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REGIONAL FREQUENCY ANALYSIS OF LOW FLOWS USING L.MOMENTS FOR INDUS BASIN, IN PAKISTAN.

Byline: I. Ahmad, M. Yasin, M. Fawad and A. Saghir

ABSTRACT:

The regional frequency analysis (RFA) for 10 days low flow series was conducted on 09 sites of Indus basin by using method of linear moments (L-Moments). Out of nine sites, one site was found to be discordant on the basis of discordant measure. The remaining eight sites formed two homogenous regions, each of the regions i.e. Region-I and Region-II consisted of four sites. Best distribution was selected on the basis of L-Moments ratio diagram (LMRD) and goodness-of-fit criterion. Generalized Normal (GNO) was found to be the most appropriate probability distribution for Region-I while GPA for Region-II. The low flow quantiles of regional frequency distribution (growth curves) were estimated for different return periods and accuracy of these estimates were assessed under Relative Root Mean Square Error (RRMSE), Absolute Bias (AB), Relative Absolute Bias (RAB) together with Lower Error Bound (LEB) and Upper Error Bounds (UEB).

For Region II, RRMSE of GPA distribution was smaller for shorter return periods of 2, 5, 10 and 20 years and was higher for larger return periods of 50, 100, 500 and 1000 years. The lower and upper error bounds were narrow at low return periods as compared to error bounds for larger return periods. This regional study showed that there was a good agreement in general between observed and simulated low flows by regional frequency distribution and the results would be suitable for practical applications.

Key words: Indus basin, L-Moments, Relative Root Mean Square Error, Absolute Bias. Goodness-of-Fit.

INTRODUCTION

Pakistan is an agro-based country and the agriculture sector being the back bone of its economy, 24 % of G.D.P of Pakistan is based on its agriculture. The agriculture of Pakistan mainly relies on waters of Indus Basin (Usman et al., 2016). It stretches from Himalaya in north and Arabian Sea in the south. The irrigation system of Pakistan is well-developed. However, the agriculture and hydro-electric generation of Pakistan suffer heavy water loss due to low flows of water of Indus River Basin especially in the Rabi Season (October to March). The last decade has been a period of drought conditions due to the low flows of water of Indus River Basin. The flows below the normal are extremely hazardous and perilous for the country (Rasul et al., 2012). The drought and famine conditions have been distressing the Indus Basin since many years.

In the history, Pakistan faced severe droughts in some area, so robust and reliable estimation of low flows is of immense importance in water resources research and its water management should include designing of irrigation system, hydropower generation, and impact of prolonged droughts on aquatic ecosystems in the country. In environmental sciences, use of one kind of data from different sites to form homogeneous regions from these sites with respect to some common hydrological and geographical features and further to estimate the suitable frequency distribution of these homogenous regions gives shape to RFA (Hosking and Wallis, 1997). RFA of low flow at non gauged stations has always been an issue of keen interest (Modarres, 2008). Due to lack of homogeneity in the regions and accuracy in the quantities of the basins near regional boundaries, these methods are not now considered as good choice in hydrology.

A more robust methodology developed by Hosking and Wallis (1997; 1993) for regional frequency analysis using the concept of L.Moments has been in practice by many researchers. In India Parida et al. (1998) and Kumar et al. (2003), explored regional flood frequency analysis using-moments and Index flood procedure. Lim and Lye (2003) investigated that GEV and GLO were appropriate choice for extreme flood events in the Sarawak region of Malaysia. Hussain and Pasha (2009) carried out RFFA for seven sites of Punjab province in Pakistan. Ahmad et al. (2016) carried out the regional flood frequency analysis using L.Moments and its variants. Hassan and Ping (2012) used L-moments together with hierarchical cluster techniques for formation of homogeneous regions of Luanhe.

Similar work is available such as Shahzadi (2013), Yurekli et al. (2005), Chen et al. (2006), Modarres (2008), Abolverdi and Khalili (2010), Osman et al. (2013), but no any work is available for regional frequency analysis of low flows in the country.

MATERIALS AND METHODS

Study area and data: This study investigated RFA of 10 days average of low flows at Indus Basin of Pakistan using Method of L-moments. The 10 days average was used to avoid zero values in dry seasons. The sites for the proposed study were selected on Indus Basin which included the main River Indus and its tributaries namely River Jhelum, Chenab, Ravi, Sutlej and Kabul using standard criteria such as length of the data, urbanization, quality, variability, and climate change. For the current study the data of nine sites was collected on Western Rivers of Indus Basin controlled by Pakistan. The data was retrieved in collaboration with Federal Flood Commission, WAPDA, Indus River System Authority and Provincial Irrigation Departments.

L-Moments theory and regional frequency analysis based on L. moments: Method of L-moments is alike to method of conventional moments with some advantages over conventional moments. The method of L-moments presents almost unbiased and efficient estimates relative to the other estimation methods. In practice L-moments were estimated from sample observations. It would be appropriate to use an unbiased estimator of Probability Weighted Moments (PWMs) b as:

After the initial screening of the data, i.e. checking basic assumption of hydrological data, assumption of independence, stationarity and consistency or homogeneity, RFA was performed using L. Moments theory for the current data.

1): Screening of the data for regional analysis: We used the Discordancy Statistic to screen out site i from analysis. The critical values for Discordancy Statistic, which was dependent on number of sites included in the study at initial stage. If the value of statistic ( ) for site i exceeded the critical value, site was being discarded from the data. It was useful as it provided the initial guidance to formulate homogeneous regions.

2): Identification of homogeneous regions: The most important step in RFA was formulation of homogeneous regions of selected sites. A region was considered to be homogeneous if the sites included in the region had some common characteristics. To assess the amount of heterogeneity in a set of hydrological sites and to judge whether regions might be considered as homogeneous regions, we used Heterogeneity measure H as: (Eq.) where = weighted standard deviation of sample Lcv based on all N sites. Mathematically,

(Eq.) where tR = Regional Average of sample Lcv defined as: (Eq.), Similarly define tR3 = regional average of sample L-skewness, tR4 = regional average of sample L-kurtosis, which would be further used in simulation study fitting Kappa distribution. If H < 1 Region possibly homogeneous; If [?] H < 2 Region possibly heterogeneous; If H [greater than or equal to] Region definitely declared as heterogeneous.

3): Choice of frequency distribution for homogenous regions: After formulation of homogeneous regions, the next step was to choose the most robust frequency distribution for the homogenous regions. Such a frequency distribution which was capable of giving reasonable quantile estimates even though future data values might come from a different distribution than the fitted distribution. The objective was not to find a distribution, which gave only best fit for a particular data set at hand, but also to find a robust distribution giving good quantile estimates from which future data values would arise.

The approach with regional L-moment statistics i.e. ZDist was used. Let regional average L-moments ratios are (tR, TR3, TR4). d4 the standard deviation of tR4, which could be achieved by simulation of homogeneous region, whose sites supposed to show a particular frequency distribution. Initially, ZDist could be defined as:

(EQUATION)

The fit was considered to be good if (Eq.) indicated the smaller value or sufficiently near to zero or reasonable criteria might be adopted as if (Eq.), then the fit might be considered as adequate.

4): Estimation of frequency distribution: Assuming the region to be homogenous, the frequency distributions of all sites in this region were more or less similar except scale factor and single distribution was selected for fitting the data in the entire region. The theoretical relationship among various frequency distributions was the justification for regional frequency analysis. The parameters and ultimately quantiles of this single distribution could be obtained estimated by more precise way after combining the data from different sites rather than performing at site frequency analysis.

Index flood procedure was used to provide the summary statistic of the data at each site and combines them to form regional estimates but with implementation of L-moments. The resulting procedure as suggested by (Hosking and Wallis, 1997) was known as "Regional L-Moments Algorithm" In this study, the parameters of regional frequency distribution were estimated using method of L-moments.

Accuracy of estimated quantiles: Results of statistical analysis were inherently uncertain, and to make the results useful certain criteria for measuring the extent of uncertainty should be used. Here Regional L-moment algorithm suggested by Hosking and Wallis (1997) was applied with some advantages. In the simulation procedure, quantile estimates were calculated for various non-exceedances probabilities. At m th repetition, for site "i" the quantile estimate was (Eq.) (F) for non-exceedance probability F. The relative error of this quantity was (Eq.). The quantity was squared and averaged over a total number of M repetitions to approximate BIAS and relative RMSE as given by

(EQUATIONS)

The corresponding quantities for assessment of all sites in the region were given by:

(EQUATIONS)

RESULTS AND DISCUSSION

Initial screening of the hydrological data: The fundamental assumptions of stationarity, independence and homogeneity were important before the analysis. Visual inspection initial action through time series plots, Ljung box Test, Mann Kendall Test conformed that data was stationary, independent and homogenous and further used for regional analysis.

Initially after checking the basic assumptions of the data of the study area, the first step carried out was to screen the data with the help of discordancy measure. All stations/sites were considered as one single hydrological region consisting of nine (09) sites of the Indus Basin. The discordancy statistic was computed for each site in the whole group. The critical value for the discordancy statistic was 2.329. On the basis of said statistic, the critical value of only one site namely Indus at Sukkur (site 07) was discordant from the study, which has the computed value 2.46. It was greater than the critical value 2.329 of the table of critical values given by Hosking and Wallis (1997). The all remaining eight (08) sites have the value of statistic less than the tabulated value. It indicates that the remaining sites do not have outlier and any discordancy. The L-CV, L-Skewness and L-Kurtosis together with discordancy measures are shown in table 3 below:

From table 1, it was clear that the discordant site 07 has the lager L-CV, L-Skewness and L-Kurtosis in comparison with other sites, which was also the indicator that the site does not consistent with the other sites of the region. Moreover, the statistics Di, which was based on L-moment ratios, were useful guide for screening the data and also provide sufficient evidence to stand out the site 07. To further investigate the discordant sites, the following Figure 3 was drawn, comparing L-Skewness versus L-CV and L-Skewness versus L-Kurtosis:

Table 1. L-Moments and Discordancy Measure for all sites.

###Site No###Site Name###ni###l1###t###t3###t4###Di

###1###Indus at Tarbela###30###13650.0000###0.2061###-0.0975###0.0603###0.42

###2###Kabul at Nowsahera###53###6960.37740###0.1591###0.0559###0.1610###0.64

###3###Indus at Kalabagh###52###21136.5385###0.1786###0.1730###0.1033###1.33

###4###Indus at Chashma###43###12609.3023###0.3549###-0.0410###0.0611###0.70

###5###Indus at Taunsa###51###5115458.82###0.2011###-0.0109###0.2178###1.61

###6###Indus at Guddu###30###15556.6667###0.3032###0.1673###0.0596###0.68

###7###Indus at Sukkur###74###823.513500###0.7462###0.5728###0.2618###2.46 *

###8###Jhelum at Mangla###47###5161.70210###0.3881###0.1477###0.0927###0.10

###9###Chenab at Marala###26###1465.38460###0.4075###0.0131###0.0010###1.05

The site 7 with high L-CV, L-Skewness and L-Kurtosis, does not possess good agreement to any other site of the region as it was too distant from other sites. Although the site 5 (Indus at Taunsa) yields a bit high L-Kurtosis value of 0.2178, however inspecting the other statistics like L-Skewness and L-CV used for screen analysis, was found in line with L-moment ratios of other sites, which advocates the site 5 and hence it cannot be discorded. Figure 2 and discordancy measure shown in table 1, was supporting to keep the site 5 intact for analysis. Hence, on graphical and numerical base the site 5 was retained for further regional frequency analysis, except site 7 with prominently different behavior from other sites. The results of discordancy measure of remaining sites indicated that there was no inconsistency or gross errors reported in the data over time. Thus the data of remaining eight (08) sites was consistent and applicable for further Regional Frequency Analysis.

After applying the discordancy measure, the second step was formulation of homogeneous regions. Although it was a tedious work to formulate homogeneous region, however, Hosking and Wallis (1993) provided a useful statistical measure based on degree of heterogeneity by using statistic H to identify such homogeneous regions. Initially, all sites under study were considered as a single region. The results of various heterogeneity statistics H, based on 500 simulations using four parameters Kappa distribution are:

Table 2. Values of three Heterogeneity Statistics.

Heterogeneity Statistics###H1###H2###H3

###18.85**###13.33**###7.71**

The table 2 presents three different types of heterogeneity statistics i.e. H1= 18.85, H2=13.33 and H3=7.71, any of them do not fulfill the criterion of homogeneous regions. From the above table 2 it is revealed that the values of H statistic were greater than 2, which implies that a single homogeneous region cannot be constructed. The formulation of homogeneous regions is based on hydrological characteristics. Two hydrological characteristics i.e. Basin Drainage Area in sq. miles and Mean Annual Minimum Flow were utilized for this purpose, plotting Basin Drainage Area along horizontal axis and Mean Annual Minimum Flow / Area along vertical axis see Figure 05. It is suggested two homogenous regions.

The corresponding location of the sites was shown in the map of Pakistan together with river flows on Indus Basin at Figure 4 shown as below:

Table 3. Heterogeneity Statistic for Homogeneous Region-I.

Sites###Site Name###Di###Heterogeneity Statistic###Regional L-moments

1###Indus at Tarbela###1.00###H1= 0.16###tR= 0.1837

2###Kabul at Nowshera###1.00###H2= 1.39###t3R= 0.0456

3###Indus at Kalabagh###1.00###H3= 1.37###t4R=0.1442

5###Indus at Taunsa###1.00

Table 4. Heterogeneity Statistic for Homogeneous Region-II.

Sites###Site Name###Di###Heterogeneity Statistic###Regional L-moments

4###Indus at Chashma###1.00###H1= 0.07###tR= 0.3612

6###Indus at Guddu###1.00###H2= 1.65###t3R= 0.0670

8###Jhelum at Mangla###1.00###H3= 1.43###t4R= 0.0604

9###Chenab at Marala###1.00

The results of Table 5 for Region-I indicates that the values of discordancy measure Di for four sites were smaller than the critical value of 3.00, moreover the value of heterogeneity statistic H1 was 0.16 was also less than corresponding critical value of 2. Keeping in view of these results, a homogeneous region (Region-I) consisting of four sites was formed. Similarly, the results of Table 6 for Region-II indicate the values of discordancy measure Di for four sites included in this region are less than the critical value i.e. 3.00, moreover the value of heterogeneity statistic H1 is 0.07, that also suggests the region is homogenous. Keeping in view of these results second homogeneous region (Region-II) also consists of four sites.

Selection of frequency distribution for regional analysis: After successful formulation of homogeneous regions, the next step was the selection of robust and suitable statistical model. By suitable distribution did not mean only the distribution provided best fit to the selected region but also the distribution which provided accurate quantile estimates for different return periods. The appropriate distribution would be selected amongst family of three parameter distributions i.e. Generalized Logistic (GLO), Generalized Pearson type III (PE3), Generalized Extreme Value (GEV), Generalized Normal (GNO) and Generalized Pareto (GPA). Apart from this, Wake-by (WAK) distribution had also been considered as a candidate distribution because of its five parameters, it had broader range of distributional shapes.

The calculation of |ZDIST| for the two selected regions was listed in the following table5:

From the above mentioned results, it was concluded that GLO, GEV, GNO and PE Type III were possible candidate distributions for Region I. However, (Generalized) Pearson Type III and GNO had the least value which was near to zero. GNO was considered as the final distribution for Region -I. So far as Region II was concerned only a Generalized Pareto was found the most suitable probability distribution.

Table 5. ZDIST Statistic for Various Probability Distributions.

Region###ZGLO###ZGEV###ZGNO###ZPE3###ZGPA

1###0.93*###-1.17*###-0.77*###-0.81*###-5.15

2###4.29###2.30###2.59###2.51###-1.55*

For Region-I ratio of regional average L-Skewness and L-Kurtosis average lie nearest to the GNO distribution and similarly for Region-II regional average L-Skewness and L-Kurtosis average lie closest to GPA. The results of L-moment ratio diagram coincided with our previous results of ZDIST statistic. (Fig.5a and 5b).

Estimation of frequency distribution for regional analysis: The low flow regional quantile estimates (estimates of growth curves i.e. ^(F)) for return periods of 2, 5,10,20,50,100,500 and 1000 years of Region I and Region II were shown in table 6(a) and table 6(b) respectively. 10 days low flow quantile estimates might be obtained for each site (Eq.) for different periods by simply multiplying regional quantile estimates q(F) to sample average (Eq.) of each site in the respective region.

The graphic representation of estimates of growth curves were shown in figure 6 (a) and 6 (b) by measuring return periods based on non-exceedance probability along horizontal axis and quantiles of regional frequency distribution (growth curves) on vertical axis. The table 6 (a) and figure 6 (a) indicate that growth curves for all the five distributions for return period up to 50 years almost reflect a close behavior. However, for the higher return periods the quantiles estimate of GLO and WAK were higher than other candidate distributions of the Region-I. Moreover, GNO and PE3 gives almost the close quantiles for low as well as high return periods, finally GNO can be used as the robust distribution for Region-I. From table 6(b) and figure 6(b) for Region-II, the quantile estimates for lower return periods were almost in close agreement for both GPA and WAK, however for high return period the quantile estimates for WAK are high.

Table 6(a). Regional Quantile Estimates for Region-I.

###Parameters###Regional quantile estimates with non-exceedance probability F

Dist###e###a###k###0.10###0.500###0.800###0.900###0.950###0.980###0.990###0.998###0.999

###1###2###5###10###20###50###100###500###1000

GLO###0.986###0.183###-0.04###0.604###0.740###0.986###1.248###1.409###1.563###1.766###1.922###2.472

GEV###0.874###0.309###0.203###0.593###0.719###0.983###1.274###1.433###1.564###1.708###1.799###2.023

GNO###0.984###0.324###-0.09###0.593###0.722###0.985###1.269###1.426###1.561###1.719###1.828###2.147

PE3###1.000###0.326###0.279###0.593###0.722###0.985###1.269###1.427###1.561###1.718###1.825###2.139

WAK###0.33###2.33###4.85###0.561###0.720###0.995###1.248###1.414###1.570###1.762###1.898###2.291

###g0.28###d-.08

Table 6 (b). Regional Quantile Estimates for Region-II.

###Parameters###Regional quantile estimates with non-exceedance probability F

Dist###e###a###k###0.10###0.500###0.800###0.900###0.950###0.980###0.990###0.998###0.999

###1###2###5###10###20###50###100###500###1000

GPA###0.007###1.736###0.749###0.183###0.364###0.946###1.631###1.913###2.080###2.202###2.252###2.313

WAK###-0.03###1.97###1.14###0.165###0.368###0.966###1.578###1.833###2.035###2.309###2.559###3.963

###g0.07###d0.35

Accuracy of growth curve and at-site quantiles: For Region-I there were four candidates distributions namely GLO, GEV, GNO and PE3. For all these distributions simulation study was conducted using Kappa distribution. For Region-I, [Table 8 a], it is revealed that for low return periods up to 20 years with non-exceedance probability 0.950 the AB and RAB were lower as compared to high return periods up to 1000 years with non-exceedance probability 0.999. RRMSE of all four distributions were smaller for lower return periods 2, 5, 10 and 20 years and were high for return periods 50, 100, 500 and 1000 years. It was found that GNO had smaller RRMSE for lower as well as higher return periods amongst all the candidate distributions. So far as error bounds were concerned, it was also found that GNO has narrow error bounds for both lower as well as higher return periods as compared to GLO, GEV and PE3.

Hence the GNO was found the most appropriate distribution for growth curve estimation for smaller as well as larger return periods for Region-I.

Table 8 (a). Regional growth curves simulation results for Region-I.

Distribution###F###0.100###0.500###0.800###0.900###0.950###0.980###0.990###0.998###0.9

###1###2###5###10###20###50###100###500###1000

GLO###RRMSE(F)###0.0543###0.0049###0.0175###0.0243###0.0301###0.0372###0.0421###0.0531###0.0578

###AB(F)###0.0016###0.0049###-###-###-###0.0086###0.0320###0.1178###0.1687

###0.0111###0.0146###0.0100

###RAB(F)###0.0269###0.0084###0.0142###0.0224###0.0289###0.0411###0.0577###0.1306###0.1781

###LEB*###0.9166###0.9938###0.9759###0.9640###0.9538###0.9437###0.9349###0.9164###0.9079

###UEB*###1.0088###1.0086###1.0277###1.0403###1.0505###1.0625###1.0703###1.0890###1.0960

GEV###RRMSE(F)###0.0561###0.0044###0.0174###0.0247###0.0300###0.0354###0.0388###0.0453###0.0477

###AB(F)###0.0033###0.0001###0.0004###0.0008###0.0014###0.0025###0.0034###0.0060###0.0073

###RAB(F)###0.0645###0.0087###0.0201###0.0285###0.0356###0.0440###0.0500###0.0633###0.0687

###LEB*###0.9184###0.9936###0.9732###0.9620###0.9542###0.9453###0.9395###0.9297###0.9252

###UEB*###1.0944###1.0072###1.0281###1.0397###1.0489###1.0577###1.0639###1.0752###1.0803

GNO###RRMSE(F)###0.0506###0.0040###0.0170###0.0240###0.0293###0.0347###0.0381###0.0446###0.0464

###AB(F)###0.0037###0.0002###0.0003###0.0008###0.0013###0.0021###0.0028###0.0045###0.0053

###RAB(F)###0.0645###0.0092###0.0198###0.0283###0.0354###0.0436###0.0492###0.0613###0.0662

###LEB*###0.9197###0.9934###0.9741###0.9638###0.9564###0.9484###0.9435###0.9334###0.9293

###UEB*###1.0934###1.0066###1.0274###1.0394###1.0482###1.0574###1.0632###1.0757###1.0804

PE3###RRMSE(F)###0.0560###0.0043###0.0172###0.0243###0.0296###0.0350###0.0383###0.04468###0.0469

###AB(F)###0.0035###0.0001###0.0005###0.0010###0.0015###0.0021###0.0025###0.0033###0.0037

###RAB(F)###0.0638###0.0091###0.0195###0.0281###0.0350###0.0428###0.0481###0.0589###0.0631

###LEB*###0.9184###0.9928###0.9739###0.9634###0.9553###0.9474###0.9422###0.9323###0.9830

###UEB*###1.0941###1.0074###1.0278###1.0399###1.0487###1.0581###1.0642###1.0754###1.0793

For Region-II, RRMSE, AB, RAB, together with Lower and Upper Bounds were shown in Table 8 (b). The GPA found the only suitable candidate distribution for the region-II. It was shown for low return periods up to 20 years the AB and RAB were higher for shorter return periods from 2 to 20 years as compared to high return periods. RRMSE of GPA distribution were also smaller for shorter return periods 2, 5, 10 and 20 years and were high for return periods 50, 100, 500 and 1000 years. The lower and upper error bounds were narrow at low return periods as compared to error bounds for larger return periods.

Table 8(b). Regional growth curves simulation results for Region-II.

Distribution###F###0.100###0.500###0.800###0.900###0.950###0.980###0.990###0.998###0.999

###1###2###5###10###20###50###100###500###1000

GPA###RRMSE(F)###1.0516###0.0322###0.0466###0.0639###0.0771###0.0898###0.0965###0.1059###0.1081

###AB(F)###0.3515###-0.0011###0.0033###0.0062###0.0090###0.0126###0.0150###0.0192###0.0204

###RAB(F)###0.6858###0.0276###0.0392###0.0539###0.0657###0.0777###0.0843###0.0937###0.0960

###LEB*###0.4447###0.9539###0.9277###0.8986###0.8747###0.8521###0.8400###0.8275###0.8251

###UEB*###3.5803###1.0619###1.0760###1.1093###1.1357###1.1488###1.1578###1.1812###1.1859

Conclusion: Regional Frequency Analysis was performed on 09 sites located on the Indus Basin, Pakistan. Two regions were formed in this study. After checking their homogeneity, we selected best distribution for each region. GNO and GPA were found the most appropriate distributions for growth curve and quantiles estimation of Region-I and Region-II respectively. On the basis of RB, RAB, RRMSE, LEB and UEB, the simulation results of Region-I and Region-II indicated that regional quantile estimates are more accurate using GNO and GPA respectively. The estimates of the study could be used to assess the feasibility of construction of new water structures in future. Quantile estimates of low flow would help the managers to combat the drought like situation and to minimize the losses in future by better planning in light of such estimates.

Acknowledge: Authors are grateful to Higher Education Commission for financial support under the project Number: NO: 20-3954RandD/HEC/305 and also WAPDA and Federal Flood Commission for providing data.

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