23268
08848
09261
08844
08568
08301
08139
08000
07833
07722
119880119861119864lowast
15168
11542
10942
11489
11851
12197
1240
112
575
12779
12925
L-mom
ents
GPA
119861119877 (119865)
49081
minus192995
minus192417
minus192865
minus193078
minus193175
minus193195
minus193209
minus193256
minus193330
119877(119865)
01234
00571
00935
02081
03365
05322
07054
09053
12180
14975
119877119877 (119865)
246
8800882
00576
00870
01054
01239
01364
01489
01660
01798
119871119861119864lowast
minus45353
08783
09210
08838
08594
08326
08141
07961
07731
07557
119880119861119864lowast
14833
11574
10996
11532
11870
12223
1246
712
711
1304
413
312
PL-m
oments
GEV
119861119877 (119865)
minus00877
minus194145
minus193060
minus193062
minus192847
minus192278
minus191698
minus191038
minus190100
minus189374
119877(119865)
05458
00565
00855
01963
03338
05712
08090
11156
16650
22276
119877119877 (119865)
39079
00784
00585
00934
0117
201406
01558
01703
01897
02050
119871119861119864lowast
minus10190
08983
09236
08765
08431
08067
07818
07579
07271
07041
119880119861119864lowast
58730
1144
811036
11652
12081
12504
12771
13026
13364
13623
PL-m
oments
GNO
119861119877 (119865)
38414
minus193665
minus192536
minus192859
minus193044
minus193102
minus193057
minus192966
minus192804
minus192665
119877(119865)
06308
00578
00896
02088
03488
05690
07659
09926
13434
16524
119877119877 (119865)
44651
00819
00596
00961
0119
301405
01526
01628
01744
01822
119871119861119864lowast
minus13011
08910
09192
08716
08411
08122
07949
07801
07634
07519
119880119861119864lowast
56756
11491
11030
11693
12128
12539
12775
12974
13197
13347
PL-m
oments
GPA
119861119877 (119865)
126398
minus193142
minus192484
minus193077
minus193404
minus193589
minus193637
minus193655
minus193680
minus193720
119877(119865)
08641
00588
00999
02220
03558
05535
07227
09125
1200
914
521
119877119877 (119865)
60116
00855
006
4000985
0119
901400
01529
01651
01812
01937
119871119861119864lowast
minus19889
08842
09130
08700
08435
08156
07970
07789
07566
07399
119880119861119864lowast
53743
11528
11103
11755
12158
12552
12807
13055
13385
13636
Advances in Meteorology 15
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
0 2 4 6
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
Regional growth curve for GPA
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F))
0
5
10
15
20
Qua
ntile
2 4 60Reduced variate minuslog (minus log (F))
0
5
10
15
Qua
ntile
(based on L-moments) Regional growth curve for GPA
(based on PL-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30
16 Advances in Meteorology
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60
Advances in Meteorology 17
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90
18 Advances in Meteorology
The first four PL-moments of the GEV are defined as
12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)
(A5)
In previous equation 119875(sdot sdot) is an Incomplete Gammafunction
119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500
120579119896expminus120579119889Θ (A6)
Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution
The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by
119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)
The partial PWMs of the GLO are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)
(A9)
where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function
1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geological ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geology Advances in
Advances in Meteorology 5
where 119905 1199053 and 1199054 represent the sample partial L-momentsratios of the PL-Cv PL-Cs and PL-Ck respectively Thederivation is L-moment and PL-moments are given inAppendix In the present study different level of censoringthreshold is selected
23 Regional Frequency Analysis Hosking and Wallis [7 30]identified the following four steps to explain the procedure ofthe 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
231 Data Screening We screened data anomalies beforeapplying any statistical analysis
232 Discordance Test Hosking and Wallis [7] suggested adiscordancy measure (119863119894) test that recognizes the locationswhere sample L-moments are marked contrarily from themost other locations Locations with the large flaws in thedata will be flagged as discordant
The discordancy test for a region containing119873 locationsfor site 119894 is proposed by Hosking and Wallis [7] as follows
119863119894 = 13119873 (119906119894 minus 119906)119879 119878minus1 (119906119894 minus 119906) 119894 = 1 2 119873 (18)
where 119906119894 is the vector containing the three sample L-momentsratios for the site 119894 expressed as
119906119894 = [119905(119894)2 119905(119894)3 119905(119894)4 ]119879 (19)
119906 is the average vector of 119906119894 for the overall region that is
119906 = 1119873119873sum119894=1
119906119894 (20)
and 119878 is the covariance matrix for the sample that can beexpressed as
119878 = 119873sum119894=1
(119906119894 minus 119906) (119906119894 minus 119906)119879 (21)
Broadly speaking a location or a site is considered to bediscordant from the whole region or group if the value of 119863119894is larger than the critical value
233 Heterogeneity Test The homogeneity measure (119867119895)identifies homogenous regions Hosking and Wallis [7] Itis also useful to tag the locations if they are plausible tohandle as a homogeneous region It estimates the amountof heterogeneity in the overall region The heterogeneity test(119867119895) is computed as
119867119895 = 119881119895 minus 120583V119895120590V119895 119895 = 1 2 3 (22)
where 120583V119895 and 120590V119895 are representing the mean and standarddeviation of the simulated 119881119895 values Also
1198811 = 119873sum119894=1
119899119894 (119905(119894)2 minus 1199051198772 )sum119873119894=1 119899119894
12
1198812 =sum119873119894=1 [119899119894 (119905(119894)2 minus 1199051198772 )2 + (119905(119894)3 minus 1199051198773 )212]
sum119873119894=1 119899119894
1198813 =sum119873119894=1 [119899119894 (119905(119894)3 minus 1199051198773 )2 + (119905(119894)4 minus 1199051198774 )212]
sum119873119894=1 119899119894
(23)
Here 1199051198772 1199051198773 and 1199051198774 are region average L-moments or PL-moments ratios We assessed the heterogeneity of a region assuggested by Hosking and Wallis [7]
Region is acceptably homogeneous if119867 lt 1Region is possibly heterogeneous if 1 le 119867 lt 2Region is definitely heterogeneous if119867 ge 2
24 Selection of the Appropriate Probability DistributionHosking andWallis [7] proposed two approaches to select thedistribution that fitted best the data the L-moment ratios dia-gram and the 119885-test The L-moment ratios diagram is usingthe unbiased estimators Hosking [28] Stedinger et al [31]Vogel and Fennessey [32] and Hosking [33]The L-momentsratio diagram is a plot of the computed values L-Cs andthe observed values L-Ck of the distribution function Thecurves indicate the hypothetical connections between L-Csand L-Ck of the candidate distribution The L-moment ratiodiagrams have been proposed for discriminating between thecandidate probability distributions in describing the regionalinformation (Hosking [28] Stedinger et al [31] Hoskingand Wallis [7]) The L-moments ratio diagrams have beenused as a component of probability distribution process forregional information (Schaefer [34] Pearson [35] Vogel andFennessey [32] Vogel et al [36] Chow and Watt [37] OnOzand Bayazit [38] Vogel and Wilson [39] Peel et al [40])
Hosking and Wallis [7] suggested a measure to see howwell the L-Cs and L-Ck of the fitted probability distributionmatch the regional average L-Cs and L-Ck of the observedinformation
The measure goodness of fit for every single selectedprobability distribution is computed as follows
119885Dis = (120591Dis4 minus 1199051198774 )1205904 (24)
where 120591Dis4 represents the value of the L-Ck of the fitteddistribution 1199051198774 represents the weighted regional averageL-Ck and 1205904 represents standard deviation of the 1199051198774 whichis obtained from the simulation of the Kappa probabilitydistribution
If the computed value of 119885Dis is equal to zero the proba-bility distributionwill be themost suitable fit If the computed
6 Advances in Meteorology
Table 1 Statistics of annual extreme monthly precipitation for study region based on L-moments and PL-moments
Sites L-moments PL-momentsMean 119905 1199053 1199054 Mean 119905 1199053 1199054
Astore 37624 0518 0366 0193 41752 0467 0377 0183Balakot 120450 0478 0322 0192 133259 0428 0335 0198Bunji 14297 0609 0456 0259 16339 0553 0450 0257Chilas 15091 0585 0392 0194 17603 0516 0375 0200Cherat 49038 0580 0375 0162 55289 0526 0355 0161DI khan 28885 0689 0516 0262 32351 0651 0487 0250Dir 105694 0432 0216 0098 116741 0378 0222 0097Drosh 45284 0525 0317 0136 50316 0474 0307 0137Garhi Dupatta 111352 0454 0227 0111 123374 0398 0229 0120Gilgit 12677 0591 0412 0228 14290 0539 0399 0239Gupis 23446 0678 0529 0331 30299 0583 0506 0350Kakul 99431 0453 0247 0102 110068 0399 0254 0093Kotli 95070 0542 0332 0136 106903 0487 0314 0138Muzaffarabad 118439 0459 0284 0182 131122 0405 0301 0194Peshawar 46216 0572 0398 0216 51761 0520 0392 0222Saidu Sharif 85667 0455 0265 0181 94978 0399 0283 0200Skardu 20438 0616 04564 0238 22885 0570 0444 0231
value of 119885-statistic is less than 164 at 90 confidence level(ie |119885Dis| le 164) it will indicate that the distributionqualifies the goodness of fit criteria If there are more thanone distribution that qualify the criteria the most suitabledistribution has the minimum |119885Dis| value3 Estimation
The sitesrsquo information and statistic by using L-moments forthe present study are presented in Table 1 In Table 1 meanrepresents the first sample LPL-moments and 119905 1199053 and 1199054are the sample LPL-moments ratios of the L-CvPL-CvL-CsPL-Cs and L-CkPL-Ck respectively The lower levelcensoring threshold is selected from 10 to 23 Table 2expresses the feasible threshold values according to thepercentile technique along with Average Annual OccurrenceNumber (AAON) Jiang et al [41] and Yuguo [42] suggestedthat the optimal threshold can be obtained if the values ofAAON lie between 1 and 2 Table 2 shows that 90th percentileobservations are suitable for the optimum threshold selectionof most of areas in the present study We have 168 valuesin each station according to the above table Astore stationhas 32 threshold values due to which 16 values are beingcensored in 168 According to censored level 102 censoredlevel was selected Similarly 105 was selected for Balakotand Muzaffarabad According to this process maximumthreshold level 223 was selected for Gupis By using theabove process for each station 17 different censoring levelswere selected So we decide that the range from 10 to 23of censoring level should be kept for selecting thresholdvalues
31 Regional Frequency Analysis The following four steps areconsidered as prerequisite for frequency analysis Hoskingand 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
311 Data Screening In this study we use secondary dataafter carefully examining all locations for abnormalities andmissing observations Therefore we use 14 years of data forRFA that were obtained from seventeen locations
312 Discordance Test Table 3 shows 119863119894 result of (18) for17 locations of this study region It can be observed fromthe results of LPL-moments in Table 3 that the value of 119863-statistic varies from 007 to 244 If 119863119894 is greater than 3 thelocation is considered to be discordant from the rest of theregional data Hosking and Wallis [7] In this study regionno location is diagnosed as discordant (119863119894 ge 3) Thereforewe use all data for the development of the RFA based on L-moments and PL-moments
313 Regional Heterogeneity Measure The next step is theformation of the homogeneous region which is conven-tionally tougher and needs the higher number of subjectivejudgments The homogeneity conditions are defined as thelocations that have the same frequency distributions
Advances in Meteorology 7
Table 2 Precipitation threshold selection in GPA distribution for 17 stations
Station 80th 90th 95th 975th 99thAstore
Threshold 98 32 02 0 0AAON 243 121 064 043 043
BalakotThreshold 325 148 51 3 0AAON 243 129 057 029 014
MuzaffarabadThreshold 304 118 5 1 0AAON 25 121 057 029 021
Garhi DupattaThreshold 288 10 38 0 0AAON 243 121 057 029 029
DirThreshold 31 20 5 005 0AAON 243 121 057 029 021
KakulThreshold 289 134 21 05 0AAON 164 129 057 029 021
KotliThreshold 13 6 005 0 0AAON 243 135 057 036 036
Saidu SharifThreshold 256 83 2 0 0AAON 243 121 057 036 036
cheratThreshold 5 005 0 0 0AAON 243 143 093 093 093
PeshawarThreshold 6 005 0 0 0AAON 243 129 057 057 057
DroshThreshold 76 15 02 005 0AAON 243 121 057 05 021
D I KhanThreshold 03 0 0 0 0AAON 257 129 129 129 129
GupisThreshold 0 0 0 0 0AAON 271 271 271 271 271
SkarduThreshold 2 03 0 0 0AAON 243 071 071 071 071
ChilasThreshold 1 05 0 0 0AAON 171 107 107 107 107
BunjiThreshold 13 0 0 0 0AAON 15 15 15 15 15
GilgitThreshold 12 005 0 0 0AAON 243 135 071 071 071
8 Advances in Meteorology
04
03
02
01
00
04
03
02
01
00
00 01 02 03 04 05 06 00 01 02 03 04 05 06
GLOGEVGPA
GNOAverageData
GLOGEVGPA
GNOAverageData
L-ku
rtos
is
L㰀-k
urto
sis
L-skewness L-skewness
Figure 2 L-diagram and PL-diagram of the GLO GEV GPA and GNO distributions
Table 3 Discordance test result based on L-moments and PL-moments
Name of site L-moments (119863119894) PL-moments (119863119894)Astore 205 244Balakot 142 085Bunji 055 060Chilas 029 007Cherat 079 064DI khan 129 190Dir 082 086Drosh 051 043Garhi Dupatta 143 133Gilgit 026 059Gupis 237 231Kakul 082 122Kotli 077 059Muzaffarabad 099 085Peshawar 007 008Saidu Sharif 191 157Skardu 067 065
For the present study area realization of the Kappaprobability distribution is used to conduct the heterogeneitytest based on the L-moments and PL-moments
Number of simulations are 10000 for computing theheterogeneityWe computed the regional average L-momentsratios the regional PL-moments ratios and the correspond-ing parameter values of the fitted Kappa probability dis-tribution (see Table 4) Table 4 shows the results of theheterogeneity measure using L-moments and PL-momentsmethods It can be observed from Table 4 that the different
values for the119867-statistic are minus041 minus160 and minus289 based onL-moments andminus006minus18 andminus317 based onPL-momentsTherefore we concluded that by comparing these resultsand the heterogeneity conditions study region is acceptablyhomogeneous for L-moments and PL-moments No furthersubdivisions of the present study are necessary
32 Fitting Appropriate Probability Distribution After homo-geneity analysis of the study area a suitable probabilitydistribution is required for the RFAThe objective is not onlyto recognize a suitable probability distribution for RFA butalso to observe a probability distribution that will providerobust quantile estimate for each location and for the regionalgrowth cure List of candidate probability distributions forRFA is GLO GEV GPA and GNO
We plotted L-moments and PL-moments diagrams forpreliminary evaluation of the probability distribution for thestudy area
Figure 2 illustrates an analogy of the observed and hypo-thetical relationships of the probability distribution Figure 2shows that GLO distribution is not a suitable candidate forthe L-moments and PL-moments
Interestingly both analyses of the L-moments and the PL-moments diagram show that the sample average values areappropriately distinguished by the hypothetical L-momentsand PL-moments for GPA and GNO distributions
However it is hard to find a suitable probability distri-bution that fits most of the regional observed data Table 5shows the goodness of fit test results for candidate probabilitydistributions
Table 5 shows that GLO distribution failed the goodnessof fit test for both L-moments and for PL-moments methodsas the calculated value of the 119885-test for the GLO distribution
Advances in Meteorology 9
Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments
Heterogeneity measures L-moment PL-momentHeterogeneity measure1198671
Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060
Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800
Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170
Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments
Method Distribution Parameters120585 120572 119870L-moments
GEV 04660 05644 minus02751GNO 06671 07565 minus07594GPA minus00551 09944 minus00575
PL-momentsGEV 05220 05116 minus02686GNO 07046 06837 minus07487GPA 00478 09073 minus00472
is larger than the critical value of 164 (at 90 confidencelevel)
It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile
Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution
33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7
based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)
Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is
119876 (119865) = 1198971198941119902 (119865) (25)
The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3
Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions
Therefore it is necessary to assess the performance ofregional quantile estimates
4 Accuracy of the Estimated Quantiles and theRegional Growth Curve
A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten
10 Advances in Meteorology
Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865
01000 05000 08000 09000 09500 09800 09900 09975 099875 09990
L-momentsGEV 00454 06836 15139 22247 30591 44161 56871 90748 113165 121338GNO 00473 06671 15585 23072 31448 44098 54997 80676 95666 100823GPA 00500 06481 16217 23931 31958 43073 51878 70578 80502 83782
PL-momentsGEV 01397 07190 14670 21034 28469 40497 51703 81363 10086 107950GNO 01412 07046 15062 21752 29203 40410 50032 72608 85739 90249GPA 01436 06871 15649 22547 29674 39460 47149 63304 71786 74577
Qua
ntile
Regional growth curve (L-moments)
GPAGNOGEV
GPAGNOGEV
Qua
ntile
Return period
Regional growth gurve (PL-moments)
5
4
3
2
1
0
5
4
3
2
1
0
minus2 minus1 0 1 2 3 4 5
2 5 10 20 50 100
Return period2 5 10 20 50 100
Reduced variate minuslog (minus log (F))
minus2 minus1 0 1 2 3 4 5
Reduced variate minuslog (minus log (F))
Figure 3 L-moments and PL-moments regional growth curves
as 119876(119898)119894 (119865) for the nonexceedance probability 119865 The relativeerror for this estimator is
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) (26)
The bias and the RMSE of the above quantity over all 119872repetition are
Bias = 119861119894 (119865) = 1119872119872sum119898=1
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)
= 1119872 [[119872sum119898=1
(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12
(27)
Also for the estimated quantile the regional average bias andthe relative RMSE are
119861119877 (119865) = 1119873119873sum119894=1
119861119894 (119865)
119877119877 (119865) = 1119873119873sum119894=1
119877119894 (119865) (28)
We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval
119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)
Advances in Meteorology 11
Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le
119902 (119865)119871005 (119865) (30)
The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves
We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]
We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively
Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds
Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion
This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail
GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments
Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents
Appendix
The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows
PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by
119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (minus ln119865)119896 119896 = 0 (A2)
Wang [10] developed the partial probability-weightedmoments (PWMs) of the GEV as
(119903 + 1) 1205731015840119903 = 120577 + 120572119867 (120574 1198650 119896) (A3)
where
119867(120574 1198650 119896) = 1119896 [1 minus119875 1 + 119896 minus (119903 + 1) ln1198650(1 minus 119865119903+10 ) (119903 + 1)119896 ] (A4)
12 Advances in Meteorology
Table8Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f30
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus0540
0minus19
7674
minus195328
minus193602
minus191579
minus188659
minus186376
minus1840
97minus18
1165
minus179056
119877(119865)
00900
00695
00963
01952
03334
06038
09057
13261
21337
300
48119877119877 (119865)
19873
01019
00635
00876
01090
01369
01595
01839
02191
02478
119871119861119864lowast
minus20937
08829
09296
08842
08411
07862
07460
07065
06557
06177
119880119861119864lowast
18714
11949
11208
11580
11884
12244
12516
12818
13285
13683
L-mom
ents
GNO
119861119877 (119865)
08124
minus196279
minus193846
minus192837
minus191816
minus190506
minus189573
minus188699
minus187636
minus186899
119877(119865)
00986
00696
00917
02051
03520
06027
08421
11312
16008
20312
119877119877 (119865)
20766
01045
00587
00889
01120
01368
01533
01686
01878
02016
119871119861119864lowast
minus21331
08683
09271
08780
08379
07958
07691
07453
07172
06978
119880119861119864lowast
16569
11974
11077
11569
11954
12355
12615
12851
13149
13369
L-mom
ents
GPA
119861119877 (119865)
38654
minus194351
minus193564
minus193226
minus192616
minus191706
minus191060
minus190506
minus189966
minus189727
119877(119865)
01242
006
6900998
02136
03552
05968
08313
11188
15941
20368
119877119877 (119865)
24785
01033
00614
00892
01111
01386
01602
01833
02162
02431
119871119861119864lowast
minus25192
08591
09215
08817
08444
07979
07646
07327
06919
06623
119880119861119864lowast
14272
11911
11115
11588
11976
12459
12841
13257
13859
14367
PL-m
oments
GEV
119861119877 (119865)
minus02662
minus197059
minus194770
minus193340
minus191600
minus188996
minus186909
minus184797
minus182052
minus1800
66119877(119865)
05379
006
8700947
02023
03467
06171
09087
13058
20551
2854
119877119877 (119865)
38117
00948
006
4900960
01210
01508
01736
01977
02324
02609
119871119861119864lowast
minus77814
08892
09252
08725
08259
07697
07296
06911
064
1706050
119880119861119864lowast
45516
11827
11207
11716
12117
12554
12881
13211
13693
14104
PL-m
oments
GNO
119861119877 (119865)
35142
minus195922
minus193485
minus192704
minus191894
minus190806
minus1900
03minus18
9235
minus188284
minus187617
119877(119865)
06203
006
8900939
02141
03644
06143
08483
11272
15747
19809
119877119877 (119865)
43522
00971
00626
00980
01238
01504
01675
01831
02024
02162
119871119861119864lowast
minus10618
08772
09204
08655
08233
07800
07530
07290
07012
06822
119880119861119864lowast
46353
11849
11122
11734
12189
12653
12949
13218
13546
13780
PL-m
oments
GPA
119861119877 (119865)
1212
3minus19
4300
minus193238
minus193113
minus192716
minus192046
minus191555
minus19114
1minus19
0768
minus1906
49119877(119865)
08513
006
6601041
02250
03684
06021
08215
10853
15144
19094
119877119877 (119865)
58811
00962
006
6900996
01233
01509
01720
01941
02258
02518
119871119861119864lowast
minus17507
08698
09130
08680
08299
07846
07524
07220
06831
06552
119880119861119864lowast
47307
11791
11189
11774
12215
12739
1314
13567
14195
14722
Advances in Meteorology 13
Table9Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f60
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus13271
minus194230
minus193551
minus193241
minus192649
minus191612
minus190736
minus189839
minus188690
minus187888
119877(119865)
00846
00624
00851
01873
03250
05841
08652
12505
19839
27721
119877119877 (119865)
19155
00833
00612
00855
01024
01213
01360
01523
01770
01983
119871119861119864lowast
minus19593
08756
09254
08851
08491
08074
07796
07501
07118
06834
119880119861119864lowast
16242
11675
11015
11472
11879
12309
12678
13038
13659
14141
L-mom
ents
GNO
119861119877 (119865)
08417
minus193335
minus192795
minus192984
minus192999
minus192879
minus192742
minus192593
minus192399
minus192264
119877(119865)
00968
00634
00858
02000
03439
05850
08123
10845
15235
19239
119877119877 (119865)
21055
00881
00582
00861
01050
01235
01354
01463
01599
01699
119871119861119864lowast
minus22887
08647
09244
08794
08466
08141
07957
07798
07597
07443
119880119861119864lowast
15863
11714
1096
11511
11934
1240
512
666
12939
13253
13505
L-mom
ents
GPA
119861119877 (119865)
46371
minus192512
minus192631
minus193160
minus193417
minus193606
minus193745
minus193933
minus194306
minus194708
119877(119865)
01280
00624
00950
02094
03468
05740
07903
10536
14878
18928
119877119877 (119865)
264
4200915
00614
00882
01064
01270
01429
01599
01849
02059
119871119861119864lowast
minus38088
08628
09186
08815
08510
08167
07924
07703
07422
07178
119880119861119864lowast
14699
11721
11020
11534
11925
12454
12791
13236
13869
1440
5
PL-m
oments
GEV
119861119877 (119865)
minus16968
minus193984
minus193601
minus193623
minus193319
minus192576
minus191861
minus191081
minus190026
minus189255
119877(119865)
05431
00596
00888
01978
03366
05844
08417
11838
18179
24868
119877119877 (119865)
38656
00783
00634
0095
01166
01387
01541
01700
01928
02120
119871119861119864lowast
minus92921
08907
09201
08750
08403
07984
07714
07434
07043
06783
119880119861119864lowast
53553
11521
11099
1166
412
098
12572
12869
13216
13697
14105
PL-m
oments
GNO
119861119877 (119865)
23785
minus193306
minus192979
minus193417
minus193628
minus193678
minus193617
minus193509
minus193331
minus193185
119877(119865)
06278
006
0100911
02100
03523
05804
0788
10303
14113
17516
119877119877 (119865)
43743
00812
00623
00965
01186
01391
01513
01618
01742
01830
119871119861119864lowast
minus12083
08856
09188
08709
08389
08061
07870
07706
07517
07380
119880119861119864lowast
52707
11561
11059
11704
12152
12613
12898
13138
13361
1356
PL-m
oments
GPA
119861119877 (119865)
1118
7minus19
2590
minus192910
minus193681
minus1940
84minus19
4322
minus1944
04minus19
4463
minus194560
minus1946
73119877(119865)
08596
00597
01009
02229
03581
05615
07391
09418
12558
15341
119877119877 (119865)
58310
00840
006
6000991
0119
501391
01522
01650
01828
01971
119871119861119864lowast
minus18977
08800
09127
08699
08413
08098
07912
07682
07447
07255
119880119861119864lowast
51281
11551
11117
11769
12173
12615
12887
13227
1360
913
899
14 Advances in Meteorology
Table10A
ccuracymeasure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f90
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus00117
minus194226
minus193109
minus192926
minus192542
minus191809
minus191151
minus1904
48minus18
9497
minus188790
119877(119865)
00787
00556
00823
01850
03151
05459
07840
10989
16793
22883
119877119877 (119865)
17329
00814
00544
00832
01031
01238
01381
01525
01727
01892
119871119861119864lowast
minus19303
08935
09287
08885
08587
08246
08006
07772
07463
07229
119880119861119864lowast
15717
11499
10973
11461
11804
12156
12393
12635
12966
13242
L-mom
ents
GNO
119861119877 (119865)
15851
minus193606
minus192517
minus192695
minus192747
minus192678
minus192569
minus192437
minus192245
minus192098
119877(119865)
00917
00567
00846
01967
03305
05446
07392
09660
13223
1640
0119877119877 (119865)
19328
00850
00543
00853
01052
01237
01347
01442
01554
01632
119871119861119864lowast
minus23268
08848
09261
08844
08568
08301
08139
08000
07833
07722
119880119861119864lowast
15168
11542
10942
11489
11851
12197
1240
112
575
12779
12925
L-mom
ents
GPA
119861119877 (119865)
49081
minus192995
minus192417
minus192865
minus193078
minus193175
minus193195
minus193209
minus193256
minus193330
119877(119865)
01234
00571
00935
02081
03365
05322
07054
09053
12180
14975
119877119877 (119865)
246
8800882
00576
00870
01054
01239
01364
01489
01660
01798
119871119861119864lowast
minus45353
08783
09210
08838
08594
08326
08141
07961
07731
07557
119880119861119864lowast
14833
11574
10996
11532
11870
12223
1246
712
711
1304
413
312
PL-m
oments
GEV
119861119877 (119865)
minus00877
minus194145
minus193060
minus193062
minus192847
minus192278
minus191698
minus191038
minus190100
minus189374
119877(119865)
05458
00565
00855
01963
03338
05712
08090
11156
16650
22276
119877119877 (119865)
39079
00784
00585
00934
0117
201406
01558
01703
01897
02050
119871119861119864lowast
minus10190
08983
09236
08765
08431
08067
07818
07579
07271
07041
119880119861119864lowast
58730
1144
811036
11652
12081
12504
12771
13026
13364
13623
PL-m
oments
GNO
119861119877 (119865)
38414
minus193665
minus192536
minus192859
minus193044
minus193102
minus193057
minus192966
minus192804
minus192665
119877(119865)
06308
00578
00896
02088
03488
05690
07659
09926
13434
16524
119877119877 (119865)
44651
00819
00596
00961
0119
301405
01526
01628
01744
01822
119871119861119864lowast
minus13011
08910
09192
08716
08411
08122
07949
07801
07634
07519
119880119861119864lowast
56756
11491
11030
11693
12128
12539
12775
12974
13197
13347
PL-m
oments
GPA
119861119877 (119865)
126398
minus193142
minus192484
minus193077
minus193404
minus193589
minus193637
minus193655
minus193680
minus193720
119877(119865)
08641
00588
00999
02220
03558
05535
07227
09125
1200
914
521
119877119877 (119865)
60116
00855
006
4000985
0119
901400
01529
01651
01812
01937
119871119861119864lowast
minus19889
08842
09130
08700
08435
08156
07970
07789
07566
07399
119880119861119864lowast
53743
11528
11103
11755
12158
12552
12807
13055
13385
13636
Advances in Meteorology 15
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
0 2 4 6
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
Regional growth curve for GPA
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F))
0
5
10
15
20
Qua
ntile
2 4 60Reduced variate minuslog (minus log (F))
0
5
10
15
Qua
ntile
(based on L-moments) Regional growth curve for GPA
(based on PL-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30
16 Advances in Meteorology
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60
Advances in Meteorology 17
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90
18 Advances in Meteorology
The first four PL-moments of the GEV are defined as
12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)
(A5)
In previous equation 119875(sdot sdot) is an Incomplete Gammafunction
119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500
120579119896expminus120579119889Θ (A6)
Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution
The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by
119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)
The partial PWMs of the GLO are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)
(A9)
where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function
1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
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6 Advances in Meteorology
Table 1 Statistics of annual extreme monthly precipitation for study region based on L-moments and PL-moments
Sites L-moments PL-momentsMean 119905 1199053 1199054 Mean 119905 1199053 1199054
Astore 37624 0518 0366 0193 41752 0467 0377 0183Balakot 120450 0478 0322 0192 133259 0428 0335 0198Bunji 14297 0609 0456 0259 16339 0553 0450 0257Chilas 15091 0585 0392 0194 17603 0516 0375 0200Cherat 49038 0580 0375 0162 55289 0526 0355 0161DI khan 28885 0689 0516 0262 32351 0651 0487 0250Dir 105694 0432 0216 0098 116741 0378 0222 0097Drosh 45284 0525 0317 0136 50316 0474 0307 0137Garhi Dupatta 111352 0454 0227 0111 123374 0398 0229 0120Gilgit 12677 0591 0412 0228 14290 0539 0399 0239Gupis 23446 0678 0529 0331 30299 0583 0506 0350Kakul 99431 0453 0247 0102 110068 0399 0254 0093Kotli 95070 0542 0332 0136 106903 0487 0314 0138Muzaffarabad 118439 0459 0284 0182 131122 0405 0301 0194Peshawar 46216 0572 0398 0216 51761 0520 0392 0222Saidu Sharif 85667 0455 0265 0181 94978 0399 0283 0200Skardu 20438 0616 04564 0238 22885 0570 0444 0231
value of 119885-statistic is less than 164 at 90 confidence level(ie |119885Dis| le 164) it will indicate that the distributionqualifies the goodness of fit criteria If there are more thanone distribution that qualify the criteria the most suitabledistribution has the minimum |119885Dis| value3 Estimation
The sitesrsquo information and statistic by using L-moments forthe present study are presented in Table 1 In Table 1 meanrepresents the first sample LPL-moments and 119905 1199053 and 1199054are the sample LPL-moments ratios of the L-CvPL-CvL-CsPL-Cs and L-CkPL-Ck respectively The lower levelcensoring threshold is selected from 10 to 23 Table 2expresses the feasible threshold values according to thepercentile technique along with Average Annual OccurrenceNumber (AAON) Jiang et al [41] and Yuguo [42] suggestedthat the optimal threshold can be obtained if the values ofAAON lie between 1 and 2 Table 2 shows that 90th percentileobservations are suitable for the optimum threshold selectionof most of areas in the present study We have 168 valuesin each station according to the above table Astore stationhas 32 threshold values due to which 16 values are beingcensored in 168 According to censored level 102 censoredlevel was selected Similarly 105 was selected for Balakotand Muzaffarabad According to this process maximumthreshold level 223 was selected for Gupis By using theabove process for each station 17 different censoring levelswere selected So we decide that the range from 10 to 23of censoring level should be kept for selecting thresholdvalues
31 Regional Frequency Analysis The following four steps areconsidered as prerequisite for frequency analysis Hoskingand 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
311 Data Screening In this study we use secondary dataafter carefully examining all locations for abnormalities andmissing observations Therefore we use 14 years of data forRFA that were obtained from seventeen locations
312 Discordance Test Table 3 shows 119863119894 result of (18) for17 locations of this study region It can be observed fromthe results of LPL-moments in Table 3 that the value of 119863-statistic varies from 007 to 244 If 119863119894 is greater than 3 thelocation is considered to be discordant from the rest of theregional data Hosking and Wallis [7] In this study regionno location is diagnosed as discordant (119863119894 ge 3) Thereforewe use all data for the development of the RFA based on L-moments and PL-moments
313 Regional Heterogeneity Measure The next step is theformation of the homogeneous region which is conven-tionally tougher and needs the higher number of subjectivejudgments The homogeneity conditions are defined as thelocations that have the same frequency distributions
Advances in Meteorology 7
Table 2 Precipitation threshold selection in GPA distribution for 17 stations
Station 80th 90th 95th 975th 99thAstore
Threshold 98 32 02 0 0AAON 243 121 064 043 043
BalakotThreshold 325 148 51 3 0AAON 243 129 057 029 014
MuzaffarabadThreshold 304 118 5 1 0AAON 25 121 057 029 021
Garhi DupattaThreshold 288 10 38 0 0AAON 243 121 057 029 029
DirThreshold 31 20 5 005 0AAON 243 121 057 029 021
KakulThreshold 289 134 21 05 0AAON 164 129 057 029 021
KotliThreshold 13 6 005 0 0AAON 243 135 057 036 036
Saidu SharifThreshold 256 83 2 0 0AAON 243 121 057 036 036
cheratThreshold 5 005 0 0 0AAON 243 143 093 093 093
PeshawarThreshold 6 005 0 0 0AAON 243 129 057 057 057
DroshThreshold 76 15 02 005 0AAON 243 121 057 05 021
D I KhanThreshold 03 0 0 0 0AAON 257 129 129 129 129
GupisThreshold 0 0 0 0 0AAON 271 271 271 271 271
SkarduThreshold 2 03 0 0 0AAON 243 071 071 071 071
ChilasThreshold 1 05 0 0 0AAON 171 107 107 107 107
BunjiThreshold 13 0 0 0 0AAON 15 15 15 15 15
GilgitThreshold 12 005 0 0 0AAON 243 135 071 071 071
8 Advances in Meteorology
04
03
02
01
00
04
03
02
01
00
00 01 02 03 04 05 06 00 01 02 03 04 05 06
GLOGEVGPA
GNOAverageData
GLOGEVGPA
GNOAverageData
L-ku
rtos
is
L㰀-k
urto
sis
L-skewness L-skewness
Figure 2 L-diagram and PL-diagram of the GLO GEV GPA and GNO distributions
Table 3 Discordance test result based on L-moments and PL-moments
Name of site L-moments (119863119894) PL-moments (119863119894)Astore 205 244Balakot 142 085Bunji 055 060Chilas 029 007Cherat 079 064DI khan 129 190Dir 082 086Drosh 051 043Garhi Dupatta 143 133Gilgit 026 059Gupis 237 231Kakul 082 122Kotli 077 059Muzaffarabad 099 085Peshawar 007 008Saidu Sharif 191 157Skardu 067 065
For the present study area realization of the Kappaprobability distribution is used to conduct the heterogeneitytest based on the L-moments and PL-moments
Number of simulations are 10000 for computing theheterogeneityWe computed the regional average L-momentsratios the regional PL-moments ratios and the correspond-ing parameter values of the fitted Kappa probability dis-tribution (see Table 4) Table 4 shows the results of theheterogeneity measure using L-moments and PL-momentsmethods It can be observed from Table 4 that the different
values for the119867-statistic are minus041 minus160 and minus289 based onL-moments andminus006minus18 andminus317 based onPL-momentsTherefore we concluded that by comparing these resultsand the heterogeneity conditions study region is acceptablyhomogeneous for L-moments and PL-moments No furthersubdivisions of the present study are necessary
32 Fitting Appropriate Probability Distribution After homo-geneity analysis of the study area a suitable probabilitydistribution is required for the RFAThe objective is not onlyto recognize a suitable probability distribution for RFA butalso to observe a probability distribution that will providerobust quantile estimate for each location and for the regionalgrowth cure List of candidate probability distributions forRFA is GLO GEV GPA and GNO
We plotted L-moments and PL-moments diagrams forpreliminary evaluation of the probability distribution for thestudy area
Figure 2 illustrates an analogy of the observed and hypo-thetical relationships of the probability distribution Figure 2shows that GLO distribution is not a suitable candidate forthe L-moments and PL-moments
Interestingly both analyses of the L-moments and the PL-moments diagram show that the sample average values areappropriately distinguished by the hypothetical L-momentsand PL-moments for GPA and GNO distributions
However it is hard to find a suitable probability distri-bution that fits most of the regional observed data Table 5shows the goodness of fit test results for candidate probabilitydistributions
Table 5 shows that GLO distribution failed the goodnessof fit test for both L-moments and for PL-moments methodsas the calculated value of the 119885-test for the GLO distribution
Advances in Meteorology 9
Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments
Heterogeneity measures L-moment PL-momentHeterogeneity measure1198671
Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060
Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800
Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170
Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments
Method Distribution Parameters120585 120572 119870L-moments
GEV 04660 05644 minus02751GNO 06671 07565 minus07594GPA minus00551 09944 minus00575
PL-momentsGEV 05220 05116 minus02686GNO 07046 06837 minus07487GPA 00478 09073 minus00472
is larger than the critical value of 164 (at 90 confidencelevel)
It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile
Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution
33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7
based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)
Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is
119876 (119865) = 1198971198941119902 (119865) (25)
The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3
Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions
Therefore it is necessary to assess the performance ofregional quantile estimates
4 Accuracy of the Estimated Quantiles and theRegional Growth Curve
A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten
10 Advances in Meteorology
Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865
01000 05000 08000 09000 09500 09800 09900 09975 099875 09990
L-momentsGEV 00454 06836 15139 22247 30591 44161 56871 90748 113165 121338GNO 00473 06671 15585 23072 31448 44098 54997 80676 95666 100823GPA 00500 06481 16217 23931 31958 43073 51878 70578 80502 83782
PL-momentsGEV 01397 07190 14670 21034 28469 40497 51703 81363 10086 107950GNO 01412 07046 15062 21752 29203 40410 50032 72608 85739 90249GPA 01436 06871 15649 22547 29674 39460 47149 63304 71786 74577
Qua
ntile
Regional growth curve (L-moments)
GPAGNOGEV
GPAGNOGEV
Qua
ntile
Return period
Regional growth gurve (PL-moments)
5
4
3
2
1
0
5
4
3
2
1
0
minus2 minus1 0 1 2 3 4 5
2 5 10 20 50 100
Return period2 5 10 20 50 100
Reduced variate minuslog (minus log (F))
minus2 minus1 0 1 2 3 4 5
Reduced variate minuslog (minus log (F))
Figure 3 L-moments and PL-moments regional growth curves
as 119876(119898)119894 (119865) for the nonexceedance probability 119865 The relativeerror for this estimator is
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) (26)
The bias and the RMSE of the above quantity over all 119872repetition are
Bias = 119861119894 (119865) = 1119872119872sum119898=1
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)
= 1119872 [[119872sum119898=1
(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12
(27)
Also for the estimated quantile the regional average bias andthe relative RMSE are
119861119877 (119865) = 1119873119873sum119894=1
119861119894 (119865)
119877119877 (119865) = 1119873119873sum119894=1
119877119894 (119865) (28)
We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval
119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)
Advances in Meteorology 11
Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le
119902 (119865)119871005 (119865) (30)
The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves
We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]
We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively
Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds
Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion
This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail
GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments
Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents
Appendix
The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows
PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by
119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (minus ln119865)119896 119896 = 0 (A2)
Wang [10] developed the partial probability-weightedmoments (PWMs) of the GEV as
(119903 + 1) 1205731015840119903 = 120577 + 120572119867 (120574 1198650 119896) (A3)
where
119867(120574 1198650 119896) = 1119896 [1 minus119875 1 + 119896 minus (119903 + 1) ln1198650(1 minus 119865119903+10 ) (119903 + 1)119896 ] (A4)
12 Advances in Meteorology
Table8Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f30
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus0540
0minus19
7674
minus195328
minus193602
minus191579
minus188659
minus186376
minus1840
97minus18
1165
minus179056
119877(119865)
00900
00695
00963
01952
03334
06038
09057
13261
21337
300
48119877119877 (119865)
19873
01019
00635
00876
01090
01369
01595
01839
02191
02478
119871119861119864lowast
minus20937
08829
09296
08842
08411
07862
07460
07065
06557
06177
119880119861119864lowast
18714
11949
11208
11580
11884
12244
12516
12818
13285
13683
L-mom
ents
GNO
119861119877 (119865)
08124
minus196279
minus193846
minus192837
minus191816
minus190506
minus189573
minus188699
minus187636
minus186899
119877(119865)
00986
00696
00917
02051
03520
06027
08421
11312
16008
20312
119877119877 (119865)
20766
01045
00587
00889
01120
01368
01533
01686
01878
02016
119871119861119864lowast
minus21331
08683
09271
08780
08379
07958
07691
07453
07172
06978
119880119861119864lowast
16569
11974
11077
11569
11954
12355
12615
12851
13149
13369
L-mom
ents
GPA
119861119877 (119865)
38654
minus194351
minus193564
minus193226
minus192616
minus191706
minus191060
minus190506
minus189966
minus189727
119877(119865)
01242
006
6900998
02136
03552
05968
08313
11188
15941
20368
119877119877 (119865)
24785
01033
00614
00892
01111
01386
01602
01833
02162
02431
119871119861119864lowast
minus25192
08591
09215
08817
08444
07979
07646
07327
06919
06623
119880119861119864lowast
14272
11911
11115
11588
11976
12459
12841
13257
13859
14367
PL-m
oments
GEV
119861119877 (119865)
minus02662
minus197059
minus194770
minus193340
minus191600
minus188996
minus186909
minus184797
minus182052
minus1800
66119877(119865)
05379
006
8700947
02023
03467
06171
09087
13058
20551
2854
119877119877 (119865)
38117
00948
006
4900960
01210
01508
01736
01977
02324
02609
119871119861119864lowast
minus77814
08892
09252
08725
08259
07697
07296
06911
064
1706050
119880119861119864lowast
45516
11827
11207
11716
12117
12554
12881
13211
13693
14104
PL-m
oments
GNO
119861119877 (119865)
35142
minus195922
minus193485
minus192704
minus191894
minus190806
minus1900
03minus18
9235
minus188284
minus187617
119877(119865)
06203
006
8900939
02141
03644
06143
08483
11272
15747
19809
119877119877 (119865)
43522
00971
00626
00980
01238
01504
01675
01831
02024
02162
119871119861119864lowast
minus10618
08772
09204
08655
08233
07800
07530
07290
07012
06822
119880119861119864lowast
46353
11849
11122
11734
12189
12653
12949
13218
13546
13780
PL-m
oments
GPA
119861119877 (119865)
1212
3minus19
4300
minus193238
minus193113
minus192716
minus192046
minus191555
minus19114
1minus19
0768
minus1906
49119877(119865)
08513
006
6601041
02250
03684
06021
08215
10853
15144
19094
119877119877 (119865)
58811
00962
006
6900996
01233
01509
01720
01941
02258
02518
119871119861119864lowast
minus17507
08698
09130
08680
08299
07846
07524
07220
06831
06552
119880119861119864lowast
47307
11791
11189
11774
12215
12739
1314
13567
14195
14722
Advances in Meteorology 13
Table9Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f60
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus13271
minus194230
minus193551
minus193241
minus192649
minus191612
minus190736
minus189839
minus188690
minus187888
119877(119865)
00846
00624
00851
01873
03250
05841
08652
12505
19839
27721
119877119877 (119865)
19155
00833
00612
00855
01024
01213
01360
01523
01770
01983
119871119861119864lowast
minus19593
08756
09254
08851
08491
08074
07796
07501
07118
06834
119880119861119864lowast
16242
11675
11015
11472
11879
12309
12678
13038
13659
14141
L-mom
ents
GNO
119861119877 (119865)
08417
minus193335
minus192795
minus192984
minus192999
minus192879
minus192742
minus192593
minus192399
minus192264
119877(119865)
00968
00634
00858
02000
03439
05850
08123
10845
15235
19239
119877119877 (119865)
21055
00881
00582
00861
01050
01235
01354
01463
01599
01699
119871119861119864lowast
minus22887
08647
09244
08794
08466
08141
07957
07798
07597
07443
119880119861119864lowast
15863
11714
1096
11511
11934
1240
512
666
12939
13253
13505
L-mom
ents
GPA
119861119877 (119865)
46371
minus192512
minus192631
minus193160
minus193417
minus193606
minus193745
minus193933
minus194306
minus194708
119877(119865)
01280
00624
00950
02094
03468
05740
07903
10536
14878
18928
119877119877 (119865)
264
4200915
00614
00882
01064
01270
01429
01599
01849
02059
119871119861119864lowast
minus38088
08628
09186
08815
08510
08167
07924
07703
07422
07178
119880119861119864lowast
14699
11721
11020
11534
11925
12454
12791
13236
13869
1440
5
PL-m
oments
GEV
119861119877 (119865)
minus16968
minus193984
minus193601
minus193623
minus193319
minus192576
minus191861
minus191081
minus190026
minus189255
119877(119865)
05431
00596
00888
01978
03366
05844
08417
11838
18179
24868
119877119877 (119865)
38656
00783
00634
0095
01166
01387
01541
01700
01928
02120
119871119861119864lowast
minus92921
08907
09201
08750
08403
07984
07714
07434
07043
06783
119880119861119864lowast
53553
11521
11099
1166
412
098
12572
12869
13216
13697
14105
PL-m
oments
GNO
119861119877 (119865)
23785
minus193306
minus192979
minus193417
minus193628
minus193678
minus193617
minus193509
minus193331
minus193185
119877(119865)
06278
006
0100911
02100
03523
05804
0788
10303
14113
17516
119877119877 (119865)
43743
00812
00623
00965
01186
01391
01513
01618
01742
01830
119871119861119864lowast
minus12083
08856
09188
08709
08389
08061
07870
07706
07517
07380
119880119861119864lowast
52707
11561
11059
11704
12152
12613
12898
13138
13361
1356
PL-m
oments
GPA
119861119877 (119865)
1118
7minus19
2590
minus192910
minus193681
minus1940
84minus19
4322
minus1944
04minus19
4463
minus194560
minus1946
73119877(119865)
08596
00597
01009
02229
03581
05615
07391
09418
12558
15341
119877119877 (119865)
58310
00840
006
6000991
0119
501391
01522
01650
01828
01971
119871119861119864lowast
minus18977
08800
09127
08699
08413
08098
07912
07682
07447
07255
119880119861119864lowast
51281
11551
11117
11769
12173
12615
12887
13227
1360
913
899
14 Advances in Meteorology
Table10A
ccuracymeasure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f90
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus00117
minus194226
minus193109
minus192926
minus192542
minus191809
minus191151
minus1904
48minus18
9497
minus188790
119877(119865)
00787
00556
00823
01850
03151
05459
07840
10989
16793
22883
119877119877 (119865)
17329
00814
00544
00832
01031
01238
01381
01525
01727
01892
119871119861119864lowast
minus19303
08935
09287
08885
08587
08246
08006
07772
07463
07229
119880119861119864lowast
15717
11499
10973
11461
11804
12156
12393
12635
12966
13242
L-mom
ents
GNO
119861119877 (119865)
15851
minus193606
minus192517
minus192695
minus192747
minus192678
minus192569
minus192437
minus192245
minus192098
119877(119865)
00917
00567
00846
01967
03305
05446
07392
09660
13223
1640
0119877119877 (119865)
19328
00850
00543
00853
01052
01237
01347
01442
01554
01632
119871119861119864lowast
minus23268
08848
09261
08844
08568
08301
08139
08000
07833
07722
119880119861119864lowast
15168
11542
10942
11489
11851
12197
1240
112
575
12779
12925
L-mom
ents
GPA
119861119877 (119865)
49081
minus192995
minus192417
minus192865
minus193078
minus193175
minus193195
minus193209
minus193256
minus193330
119877(119865)
01234
00571
00935
02081
03365
05322
07054
09053
12180
14975
119877119877 (119865)
246
8800882
00576
00870
01054
01239
01364
01489
01660
01798
119871119861119864lowast
minus45353
08783
09210
08838
08594
08326
08141
07961
07731
07557
119880119861119864lowast
14833
11574
10996
11532
11870
12223
1246
712
711
1304
413
312
PL-m
oments
GEV
119861119877 (119865)
minus00877
minus194145
minus193060
minus193062
minus192847
minus192278
minus191698
minus191038
minus190100
minus189374
119877(119865)
05458
00565
00855
01963
03338
05712
08090
11156
16650
22276
119877119877 (119865)
39079
00784
00585
00934
0117
201406
01558
01703
01897
02050
119871119861119864lowast
minus10190
08983
09236
08765
08431
08067
07818
07579
07271
07041
119880119861119864lowast
58730
1144
811036
11652
12081
12504
12771
13026
13364
13623
PL-m
oments
GNO
119861119877 (119865)
38414
minus193665
minus192536
minus192859
minus193044
minus193102
minus193057
minus192966
minus192804
minus192665
119877(119865)
06308
00578
00896
02088
03488
05690
07659
09926
13434
16524
119877119877 (119865)
44651
00819
00596
00961
0119
301405
01526
01628
01744
01822
119871119861119864lowast
minus13011
08910
09192
08716
08411
08122
07949
07801
07634
07519
119880119861119864lowast
56756
11491
11030
11693
12128
12539
12775
12974
13197
13347
PL-m
oments
GPA
119861119877 (119865)
126398
minus193142
minus192484
minus193077
minus193404
minus193589
minus193637
minus193655
minus193680
minus193720
119877(119865)
08641
00588
00999
02220
03558
05535
07227
09125
1200
914
521
119877119877 (119865)
60116
00855
006
4000985
0119
901400
01529
01651
01812
01937
119871119861119864lowast
minus19889
08842
09130
08700
08435
08156
07970
07789
07566
07399
119880119861119864lowast
53743
11528
11103
11755
12158
12552
12807
13055
13385
13636
Advances in Meteorology 15
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
0 2 4 6
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
Regional growth curve for GPA
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F))
0
5
10
15
20
Qua
ntile
2 4 60Reduced variate minuslog (minus log (F))
0
5
10
15
Qua
ntile
(based on L-moments) Regional growth curve for GPA
(based on PL-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30
16 Advances in Meteorology
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60
Advances in Meteorology 17
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90
18 Advances in Meteorology
The first four PL-moments of the GEV are defined as
12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)
(A5)
In previous equation 119875(sdot sdot) is an Incomplete Gammafunction
119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500
120579119896expminus120579119889Θ (A6)
Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution
The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by
119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)
The partial PWMs of the GLO are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)
(A9)
where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function
1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Geology Advances in
Advances in Meteorology 7
Table 2 Precipitation threshold selection in GPA distribution for 17 stations
Station 80th 90th 95th 975th 99thAstore
Threshold 98 32 02 0 0AAON 243 121 064 043 043
BalakotThreshold 325 148 51 3 0AAON 243 129 057 029 014
MuzaffarabadThreshold 304 118 5 1 0AAON 25 121 057 029 021
Garhi DupattaThreshold 288 10 38 0 0AAON 243 121 057 029 029
DirThreshold 31 20 5 005 0AAON 243 121 057 029 021
KakulThreshold 289 134 21 05 0AAON 164 129 057 029 021
KotliThreshold 13 6 005 0 0AAON 243 135 057 036 036
Saidu SharifThreshold 256 83 2 0 0AAON 243 121 057 036 036
cheratThreshold 5 005 0 0 0AAON 243 143 093 093 093
PeshawarThreshold 6 005 0 0 0AAON 243 129 057 057 057
DroshThreshold 76 15 02 005 0AAON 243 121 057 05 021
D I KhanThreshold 03 0 0 0 0AAON 257 129 129 129 129
GupisThreshold 0 0 0 0 0AAON 271 271 271 271 271
SkarduThreshold 2 03 0 0 0AAON 243 071 071 071 071
ChilasThreshold 1 05 0 0 0AAON 171 107 107 107 107
BunjiThreshold 13 0 0 0 0AAON 15 15 15 15 15
GilgitThreshold 12 005 0 0 0AAON 243 135 071 071 071
8 Advances in Meteorology
04
03
02
01
00
04
03
02
01
00
00 01 02 03 04 05 06 00 01 02 03 04 05 06
GLOGEVGPA
GNOAverageData
GLOGEVGPA
GNOAverageData
L-ku
rtos
is
L㰀-k
urto
sis
L-skewness L-skewness
Figure 2 L-diagram and PL-diagram of the GLO GEV GPA and GNO distributions
Table 3 Discordance test result based on L-moments and PL-moments
Name of site L-moments (119863119894) PL-moments (119863119894)Astore 205 244Balakot 142 085Bunji 055 060Chilas 029 007Cherat 079 064DI khan 129 190Dir 082 086Drosh 051 043Garhi Dupatta 143 133Gilgit 026 059Gupis 237 231Kakul 082 122Kotli 077 059Muzaffarabad 099 085Peshawar 007 008Saidu Sharif 191 157Skardu 067 065
For the present study area realization of the Kappaprobability distribution is used to conduct the heterogeneitytest based on the L-moments and PL-moments
Number of simulations are 10000 for computing theheterogeneityWe computed the regional average L-momentsratios the regional PL-moments ratios and the correspond-ing parameter values of the fitted Kappa probability dis-tribution (see Table 4) Table 4 shows the results of theheterogeneity measure using L-moments and PL-momentsmethods It can be observed from Table 4 that the different
values for the119867-statistic are minus041 minus160 and minus289 based onL-moments andminus006minus18 andminus317 based onPL-momentsTherefore we concluded that by comparing these resultsand the heterogeneity conditions study region is acceptablyhomogeneous for L-moments and PL-moments No furthersubdivisions of the present study are necessary
32 Fitting Appropriate Probability Distribution After homo-geneity analysis of the study area a suitable probabilitydistribution is required for the RFAThe objective is not onlyto recognize a suitable probability distribution for RFA butalso to observe a probability distribution that will providerobust quantile estimate for each location and for the regionalgrowth cure List of candidate probability distributions forRFA is GLO GEV GPA and GNO
We plotted L-moments and PL-moments diagrams forpreliminary evaluation of the probability distribution for thestudy area
Figure 2 illustrates an analogy of the observed and hypo-thetical relationships of the probability distribution Figure 2shows that GLO distribution is not a suitable candidate forthe L-moments and PL-moments
Interestingly both analyses of the L-moments and the PL-moments diagram show that the sample average values areappropriately distinguished by the hypothetical L-momentsand PL-moments for GPA and GNO distributions
However it is hard to find a suitable probability distri-bution that fits most of the regional observed data Table 5shows the goodness of fit test results for candidate probabilitydistributions
Table 5 shows that GLO distribution failed the goodnessof fit test for both L-moments and for PL-moments methodsas the calculated value of the 119885-test for the GLO distribution
Advances in Meteorology 9
Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments
Heterogeneity measures L-moment PL-momentHeterogeneity measure1198671
Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060
Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800
Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170
Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments
Method Distribution Parameters120585 120572 119870L-moments
GEV 04660 05644 minus02751GNO 06671 07565 minus07594GPA minus00551 09944 minus00575
PL-momentsGEV 05220 05116 minus02686GNO 07046 06837 minus07487GPA 00478 09073 minus00472
is larger than the critical value of 164 (at 90 confidencelevel)
It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile
Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution
33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7
based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)
Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is
119876 (119865) = 1198971198941119902 (119865) (25)
The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3
Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions
Therefore it is necessary to assess the performance ofregional quantile estimates
4 Accuracy of the Estimated Quantiles and theRegional Growth Curve
A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten
10 Advances in Meteorology
Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865
01000 05000 08000 09000 09500 09800 09900 09975 099875 09990
L-momentsGEV 00454 06836 15139 22247 30591 44161 56871 90748 113165 121338GNO 00473 06671 15585 23072 31448 44098 54997 80676 95666 100823GPA 00500 06481 16217 23931 31958 43073 51878 70578 80502 83782
PL-momentsGEV 01397 07190 14670 21034 28469 40497 51703 81363 10086 107950GNO 01412 07046 15062 21752 29203 40410 50032 72608 85739 90249GPA 01436 06871 15649 22547 29674 39460 47149 63304 71786 74577
Qua
ntile
Regional growth curve (L-moments)
GPAGNOGEV
GPAGNOGEV
Qua
ntile
Return period
Regional growth gurve (PL-moments)
5
4
3
2
1
0
5
4
3
2
1
0
minus2 minus1 0 1 2 3 4 5
2 5 10 20 50 100
Return period2 5 10 20 50 100
Reduced variate minuslog (minus log (F))
minus2 minus1 0 1 2 3 4 5
Reduced variate minuslog (minus log (F))
Figure 3 L-moments and PL-moments regional growth curves
as 119876(119898)119894 (119865) for the nonexceedance probability 119865 The relativeerror for this estimator is
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) (26)
The bias and the RMSE of the above quantity over all 119872repetition are
Bias = 119861119894 (119865) = 1119872119872sum119898=1
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)
= 1119872 [[119872sum119898=1
(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12
(27)
Also for the estimated quantile the regional average bias andthe relative RMSE are
119861119877 (119865) = 1119873119873sum119894=1
119861119894 (119865)
119877119877 (119865) = 1119873119873sum119894=1
119877119894 (119865) (28)
We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval
119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)
Advances in Meteorology 11
Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le
119902 (119865)119871005 (119865) (30)
The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves
We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]
We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively
Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds
Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion
This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail
GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments
Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents
Appendix
The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows
PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by
119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (minus ln119865)119896 119896 = 0 (A2)
Wang [10] developed the partial probability-weightedmoments (PWMs) of the GEV as
(119903 + 1) 1205731015840119903 = 120577 + 120572119867 (120574 1198650 119896) (A3)
where
119867(120574 1198650 119896) = 1119896 [1 minus119875 1 + 119896 minus (119903 + 1) ln1198650(1 minus 119865119903+10 ) (119903 + 1)119896 ] (A4)
12 Advances in Meteorology
Table8Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f30
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus0540
0minus19
7674
minus195328
minus193602
minus191579
minus188659
minus186376
minus1840
97minus18
1165
minus179056
119877(119865)
00900
00695
00963
01952
03334
06038
09057
13261
21337
300
48119877119877 (119865)
19873
01019
00635
00876
01090
01369
01595
01839
02191
02478
119871119861119864lowast
minus20937
08829
09296
08842
08411
07862
07460
07065
06557
06177
119880119861119864lowast
18714
11949
11208
11580
11884
12244
12516
12818
13285
13683
L-mom
ents
GNO
119861119877 (119865)
08124
minus196279
minus193846
minus192837
minus191816
minus190506
minus189573
minus188699
minus187636
minus186899
119877(119865)
00986
00696
00917
02051
03520
06027
08421
11312
16008
20312
119877119877 (119865)
20766
01045
00587
00889
01120
01368
01533
01686
01878
02016
119871119861119864lowast
minus21331
08683
09271
08780
08379
07958
07691
07453
07172
06978
119880119861119864lowast
16569
11974
11077
11569
11954
12355
12615
12851
13149
13369
L-mom
ents
GPA
119861119877 (119865)
38654
minus194351
minus193564
minus193226
minus192616
minus191706
minus191060
minus190506
minus189966
minus189727
119877(119865)
01242
006
6900998
02136
03552
05968
08313
11188
15941
20368
119877119877 (119865)
24785
01033
00614
00892
01111
01386
01602
01833
02162
02431
119871119861119864lowast
minus25192
08591
09215
08817
08444
07979
07646
07327
06919
06623
119880119861119864lowast
14272
11911
11115
11588
11976
12459
12841
13257
13859
14367
PL-m
oments
GEV
119861119877 (119865)
minus02662
minus197059
minus194770
minus193340
minus191600
minus188996
minus186909
minus184797
minus182052
minus1800
66119877(119865)
05379
006
8700947
02023
03467
06171
09087
13058
20551
2854
119877119877 (119865)
38117
00948
006
4900960
01210
01508
01736
01977
02324
02609
119871119861119864lowast
minus77814
08892
09252
08725
08259
07697
07296
06911
064
1706050
119880119861119864lowast
45516
11827
11207
11716
12117
12554
12881
13211
13693
14104
PL-m
oments
GNO
119861119877 (119865)
35142
minus195922
minus193485
minus192704
minus191894
minus190806
minus1900
03minus18
9235
minus188284
minus187617
119877(119865)
06203
006
8900939
02141
03644
06143
08483
11272
15747
19809
119877119877 (119865)
43522
00971
00626
00980
01238
01504
01675
01831
02024
02162
119871119861119864lowast
minus10618
08772
09204
08655
08233
07800
07530
07290
07012
06822
119880119861119864lowast
46353
11849
11122
11734
12189
12653
12949
13218
13546
13780
PL-m
oments
GPA
119861119877 (119865)
1212
3minus19
4300
minus193238
minus193113
minus192716
minus192046
minus191555
minus19114
1minus19
0768
minus1906
49119877(119865)
08513
006
6601041
02250
03684
06021
08215
10853
15144
19094
119877119877 (119865)
58811
00962
006
6900996
01233
01509
01720
01941
02258
02518
119871119861119864lowast
minus17507
08698
09130
08680
08299
07846
07524
07220
06831
06552
119880119861119864lowast
47307
11791
11189
11774
12215
12739
1314
13567
14195
14722
Advances in Meteorology 13
Table9Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f60
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus13271
minus194230
minus193551
minus193241
minus192649
minus191612
minus190736
minus189839
minus188690
minus187888
119877(119865)
00846
00624
00851
01873
03250
05841
08652
12505
19839
27721
119877119877 (119865)
19155
00833
00612
00855
01024
01213
01360
01523
01770
01983
119871119861119864lowast
minus19593
08756
09254
08851
08491
08074
07796
07501
07118
06834
119880119861119864lowast
16242
11675
11015
11472
11879
12309
12678
13038
13659
14141
L-mom
ents
GNO
119861119877 (119865)
08417
minus193335
minus192795
minus192984
minus192999
minus192879
minus192742
minus192593
minus192399
minus192264
119877(119865)
00968
00634
00858
02000
03439
05850
08123
10845
15235
19239
119877119877 (119865)
21055
00881
00582
00861
01050
01235
01354
01463
01599
01699
119871119861119864lowast
minus22887
08647
09244
08794
08466
08141
07957
07798
07597
07443
119880119861119864lowast
15863
11714
1096
11511
11934
1240
512
666
12939
13253
13505
L-mom
ents
GPA
119861119877 (119865)
46371
minus192512
minus192631
minus193160
minus193417
minus193606
minus193745
minus193933
minus194306
minus194708
119877(119865)
01280
00624
00950
02094
03468
05740
07903
10536
14878
18928
119877119877 (119865)
264
4200915
00614
00882
01064
01270
01429
01599
01849
02059
119871119861119864lowast
minus38088
08628
09186
08815
08510
08167
07924
07703
07422
07178
119880119861119864lowast
14699
11721
11020
11534
11925
12454
12791
13236
13869
1440
5
PL-m
oments
GEV
119861119877 (119865)
minus16968
minus193984
minus193601
minus193623
minus193319
minus192576
minus191861
minus191081
minus190026
minus189255
119877(119865)
05431
00596
00888
01978
03366
05844
08417
11838
18179
24868
119877119877 (119865)
38656
00783
00634
0095
01166
01387
01541
01700
01928
02120
119871119861119864lowast
minus92921
08907
09201
08750
08403
07984
07714
07434
07043
06783
119880119861119864lowast
53553
11521
11099
1166
412
098
12572
12869
13216
13697
14105
PL-m
oments
GNO
119861119877 (119865)
23785
minus193306
minus192979
minus193417
minus193628
minus193678
minus193617
minus193509
minus193331
minus193185
119877(119865)
06278
006
0100911
02100
03523
05804
0788
10303
14113
17516
119877119877 (119865)
43743
00812
00623
00965
01186
01391
01513
01618
01742
01830
119871119861119864lowast
minus12083
08856
09188
08709
08389
08061
07870
07706
07517
07380
119880119861119864lowast
52707
11561
11059
11704
12152
12613
12898
13138
13361
1356
PL-m
oments
GPA
119861119877 (119865)
1118
7minus19
2590
minus192910
minus193681
minus1940
84minus19
4322
minus1944
04minus19
4463
minus194560
minus1946
73119877(119865)
08596
00597
01009
02229
03581
05615
07391
09418
12558
15341
119877119877 (119865)
58310
00840
006
6000991
0119
501391
01522
01650
01828
01971
119871119861119864lowast
minus18977
08800
09127
08699
08413
08098
07912
07682
07447
07255
119880119861119864lowast
51281
11551
11117
11769
12173
12615
12887
13227
1360
913
899
14 Advances in Meteorology
Table10A
ccuracymeasure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f90
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus00117
minus194226
minus193109
minus192926
minus192542
minus191809
minus191151
minus1904
48minus18
9497
minus188790
119877(119865)
00787
00556
00823
01850
03151
05459
07840
10989
16793
22883
119877119877 (119865)
17329
00814
00544
00832
01031
01238
01381
01525
01727
01892
119871119861119864lowast
minus19303
08935
09287
08885
08587
08246
08006
07772
07463
07229
119880119861119864lowast
15717
11499
10973
11461
11804
12156
12393
12635
12966
13242
L-mom
ents
GNO
119861119877 (119865)
15851
minus193606
minus192517
minus192695
minus192747
minus192678
minus192569
minus192437
minus192245
minus192098
119877(119865)
00917
00567
00846
01967
03305
05446
07392
09660
13223
1640
0119877119877 (119865)
19328
00850
00543
00853
01052
01237
01347
01442
01554
01632
119871119861119864lowast
minus23268
08848
09261
08844
08568
08301
08139
08000
07833
07722
119880119861119864lowast
15168
11542
10942
11489
11851
12197
1240
112
575
12779
12925
L-mom
ents
GPA
119861119877 (119865)
49081
minus192995
minus192417
minus192865
minus193078
minus193175
minus193195
minus193209
minus193256
minus193330
119877(119865)
01234
00571
00935
02081
03365
05322
07054
09053
12180
14975
119877119877 (119865)
246
8800882
00576
00870
01054
01239
01364
01489
01660
01798
119871119861119864lowast
minus45353
08783
09210
08838
08594
08326
08141
07961
07731
07557
119880119861119864lowast
14833
11574
10996
11532
11870
12223
1246
712
711
1304
413
312
PL-m
oments
GEV
119861119877 (119865)
minus00877
minus194145
minus193060
minus193062
minus192847
minus192278
minus191698
minus191038
minus190100
minus189374
119877(119865)
05458
00565
00855
01963
03338
05712
08090
11156
16650
22276
119877119877 (119865)
39079
00784
00585
00934
0117
201406
01558
01703
01897
02050
119871119861119864lowast
minus10190
08983
09236
08765
08431
08067
07818
07579
07271
07041
119880119861119864lowast
58730
1144
811036
11652
12081
12504
12771
13026
13364
13623
PL-m
oments
GNO
119861119877 (119865)
38414
minus193665
minus192536
minus192859
minus193044
minus193102
minus193057
minus192966
minus192804
minus192665
119877(119865)
06308
00578
00896
02088
03488
05690
07659
09926
13434
16524
119877119877 (119865)
44651
00819
00596
00961
0119
301405
01526
01628
01744
01822
119871119861119864lowast
minus13011
08910
09192
08716
08411
08122
07949
07801
07634
07519
119880119861119864lowast
56756
11491
11030
11693
12128
12539
12775
12974
13197
13347
PL-m
oments
GPA
119861119877 (119865)
126398
minus193142
minus192484
minus193077
minus193404
minus193589
minus193637
minus193655
minus193680
minus193720
119877(119865)
08641
00588
00999
02220
03558
05535
07227
09125
1200
914
521
119877119877 (119865)
60116
00855
006
4000985
0119
901400
01529
01651
01812
01937
119871119861119864lowast
minus19889
08842
09130
08700
08435
08156
07970
07789
07566
07399
119880119861119864lowast
53743
11528
11103
11755
12158
12552
12807
13055
13385
13636
Advances in Meteorology 15
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
0 2 4 6
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
Regional growth curve for GPA
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F))
0
5
10
15
20
Qua
ntile
2 4 60Reduced variate minuslog (minus log (F))
0
5
10
15
Qua
ntile
(based on L-moments) Regional growth curve for GPA
(based on PL-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30
16 Advances in Meteorology
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60
Advances in Meteorology 17
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90
18 Advances in Meteorology
The first four PL-moments of the GEV are defined as
12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)
(A5)
In previous equation 119875(sdot sdot) is an Incomplete Gammafunction
119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500
120579119896expminus120579119889Θ (A6)
Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution
The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by
119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)
The partial PWMs of the GLO are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)
(A9)
where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function
1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
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Geology Advances in
8 Advances in Meteorology
04
03
02
01
00
04
03
02
01
00
00 01 02 03 04 05 06 00 01 02 03 04 05 06
GLOGEVGPA
GNOAverageData
GLOGEVGPA
GNOAverageData
L-ku
rtos
is
L㰀-k
urto
sis
L-skewness L-skewness
Figure 2 L-diagram and PL-diagram of the GLO GEV GPA and GNO distributions
Table 3 Discordance test result based on L-moments and PL-moments
Name of site L-moments (119863119894) PL-moments (119863119894)Astore 205 244Balakot 142 085Bunji 055 060Chilas 029 007Cherat 079 064DI khan 129 190Dir 082 086Drosh 051 043Garhi Dupatta 143 133Gilgit 026 059Gupis 237 231Kakul 082 122Kotli 077 059Muzaffarabad 099 085Peshawar 007 008Saidu Sharif 191 157Skardu 067 065
For the present study area realization of the Kappaprobability distribution is used to conduct the heterogeneitytest based on the L-moments and PL-moments
Number of simulations are 10000 for computing theheterogeneityWe computed the regional average L-momentsratios the regional PL-moments ratios and the correspond-ing parameter values of the fitted Kappa probability dis-tribution (see Table 4) Table 4 shows the results of theheterogeneity measure using L-moments and PL-momentsmethods It can be observed from Table 4 that the different
values for the119867-statistic are minus041 minus160 and minus289 based onL-moments andminus006minus18 andminus317 based onPL-momentsTherefore we concluded that by comparing these resultsand the heterogeneity conditions study region is acceptablyhomogeneous for L-moments and PL-moments No furthersubdivisions of the present study are necessary
32 Fitting Appropriate Probability Distribution After homo-geneity analysis of the study area a suitable probabilitydistribution is required for the RFAThe objective is not onlyto recognize a suitable probability distribution for RFA butalso to observe a probability distribution that will providerobust quantile estimate for each location and for the regionalgrowth cure List of candidate probability distributions forRFA is GLO GEV GPA and GNO
We plotted L-moments and PL-moments diagrams forpreliminary evaluation of the probability distribution for thestudy area
Figure 2 illustrates an analogy of the observed and hypo-thetical relationships of the probability distribution Figure 2shows that GLO distribution is not a suitable candidate forthe L-moments and PL-moments
Interestingly both analyses of the L-moments and the PL-moments diagram show that the sample average values areappropriately distinguished by the hypothetical L-momentsand PL-moments for GPA and GNO distributions
However it is hard to find a suitable probability distri-bution that fits most of the regional observed data Table 5shows the goodness of fit test results for candidate probabilitydistributions
Table 5 shows that GLO distribution failed the goodnessof fit test for both L-moments and for PL-moments methodsas the calculated value of the 119885-test for the GLO distribution
Advances in Meteorology 9
Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments
Heterogeneity measures L-moment PL-momentHeterogeneity measure1198671
Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060
Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800
Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170
Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments
Method Distribution Parameters120585 120572 119870L-moments
GEV 04660 05644 minus02751GNO 06671 07565 minus07594GPA minus00551 09944 minus00575
PL-momentsGEV 05220 05116 minus02686GNO 07046 06837 minus07487GPA 00478 09073 minus00472
is larger than the critical value of 164 (at 90 confidencelevel)
It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile
Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution
33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7
based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)
Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is
119876 (119865) = 1198971198941119902 (119865) (25)
The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3
Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions
Therefore it is necessary to assess the performance ofregional quantile estimates
4 Accuracy of the Estimated Quantiles and theRegional Growth Curve
A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten
10 Advances in Meteorology
Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865
01000 05000 08000 09000 09500 09800 09900 09975 099875 09990
L-momentsGEV 00454 06836 15139 22247 30591 44161 56871 90748 113165 121338GNO 00473 06671 15585 23072 31448 44098 54997 80676 95666 100823GPA 00500 06481 16217 23931 31958 43073 51878 70578 80502 83782
PL-momentsGEV 01397 07190 14670 21034 28469 40497 51703 81363 10086 107950GNO 01412 07046 15062 21752 29203 40410 50032 72608 85739 90249GPA 01436 06871 15649 22547 29674 39460 47149 63304 71786 74577
Qua
ntile
Regional growth curve (L-moments)
GPAGNOGEV
GPAGNOGEV
Qua
ntile
Return period
Regional growth gurve (PL-moments)
5
4
3
2
1
0
5
4
3
2
1
0
minus2 minus1 0 1 2 3 4 5
2 5 10 20 50 100
Return period2 5 10 20 50 100
Reduced variate minuslog (minus log (F))
minus2 minus1 0 1 2 3 4 5
Reduced variate minuslog (minus log (F))
Figure 3 L-moments and PL-moments regional growth curves
as 119876(119898)119894 (119865) for the nonexceedance probability 119865 The relativeerror for this estimator is
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) (26)
The bias and the RMSE of the above quantity over all 119872repetition are
Bias = 119861119894 (119865) = 1119872119872sum119898=1
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)
= 1119872 [[119872sum119898=1
(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12
(27)
Also for the estimated quantile the regional average bias andthe relative RMSE are
119861119877 (119865) = 1119873119873sum119894=1
119861119894 (119865)
119877119877 (119865) = 1119873119873sum119894=1
119877119894 (119865) (28)
We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval
119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)
Advances in Meteorology 11
Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le
119902 (119865)119871005 (119865) (30)
The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves
We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]
We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively
Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds
Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion
This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail
GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments
Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents
Appendix
The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows
PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by
119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (minus ln119865)119896 119896 = 0 (A2)
Wang [10] developed the partial probability-weightedmoments (PWMs) of the GEV as
(119903 + 1) 1205731015840119903 = 120577 + 120572119867 (120574 1198650 119896) (A3)
where
119867(120574 1198650 119896) = 1119896 [1 minus119875 1 + 119896 minus (119903 + 1) ln1198650(1 minus 119865119903+10 ) (119903 + 1)119896 ] (A4)
12 Advances in Meteorology
Table8Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f30
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus0540
0minus19
7674
minus195328
minus193602
minus191579
minus188659
minus186376
minus1840
97minus18
1165
minus179056
119877(119865)
00900
00695
00963
01952
03334
06038
09057
13261
21337
300
48119877119877 (119865)
19873
01019
00635
00876
01090
01369
01595
01839
02191
02478
119871119861119864lowast
minus20937
08829
09296
08842
08411
07862
07460
07065
06557
06177
119880119861119864lowast
18714
11949
11208
11580
11884
12244
12516
12818
13285
13683
L-mom
ents
GNO
119861119877 (119865)
08124
minus196279
minus193846
minus192837
minus191816
minus190506
minus189573
minus188699
minus187636
minus186899
119877(119865)
00986
00696
00917
02051
03520
06027
08421
11312
16008
20312
119877119877 (119865)
20766
01045
00587
00889
01120
01368
01533
01686
01878
02016
119871119861119864lowast
minus21331
08683
09271
08780
08379
07958
07691
07453
07172
06978
119880119861119864lowast
16569
11974
11077
11569
11954
12355
12615
12851
13149
13369
L-mom
ents
GPA
119861119877 (119865)
38654
minus194351
minus193564
minus193226
minus192616
minus191706
minus191060
minus190506
minus189966
minus189727
119877(119865)
01242
006
6900998
02136
03552
05968
08313
11188
15941
20368
119877119877 (119865)
24785
01033
00614
00892
01111
01386
01602
01833
02162
02431
119871119861119864lowast
minus25192
08591
09215
08817
08444
07979
07646
07327
06919
06623
119880119861119864lowast
14272
11911
11115
11588
11976
12459
12841
13257
13859
14367
PL-m
oments
GEV
119861119877 (119865)
minus02662
minus197059
minus194770
minus193340
minus191600
minus188996
minus186909
minus184797
minus182052
minus1800
66119877(119865)
05379
006
8700947
02023
03467
06171
09087
13058
20551
2854
119877119877 (119865)
38117
00948
006
4900960
01210
01508
01736
01977
02324
02609
119871119861119864lowast
minus77814
08892
09252
08725
08259
07697
07296
06911
064
1706050
119880119861119864lowast
45516
11827
11207
11716
12117
12554
12881
13211
13693
14104
PL-m
oments
GNO
119861119877 (119865)
35142
minus195922
minus193485
minus192704
minus191894
minus190806
minus1900
03minus18
9235
minus188284
minus187617
119877(119865)
06203
006
8900939
02141
03644
06143
08483
11272
15747
19809
119877119877 (119865)
43522
00971
00626
00980
01238
01504
01675
01831
02024
02162
119871119861119864lowast
minus10618
08772
09204
08655
08233
07800
07530
07290
07012
06822
119880119861119864lowast
46353
11849
11122
11734
12189
12653
12949
13218
13546
13780
PL-m
oments
GPA
119861119877 (119865)
1212
3minus19
4300
minus193238
minus193113
minus192716
minus192046
minus191555
minus19114
1minus19
0768
minus1906
49119877(119865)
08513
006
6601041
02250
03684
06021
08215
10853
15144
19094
119877119877 (119865)
58811
00962
006
6900996
01233
01509
01720
01941
02258
02518
119871119861119864lowast
minus17507
08698
09130
08680
08299
07846
07524
07220
06831
06552
119880119861119864lowast
47307
11791
11189
11774
12215
12739
1314
13567
14195
14722
Advances in Meteorology 13
Table9Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f60
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus13271
minus194230
minus193551
minus193241
minus192649
minus191612
minus190736
minus189839
minus188690
minus187888
119877(119865)
00846
00624
00851
01873
03250
05841
08652
12505
19839
27721
119877119877 (119865)
19155
00833
00612
00855
01024
01213
01360
01523
01770
01983
119871119861119864lowast
minus19593
08756
09254
08851
08491
08074
07796
07501
07118
06834
119880119861119864lowast
16242
11675
11015
11472
11879
12309
12678
13038
13659
14141
L-mom
ents
GNO
119861119877 (119865)
08417
minus193335
minus192795
minus192984
minus192999
minus192879
minus192742
minus192593
minus192399
minus192264
119877(119865)
00968
00634
00858
02000
03439
05850
08123
10845
15235
19239
119877119877 (119865)
21055
00881
00582
00861
01050
01235
01354
01463
01599
01699
119871119861119864lowast
minus22887
08647
09244
08794
08466
08141
07957
07798
07597
07443
119880119861119864lowast
15863
11714
1096
11511
11934
1240
512
666
12939
13253
13505
L-mom
ents
GPA
119861119877 (119865)
46371
minus192512
minus192631
minus193160
minus193417
minus193606
minus193745
minus193933
minus194306
minus194708
119877(119865)
01280
00624
00950
02094
03468
05740
07903
10536
14878
18928
119877119877 (119865)
264
4200915
00614
00882
01064
01270
01429
01599
01849
02059
119871119861119864lowast
minus38088
08628
09186
08815
08510
08167
07924
07703
07422
07178
119880119861119864lowast
14699
11721
11020
11534
11925
12454
12791
13236
13869
1440
5
PL-m
oments
GEV
119861119877 (119865)
minus16968
minus193984
minus193601
minus193623
minus193319
minus192576
minus191861
minus191081
minus190026
minus189255
119877(119865)
05431
00596
00888
01978
03366
05844
08417
11838
18179
24868
119877119877 (119865)
38656
00783
00634
0095
01166
01387
01541
01700
01928
02120
119871119861119864lowast
minus92921
08907
09201
08750
08403
07984
07714
07434
07043
06783
119880119861119864lowast
53553
11521
11099
1166
412
098
12572
12869
13216
13697
14105
PL-m
oments
GNO
119861119877 (119865)
23785
minus193306
minus192979
minus193417
minus193628
minus193678
minus193617
minus193509
minus193331
minus193185
119877(119865)
06278
006
0100911
02100
03523
05804
0788
10303
14113
17516
119877119877 (119865)
43743
00812
00623
00965
01186
01391
01513
01618
01742
01830
119871119861119864lowast
minus12083
08856
09188
08709
08389
08061
07870
07706
07517
07380
119880119861119864lowast
52707
11561
11059
11704
12152
12613
12898
13138
13361
1356
PL-m
oments
GPA
119861119877 (119865)
1118
7minus19
2590
minus192910
minus193681
minus1940
84minus19
4322
minus1944
04minus19
4463
minus194560
minus1946
73119877(119865)
08596
00597
01009
02229
03581
05615
07391
09418
12558
15341
119877119877 (119865)
58310
00840
006
6000991
0119
501391
01522
01650
01828
01971
119871119861119864lowast
minus18977
08800
09127
08699
08413
08098
07912
07682
07447
07255
119880119861119864lowast
51281
11551
11117
11769
12173
12615
12887
13227
1360
913
899
14 Advances in Meteorology
Table10A
ccuracymeasure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f90
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus00117
minus194226
minus193109
minus192926
minus192542
minus191809
minus191151
minus1904
48minus18
9497
minus188790
119877(119865)
00787
00556
00823
01850
03151
05459
07840
10989
16793
22883
119877119877 (119865)
17329
00814
00544
00832
01031
01238
01381
01525
01727
01892
119871119861119864lowast
minus19303
08935
09287
08885
08587
08246
08006
07772
07463
07229
119880119861119864lowast
15717
11499
10973
11461
11804
12156
12393
12635
12966
13242
L-mom
ents
GNO
119861119877 (119865)
15851
minus193606
minus192517
minus192695
minus192747
minus192678
minus192569
minus192437
minus192245
minus192098
119877(119865)
00917
00567
00846
01967
03305
05446
07392
09660
13223
1640
0119877119877 (119865)
19328
00850
00543
00853
01052
01237
01347
01442
01554
01632
119871119861119864lowast
minus23268
08848
09261
08844
08568
08301
08139
08000
07833
07722
119880119861119864lowast
15168
11542
10942
11489
11851
12197
1240
112
575
12779
12925
L-mom
ents
GPA
119861119877 (119865)
49081
minus192995
minus192417
minus192865
minus193078
minus193175
minus193195
minus193209
minus193256
minus193330
119877(119865)
01234
00571
00935
02081
03365
05322
07054
09053
12180
14975
119877119877 (119865)
246
8800882
00576
00870
01054
01239
01364
01489
01660
01798
119871119861119864lowast
minus45353
08783
09210
08838
08594
08326
08141
07961
07731
07557
119880119861119864lowast
14833
11574
10996
11532
11870
12223
1246
712
711
1304
413
312
PL-m
oments
GEV
119861119877 (119865)
minus00877
minus194145
minus193060
minus193062
minus192847
minus192278
minus191698
minus191038
minus190100
minus189374
119877(119865)
05458
00565
00855
01963
03338
05712
08090
11156
16650
22276
119877119877 (119865)
39079
00784
00585
00934
0117
201406
01558
01703
01897
02050
119871119861119864lowast
minus10190
08983
09236
08765
08431
08067
07818
07579
07271
07041
119880119861119864lowast
58730
1144
811036
11652
12081
12504
12771
13026
13364
13623
PL-m
oments
GNO
119861119877 (119865)
38414
minus193665
minus192536
minus192859
minus193044
minus193102
minus193057
minus192966
minus192804
minus192665
119877(119865)
06308
00578
00896
02088
03488
05690
07659
09926
13434
16524
119877119877 (119865)
44651
00819
00596
00961
0119
301405
01526
01628
01744
01822
119871119861119864lowast
minus13011
08910
09192
08716
08411
08122
07949
07801
07634
07519
119880119861119864lowast
56756
11491
11030
11693
12128
12539
12775
12974
13197
13347
PL-m
oments
GPA
119861119877 (119865)
126398
minus193142
minus192484
minus193077
minus193404
minus193589
minus193637
minus193655
minus193680
minus193720
119877(119865)
08641
00588
00999
02220
03558
05535
07227
09125
1200
914
521
119877119877 (119865)
60116
00855
006
4000985
0119
901400
01529
01651
01812
01937
119871119861119864lowast
minus19889
08842
09130
08700
08435
08156
07970
07789
07566
07399
119880119861119864lowast
53743
11528
11103
11755
12158
12552
12807
13055
13385
13636
Advances in Meteorology 15
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
0 2 4 6
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
Regional growth curve for GPA
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F))
0
5
10
15
20
Qua
ntile
2 4 60Reduced variate minuslog (minus log (F))
0
5
10
15
Qua
ntile
(based on L-moments) Regional growth curve for GPA
(based on PL-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30
16 Advances in Meteorology
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60
Advances in Meteorology 17
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90
18 Advances in Meteorology
The first four PL-moments of the GEV are defined as
12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)
(A5)
In previous equation 119875(sdot sdot) is an Incomplete Gammafunction
119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500
120579119896expminus120579119889Θ (A6)
Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution
The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by
119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)
The partial PWMs of the GLO are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)
(A9)
where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function
1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mining
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GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geological ResearchJournal of
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Geology Advances in
Advances in Meteorology 9
Table 4 Heterogeneity measures for the study region based on L-moments and PL-moments
Heterogeneity measures L-moment PL-momentHeterogeneity measure1198671
Observed standard deviation of group L-Cv 0078 0077Simulated mean of standard deviation of group L-Cv 0084 0077Simulated standard deviation of standard deviation of group L-Cv 0015 0014Value of the heterogeneity measure1198671 minus0410 minus0060
Heterogeneity measure1198672Observed average of L-CvL-skewness distance 0105 0139Simulated mean of average L-CvL-skewness distance 0141 0139Simulated standard deviation of average L-CvL-skewness distance 0023 0022Value of the heterogeneity measure1198672 minus1600 minus1800
Heterogeneity measure1198673Observed average of L-skewnessL-kurtosis distance 0094 0088Simulated mean of average L-skewnessL-kurtosis distance 0175 0176Simulated standard deviation of average L-skewnessL-kurtosis distance 0028 0028Value of the heterogeneity measure1198673 minus2890 minus3170
Table 5 119885-test result for the goodness of fitMethod GLO GEV GNO GPAL-moments 196 138 061 minus042PL-moments 177 119 043 minus061Table 6 Regional parameters for the three candidate distributionsfor L-moments and PL-moments
Method Distribution Parameters120585 120572 119870L-moments
GEV 04660 05644 minus02751GNO 06671 07565 minus07594GPA minus00551 09944 minus00575
PL-momentsGEV 05220 05116 minus02686GNO 07046 06837 minus07487GPA 00478 09073 minus00472
is larger than the critical value of 164 (at 90 confidencelevel)
It has been observed that the computed values of |119885Dis|are less than 164 (at 90 confidence level) namely GEVGNO andGPA distributions However GEV GNO andGPAdistributions are suitable for regional distribution based onL-moments and PL-moments methods and for obtaining thefuture estimates of the quantile
Further it can be noted that GPA distribution is suitablefor L-moments method (lowest critical |119885Dis| value) Simi-larly GNO distribution is suitable for PL-moments method(lowest critical value) Table 6 shows the estimates of theregional parameters for L-moments and PL-moments for thesuitable probability distribution
33 Estimation of the Quantiles The regional quantile esti-mates 119902(119865) with varying nonexceedance probability119865 for theGNO GEV and GPA distributions are presented in Table 7
based on L-moments and PL-moments Quantile functionis normally represented as 119902(sdot) for fitted regional frequencydistribution The quantile estimate at location 119894 is establishedby joining the estimate of 120583119894 and 119902(sdot)
Mathematical form of the quantile estimate with nonex-ceedance probability 119865 is
119876 (119865) = 1198971198941119902 (119865) (25)
The regional growth curves for the GEV GNO and GPAdistributions are shown in Figure 3
Figure 3 shows the regional growth curves of each can-didate distribution for L-moments and PL-moments GEVGNO and GPA distributions are approximately identicalup until 100-year return period (119865 = 099) for both L-moments and PL-moments However afterward the growthcurves of the GPA distribution lie below the GEV and GNOdistributions
Therefore it is necessary to assess the performance ofregional quantile estimates
4 Accuracy of the Estimated Quantiles and theRegional Growth Curve
A Monte Carlo simulation is designed to assess accuracyof the regional quantile estimates that are obtained by theRFA We use logical L-moments algorithm that has beenreported by Hosking and Wallis [7] in Section 64 Thisalgorithm takes samples from a region that has comparablecharacteristics as of the actual region such as having the samerecord length same number of locations and the regionalL-moments ratios The area used for simulation shouldreport the plausible heterogeneity in the area and intersitedependency if exist (Hosking andWallis [7]) In the repeatedsampling procedure the quantile estimates are computed forthe different nonexceedance probabilities Suppose that at119898th repetition and location 119894 quantile estimate can bewritten
10 Advances in Meteorology
Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865
01000 05000 08000 09000 09500 09800 09900 09975 099875 09990
L-momentsGEV 00454 06836 15139 22247 30591 44161 56871 90748 113165 121338GNO 00473 06671 15585 23072 31448 44098 54997 80676 95666 100823GPA 00500 06481 16217 23931 31958 43073 51878 70578 80502 83782
PL-momentsGEV 01397 07190 14670 21034 28469 40497 51703 81363 10086 107950GNO 01412 07046 15062 21752 29203 40410 50032 72608 85739 90249GPA 01436 06871 15649 22547 29674 39460 47149 63304 71786 74577
Qua
ntile
Regional growth curve (L-moments)
GPAGNOGEV
GPAGNOGEV
Qua
ntile
Return period
Regional growth gurve (PL-moments)
5
4
3
2
1
0
5
4
3
2
1
0
minus2 minus1 0 1 2 3 4 5
2 5 10 20 50 100
Return period2 5 10 20 50 100
Reduced variate minuslog (minus log (F))
minus2 minus1 0 1 2 3 4 5
Reduced variate minuslog (minus log (F))
Figure 3 L-moments and PL-moments regional growth curves
as 119876(119898)119894 (119865) for the nonexceedance probability 119865 The relativeerror for this estimator is
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) (26)
The bias and the RMSE of the above quantity over all 119872repetition are
Bias = 119861119894 (119865) = 1119872119872sum119898=1
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)
= 1119872 [[119872sum119898=1
(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12
(27)
Also for the estimated quantile the regional average bias andthe relative RMSE are
119861119877 (119865) = 1119873119873sum119894=1
119861119894 (119865)
119877119877 (119865) = 1119873119873sum119894=1
119877119894 (119865) (28)
We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval
119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)
Advances in Meteorology 11
Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le
119902 (119865)119871005 (119865) (30)
The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves
We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]
We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively
Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds
Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion
This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail
GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments
Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents
Appendix
The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows
PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by
119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (minus ln119865)119896 119896 = 0 (A2)
Wang [10] developed the partial probability-weightedmoments (PWMs) of the GEV as
(119903 + 1) 1205731015840119903 = 120577 + 120572119867 (120574 1198650 119896) (A3)
where
119867(120574 1198650 119896) = 1119896 [1 minus119875 1 + 119896 minus (119903 + 1) ln1198650(1 minus 119865119903+10 ) (119903 + 1)119896 ] (A4)
12 Advances in Meteorology
Table8Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f30
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus0540
0minus19
7674
minus195328
minus193602
minus191579
minus188659
minus186376
minus1840
97minus18
1165
minus179056
119877(119865)
00900
00695
00963
01952
03334
06038
09057
13261
21337
300
48119877119877 (119865)
19873
01019
00635
00876
01090
01369
01595
01839
02191
02478
119871119861119864lowast
minus20937
08829
09296
08842
08411
07862
07460
07065
06557
06177
119880119861119864lowast
18714
11949
11208
11580
11884
12244
12516
12818
13285
13683
L-mom
ents
GNO
119861119877 (119865)
08124
minus196279
minus193846
minus192837
minus191816
minus190506
minus189573
minus188699
minus187636
minus186899
119877(119865)
00986
00696
00917
02051
03520
06027
08421
11312
16008
20312
119877119877 (119865)
20766
01045
00587
00889
01120
01368
01533
01686
01878
02016
119871119861119864lowast
minus21331
08683
09271
08780
08379
07958
07691
07453
07172
06978
119880119861119864lowast
16569
11974
11077
11569
11954
12355
12615
12851
13149
13369
L-mom
ents
GPA
119861119877 (119865)
38654
minus194351
minus193564
minus193226
minus192616
minus191706
minus191060
minus190506
minus189966
minus189727
119877(119865)
01242
006
6900998
02136
03552
05968
08313
11188
15941
20368
119877119877 (119865)
24785
01033
00614
00892
01111
01386
01602
01833
02162
02431
119871119861119864lowast
minus25192
08591
09215
08817
08444
07979
07646
07327
06919
06623
119880119861119864lowast
14272
11911
11115
11588
11976
12459
12841
13257
13859
14367
PL-m
oments
GEV
119861119877 (119865)
minus02662
minus197059
minus194770
minus193340
minus191600
minus188996
minus186909
minus184797
minus182052
minus1800
66119877(119865)
05379
006
8700947
02023
03467
06171
09087
13058
20551
2854
119877119877 (119865)
38117
00948
006
4900960
01210
01508
01736
01977
02324
02609
119871119861119864lowast
minus77814
08892
09252
08725
08259
07697
07296
06911
064
1706050
119880119861119864lowast
45516
11827
11207
11716
12117
12554
12881
13211
13693
14104
PL-m
oments
GNO
119861119877 (119865)
35142
minus195922
minus193485
minus192704
minus191894
minus190806
minus1900
03minus18
9235
minus188284
minus187617
119877(119865)
06203
006
8900939
02141
03644
06143
08483
11272
15747
19809
119877119877 (119865)
43522
00971
00626
00980
01238
01504
01675
01831
02024
02162
119871119861119864lowast
minus10618
08772
09204
08655
08233
07800
07530
07290
07012
06822
119880119861119864lowast
46353
11849
11122
11734
12189
12653
12949
13218
13546
13780
PL-m
oments
GPA
119861119877 (119865)
1212
3minus19
4300
minus193238
minus193113
minus192716
minus192046
minus191555
minus19114
1minus19
0768
minus1906
49119877(119865)
08513
006
6601041
02250
03684
06021
08215
10853
15144
19094
119877119877 (119865)
58811
00962
006
6900996
01233
01509
01720
01941
02258
02518
119871119861119864lowast
minus17507
08698
09130
08680
08299
07846
07524
07220
06831
06552
119880119861119864lowast
47307
11791
11189
11774
12215
12739
1314
13567
14195
14722
Advances in Meteorology 13
Table9Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f60
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus13271
minus194230
minus193551
minus193241
minus192649
minus191612
minus190736
minus189839
minus188690
minus187888
119877(119865)
00846
00624
00851
01873
03250
05841
08652
12505
19839
27721
119877119877 (119865)
19155
00833
00612
00855
01024
01213
01360
01523
01770
01983
119871119861119864lowast
minus19593
08756
09254
08851
08491
08074
07796
07501
07118
06834
119880119861119864lowast
16242
11675
11015
11472
11879
12309
12678
13038
13659
14141
L-mom
ents
GNO
119861119877 (119865)
08417
minus193335
minus192795
minus192984
minus192999
minus192879
minus192742
minus192593
minus192399
minus192264
119877(119865)
00968
00634
00858
02000
03439
05850
08123
10845
15235
19239
119877119877 (119865)
21055
00881
00582
00861
01050
01235
01354
01463
01599
01699
119871119861119864lowast
minus22887
08647
09244
08794
08466
08141
07957
07798
07597
07443
119880119861119864lowast
15863
11714
1096
11511
11934
1240
512
666
12939
13253
13505
L-mom
ents
GPA
119861119877 (119865)
46371
minus192512
minus192631
minus193160
minus193417
minus193606
minus193745
minus193933
minus194306
minus194708
119877(119865)
01280
00624
00950
02094
03468
05740
07903
10536
14878
18928
119877119877 (119865)
264
4200915
00614
00882
01064
01270
01429
01599
01849
02059
119871119861119864lowast
minus38088
08628
09186
08815
08510
08167
07924
07703
07422
07178
119880119861119864lowast
14699
11721
11020
11534
11925
12454
12791
13236
13869
1440
5
PL-m
oments
GEV
119861119877 (119865)
minus16968
minus193984
minus193601
minus193623
minus193319
minus192576
minus191861
minus191081
minus190026
minus189255
119877(119865)
05431
00596
00888
01978
03366
05844
08417
11838
18179
24868
119877119877 (119865)
38656
00783
00634
0095
01166
01387
01541
01700
01928
02120
119871119861119864lowast
minus92921
08907
09201
08750
08403
07984
07714
07434
07043
06783
119880119861119864lowast
53553
11521
11099
1166
412
098
12572
12869
13216
13697
14105
PL-m
oments
GNO
119861119877 (119865)
23785
minus193306
minus192979
minus193417
minus193628
minus193678
minus193617
minus193509
minus193331
minus193185
119877(119865)
06278
006
0100911
02100
03523
05804
0788
10303
14113
17516
119877119877 (119865)
43743
00812
00623
00965
01186
01391
01513
01618
01742
01830
119871119861119864lowast
minus12083
08856
09188
08709
08389
08061
07870
07706
07517
07380
119880119861119864lowast
52707
11561
11059
11704
12152
12613
12898
13138
13361
1356
PL-m
oments
GPA
119861119877 (119865)
1118
7minus19
2590
minus192910
minus193681
minus1940
84minus19
4322
minus1944
04minus19
4463
minus194560
minus1946
73119877(119865)
08596
00597
01009
02229
03581
05615
07391
09418
12558
15341
119877119877 (119865)
58310
00840
006
6000991
0119
501391
01522
01650
01828
01971
119871119861119864lowast
minus18977
08800
09127
08699
08413
08098
07912
07682
07447
07255
119880119861119864lowast
51281
11551
11117
11769
12173
12615
12887
13227
1360
913
899
14 Advances in Meteorology
Table10A
ccuracymeasure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f90
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus00117
minus194226
minus193109
minus192926
minus192542
minus191809
minus191151
minus1904
48minus18
9497
minus188790
119877(119865)
00787
00556
00823
01850
03151
05459
07840
10989
16793
22883
119877119877 (119865)
17329
00814
00544
00832
01031
01238
01381
01525
01727
01892
119871119861119864lowast
minus19303
08935
09287
08885
08587
08246
08006
07772
07463
07229
119880119861119864lowast
15717
11499
10973
11461
11804
12156
12393
12635
12966
13242
L-mom
ents
GNO
119861119877 (119865)
15851
minus193606
minus192517
minus192695
minus192747
minus192678
minus192569
minus192437
minus192245
minus192098
119877(119865)
00917
00567
00846
01967
03305
05446
07392
09660
13223
1640
0119877119877 (119865)
19328
00850
00543
00853
01052
01237
01347
01442
01554
01632
119871119861119864lowast
minus23268
08848
09261
08844
08568
08301
08139
08000
07833
07722
119880119861119864lowast
15168
11542
10942
11489
11851
12197
1240
112
575
12779
12925
L-mom
ents
GPA
119861119877 (119865)
49081
minus192995
minus192417
minus192865
minus193078
minus193175
minus193195
minus193209
minus193256
minus193330
119877(119865)
01234
00571
00935
02081
03365
05322
07054
09053
12180
14975
119877119877 (119865)
246
8800882
00576
00870
01054
01239
01364
01489
01660
01798
119871119861119864lowast
minus45353
08783
09210
08838
08594
08326
08141
07961
07731
07557
119880119861119864lowast
14833
11574
10996
11532
11870
12223
1246
712
711
1304
413
312
PL-m
oments
GEV
119861119877 (119865)
minus00877
minus194145
minus193060
minus193062
minus192847
minus192278
minus191698
minus191038
minus190100
minus189374
119877(119865)
05458
00565
00855
01963
03338
05712
08090
11156
16650
22276
119877119877 (119865)
39079
00784
00585
00934
0117
201406
01558
01703
01897
02050
119871119861119864lowast
minus10190
08983
09236
08765
08431
08067
07818
07579
07271
07041
119880119861119864lowast
58730
1144
811036
11652
12081
12504
12771
13026
13364
13623
PL-m
oments
GNO
119861119877 (119865)
38414
minus193665
minus192536
minus192859
minus193044
minus193102
minus193057
minus192966
minus192804
minus192665
119877(119865)
06308
00578
00896
02088
03488
05690
07659
09926
13434
16524
119877119877 (119865)
44651
00819
00596
00961
0119
301405
01526
01628
01744
01822
119871119861119864lowast
minus13011
08910
09192
08716
08411
08122
07949
07801
07634
07519
119880119861119864lowast
56756
11491
11030
11693
12128
12539
12775
12974
13197
13347
PL-m
oments
GPA
119861119877 (119865)
126398
minus193142
minus192484
minus193077
minus193404
minus193589
minus193637
minus193655
minus193680
minus193720
119877(119865)
08641
00588
00999
02220
03558
05535
07227
09125
1200
914
521
119877119877 (119865)
60116
00855
006
4000985
0119
901400
01529
01651
01812
01937
119871119861119864lowast
minus19889
08842
09130
08700
08435
08156
07970
07789
07566
07399
119880119861119864lowast
53743
11528
11103
11755
12158
12552
12807
13055
13385
13636
Advances in Meteorology 15
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
0 2 4 6
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
Regional growth curve for GPA
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F))
0
5
10
15
20
Qua
ntile
2 4 60Reduced variate minuslog (minus log (F))
0
5
10
15
Qua
ntile
(based on L-moments) Regional growth curve for GPA
(based on PL-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30
16 Advances in Meteorology
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60
Advances in Meteorology 17
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90
18 Advances in Meteorology
The first four PL-moments of the GEV are defined as
12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)
(A5)
In previous equation 119875(sdot sdot) is an Incomplete Gammafunction
119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500
120579119896expminus120579119889Θ (A6)
Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution
The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by
119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)
The partial PWMs of the GLO are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)
(A9)
where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function
1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Applied ampEnvironmentalSoil Science
Volume 2014
Mining
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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OceanographyInternational Journal of
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GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Atmospheric SciencesInternational Journal of
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OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
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MineralogyInternational Journal of
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Geological ResearchJournal of
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Geology Advances in
10 Advances in Meteorology
Table 7 Regional quantile estimates with nonexceedance probability 119865Method Distribution 119865
01000 05000 08000 09000 09500 09800 09900 09975 099875 09990
L-momentsGEV 00454 06836 15139 22247 30591 44161 56871 90748 113165 121338GNO 00473 06671 15585 23072 31448 44098 54997 80676 95666 100823GPA 00500 06481 16217 23931 31958 43073 51878 70578 80502 83782
PL-momentsGEV 01397 07190 14670 21034 28469 40497 51703 81363 10086 107950GNO 01412 07046 15062 21752 29203 40410 50032 72608 85739 90249GPA 01436 06871 15649 22547 29674 39460 47149 63304 71786 74577
Qua
ntile
Regional growth curve (L-moments)
GPAGNOGEV
GPAGNOGEV
Qua
ntile
Return period
Regional growth gurve (PL-moments)
5
4
3
2
1
0
5
4
3
2
1
0
minus2 minus1 0 1 2 3 4 5
2 5 10 20 50 100
Return period2 5 10 20 50 100
Reduced variate minuslog (minus log (F))
minus2 minus1 0 1 2 3 4 5
Reduced variate minuslog (minus log (F))
Figure 3 L-moments and PL-moments regional growth curves
as 119876(119898)119894 (119865) for the nonexceedance probability 119865 The relativeerror for this estimator is
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) (26)
The bias and the RMSE of the above quantity over all 119872repetition are
Bias = 119861119894 (119865) = 1119872119872sum119898=1
119876(119898)119894 (119865) minus 119876119894 (119865)119876119894 (119865) RMSE = 119877119894 (119865)
= 1119872 [[119872sum119898=1
(119876(119898)119894 (119865) minus 119876119894 (119865))119876119894 (119865) 2]]12
(27)
Also for the estimated quantile the regional average bias andthe relative RMSE are
119861119877 (119865) = 1119873119873sum119894=1
119861119894 (119865)
119877119877 (119865) = 1119873119873sum119894=1
119877119894 (119865) (28)
We use empirical quantities of quantile distribution for theassessment analysis that can be computed by taking the ratioof estimated to true values119876119894(119865)119876119894(119865) for the quantile and119902119894(119865)119902119894(119865) for the regional growth curvesTherefore 90 ofthe regional growth curve lie in between the interval
119871005 (119865) le 119902 (119865)119902 (119865) le 119880005 (119865) (29)
Advances in Meteorology 11
Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le
119902 (119865)119871005 (119865) (30)
The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves
We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]
We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively
Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds
Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion
This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail
GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments
Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents
Appendix
The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows
PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by
119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (minus ln119865)119896 119896 = 0 (A2)
Wang [10] developed the partial probability-weightedmoments (PWMs) of the GEV as
(119903 + 1) 1205731015840119903 = 120577 + 120572119867 (120574 1198650 119896) (A3)
where
119867(120574 1198650 119896) = 1119896 [1 minus119875 1 + 119896 minus (119903 + 1) ln1198650(1 minus 119865119903+10 ) (119903 + 1)119896 ] (A4)
12 Advances in Meteorology
Table8Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f30
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus0540
0minus19
7674
minus195328
minus193602
minus191579
minus188659
minus186376
minus1840
97minus18
1165
minus179056
119877(119865)
00900
00695
00963
01952
03334
06038
09057
13261
21337
300
48119877119877 (119865)
19873
01019
00635
00876
01090
01369
01595
01839
02191
02478
119871119861119864lowast
minus20937
08829
09296
08842
08411
07862
07460
07065
06557
06177
119880119861119864lowast
18714
11949
11208
11580
11884
12244
12516
12818
13285
13683
L-mom
ents
GNO
119861119877 (119865)
08124
minus196279
minus193846
minus192837
minus191816
minus190506
minus189573
minus188699
minus187636
minus186899
119877(119865)
00986
00696
00917
02051
03520
06027
08421
11312
16008
20312
119877119877 (119865)
20766
01045
00587
00889
01120
01368
01533
01686
01878
02016
119871119861119864lowast
minus21331
08683
09271
08780
08379
07958
07691
07453
07172
06978
119880119861119864lowast
16569
11974
11077
11569
11954
12355
12615
12851
13149
13369
L-mom
ents
GPA
119861119877 (119865)
38654
minus194351
minus193564
minus193226
minus192616
minus191706
minus191060
minus190506
minus189966
minus189727
119877(119865)
01242
006
6900998
02136
03552
05968
08313
11188
15941
20368
119877119877 (119865)
24785
01033
00614
00892
01111
01386
01602
01833
02162
02431
119871119861119864lowast
minus25192
08591
09215
08817
08444
07979
07646
07327
06919
06623
119880119861119864lowast
14272
11911
11115
11588
11976
12459
12841
13257
13859
14367
PL-m
oments
GEV
119861119877 (119865)
minus02662
minus197059
minus194770
minus193340
minus191600
minus188996
minus186909
minus184797
minus182052
minus1800
66119877(119865)
05379
006
8700947
02023
03467
06171
09087
13058
20551
2854
119877119877 (119865)
38117
00948
006
4900960
01210
01508
01736
01977
02324
02609
119871119861119864lowast
minus77814
08892
09252
08725
08259
07697
07296
06911
064
1706050
119880119861119864lowast
45516
11827
11207
11716
12117
12554
12881
13211
13693
14104
PL-m
oments
GNO
119861119877 (119865)
35142
minus195922
minus193485
minus192704
minus191894
minus190806
minus1900
03minus18
9235
minus188284
minus187617
119877(119865)
06203
006
8900939
02141
03644
06143
08483
11272
15747
19809
119877119877 (119865)
43522
00971
00626
00980
01238
01504
01675
01831
02024
02162
119871119861119864lowast
minus10618
08772
09204
08655
08233
07800
07530
07290
07012
06822
119880119861119864lowast
46353
11849
11122
11734
12189
12653
12949
13218
13546
13780
PL-m
oments
GPA
119861119877 (119865)
1212
3minus19
4300
minus193238
minus193113
minus192716
minus192046
minus191555
minus19114
1minus19
0768
minus1906
49119877(119865)
08513
006
6601041
02250
03684
06021
08215
10853
15144
19094
119877119877 (119865)
58811
00962
006
6900996
01233
01509
01720
01941
02258
02518
119871119861119864lowast
minus17507
08698
09130
08680
08299
07846
07524
07220
06831
06552
119880119861119864lowast
47307
11791
11189
11774
12215
12739
1314
13567
14195
14722
Advances in Meteorology 13
Table9Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f60
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus13271
minus194230
minus193551
minus193241
minus192649
minus191612
minus190736
minus189839
minus188690
minus187888
119877(119865)
00846
00624
00851
01873
03250
05841
08652
12505
19839
27721
119877119877 (119865)
19155
00833
00612
00855
01024
01213
01360
01523
01770
01983
119871119861119864lowast
minus19593
08756
09254
08851
08491
08074
07796
07501
07118
06834
119880119861119864lowast
16242
11675
11015
11472
11879
12309
12678
13038
13659
14141
L-mom
ents
GNO
119861119877 (119865)
08417
minus193335
minus192795
minus192984
minus192999
minus192879
minus192742
minus192593
minus192399
minus192264
119877(119865)
00968
00634
00858
02000
03439
05850
08123
10845
15235
19239
119877119877 (119865)
21055
00881
00582
00861
01050
01235
01354
01463
01599
01699
119871119861119864lowast
minus22887
08647
09244
08794
08466
08141
07957
07798
07597
07443
119880119861119864lowast
15863
11714
1096
11511
11934
1240
512
666
12939
13253
13505
L-mom
ents
GPA
119861119877 (119865)
46371
minus192512
minus192631
minus193160
minus193417
minus193606
minus193745
minus193933
minus194306
minus194708
119877(119865)
01280
00624
00950
02094
03468
05740
07903
10536
14878
18928
119877119877 (119865)
264
4200915
00614
00882
01064
01270
01429
01599
01849
02059
119871119861119864lowast
minus38088
08628
09186
08815
08510
08167
07924
07703
07422
07178
119880119861119864lowast
14699
11721
11020
11534
11925
12454
12791
13236
13869
1440
5
PL-m
oments
GEV
119861119877 (119865)
minus16968
minus193984
minus193601
minus193623
minus193319
minus192576
minus191861
minus191081
minus190026
minus189255
119877(119865)
05431
00596
00888
01978
03366
05844
08417
11838
18179
24868
119877119877 (119865)
38656
00783
00634
0095
01166
01387
01541
01700
01928
02120
119871119861119864lowast
minus92921
08907
09201
08750
08403
07984
07714
07434
07043
06783
119880119861119864lowast
53553
11521
11099
1166
412
098
12572
12869
13216
13697
14105
PL-m
oments
GNO
119861119877 (119865)
23785
minus193306
minus192979
minus193417
minus193628
minus193678
minus193617
minus193509
minus193331
minus193185
119877(119865)
06278
006
0100911
02100
03523
05804
0788
10303
14113
17516
119877119877 (119865)
43743
00812
00623
00965
01186
01391
01513
01618
01742
01830
119871119861119864lowast
minus12083
08856
09188
08709
08389
08061
07870
07706
07517
07380
119880119861119864lowast
52707
11561
11059
11704
12152
12613
12898
13138
13361
1356
PL-m
oments
GPA
119861119877 (119865)
1118
7minus19
2590
minus192910
minus193681
minus1940
84minus19
4322
minus1944
04minus19
4463
minus194560
minus1946
73119877(119865)
08596
00597
01009
02229
03581
05615
07391
09418
12558
15341
119877119877 (119865)
58310
00840
006
6000991
0119
501391
01522
01650
01828
01971
119871119861119864lowast
minus18977
08800
09127
08699
08413
08098
07912
07682
07447
07255
119880119861119864lowast
51281
11551
11117
11769
12173
12615
12887
13227
1360
913
899
14 Advances in Meteorology
Table10A
ccuracymeasure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f90
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus00117
minus194226
minus193109
minus192926
minus192542
minus191809
minus191151
minus1904
48minus18
9497
minus188790
119877(119865)
00787
00556
00823
01850
03151
05459
07840
10989
16793
22883
119877119877 (119865)
17329
00814
00544
00832
01031
01238
01381
01525
01727
01892
119871119861119864lowast
minus19303
08935
09287
08885
08587
08246
08006
07772
07463
07229
119880119861119864lowast
15717
11499
10973
11461
11804
12156
12393
12635
12966
13242
L-mom
ents
GNO
119861119877 (119865)
15851
minus193606
minus192517
minus192695
minus192747
minus192678
minus192569
minus192437
minus192245
minus192098
119877(119865)
00917
00567
00846
01967
03305
05446
07392
09660
13223
1640
0119877119877 (119865)
19328
00850
00543
00853
01052
01237
01347
01442
01554
01632
119871119861119864lowast
minus23268
08848
09261
08844
08568
08301
08139
08000
07833
07722
119880119861119864lowast
15168
11542
10942
11489
11851
12197
1240
112
575
12779
12925
L-mom
ents
GPA
119861119877 (119865)
49081
minus192995
minus192417
minus192865
minus193078
minus193175
minus193195
minus193209
minus193256
minus193330
119877(119865)
01234
00571
00935
02081
03365
05322
07054
09053
12180
14975
119877119877 (119865)
246
8800882
00576
00870
01054
01239
01364
01489
01660
01798
119871119861119864lowast
minus45353
08783
09210
08838
08594
08326
08141
07961
07731
07557
119880119861119864lowast
14833
11574
10996
11532
11870
12223
1246
712
711
1304
413
312
PL-m
oments
GEV
119861119877 (119865)
minus00877
minus194145
minus193060
minus193062
minus192847
minus192278
minus191698
minus191038
minus190100
minus189374
119877(119865)
05458
00565
00855
01963
03338
05712
08090
11156
16650
22276
119877119877 (119865)
39079
00784
00585
00934
0117
201406
01558
01703
01897
02050
119871119861119864lowast
minus10190
08983
09236
08765
08431
08067
07818
07579
07271
07041
119880119861119864lowast
58730
1144
811036
11652
12081
12504
12771
13026
13364
13623
PL-m
oments
GNO
119861119877 (119865)
38414
minus193665
minus192536
minus192859
minus193044
minus193102
minus193057
minus192966
minus192804
minus192665
119877(119865)
06308
00578
00896
02088
03488
05690
07659
09926
13434
16524
119877119877 (119865)
44651
00819
00596
00961
0119
301405
01526
01628
01744
01822
119871119861119864lowast
minus13011
08910
09192
08716
08411
08122
07949
07801
07634
07519
119880119861119864lowast
56756
11491
11030
11693
12128
12539
12775
12974
13197
13347
PL-m
oments
GPA
119861119877 (119865)
126398
minus193142
minus192484
minus193077
minus193404
minus193589
minus193637
minus193655
minus193680
minus193720
119877(119865)
08641
00588
00999
02220
03558
05535
07227
09125
1200
914
521
119877119877 (119865)
60116
00855
006
4000985
0119
901400
01529
01651
01812
01937
119871119861119864lowast
minus19889
08842
09130
08700
08435
08156
07970
07789
07566
07399
119880119861119864lowast
53743
11528
11103
11755
12158
12552
12807
13055
13385
13636
Advances in Meteorology 15
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
0 2 4 6
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
Regional growth curve for GPA
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F))
0
5
10
15
20
Qua
ntile
2 4 60Reduced variate minuslog (minus log (F))
0
5
10
15
Qua
ntile
(based on L-moments) Regional growth curve for GPA
(based on PL-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30
16 Advances in Meteorology
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60
Advances in Meteorology 17
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90
18 Advances in Meteorology
The first four PL-moments of the GEV are defined as
12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)
(A5)
In previous equation 119875(sdot sdot) is an Incomplete Gammafunction
119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500
120579119896expminus120579119889Θ (A6)
Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution
The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by
119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)
The partial PWMs of the GLO are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)
(A9)
where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function
1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mining
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Journal of
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International Journal of
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Journal of
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Geological ResearchJournal of
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Geology Advances in
Advances in Meteorology 11
Inverting the expression for 119902119894(119865) we have119902 (119865)119880005 (119865) le 119902 (119865) le
119902 (119865)119871005 (119865) (30)
The 90 confidence interval limits show the measure ofvariation between the estimated and the true quantilesThese limits provide the expected magnitude of errors in theestimated quantiles and the regional growth curves
We computed L-moments ratios to find the most suitabledistribution and the precision of original growth curves Thecorrelation between the study region sites varies from minus005to 086 with an average of 040 Therefore we use algorithmfrom Table 61 of Hosking and Wallis [7]
We held out the analysis for recurrence of different yearsWe run 10000 simulations with sample size of 30 60 and 90in each case The whole process is repeated for GEV GNOand GPA distributions From these repetitions we computedseveral performance measures such as the regional averagerelative bias regional average RMSE regional average relativeRMSE and the error bounds for the estimated regionalgrowth curves for the selected nonexceedance probability 119865Overall results for the suitable probability distribution forboth methods are presented in Tables 8 9 and 10 for samplesize of 30 60 and 90 respectively
Figures 4 5 and 6 show estimated regional growth curvesfor sample sizes 30 60 and 90 respectively and also GEVGNO and GPA distributions with the 90 error bounds
Tables 8 9 and 10 show that increase in the samplesize such as 30 to 90 improved the performance particularlyin the prediction of the large nonexceedance probability 119865L-moments method provides similar performance for theGNO and GPA distributions in terms of relative bias thatis presented in Tables 8 9 and 10 We found that the GPAdistribution produced the lowest relative bias compared toGEV and GNO distribution for the PL-moment for thevarious values of the nonexceedance probability 119865 Howeverthe GPA distribution performs better in terms of RMSEthan the GEV and GNO distribution for both methods (L-moments and PL-moments) Furthermore RMSE is lowestfor PL-moments compared to L-moments In addition theerror bounds for the GPA distribution of regional quantilesare narrow compared to GEV and GNO distributions Itshows that the estimation of censored sample improvesthe prediction of extreme precipitation explicitly at largenonexceedance probability 1198655 Discussion and Conclusion
This study provides a comprehensive evaluation of theL-moments and the PL-moments First revisiting RFAon L-moments by Hosking and Wallis [7] we aimed todevelop similar connections of regional homogeneity for PL-moments The L-moments and the PL-moments for candi-date probability distributions (GLO GEV GNO and GPA)are also developed for presenting the corresponding L-ratioand PL-ratio diagrams with the goodness of fit test resultsThe regional growth curves for the selected distributionhave been shown in Figures 4 5 and 6 At the lower tail
GEV GPA and GNO distributions are approximately thesame but at the upper tail there is variation between theregional quantiles The regional homogeneity analysis startsby assuming 17 locations of Northern areas and KhyberPakhtunkhwa Pakistan as one homogeneous region basedon L-moments and PL-moments at censoring level rangingfrom 10 to 23This assumption is statistically accepted afterapplying the heterogeneity and discordancy tests The 119885-statistic provides appropriate distribution for modeling themonthly extreme precipitation in Northern areas and KhyberPakhtunkhwa Pakistan We found that GPA distribution issuitable for the L-moments and GNOdistribution for the PL-moments
Finally Monte Carlo simulation used for performanceevaluation by commonly used error functions Several accu-racy measures such as relative bias RMSE relative RMSEand error function bounds for the regional quantiles arecomputed with 10000 runs of Monte Carlo simulations Wefound that GPA distribution produced robust quantile esti-mates for both return periods and methods (L-moments andPL-moments) Our results support the finding of previousstudy (eg Cunnane [9] Bhattarai [12]) for censored sampleanalysis where PL-moments method outperformed the L-moments method for the estimation of large return periodsevents
Appendix
The partial L-moments (PL-moments) for generalized logis-tic (GLO) generalized Pareto (GPA) generalized normal(GNO) and generalized extreme value (GEV) distributionswere derived based on the formula defined by Wang (WaterResour Res 321767O 1771 1996 (In references lines from442 to 444 mentioned that in study)) The summary of thederived distributions and parameters estimation for thesedistributions is as follows
PL-moments for the GEVDistribution [10] are as followsThe CDF and quantile function of the GEV are given by
119865 (119909) = exp[minus1 minus 119896120572 (120594 minus 120577)1119896] 119896 = 0 (A1)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (minus ln119865)119896 119896 = 0 (A2)
Wang [10] developed the partial probability-weightedmoments (PWMs) of the GEV as
(119903 + 1) 1205731015840119903 = 120577 + 120572119867 (120574 1198650 119896) (A3)
where
119867(120574 1198650 119896) = 1119896 [1 minus119875 1 + 119896 minus (119903 + 1) ln1198650(1 minus 119865119903+10 ) (119903 + 1)119896 ] (A4)
12 Advances in Meteorology
Table8Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f30
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus0540
0minus19
7674
minus195328
minus193602
minus191579
minus188659
minus186376
minus1840
97minus18
1165
minus179056
119877(119865)
00900
00695
00963
01952
03334
06038
09057
13261
21337
300
48119877119877 (119865)
19873
01019
00635
00876
01090
01369
01595
01839
02191
02478
119871119861119864lowast
minus20937
08829
09296
08842
08411
07862
07460
07065
06557
06177
119880119861119864lowast
18714
11949
11208
11580
11884
12244
12516
12818
13285
13683
L-mom
ents
GNO
119861119877 (119865)
08124
minus196279
minus193846
minus192837
minus191816
minus190506
minus189573
minus188699
minus187636
minus186899
119877(119865)
00986
00696
00917
02051
03520
06027
08421
11312
16008
20312
119877119877 (119865)
20766
01045
00587
00889
01120
01368
01533
01686
01878
02016
119871119861119864lowast
minus21331
08683
09271
08780
08379
07958
07691
07453
07172
06978
119880119861119864lowast
16569
11974
11077
11569
11954
12355
12615
12851
13149
13369
L-mom
ents
GPA
119861119877 (119865)
38654
minus194351
minus193564
minus193226
minus192616
minus191706
minus191060
minus190506
minus189966
minus189727
119877(119865)
01242
006
6900998
02136
03552
05968
08313
11188
15941
20368
119877119877 (119865)
24785
01033
00614
00892
01111
01386
01602
01833
02162
02431
119871119861119864lowast
minus25192
08591
09215
08817
08444
07979
07646
07327
06919
06623
119880119861119864lowast
14272
11911
11115
11588
11976
12459
12841
13257
13859
14367
PL-m
oments
GEV
119861119877 (119865)
minus02662
minus197059
minus194770
minus193340
minus191600
minus188996
minus186909
minus184797
minus182052
minus1800
66119877(119865)
05379
006
8700947
02023
03467
06171
09087
13058
20551
2854
119877119877 (119865)
38117
00948
006
4900960
01210
01508
01736
01977
02324
02609
119871119861119864lowast
minus77814
08892
09252
08725
08259
07697
07296
06911
064
1706050
119880119861119864lowast
45516
11827
11207
11716
12117
12554
12881
13211
13693
14104
PL-m
oments
GNO
119861119877 (119865)
35142
minus195922
minus193485
minus192704
minus191894
minus190806
minus1900
03minus18
9235
minus188284
minus187617
119877(119865)
06203
006
8900939
02141
03644
06143
08483
11272
15747
19809
119877119877 (119865)
43522
00971
00626
00980
01238
01504
01675
01831
02024
02162
119871119861119864lowast
minus10618
08772
09204
08655
08233
07800
07530
07290
07012
06822
119880119861119864lowast
46353
11849
11122
11734
12189
12653
12949
13218
13546
13780
PL-m
oments
GPA
119861119877 (119865)
1212
3minus19
4300
minus193238
minus193113
minus192716
minus192046
minus191555
minus19114
1minus19
0768
minus1906
49119877(119865)
08513
006
6601041
02250
03684
06021
08215
10853
15144
19094
119877119877 (119865)
58811
00962
006
6900996
01233
01509
01720
01941
02258
02518
119871119861119864lowast
minus17507
08698
09130
08680
08299
07846
07524
07220
06831
06552
119880119861119864lowast
47307
11791
11189
11774
12215
12739
1314
13567
14195
14722
Advances in Meteorology 13
Table9Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f60
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus13271
minus194230
minus193551
minus193241
minus192649
minus191612
minus190736
minus189839
minus188690
minus187888
119877(119865)
00846
00624
00851
01873
03250
05841
08652
12505
19839
27721
119877119877 (119865)
19155
00833
00612
00855
01024
01213
01360
01523
01770
01983
119871119861119864lowast
minus19593
08756
09254
08851
08491
08074
07796
07501
07118
06834
119880119861119864lowast
16242
11675
11015
11472
11879
12309
12678
13038
13659
14141
L-mom
ents
GNO
119861119877 (119865)
08417
minus193335
minus192795
minus192984
minus192999
minus192879
minus192742
minus192593
minus192399
minus192264
119877(119865)
00968
00634
00858
02000
03439
05850
08123
10845
15235
19239
119877119877 (119865)
21055
00881
00582
00861
01050
01235
01354
01463
01599
01699
119871119861119864lowast
minus22887
08647
09244
08794
08466
08141
07957
07798
07597
07443
119880119861119864lowast
15863
11714
1096
11511
11934
1240
512
666
12939
13253
13505
L-mom
ents
GPA
119861119877 (119865)
46371
minus192512
minus192631
minus193160
minus193417
minus193606
minus193745
minus193933
minus194306
minus194708
119877(119865)
01280
00624
00950
02094
03468
05740
07903
10536
14878
18928
119877119877 (119865)
264
4200915
00614
00882
01064
01270
01429
01599
01849
02059
119871119861119864lowast
minus38088
08628
09186
08815
08510
08167
07924
07703
07422
07178
119880119861119864lowast
14699
11721
11020
11534
11925
12454
12791
13236
13869
1440
5
PL-m
oments
GEV
119861119877 (119865)
minus16968
minus193984
minus193601
minus193623
minus193319
minus192576
minus191861
minus191081
minus190026
minus189255
119877(119865)
05431
00596
00888
01978
03366
05844
08417
11838
18179
24868
119877119877 (119865)
38656
00783
00634
0095
01166
01387
01541
01700
01928
02120
119871119861119864lowast
minus92921
08907
09201
08750
08403
07984
07714
07434
07043
06783
119880119861119864lowast
53553
11521
11099
1166
412
098
12572
12869
13216
13697
14105
PL-m
oments
GNO
119861119877 (119865)
23785
minus193306
minus192979
minus193417
minus193628
minus193678
minus193617
minus193509
minus193331
minus193185
119877(119865)
06278
006
0100911
02100
03523
05804
0788
10303
14113
17516
119877119877 (119865)
43743
00812
00623
00965
01186
01391
01513
01618
01742
01830
119871119861119864lowast
minus12083
08856
09188
08709
08389
08061
07870
07706
07517
07380
119880119861119864lowast
52707
11561
11059
11704
12152
12613
12898
13138
13361
1356
PL-m
oments
GPA
119861119877 (119865)
1118
7minus19
2590
minus192910
minus193681
minus1940
84minus19
4322
minus1944
04minus19
4463
minus194560
minus1946
73119877(119865)
08596
00597
01009
02229
03581
05615
07391
09418
12558
15341
119877119877 (119865)
58310
00840
006
6000991
0119
501391
01522
01650
01828
01971
119871119861119864lowast
minus18977
08800
09127
08699
08413
08098
07912
07682
07447
07255
119880119861119864lowast
51281
11551
11117
11769
12173
12615
12887
13227
1360
913
899
14 Advances in Meteorology
Table10A
ccuracymeasure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f90
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus00117
minus194226
minus193109
minus192926
minus192542
minus191809
minus191151
minus1904
48minus18
9497
minus188790
119877(119865)
00787
00556
00823
01850
03151
05459
07840
10989
16793
22883
119877119877 (119865)
17329
00814
00544
00832
01031
01238
01381
01525
01727
01892
119871119861119864lowast
minus19303
08935
09287
08885
08587
08246
08006
07772
07463
07229
119880119861119864lowast
15717
11499
10973
11461
11804
12156
12393
12635
12966
13242
L-mom
ents
GNO
119861119877 (119865)
15851
minus193606
minus192517
minus192695
minus192747
minus192678
minus192569
minus192437
minus192245
minus192098
119877(119865)
00917
00567
00846
01967
03305
05446
07392
09660
13223
1640
0119877119877 (119865)
19328
00850
00543
00853
01052
01237
01347
01442
01554
01632
119871119861119864lowast
minus23268
08848
09261
08844
08568
08301
08139
08000
07833
07722
119880119861119864lowast
15168
11542
10942
11489
11851
12197
1240
112
575
12779
12925
L-mom
ents
GPA
119861119877 (119865)
49081
minus192995
minus192417
minus192865
minus193078
minus193175
minus193195
minus193209
minus193256
minus193330
119877(119865)
01234
00571
00935
02081
03365
05322
07054
09053
12180
14975
119877119877 (119865)
246
8800882
00576
00870
01054
01239
01364
01489
01660
01798
119871119861119864lowast
minus45353
08783
09210
08838
08594
08326
08141
07961
07731
07557
119880119861119864lowast
14833
11574
10996
11532
11870
12223
1246
712
711
1304
413
312
PL-m
oments
GEV
119861119877 (119865)
minus00877
minus194145
minus193060
minus193062
minus192847
minus192278
minus191698
minus191038
minus190100
minus189374
119877(119865)
05458
00565
00855
01963
03338
05712
08090
11156
16650
22276
119877119877 (119865)
39079
00784
00585
00934
0117
201406
01558
01703
01897
02050
119871119861119864lowast
minus10190
08983
09236
08765
08431
08067
07818
07579
07271
07041
119880119861119864lowast
58730
1144
811036
11652
12081
12504
12771
13026
13364
13623
PL-m
oments
GNO
119861119877 (119865)
38414
minus193665
minus192536
minus192859
minus193044
minus193102
minus193057
minus192966
minus192804
minus192665
119877(119865)
06308
00578
00896
02088
03488
05690
07659
09926
13434
16524
119877119877 (119865)
44651
00819
00596
00961
0119
301405
01526
01628
01744
01822
119871119861119864lowast
minus13011
08910
09192
08716
08411
08122
07949
07801
07634
07519
119880119861119864lowast
56756
11491
11030
11693
12128
12539
12775
12974
13197
13347
PL-m
oments
GPA
119861119877 (119865)
126398
minus193142
minus192484
minus193077
minus193404
minus193589
minus193637
minus193655
minus193680
minus193720
119877(119865)
08641
00588
00999
02220
03558
05535
07227
09125
1200
914
521
119877119877 (119865)
60116
00855
006
4000985
0119
901400
01529
01651
01812
01937
119871119861119864lowast
minus19889
08842
09130
08700
08435
08156
07970
07789
07566
07399
119880119861119864lowast
53743
11528
11103
11755
12158
12552
12807
13055
13385
13636
Advances in Meteorology 15
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
0 2 4 6
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
Regional growth curve for GPA
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F))
0
5
10
15
20
Qua
ntile
2 4 60Reduced variate minuslog (minus log (F))
0
5
10
15
Qua
ntile
(based on L-moments) Regional growth curve for GPA
(based on PL-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30
16 Advances in Meteorology
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60
Advances in Meteorology 17
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90
18 Advances in Meteorology
The first four PL-moments of the GEV are defined as
12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)
(A5)
In previous equation 119875(sdot sdot) is an Incomplete Gammafunction
119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500
120579119896expminus120579119889Θ (A6)
Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution
The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by
119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)
The partial PWMs of the GLO are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)
(A9)
where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function
1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EarthquakesJournal of
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Applied ampEnvironmentalSoil Science
Volume 2014
Mining
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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International Journal of
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OceanographyInternational Journal of
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Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Atmospheric SciencesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
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MineralogyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MeteorologyAdvances in
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Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geological ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geology Advances in
12 Advances in Meteorology
Table8Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f30
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus0540
0minus19
7674
minus195328
minus193602
minus191579
minus188659
minus186376
minus1840
97minus18
1165
minus179056
119877(119865)
00900
00695
00963
01952
03334
06038
09057
13261
21337
300
48119877119877 (119865)
19873
01019
00635
00876
01090
01369
01595
01839
02191
02478
119871119861119864lowast
minus20937
08829
09296
08842
08411
07862
07460
07065
06557
06177
119880119861119864lowast
18714
11949
11208
11580
11884
12244
12516
12818
13285
13683
L-mom
ents
GNO
119861119877 (119865)
08124
minus196279
minus193846
minus192837
minus191816
minus190506
minus189573
minus188699
minus187636
minus186899
119877(119865)
00986
00696
00917
02051
03520
06027
08421
11312
16008
20312
119877119877 (119865)
20766
01045
00587
00889
01120
01368
01533
01686
01878
02016
119871119861119864lowast
minus21331
08683
09271
08780
08379
07958
07691
07453
07172
06978
119880119861119864lowast
16569
11974
11077
11569
11954
12355
12615
12851
13149
13369
L-mom
ents
GPA
119861119877 (119865)
38654
minus194351
minus193564
minus193226
minus192616
minus191706
minus191060
minus190506
minus189966
minus189727
119877(119865)
01242
006
6900998
02136
03552
05968
08313
11188
15941
20368
119877119877 (119865)
24785
01033
00614
00892
01111
01386
01602
01833
02162
02431
119871119861119864lowast
minus25192
08591
09215
08817
08444
07979
07646
07327
06919
06623
119880119861119864lowast
14272
11911
11115
11588
11976
12459
12841
13257
13859
14367
PL-m
oments
GEV
119861119877 (119865)
minus02662
minus197059
minus194770
minus193340
minus191600
minus188996
minus186909
minus184797
minus182052
minus1800
66119877(119865)
05379
006
8700947
02023
03467
06171
09087
13058
20551
2854
119877119877 (119865)
38117
00948
006
4900960
01210
01508
01736
01977
02324
02609
119871119861119864lowast
minus77814
08892
09252
08725
08259
07697
07296
06911
064
1706050
119880119861119864lowast
45516
11827
11207
11716
12117
12554
12881
13211
13693
14104
PL-m
oments
GNO
119861119877 (119865)
35142
minus195922
minus193485
minus192704
minus191894
minus190806
minus1900
03minus18
9235
minus188284
minus187617
119877(119865)
06203
006
8900939
02141
03644
06143
08483
11272
15747
19809
119877119877 (119865)
43522
00971
00626
00980
01238
01504
01675
01831
02024
02162
119871119861119864lowast
minus10618
08772
09204
08655
08233
07800
07530
07290
07012
06822
119880119861119864lowast
46353
11849
11122
11734
12189
12653
12949
13218
13546
13780
PL-m
oments
GPA
119861119877 (119865)
1212
3minus19
4300
minus193238
minus193113
minus192716
minus192046
minus191555
minus19114
1minus19
0768
minus1906
49119877(119865)
08513
006
6601041
02250
03684
06021
08215
10853
15144
19094
119877119877 (119865)
58811
00962
006
6900996
01233
01509
01720
01941
02258
02518
119871119861119864lowast
minus17507
08698
09130
08680
08299
07846
07524
07220
06831
06552
119880119861119864lowast
47307
11791
11189
11774
12215
12739
1314
13567
14195
14722
Advances in Meteorology 13
Table9Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f60
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus13271
minus194230
minus193551
minus193241
minus192649
minus191612
minus190736
minus189839
minus188690
minus187888
119877(119865)
00846
00624
00851
01873
03250
05841
08652
12505
19839
27721
119877119877 (119865)
19155
00833
00612
00855
01024
01213
01360
01523
01770
01983
119871119861119864lowast
minus19593
08756
09254
08851
08491
08074
07796
07501
07118
06834
119880119861119864lowast
16242
11675
11015
11472
11879
12309
12678
13038
13659
14141
L-mom
ents
GNO
119861119877 (119865)
08417
minus193335
minus192795
minus192984
minus192999
minus192879
minus192742
minus192593
minus192399
minus192264
119877(119865)
00968
00634
00858
02000
03439
05850
08123
10845
15235
19239
119877119877 (119865)
21055
00881
00582
00861
01050
01235
01354
01463
01599
01699
119871119861119864lowast
minus22887
08647
09244
08794
08466
08141
07957
07798
07597
07443
119880119861119864lowast
15863
11714
1096
11511
11934
1240
512
666
12939
13253
13505
L-mom
ents
GPA
119861119877 (119865)
46371
minus192512
minus192631
minus193160
minus193417
minus193606
minus193745
minus193933
minus194306
minus194708
119877(119865)
01280
00624
00950
02094
03468
05740
07903
10536
14878
18928
119877119877 (119865)
264
4200915
00614
00882
01064
01270
01429
01599
01849
02059
119871119861119864lowast
minus38088
08628
09186
08815
08510
08167
07924
07703
07422
07178
119880119861119864lowast
14699
11721
11020
11534
11925
12454
12791
13236
13869
1440
5
PL-m
oments
GEV
119861119877 (119865)
minus16968
minus193984
minus193601
minus193623
minus193319
minus192576
minus191861
minus191081
minus190026
minus189255
119877(119865)
05431
00596
00888
01978
03366
05844
08417
11838
18179
24868
119877119877 (119865)
38656
00783
00634
0095
01166
01387
01541
01700
01928
02120
119871119861119864lowast
minus92921
08907
09201
08750
08403
07984
07714
07434
07043
06783
119880119861119864lowast
53553
11521
11099
1166
412
098
12572
12869
13216
13697
14105
PL-m
oments
GNO
119861119877 (119865)
23785
minus193306
minus192979
minus193417
minus193628
minus193678
minus193617
minus193509
minus193331
minus193185
119877(119865)
06278
006
0100911
02100
03523
05804
0788
10303
14113
17516
119877119877 (119865)
43743
00812
00623
00965
01186
01391
01513
01618
01742
01830
119871119861119864lowast
minus12083
08856
09188
08709
08389
08061
07870
07706
07517
07380
119880119861119864lowast
52707
11561
11059
11704
12152
12613
12898
13138
13361
1356
PL-m
oments
GPA
119861119877 (119865)
1118
7minus19
2590
minus192910
minus193681
minus1940
84minus19
4322
minus1944
04minus19
4463
minus194560
minus1946
73119877(119865)
08596
00597
01009
02229
03581
05615
07391
09418
12558
15341
119877119877 (119865)
58310
00840
006
6000991
0119
501391
01522
01650
01828
01971
119871119861119864lowast
minus18977
08800
09127
08699
08413
08098
07912
07682
07447
07255
119880119861119864lowast
51281
11551
11117
11769
12173
12615
12887
13227
1360
913
899
14 Advances in Meteorology
Table10A
ccuracymeasure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f90
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus00117
minus194226
minus193109
minus192926
minus192542
minus191809
minus191151
minus1904
48minus18
9497
minus188790
119877(119865)
00787
00556
00823
01850
03151
05459
07840
10989
16793
22883
119877119877 (119865)
17329
00814
00544
00832
01031
01238
01381
01525
01727
01892
119871119861119864lowast
minus19303
08935
09287
08885
08587
08246
08006
07772
07463
07229
119880119861119864lowast
15717
11499
10973
11461
11804
12156
12393
12635
12966
13242
L-mom
ents
GNO
119861119877 (119865)
15851
minus193606
minus192517
minus192695
minus192747
minus192678
minus192569
minus192437
minus192245
minus192098
119877(119865)
00917
00567
00846
01967
03305
05446
07392
09660
13223
1640
0119877119877 (119865)
19328
00850
00543
00853
01052
01237
01347
01442
01554
01632
119871119861119864lowast
minus23268
08848
09261
08844
08568
08301
08139
08000
07833
07722
119880119861119864lowast
15168
11542
10942
11489
11851
12197
1240
112
575
12779
12925
L-mom
ents
GPA
119861119877 (119865)
49081
minus192995
minus192417
minus192865
minus193078
minus193175
minus193195
minus193209
minus193256
minus193330
119877(119865)
01234
00571
00935
02081
03365
05322
07054
09053
12180
14975
119877119877 (119865)
246
8800882
00576
00870
01054
01239
01364
01489
01660
01798
119871119861119864lowast
minus45353
08783
09210
08838
08594
08326
08141
07961
07731
07557
119880119861119864lowast
14833
11574
10996
11532
11870
12223
1246
712
711
1304
413
312
PL-m
oments
GEV
119861119877 (119865)
minus00877
minus194145
minus193060
minus193062
minus192847
minus192278
minus191698
minus191038
minus190100
minus189374
119877(119865)
05458
00565
00855
01963
03338
05712
08090
11156
16650
22276
119877119877 (119865)
39079
00784
00585
00934
0117
201406
01558
01703
01897
02050
119871119861119864lowast
minus10190
08983
09236
08765
08431
08067
07818
07579
07271
07041
119880119861119864lowast
58730
1144
811036
11652
12081
12504
12771
13026
13364
13623
PL-m
oments
GNO
119861119877 (119865)
38414
minus193665
minus192536
minus192859
minus193044
minus193102
minus193057
minus192966
minus192804
minus192665
119877(119865)
06308
00578
00896
02088
03488
05690
07659
09926
13434
16524
119877119877 (119865)
44651
00819
00596
00961
0119
301405
01526
01628
01744
01822
119871119861119864lowast
minus13011
08910
09192
08716
08411
08122
07949
07801
07634
07519
119880119861119864lowast
56756
11491
11030
11693
12128
12539
12775
12974
13197
13347
PL-m
oments
GPA
119861119877 (119865)
126398
minus193142
minus192484
minus193077
minus193404
minus193589
minus193637
minus193655
minus193680
minus193720
119877(119865)
08641
00588
00999
02220
03558
05535
07227
09125
1200
914
521
119877119877 (119865)
60116
00855
006
4000985
0119
901400
01529
01651
01812
01937
119871119861119864lowast
minus19889
08842
09130
08700
08435
08156
07970
07789
07566
07399
119880119861119864lowast
53743
11528
11103
11755
12158
12552
12807
13055
13385
13636
Advances in Meteorology 15
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
0 2 4 6
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
Regional growth curve for GPA
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F))
0
5
10
15
20
Qua
ntile
2 4 60Reduced variate minuslog (minus log (F))
0
5
10
15
Qua
ntile
(based on L-moments) Regional growth curve for GPA
(based on PL-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30
16 Advances in Meteorology
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60
Advances in Meteorology 17
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90
18 Advances in Meteorology
The first four PL-moments of the GEV are defined as
12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)
(A5)
In previous equation 119875(sdot sdot) is an Incomplete Gammafunction
119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500
120579119896expminus120579119889Θ (A6)
Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution
The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by
119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)
The partial PWMs of the GLO are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)
(A9)
where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function
1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mining
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GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
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MineralogyInternational Journal of
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Geological ResearchJournal of
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Geology Advances in
Advances in Meteorology 13
Table9Ac
curacy
measure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f60
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus13271
minus194230
minus193551
minus193241
minus192649
minus191612
minus190736
minus189839
minus188690
minus187888
119877(119865)
00846
00624
00851
01873
03250
05841
08652
12505
19839
27721
119877119877 (119865)
19155
00833
00612
00855
01024
01213
01360
01523
01770
01983
119871119861119864lowast
minus19593
08756
09254
08851
08491
08074
07796
07501
07118
06834
119880119861119864lowast
16242
11675
11015
11472
11879
12309
12678
13038
13659
14141
L-mom
ents
GNO
119861119877 (119865)
08417
minus193335
minus192795
minus192984
minus192999
minus192879
minus192742
minus192593
minus192399
minus192264
119877(119865)
00968
00634
00858
02000
03439
05850
08123
10845
15235
19239
119877119877 (119865)
21055
00881
00582
00861
01050
01235
01354
01463
01599
01699
119871119861119864lowast
minus22887
08647
09244
08794
08466
08141
07957
07798
07597
07443
119880119861119864lowast
15863
11714
1096
11511
11934
1240
512
666
12939
13253
13505
L-mom
ents
GPA
119861119877 (119865)
46371
minus192512
minus192631
minus193160
minus193417
minus193606
minus193745
minus193933
minus194306
minus194708
119877(119865)
01280
00624
00950
02094
03468
05740
07903
10536
14878
18928
119877119877 (119865)
264
4200915
00614
00882
01064
01270
01429
01599
01849
02059
119871119861119864lowast
minus38088
08628
09186
08815
08510
08167
07924
07703
07422
07178
119880119861119864lowast
14699
11721
11020
11534
11925
12454
12791
13236
13869
1440
5
PL-m
oments
GEV
119861119877 (119865)
minus16968
minus193984
minus193601
minus193623
minus193319
minus192576
minus191861
minus191081
minus190026
minus189255
119877(119865)
05431
00596
00888
01978
03366
05844
08417
11838
18179
24868
119877119877 (119865)
38656
00783
00634
0095
01166
01387
01541
01700
01928
02120
119871119861119864lowast
minus92921
08907
09201
08750
08403
07984
07714
07434
07043
06783
119880119861119864lowast
53553
11521
11099
1166
412
098
12572
12869
13216
13697
14105
PL-m
oments
GNO
119861119877 (119865)
23785
minus193306
minus192979
minus193417
minus193628
minus193678
minus193617
minus193509
minus193331
minus193185
119877(119865)
06278
006
0100911
02100
03523
05804
0788
10303
14113
17516
119877119877 (119865)
43743
00812
00623
00965
01186
01391
01513
01618
01742
01830
119871119861119864lowast
minus12083
08856
09188
08709
08389
08061
07870
07706
07517
07380
119880119861119864lowast
52707
11561
11059
11704
12152
12613
12898
13138
13361
1356
PL-m
oments
GPA
119861119877 (119865)
1118
7minus19
2590
minus192910
minus193681
minus1940
84minus19
4322
minus1944
04minus19
4463
minus194560
minus1946
73119877(119865)
08596
00597
01009
02229
03581
05615
07391
09418
12558
15341
119877119877 (119865)
58310
00840
006
6000991
0119
501391
01522
01650
01828
01971
119871119861119864lowast
minus18977
08800
09127
08699
08413
08098
07912
07682
07447
07255
119880119861119864lowast
51281
11551
11117
11769
12173
12615
12887
13227
1360
913
899
14 Advances in Meteorology
Table10A
ccuracymeasure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f90
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus00117
minus194226
minus193109
minus192926
minus192542
minus191809
minus191151
minus1904
48minus18
9497
minus188790
119877(119865)
00787
00556
00823
01850
03151
05459
07840
10989
16793
22883
119877119877 (119865)
17329
00814
00544
00832
01031
01238
01381
01525
01727
01892
119871119861119864lowast
minus19303
08935
09287
08885
08587
08246
08006
07772
07463
07229
119880119861119864lowast
15717
11499
10973
11461
11804
12156
12393
12635
12966
13242
L-mom
ents
GNO
119861119877 (119865)
15851
minus193606
minus192517
minus192695
minus192747
minus192678
minus192569
minus192437
minus192245
minus192098
119877(119865)
00917
00567
00846
01967
03305
05446
07392
09660
13223
1640
0119877119877 (119865)
19328
00850
00543
00853
01052
01237
01347
01442
01554
01632
119871119861119864lowast
minus23268
08848
09261
08844
08568
08301
08139
08000
07833
07722
119880119861119864lowast
15168
11542
10942
11489
11851
12197
1240
112
575
12779
12925
L-mom
ents
GPA
119861119877 (119865)
49081
minus192995
minus192417
minus192865
minus193078
minus193175
minus193195
minus193209
minus193256
minus193330
119877(119865)
01234
00571
00935
02081
03365
05322
07054
09053
12180
14975
119877119877 (119865)
246
8800882
00576
00870
01054
01239
01364
01489
01660
01798
119871119861119864lowast
minus45353
08783
09210
08838
08594
08326
08141
07961
07731
07557
119880119861119864lowast
14833
11574
10996
11532
11870
12223
1246
712
711
1304
413
312
PL-m
oments
GEV
119861119877 (119865)
minus00877
minus194145
minus193060
minus193062
minus192847
minus192278
minus191698
minus191038
minus190100
minus189374
119877(119865)
05458
00565
00855
01963
03338
05712
08090
11156
16650
22276
119877119877 (119865)
39079
00784
00585
00934
0117
201406
01558
01703
01897
02050
119871119861119864lowast
minus10190
08983
09236
08765
08431
08067
07818
07579
07271
07041
119880119861119864lowast
58730
1144
811036
11652
12081
12504
12771
13026
13364
13623
PL-m
oments
GNO
119861119877 (119865)
38414
minus193665
minus192536
minus192859
minus193044
minus193102
minus193057
minus192966
minus192804
minus192665
119877(119865)
06308
00578
00896
02088
03488
05690
07659
09926
13434
16524
119877119877 (119865)
44651
00819
00596
00961
0119
301405
01526
01628
01744
01822
119871119861119864lowast
minus13011
08910
09192
08716
08411
08122
07949
07801
07634
07519
119880119861119864lowast
56756
11491
11030
11693
12128
12539
12775
12974
13197
13347
PL-m
oments
GPA
119861119877 (119865)
126398
minus193142
minus192484
minus193077
minus193404
minus193589
minus193637
minus193655
minus193680
minus193720
119877(119865)
08641
00588
00999
02220
03558
05535
07227
09125
1200
914
521
119877119877 (119865)
60116
00855
006
4000985
0119
901400
01529
01651
01812
01937
119871119861119864lowast
minus19889
08842
09130
08700
08435
08156
07970
07789
07566
07399
119880119861119864lowast
53743
11528
11103
11755
12158
12552
12807
13055
13385
13636
Advances in Meteorology 15
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
0 2 4 6
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
Regional growth curve for GPA
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F))
0
5
10
15
20
Qua
ntile
2 4 60Reduced variate minuslog (minus log (F))
0
5
10
15
Qua
ntile
(based on L-moments) Regional growth curve for GPA
(based on PL-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30
16 Advances in Meteorology
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60
Advances in Meteorology 17
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90
18 Advances in Meteorology
The first four PL-moments of the GEV are defined as
12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)
(A5)
In previous equation 119875(sdot sdot) is an Incomplete Gammafunction
119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500
120579119896expminus120579119889Θ (A6)
Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution
The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by
119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)
The partial PWMs of the GLO are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)
(A9)
where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function
1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Applied ampEnvironmentalSoil Science
Volume 2014
Mining
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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International Journal of
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OceanographyInternational Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Atmospheric SciencesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
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MineralogyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geological ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geology Advances in
14 Advances in Meteorology
Table10A
ccuracymeasure
ofthee
stim
ated
grow
thcurves
ofther
egionforsam
ples
izeo
f90
Metho
dDistrib
ution
11986501
05
08
09
095
098
099
0995
0998
0999
L-mom
ents
GEV
119861119877 (119865)
minus00117
minus194226
minus193109
minus192926
minus192542
minus191809
minus191151
minus1904
48minus18
9497
minus188790
119877(119865)
00787
00556
00823
01850
03151
05459
07840
10989
16793
22883
119877119877 (119865)
17329
00814
00544
00832
01031
01238
01381
01525
01727
01892
119871119861119864lowast
minus19303
08935
09287
08885
08587
08246
08006
07772
07463
07229
119880119861119864lowast
15717
11499
10973
11461
11804
12156
12393
12635
12966
13242
L-mom
ents
GNO
119861119877 (119865)
15851
minus193606
minus192517
minus192695
minus192747
minus192678
minus192569
minus192437
minus192245
minus192098
119877(119865)
00917
00567
00846
01967
03305
05446
07392
09660
13223
1640
0119877119877 (119865)
19328
00850
00543
00853
01052
01237
01347
01442
01554
01632
119871119861119864lowast
minus23268
08848
09261
08844
08568
08301
08139
08000
07833
07722
119880119861119864lowast
15168
11542
10942
11489
11851
12197
1240
112
575
12779
12925
L-mom
ents
GPA
119861119877 (119865)
49081
minus192995
minus192417
minus192865
minus193078
minus193175
minus193195
minus193209
minus193256
minus193330
119877(119865)
01234
00571
00935
02081
03365
05322
07054
09053
12180
14975
119877119877 (119865)
246
8800882
00576
00870
01054
01239
01364
01489
01660
01798
119871119861119864lowast
minus45353
08783
09210
08838
08594
08326
08141
07961
07731
07557
119880119861119864lowast
14833
11574
10996
11532
11870
12223
1246
712
711
1304
413
312
PL-m
oments
GEV
119861119877 (119865)
minus00877
minus194145
minus193060
minus193062
minus192847
minus192278
minus191698
minus191038
minus190100
minus189374
119877(119865)
05458
00565
00855
01963
03338
05712
08090
11156
16650
22276
119877119877 (119865)
39079
00784
00585
00934
0117
201406
01558
01703
01897
02050
119871119861119864lowast
minus10190
08983
09236
08765
08431
08067
07818
07579
07271
07041
119880119861119864lowast
58730
1144
811036
11652
12081
12504
12771
13026
13364
13623
PL-m
oments
GNO
119861119877 (119865)
38414
minus193665
minus192536
minus192859
minus193044
minus193102
minus193057
minus192966
minus192804
minus192665
119877(119865)
06308
00578
00896
02088
03488
05690
07659
09926
13434
16524
119877119877 (119865)
44651
00819
00596
00961
0119
301405
01526
01628
01744
01822
119871119861119864lowast
minus13011
08910
09192
08716
08411
08122
07949
07801
07634
07519
119880119861119864lowast
56756
11491
11030
11693
12128
12539
12775
12974
13197
13347
PL-m
oments
GPA
119861119877 (119865)
126398
minus193142
minus192484
minus193077
minus193404
minus193589
minus193637
minus193655
minus193680
minus193720
119877(119865)
08641
00588
00999
02220
03558
05535
07227
09125
1200
914
521
119877119877 (119865)
60116
00855
006
4000985
0119
901400
01529
01651
01812
01937
119871119861119864lowast
minus19889
08842
09130
08700
08435
08156
07970
07789
07566
07399
119880119861119864lowast
53743
11528
11103
11755
12158
12552
12807
13055
13385
13636
Advances in Meteorology 15
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
0 2 4 6
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
Regional growth curve for GPA
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F))
0
5
10
15
20
Qua
ntile
2 4 60Reduced variate minuslog (minus log (F))
0
5
10
15
Qua
ntile
(based on L-moments) Regional growth curve for GPA
(based on PL-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30
16 Advances in Meteorology
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60
Advances in Meteorology 17
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90
18 Advances in Meteorology
The first four PL-moments of the GEV are defined as
12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)
(A5)
In previous equation 119875(sdot sdot) is an Incomplete Gammafunction
119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500
120579119896expminus120579119889Θ (A6)
Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution
The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by
119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)
The partial PWMs of the GLO are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)
(A9)
where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function
1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ClimatologyJournal of
EcologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EarthquakesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom
Applied ampEnvironmentalSoil Science
Volume 2014
Mining
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal of
Geophysics
OceanographyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Atmospheric SciencesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MineralogyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MeteorologyAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geological ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geology Advances in
Advances in Meteorology 15
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
0 2 4 6
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
Regional growth curve for GPA
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F))
0
5
10
15
20
Qua
ntile
2 4 60Reduced variate minuslog (minus log (F))
0
5
10
15
Qua
ntile
(based on L-moments) Regional growth curve for GPA
(based on PL-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 4 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of30
16 Advances in Meteorology
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60
Advances in Meteorology 17
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90
18 Advances in Meteorology
The first four PL-moments of the GEV are defined as
12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)
(A5)
In previous equation 119875(sdot sdot) is an Incomplete Gammafunction
119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500
120579119896expminus120579119889Θ (A6)
Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution
The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by
119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)
The partial PWMs of the GLO are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)
(A9)
where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function
1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ClimatologyJournal of
EcologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EarthquakesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom
Applied ampEnvironmentalSoil Science
Volume 2014
Mining
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal of
Geophysics
OceanographyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Atmospheric SciencesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MineralogyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MeteorologyAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geological ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geology Advances in
16 Advances in Meteorology
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
Figure 5 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of60
Advances in Meteorology 17
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90
18 Advances in Meteorology
The first four PL-moments of the GEV are defined as
12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)
(A5)
In previous equation 119875(sdot sdot) is an Incomplete Gammafunction
119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500
120579119896expminus120579119889Θ (A6)
Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution
The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by
119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)
The partial PWMs of the GLO are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)
(A9)
where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function
1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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EcologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EarthquakesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom
Applied ampEnvironmentalSoil Science
Volume 2014
Mining
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal of
Geophysics
OceanographyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Atmospheric SciencesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MineralogyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MeteorologyAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geological ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geology Advances in
Advances in Meteorology 17
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
5
10
15
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12
Qua
ntile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
12Q
uant
ile
2 5 10 20 50 100 500
Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500Return period
0 2 4 6
0
2
4
6
8
10
Qua
ntile
2 5 10 20 50 100 500
Return period
Regional growth curve for GPA(based on L-moments)
Regional growth curve for GNO(based on L-moments)
Regional growth curve for GEV(based on L-moments)
Regional growth curve for GPA(based on PL-moments)
Regional growth curve for GNO(based on PL-moments)
Regional growth curve for GEV(based on PL-moments)
GEVGEV lower boundGEV upper bound
GEVGEV lower boundGEV upper bound
GNOGNO lower boundGNO upper bound
GNOGNO lower boundGNO upper bound
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
GPAGPA lower boundGPA upper bound
Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Reduced variate minuslog (minus log (F)) Reduced variate minuslog (minus log (F))
Figure 6 Regional growth curves of the three distributions for L-moments and PL-moments with their 90 error bounds for sample size of90
18 Advances in Meteorology
The first four PL-moments of the GEV are defined as
12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)
(A5)
In previous equation 119875(sdot sdot) is an Incomplete Gammafunction
119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500
120579119896expminus120579119889Θ (A6)
Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution
The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by
119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)
The partial PWMs of the GLO are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)
(A9)
where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function
1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ClimatologyJournal of
EcologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EarthquakesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom
Applied ampEnvironmentalSoil Science
Volume 2014
Mining
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal of
Geophysics
OceanographyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Atmospheric SciencesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MineralogyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MeteorologyAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geological ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geology Advances in
18 Advances in Meteorology
The first four PL-moments of the GEV are defined as
12058210158401 = 120577 + 120572119867 (0 1198650 119896)12058210158402 = 120572 119867 (1 1198650 119896) minus 119867 (0 1198650 119896)12058210158403 = 120572 2119867 (2 1198650 119896) minus 3119867 (1 1198650 119896) + 119867 (0 1198650 119896)12058210158404 = 120572 5119867 (3 1198650 119896) minus 10119867 (2 1198650 119896)+ 6119867 (1 1198650 119896) minus 119867 (0 1198650 119896)
(A5)
In previous equation 119875(sdot sdot) is an Incomplete Gammafunction
119875 (1 + 119896 minus (119903 + 1) ln1198650) = intminus(119903+1) ln11986500
120579119896expminus120579119889Θ (A6)
Then the first four PL-moments are computed to develop thePL-moment ratios (PL-Cv PL-Cs and PL-Ck) for the GEVdistribution
The PL-moments for the GLODistribution are as followsThe CDF and quantile function of the GLO are given by
119865 (119909) = [1 + 1 minus 119896120572 (120594 minus 120577)1119896]minus1 119896 = 0 (A7)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865119865 )119896 119896 = 0 (A8)
The partial PWMs of the GLO are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 )1198611minus1198650 (1 + 119896 119903 minus 119896 + 1)
(A9)
where 1198611minus1198650 (sdot sdot) is an Incomplete Beta function
1198611minus1198650 (1 + 119896 119903 minus 119896 + 1) = int1minus1198650
0Θ119896 (1 minus Θ)119903minus119896 119889Θ (A10)
The first four PL-moments of the GLO are defined as
12058210158401 = 120577 + 120572119896 1 minus1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158402 = minus120572119896 21198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520
minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
12058210158403 = minus120572119896 61198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530
minus 61198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520 minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650 12058210158404 = minus120572119896
201198611minus1198650 (1 + 119896 4 minus 119870)1 minus 11986540minus 301198611minus1198650 (1 + 119896 3 minus 119870)1 minus 11986530+ 121198611minus1198650 (1 + 119896 2 minus 119870)1 minus 11986520minus 1198611minus1198650 (1 + 119896 1 minus 119870)1 minus 1198650
(A11)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGLO distribution
The PL-moments for the GPADistribution are as followsThe CDF and quantile function of the GPA are given by
119865 (119909) = 1 minus 1 minus 119896120572 (120594 minus 120577)1119896 119896 = 0 (A12)
and quantile function
119909 (119865) = 120577 + 120572119896 1 minus (1 minus 119865)119896 119896 = 0 (A13)
The partial PWMs of the GPA are developed as follows
(119903 + 1) 1205731015840119903 = 120577 + 120572119896minus 120572 (119903 + 1)119896 (1 minus 119865119903+10 ) int
1
1198650
(1 minus 119865)119896 119865119903119889119865 (A14)
The first four PL-moments of the GPA are defined as
12058210158401 = 120577 + 120572119896 (1 minus 11989211)12058210158402 = minus120572119896 (211989221 minus 2119892221 minus 11989211)12058210158403 = minus120572119896 (611989231 minus 1211989232 + 611989233 minus 611989221 + 611989222+ 11989211)
12058210158404 = minus120572119896 (2011989241 minus 6011989242 + 6011989243 minus 6011989244 minus 3011989231+ 6011989232 minus 3011989233 minus 1211989221 minus 1211989222 minus 11989211)
(A15)
where
119892119904119903 = (1 minus 1198650)119896+119903(119896 + 119903) (1 minus 1198651199040) (A16)
Then the first four PL-moments are computed to developthe PL-moment ratios (PL-Cv PL-Cs and PL-Ck) for theGPA distribution
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ClimatologyJournal of
EcologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EarthquakesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom
Applied ampEnvironmentalSoil Science
Volume 2014
Mining
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal of
Geophysics
OceanographyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Atmospheric SciencesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MineralogyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MeteorologyAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geological ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geology Advances in
Advances in Meteorology 19
Ethical Approval
The manuscript is prepared in accordance with the ethicalstandards of the responsible committee on human exper-imentation and with the latest (2008) version of HelsinkiDeclaration of 1975
Conflicts of Interest
The manuscript is prepared by using secondary data andauthors declared that there are no conflicts of interest
Acknowledgments
The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1437-027
References
[1] T Dalrymple ldquoFlood-frequency analysesrdquo Water Supply Paper1543-A USGeological Survey Reston Va USA 1960
[2] J R M Hosking J R Wallis and E F Wood ldquoAn appraisal ofthe regional flood frequency procedure in the UK flood studiesreportrdquo Hydrological Sciences Journal vol 30 no 1 pp 85ndash1091985
[3] N P Greis and E F Wood ldquoRegional flood frequency estima-tion and network designrdquoWater Resources Research vol 17 no4 pp 1167ndash1177 1981
[4] J R Wallis ldquoHydrologic problems associated with oilshaledevel opmentrdquo in Environmental Systems and Management SRinaldi Ed pp 85ndash102 1982
[5] C Cunnane ldquoMethods and merits of regional flood frequencyanalysisrdquo Journal of Hydrology vol 100 no 1ndash3 pp 269ndash2901988
[6] K Subramanya Engineering Hydrology McGraw-Hill Singa-pore 2nd edition 2007
[7] J R Hosking and J R Wallis Regional Frequency Analysis AnApproach Based on L-Moments Cambridge University PressNew York NY USA 1997
[8] Q J Wang ldquoEstimation of the GEV distribution from censoredsamples by method of partial probability weighted momentsrdquoJournal of Hydrology vol 120 no 1ndash4 pp 103ndash114 1990
[9] C Cunnane ldquoStatistical distributions for flood frequency anal-ysisrdquo World Meteorological Organization Operational Hydrol-ogy Report 33 1987
[10] Q J Wang ldquoUsing partial probability weighted moments tofit the extreme value distributions to censored samplesrdquo WaterResources Research vol 32 no 6 pp 1767ndash1771 1996
[11] Q J Wang ldquoUnbiased estimation of probability weightedmoments and partial probability weighted moments from sys-tematic and historical flood information and their applicationto estimating the GEV distributionrdquo Journal of Hydrology vol120 no 1-4 pp 115ndash124 1990
[12] K P Bhattarai ldquoPartial L-moments for the analysis of censoredflood samplesrdquo Hydrological Sciences Journal vol 49 no 5 pp855ndash868 2004
[13] A B Shabri Z M Daud and N M Ariff ldquoRegional analysisof annual maximum rainfall using TL-moments methodrdquoTheoretical and Applied Climatology vol 104 no 3-4 pp 561ndash570 2011
[14] B Saf ldquoRegional flood frequency analysis using L-momentsfor the West Mediterranean region of TurkeyrdquoWater ResourcesManagement vol 23 no 3 pp 531ndash551 2009
[15] J Abolverdi and D Khalili ldquoDevelopment of regional rainfallannual maxima for Southwestern Iran by L-momentsrdquo WaterResources Management vol 24 no 11 pp 2501ndash2526 2010
[16] Z Hussain and G R Pasha ldquoRegional flood frequency analysisof the seven sites of Punjab Pakistan using L-momentsrdquoWaterResources Management vol 23 no 10 pp 1917ndash1933 2009
[17] ZA Zakaria A Shabri andUNAhmad ldquoRegional FrequencyAnalysis of Extreme Rainfalls in the West Coast of PeninsularMalaysia using Partial L-Momentsrdquo Water Resources Manage-ment vol 26 no 15 pp 4417ndash4433 2012
[18] A Shahzadi A S Akhter and B Saf ldquoRegional frequencyanalysis of annual maximum rainfall in monsoon region ofpakistan using L-momentsrdquo Pakistan Journal of Statistics andOperation Research vol 9 no 1 pp 111ndash136 2013
[19] S Coles An Introduction to Statistical Modeling of ExtremeValues Springer Series in Statistics Springer-Verlag LondonEngland 2001
[20] RW KatzM B Parlange and P Naveau ldquoStatistics of extremesin hydrologyrdquoAdvances inWater Resources vol 25 no 8ndash12 pp1287ndash1304 2002
[21] H Abida and M Ellouze ldquoProbability distribution of floodflows in Tunisiardquo Hydrology and Earth System Sciences vol 12no 3 pp 703ndash714 2008
[22] S Feng S Nadarajah and Q Hu ldquoModeling annual extremeprecipitation in China using the generalized extreme valuedistributionrdquo Journal of the Meteorological Society of Japan vol85 no 5 pp 599ndash613 2007
[23] T Yang Q Shao Z-C Hao et al ldquoRegional frequency anal-ysis and spatio-temporal pattern characterization of rainfallextremes in the Pearl River Basin Chinardquo Journal of Hydrologyvol 380 no 3-4 pp 386ndash405 2010
[24] GVillarini J A SmithM L Baeck RVitoloD B StephensonandW F Krajewski ldquoOn the frequency of heavy rainfall for theMidwest of theUnited Statesrdquo Journal of Hydrology vol 400 no1-2 pp 103ndash120 2011
[25] D She J Xia J Song H Du J Chen and L Wan ldquoSpatio-temporal variation and statistical characteristic of extreme dryspell in Yellow River Basin Chinardquo Theoretical and AppliedClimatology vol 112 no 1-2 pp 201ndash213 2013
[26] J R Wallis N C Matalas and J R Slack ldquoJust a momentrdquoWater Resources Research vol 10 no 2 pp 211ndash219 1974
[27] C P Pearson ldquoNew Zealand regional flood frequency analysisusing L-momentsrdquo Journal of Hydrology vol 30 no 2 pp 53ndash64 1991
[28] J R Hosking ldquoL-moments analysis and estimation of distri-butions using linear combinations of order statisticsrdquo Journal ofthe Royal Statistical Society Series B Methodological vol 52 no1 pp 105ndash124 1990
[29] J A Greenwood J M Landwehr N C Matalas and J RWallis ldquoProbability weighted moments definition and relationto parameters of several distributions expressable in inverseformrdquo Water Resources Research vol 15 no 5 pp 1049ndash10541979
[30] J R M Hosking and J R Wallis ldquoSome statistics useful inregional frequency analysisrdquoWater Resources Research vol 29no 2 pp 271ndash281 1993
[31] J R Stedinger R M Vogel and E Foufoula-Georgiou ldquoFre-quency analysis of extreme eventsrdquo inHand Book of HydrologyD R Maidment Ed McGraw-Hill New York NY USA 1993
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ClimatologyJournal of
EcologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EarthquakesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom
Applied ampEnvironmentalSoil Science
Volume 2014
Mining
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal of
Geophysics
OceanographyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Atmospheric SciencesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MineralogyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MeteorologyAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geological ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geology Advances in
20 Advances in Meteorology
[32] R M Vogel and N M Fennessey ldquoL-moment diagrams shouldreplace product moment diagramsrdquo Water Resources Researchvol 29 no 6 pp 1745ndash1752 1993
[33] J R M Hosking ldquoThe use of L-moments in the analysis ofcensored datardquo inRecent Advances in Life-Testing andReliabilityN Balakrishnan Ed pp 545ndash564 CRC Press Boca Raton FlaUSA 1995
[34] M G Schaefer ldquoRegional analyses of precipitation annualmaxima inWashington StaterdquoWater Resources Research vol 26no 1 pp 119ndash131 1990
[35] C Pearson ldquoApplication of l-moments to maximum river owsrdquoThe New Zealand Statistician vol 28 no 1 pp 2ndash10 1993
[36] R M Vogel W O Thomas and T A McMahon ldquoFlood-flow frequency model selection in southwestern united statesrdquoJournal of Water Resources Planning and Management vol 119no 3 pp 353ndash366 1993
[37] K C A Chow and W E Watt ldquoPractical use of the L-momentsrdquo Stochastic and Statistical Methods in Hydrology andEnvironmental Engineering vol 1 no 3 pp 55ndash69 1994
[38] B OnOz and M Bayazit ldquoBest-fit distributions of largestavailable flood samplesrdquo Journal of Hydrology vol 167 no 1-4pp 195ndash208 1995
[39] R M Vogel and I Wilson ldquoProbability distribution of annualmaximum mean and minimum streamflows in the UnitedStatesrdquo Journal of Hydrologic Engineering vol 1 no 2 pp 69ndash761996
[40] M C Peel Q J Wang R M Vogel and T A McMahonldquoThe utility of L-moment ratio diagrams for selecting a regionalprobability distributionrdquo Hydrological Sciences Journal vol 46no 1 pp 147ndash155 2001
[41] Z Jiang Y Ding L Zhu and J Zhang ldquoExtreme precipitationexperimentation over eastern china based on generalized paretodistributionrdquo Plateau Meteorology vol 28 no 3 pp 573ndash5802009
[42] S B Y J D Yuguo ldquoResearch on extreme value distributionof short-duration heavy precipitation in the Sichuan basinrdquoJournal of the Meteorological Sciences vol 4 article 007 2012
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ClimatologyJournal of
EcologyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EarthquakesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom
Applied ampEnvironmentalSoil Science
Volume 2014
Mining
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal of
Geophysics
OceanographyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geological ResearchJournal of
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Geology Advances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Atmospheric SciencesInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MineralogyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MeteorologyAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geological ResearchJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Geology Advances in