# PROBABILITY MODELING OF LOW FLOWS AT DIFFERENT SITES OF INDUS BASIN IN PAKISTAN USING L-MOMENTS AND TL-MOMENTS.

Byline: I. Ahmad, M. Fawad, A. Abbas and A. SaghirABSTRACT: At-site Frequency Analysis (ASFA) of low flow was carried out for nine sites of Indus basin in Pakistan. In the present study, 10-day annual low flow series were analyzed by robust estimation methods such as Method of L-moment (ML) and TL-moment (MTL) to identify best fit probability distributions for each site. Best distribution for each site was identified using different goodness-of-fit Tests (GFT). No single probability distribution was declared as the best-fit distribution for all sites included in the plan. The GFT results indicated GPA was the most appropriate distribution for most of the sites followed by GLO and GEV distributions. On comparison, it was found that for most of the sites ML was best estimation method and for others MTL. For ASFA, the quantiles of best fit distribution were also estimated. It was found that estimated low flows based on fitted distribution were in close agreement with observed flows.

Key words: Goodness-of-fit Test, L-Moments, Probability Distributions, Quantile Estimates, Return Period, TL-moments.

INTRODUCTION

Pakistan is an agro-based country and its agriculture mainly depends on waters of Indus basin. The basin is mainly irrigated by Indus River itself and its tributaries viz-a-viz River Jhelum, Chenab, Ravi, Sutlej and Beas. From 1998-2002 Pakistan faced severe drought conditions in the Indus plain, which proved to the worst drought in the history of Pakistan. ASFA of low flows is of great concern in water resources research including water quality management, determination of downstream flow requirement for hydropower generation, designing of irrigation system and impact of prolonged droughts on aquatic ecosystems in the country (Richter et al, 2003). Low flow in Indus basin adversely affects agriculture, environment, economy and ecosystem of Pakistan. So there is a dire need of Frequency Analysis (FA) of low flow at Indus basin in Pakistan. The procedure for estimating frequency of occurrence of hydrological events is known as FA (Noto and Loggia, 2009).

Various aspects for low flow have been discussed to determine the type of probability distribution across the world as reported by (Gubareva and Gartsman, 2010; Yurekli et al, 2005; Zaidman et al, 2002; Kroll and Vogel, 2002; Onoz and Bayazit, 2001; Caruso, 2000; Durrans and Tomic, 1996; Vogel and Wilson, 1996; Clausen and Pearson, 1995). Most of these studies considered GEV, GPA, PE3, LN3 and GLO for best fit candidate distributions. Among other estimation methods, method of L-moments was mostly used developed by (Hosking, 1990). Estimates based on simple moments methods are influenced by extreme events. While the estimates based on L-Moments are less effected from such extreme observations without removing them from the data set. The estimates from such methods are more reliable as compared to conventional methods.

LM not only outperforms the conventional moments but also often more efficient than small and moderate sample sizes for meteorological data as has been reported by (Ahmad et al, 2013; Ahmad et al, 2014; Hosking and Wallis, 1987 a and Hosking et al, 1985b).A modified version of L-moments, i.e. TL-moments, developed by (Elamir and Seheult, 2003) may be used when our concern is to show extreme evets having undue influence.

MATERIALS AND METHODS

Data Description and its Initial Screening: The annual minima of10-day average series (measured in cusecs) of nine sites of Indus basin in Pakistan, located on the four rivers namely Indus, Kabul, Jhelum and Chenab were included in this study. These sites were selected based on quality of data, climate variability and change, record length and urbanization (Figure 1). The data of these sites were collected from Water and Power Development Authority (WAPDA) and Federal Flood Commission (FFC) (Table.1).

Table 1. Basic Information of nine sites used in the study

Names of###River###Latitude###Longitude###Sample###Mean###Standard###Skewness###Coefficient

sites###(North)###(East)###size (n)###deviation###of

###variation

Tarbela###Indus###33.99###72.61###52###13543.08###4704.093###-0.5194###0.3473

Nowshera###Indus###29.32###70.05###53###6960.38###1979.058###0.3804###0.2843

Kalabagh###Indus###32.95###71.50###52###21136.54###6786.088###0.8839###0.3211

Chashma###Indus###32.43###71.38###43###12644.19###7736.564###-0.1627###0.6119

Taunsa###Indus###30.50###70.80###52###15498.08###5547.990###-0.0243###0.3579

Guddu###Indus###28.30###69.50###52###17338.46###7775.052###0.5287###0.4484

Sukkur###Indus###27.72###68.79###73###871.7808###1399.412###1.7649###1.6052

Mangla###Indus###33.15###73.65###47###5353.965###3480.487###0.7684###0.6500

Marala###Indus###32.68###74.43###26###1602.404###1003.469###0.0297###0.6262

Statistical analysis of current data was based on independence, stationarity and homogeneity. The violation of these assumptions might mislead in policy implications. The assumed data was analyzed using different parametric and non-parametric tests to test these assumptions.

Population L-Moments and Trimmed L-Moments: Population L-moments measured location, dispersion, skewness, kurtosis, and other aspects of the shape of probability distributions and sample data, using linear combinations of the ordered data values. Let X1, X2,..., XR be the random sample of magnitude, "r" with cumulative distribution Function F(X) and quantile function, X(F). Let (EQUATION) be the order statistic of random sample. For the random variable X, the r th population L-moment explained by Hosking (1990) was:

(EQUATION)

In L-moments L highlighted that lr was a linear function of the expected order statistics. From first four L-moments we could find, measure of L-coefficient of variation (L-Cv), L-Skewness and L-Kurtosis as well.

In TLM the expectations of the ordered statistics of a conceptual sample (in the sense of population L-moments) were substituted by expectations of the ordered statistics of a larger conceptual sample, the size was enlarged equal to the overall trimming amount. TLM showed certain advantage over LM and conventional moments. TLM could exist even when population's mean did not exist.

For example Cauchy distribution sample TLM were unbiased to the corresponding population quantities and more robust to outliers as reported by (Elamir and Seheult, 2003). The r th TL-moments in the case of equal trimming was written below:

(EQUATION)

All the corresponding quantities could be determined as in case of L-moments.

Estimation of L-moments and TL-moments: In practice, L-moments need to be estimated after taking a random sample drawn from an anonymous distribution. Let x1, x2,..., xn be the sample and (EQUATION) was the order statistics of the samples, then sample L-moments could be defined as reported by Hosking (2007) and Asquith (2007):

The estimates of L-CV, L-Kurtosis and L-kurtosis could be determined using sample quantities from above equation (3).

The r th sample TL-moments were defined below as reported by Elamir and Seheult (2003),

(EQUATION)

The estimates of TL-CV, TL-Kurtosis and TL-kurtosis could be determined using sample quantities from above equation (4) and also reported by (Ahmad et al, 2015). The TL-skewness and TL-kurtosis were also dimensionless quantities and gave information about shape of a data set.

Comparison of the Probability distributions using goodness-of-fit criteria: The goodness of fit tests such as RMSE, AD, and KS tests using ML and further MTL were used to find out the most suitable distribution for a specific site.

Root Mean Square Error (RMSE): For the judgment of probability distributions RMSE was applied to 10-days low flows data for all distributions considered in this study. About the distribution of overall fit RMSE provided better result because it calculated every single error in proportion to the size of the observation. It reduced the effect of outliers. The RMSE of smaller value obtained for given distribution revealed the appropriateness of a distribution to the actual data.

Anderson Darling (AD) Test: The AD test was used to check whether the given sample came from a particular probability distribution at hand. The null hypothesis at chosen level of significance would be rejected if calculated value of above statistic exceeds the critical value given in the table. One of the advantage of using AD test was to show good skills when applied to heavy tailed distributions with small sizes (Onoz and Bayazit, 1999; Ahmad et al 2015).

Kolmogorov-Smirnov (KS) test: The KS test was used to check whether the sample came from hypothesized continuous distribution. It was based on the empirical distribution function. Reject Ho at chosen level of significance (a) if the test statistics, D was greater than the critical value obtained from table. Like AD test, KS test also demonstrated good skills when applied to skewed probability distributions, commonly used in hydrology (Baldassarre et al, 2009).

L-Moment Ratio Diagram (LMRD): For visual assessment, the simplest method to determine the best-fit distribution to the actual data was the use of LMRD. LMRD displayed L-moments ratios i.e. L-Skewness and L-Kurtosis of different distributions, considered in this study and data samples for individual sites.

Estimation of Quantiles of Best fit Distribution for Different return Periods: In general, ASFA needed data of large record lengths. Since the available data was of smaller length as compared to return periods of interest. For different applications such as design floods some degree of extrapolation was required as has been reported by (Rahman et al, 2013). After selection of best fit distribution and estimation of its parameters, one needed to find out the quantiles' estimates corresponding to different return periods (T). Larger extreme events normally corresponded to large return periods and less probability and vice versa.

RESULTS AND DISCUSSIONS

Initially, basic assumptions of low flow frequency analysis was tested by different statistical tests. For stationarity of the data, Ljung-Box Q test and Mann Kendall test were applied. Further for homogeneity and independence Mann-Whitney U test and Lag-1 correlation coefficient tests were applied respectively (Table 2). Initially nine probability distributions were considered in the study such as, Generalized Logistic (GLO), Generalized Extreme Value (GEV), Generalized Pareto (GPA), Generalized Normal (GNO),Pearson Type 3 (PE3), EXP (Exponential), GUM (Gumbel), NOR (Normal) and LOG (Logistic). ML was adopted for estimation of parameters. Most of these distributions were used for hydrological modeling in different countries as has been reported by (Rahman et al, 2013 and Tasker, 1987). On the basis of ratio diagram and three goodness-of-fit tests, it was found that out of nine distributions only three distributions were most suitable for 10-days annual low flows data in the present study.

These three distribution were GPA, GLO and GEV. Further, to avoid undue favor to outliers and mitigate their effects, MTL was used for estimation of parameters for these three distribution under the umbrella of these goodness-of-fit tests as reported by (Asquith, 2007; Hosking, 2007; Elamir and Seheult, 2003). Implication of different estimation methods could change the results of the goodness-of-fit tests to some extent as has been reported by (Ahmad et al, 2015). It was found that for data of all sites ML was not only the choice (Table 3).

Table 2. Results of different tests for basic assumptions.

Sites###Mann-Whitney U test###LjungBox Q###Mann Kendall Test###Lag-1 correlation

###Statistics test###coefficient

###Mann-Whitney U###P-value###LB###Pvalue###Tau###P-value###r1###P-value

Tarbela###320.500###0.749###12.054###0.123###0.0552###0.56984###0.136###0.390

Nowshera###318.000###0.606###13.091###0.519###-0.0285###0.77057###0.167###0.212

Kalabagh###334.500###0.949###26.653###0.056###-0.0762###0.43003###0.200###0.111

Chashma###163.500###0.101###23.149###0.058###-0.151###0.15764###0.143###0.331

Taunsa###259.000###0.148###13.606###0.754###-0.156###0.10567###0.204###0.130

Guddu###245.000###0.089###19.252###0.376###-0.254###0.0801###0.088###0.515

Sukkur###581.500###0.343###18.055###0.800###-0.0354###0.67693###0.065###0.571

Mangla###269.500###0.890###10.926###0.814###0.0929###0.36375###0.063###0.657

Marala###54.500###0.125###5.663###0.773###-0.21###0.14443###0.066###0.720

Table 3. Comparison of different goodness-of-fit tests.

Sites Name###RMSE###AD test###KS test###Ratio diagram###Best distribution###Best Method of Estimation

Tarbela###GLO###GLO###GEV###GLO###GLO###TL-moments

Nowshera###GLO###GLO###GLO###GLO###GLO###L-moments

Kalabagh###GPA###GPA###GPA###GPA###GPA###L-moments

Chashma###GEV###GEV###GEV###GEV###GEV###L-moments

Taunsa###GLO###GLO###GLO###GLO###GLO###L-moments

Guddu###GEV###GEV###GEV###GEV###GEV###TL-moments

Sukkur###GPA###GPA###GPA###GPA###GPA###TL-moments

Mangla###GPA###GPA###GPA###GPA###GPA###TL-moments

Marala###GPA###GPA###GPA###GPA###GPA###L-moments

RMSE revealed that, GLO was best fit distribution for three sites, while GEV and GPA for two and four sites respectively. While AD test indicated that the number of sites for which GLO, GEV and GPA were considered as best-fit were three, two and four respectively. Similarly, while using KS test number of sites for which GLO, GEV and GPA were best fit were two, three and four respectively. The results of ratio diagram were very close to all goodness-of-fit tests. Among two estimation methods, LM was found to be the appropriate method for five sites. For four sites TLM deemed to find the best results. Most of the sites approached GPA distribution followed by GLO and GEV. None of these sites followed GNO, PE3, LOG, NORM, GUM and EXP, indicating that these distributions showed poor fit. This could also be viewed from L-Moments Ratio Diagram.

One of the task in AFSA was to estimate quantiles with given return periods, which could be useful for the hydrologists in water resources management as has been reported by (Noto and Loggia, 2009; Baldassarre et al, 2009; Yurekli and Gul, 2005; Caruso, 2000; Vogel and Wilson, 1996). The quantile estimates were calculated on the basis of best-fit distributions for each site individually and it was found that these quantiles were in close agreement to the observed values of 10-days annual low flows table.4. These results were also obvious in extreme value plots of some sites figure. 4a-4d. The low flow series used in this study were recorded up to 2013 for the nine sites on Indus basin. However using the more recent data on these sites might further confirm the findings of the study.

Table 4 Quantile estimates for best fitted distributions of each site.

Sites###Best###0.500###0.800###0.900###0.950###0.980###0.990###0.998

Name###Distribution###2###5###10###20###50###100###500

Tarbela###GLO###14,196.499###15,788.021###16,577.502###17,225.167###17,949.165###18,424.263###19,345.236

Nowshera###GLO###6,858.590###8,445.708###9,432.862###10,383.003###11,646.265###12,628.372###15,039.393

Kalabagh###GPA###19,775.663###27,166.733###31,176.315###34,193.895###37,068.960###38,628.654###40,920.844

Chashma###GEV###12,918.022###19,592.026###22,753.947###25,095.951###27,368.002###28,644.119###30,630.166

Taunsa###GLO###15,579.240###19,805.887###22,235.204###24,445.915###27,211.108###29,236.943###33,810.466

Guddu###GEV###17,013.961###20,460.728###22,161.490###23,461.084###24,766.149###25,525.194###26,762.670

Sukkur###GPA###275.827###710.268###1,148.400###1,710.608###2,705.249###3,708.342###7,272.708

Mangla###GPA###4,869.899###6,938.773###7,820.617###8,362.064###8,771.353###8,945.810###9,133.895

Marala###GPA###1,602.404###2,655.183###3,006.110###3,181.573###3,286.851###3,321.944###3,350.018

Conclusions: Modeling of low flow has always been an important concern in hydrology for water resources management. Through this study, it was found that no single probability distribution could be declared as the best fit distribution for all sites in the study. It is recommended that for practical applications in future such as water quality management, planning of water supplies, hydropower, irrigation systems, and maintenance of aquatic ecosystems, at least these three distributions i.e. GPA, GLO and GEV should be compared for final selection of distributions on these sites of Indus basin, in Pakistan.

Acknowledgements: The authors are thankful to WAPDA and Federal FFC for providing the required data. The authors are also grateful to Higher Education Commission for financial assistance under project No. 20-3954RandD/HEC/14/305.

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Publication: | Pakistan Journal of Science |
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Geographic Code: | 9PAKI |

Date: | Mar 31, 2016 |

Words: | 3364 |

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