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1. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online), Volume 5, Issue 5, May (2014), pp. 76-87 © IAEME 76 TESTING THE HOMOGENEITY OF RAINFALL RECORDS FOR SOME STATIONS IN IRAQ Najm Obaid Salim Alghazali¹, Dhelal Adnan Hussein Alawadi² (1) Corresponding author, Asst. Prof. Doctor, Civil Engineering Department, Babylon University, Iraq (2) M.Sc. Student, Civil Engineering Department, Babylon University, Iraq ABSTRACT In this study three homogeneity tests were used: Pettitt, Buishand and double mass curve tests. These tests were applied to thirteen Iraqi stations of monthly rainfall observations: Mosul, Kirkuk, Khanaqin, Ramadi, Baghdad, Karbala, Hilla, Najaf, Diwaniya, Samawa,Nasiriyah, Amara and Basrah for the period (1970-2010) for all stations except Ramadi (1981-2010) and Hilla (1980- 2010). Pettitt test indicated that there was no break point at any station, while Buishand test indicated that there was a break point only in Karbala station on March-1998 and double mass curve indicated that there was a break point at Karbala on March-1998 station and Samaw station on December-1991 so, they were corrected with double mass curve and retested by each of Pettitt and Buishand tests and they showed no break point. Keywords: Rainfall in Iraq, Homogeneity of Time Series, Pettitt Test, Buishand Test, Double Mass Curve Test. 1. INTRODUCTION Homogeneous rainfall is often required in hydrologic design. However it frequently occurs that rainfall data over different periods are not comparable since the measured amount of rainfall depends on such factors as the type, height and exposure of the raingauge, which have not always been the same. Therefore many meteorological institutes maintain an archive with information on the raingauge sites and the instrument used. Unfortunately it is often not possible to specify the nature of changes in the mean amount of rainfall from the station documentation. This is partly because it is not always known how a change in the instrument or in the raingauge site may influence the INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND TECHNOLOGY (IJCIET) ISSN 0976 – 6308 (Print) ISSN 0976 – 6316(Online) Volume 5, Issue 5, May (2014), pp. 76-87 © IAEME: www.iaeme.com/ijciet.asp Journal Impact Factor (2014): 7.9290 (Calculated by GISI) www.jifactor.com IJCIET ©IAEME
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International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online), Volume 5, Issue 5, May (2014), pp. 76-87 © IAEME 77 measured amount of rainfall and partly because it is highly questionable whether the station information gives a complete picture of raingauge site during the period that the station has been in operation. Because of the uncertainty about possible changes, graphical methods are often used in climatology and hydrology to obtain some insight into the homogeneity of a record. A popular tool is a double mass curve which is obtained by plotting the cumulative amounts of the station under consideration against the cumulative amounts of a set of neighboring stations. The plotted points tend to fall along a straight line under conditions of homogeneity. Though graphs are useful for detection of shift in the mean it is usually not obvious how real changes can be distinguished from purely random fluctuations. Therefore it is always necessary to test the significance of departure from homogeneity by statistical methods [1]. Busuioc and Storch[2] studied the variability of winter mean precipitation at 14 Romanian gauge stations from 1901-1988. Pettitt’s statistics was used to detect changes of regimes in the time series. Almost all stations exhibited systemic decrease (downward shift) at about 1960. Furthermore upward were identified for the southwestern stations at 1933, and a downward shift in the mid of 1920’s in the northwest. An upward shift at about 1919 for the Bucharest station was likely determined by the urbanization effect. They found that Pettitt’s statistic was sensitive to the presence of trend and serial correlation. Tarhule and Woo [3] examined changes in several rainfall characteristics in northern Nigeria. The records at 25 locations were analyzed for the occurrence of abrupt changes using the Pettitt test.They concluded that changes in rainfall over the Sahel were driven by a reduction in the frequency of rain days of high rainfall intensities during the months of August and September Karabork et al. [4] studied the homogeneity of 212 precipitation records in Turkey for the period 1973-2002. They were checked by Pettitt Test. Stations were considered inhomogeneous if at least one of the tests rejects the homogeneity. He found that 43 out of 212 stations were found to be inhomogeneous. Kang and Yusof[5]used Buishand and Pettitt test in checking homogeneity of daily rainfall series in variables for 33 stations in Damansara, Johor and Kelantan. The results were assessed by classifying the stations into 3 categories, which are useful, doubtful and suspect. They found that are 28 stations (84.84%) are homogeneous and “useful”, 4 stations (12.12%) are inhomogeneous and “doubtful” and 1 station is considered as “suspect”. Arndt and Redmond [6] used the double-mass method to subjectively identify in homogeneities in temperature and precipitation observations for modernized cooperative observer (COOP) stations in U.S. Results showed that in most long-term records, the accumulation of values becomes so large that a double-mass plot masks significant changes in relationship between climate elements. Metzger and Sohn[7] used double mass analysis as consistency tool of rainfall data in three watersheds: Chancay Lambayeque, Chillon and Ramis in Peru from 2008-2011. This analysis was made to the stations for the three watersheds in order to determine whether there was a need for corrections to the data. They found that the double mass analysis that applied to all stations had shown a good behavior in time which meant that the data collection procedures or other local conditions did not have a significant effect on the data, therefore there was no need for correction. In this study homogeneity of thirteen Iraqi stations is checked using Pettitt, Buishand and double mass curve tests. 2. DESCRIPTION OF THE STUDY AREA The climate in Iraq is mainly of the continental, subtropical semi-arid type, with the north and north-eastern mountainous regions having a Mediterranean climate. Rainfall is very seasonal and
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International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online), Volume 5, Issue 5, May (2014), pp. 76-87 © IAEME 78 occurs in the winter from December to February, except in the north and northeast of the country, where the rainy season is from November to April. Average annual rainfall is estimated at 216 mm, but ranges from 1200 mm in the northeast to less than 100 mm over 60 percent of the country in the south[8]. In this study total monthly rainfall data for thirteen meteorological stations in Iraq: Mosul, Kirkuk, Khanaqin, Ramadi, Baghdad, Kerbala, Hilla, Najaf, Diwaniya, Samawa, Nasiriya, Amara and Basra are used. Fig. 1 shows the stations, in orange color, that are used in this study. The monthly rainfall of these thirteen stations is analyzed with periods (1970-2010) for all station except (1981-2010) for Ramadi station and (1980-2010) for Hilla station [9]. Figure (1): Iraqi map showing the thirteen stations, in orange color, which are used in this study 3. THEORY 3.1 Pettitt Test Pettit [10] developed a nonparametric test that is capable of locating the period (month or year) where a break is likely. The null hypothesis is that the data is independent, identically distributed random quantities, and the alternative is that a stepwise shift in the mean is present. The test statistic is related to the Mann–Whitney statistic [11]. To detect a break point at a continuous data [10]: 1. The observations (ܺ) are ranked from 1 to N (i.e. ܺଵ, ܺଶ, …, ܺே).
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International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online), Volume 5, Issue 5, May (2014), pp. 76-87 © IAEME 79 2. Estimate the value of ܸ,ே : ܸ=ܰ 1 െ 2ܴ ݅ ൌ 1,2,3, … , ܰ …………….………….……...…. (1) whereܴ is the rank of ܺ in the sample of ܰobservations. 3. Estimate the value of ܷ : ܷ ൌ ܷିଵ ܸ ……………………………………………… …….. (2) ܷଵ ൌ ܸଵ ………….……………………………………......……..….. (3) 4. Detecting the value of ܭே: ܭே ൌ maxଵஸஸே |ܷ| ……...…………………...……......……… ...... (4) 5. Estimate the value of ܲை : 6. ܲை ൌ 2 ݁ ሼି ల಼ಿ మ ൫ಿయశಿమ൯ ሽ ………………….……………........…..……… (5) If ܲை ൏ ߙ , where ߙ is the statistical significance of the test, then the null hypothesis is rejected. 3.2 Bushand Test This test is used to detect a change in the mean by studying the cumulative deviation from the mean. It bases on the adjusted partial sums or cumulative deviation from the mean. According to Buishand [1] the null hypothesis is that the data are homogenous and normally distributed and the alternative hypothesis is that there is a date at which a change in a mean is occurred. The adjusted partial sum is defined as: ܵ כ =0 ;ܵ כ ൌ ∑ ሺܺ െ ܺതሻ ୀଵ , ݇ ൌ 1,2, … , ܰ ….…...…………...…….. (6) where: ܺത : the mean of time series observations (ܺଵ, ܺଶ, … , ܺே) ݇ : the number of the observation at which a break point is occurred Rescaled adjusted partial sums are obtained by dividing the ܵ כ ′s by the sample standard deviation: ܵ ככ ൌ ܵ כ /ܦ , ݇ ൌ 1,2, … , ܰ ….…………...…………......………. (7) ܦ ൌ ට∑ ሺಿ సభ ିതሻ² ே …………….………..……………......…….……. (8) A statistic which can be used for testing homogeneity is: ܳ ൌ maxஸஸே|ܵ ככ| ……………………………......……………..… (9)
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International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online), Volume 5, Issue 5, May (2014), pp. 76-87 © IAEME 80 ܳ/√ܰis then compared with the critical values given by Buishand [1]. If a calculated ܳ/√ܰ is less than a critical ܳ/√ܰ the null hypothesis is accepted but if a calculated ܳ/√ܰ is greater than a critical ܳ/√ܰ the null hypothesis should be rejected. 3.3 Double mass Curve Test The theory of double mass curve is based on the fact that a graph of the accumulation of one quantity against the accumulation of another quantity during the same period will plot as a straight line so long as the data are proportional; the slope of the line will represent the constant of proportionality between the quantities. A break in the slope of the double-mass curve means that a change in the constant of proportionality between the two variables has occurred or perhaps that the proportionality is not a constant at all rates of accumulation. If the possibility of a variable ratio between the two quantities can be ignored, a break in the slope indicates the time at which a change occurs in the relation between the two quantities. The difference in the slope of the lines on either side of the break in the slope indicates the degree of change in the relation. The use of accumulations of two measured variables plotted as a double mass curve may give indefinite results. To give more definite results, the accumulation of one of the variables can be plotted against the accumulations of a pattern, which is composed of all similar records in a given area. The pattern which is composed of the average of many records is less affected by an inconsistency in the record of any one station[12]. The theory of the double-mass curve suggests the method of adjusting an inconsistent record: ܲ ൌ ܾ/ܾܲ …………………………………………………….. (10) where: ܲ: the adjusted data ܲ: the observed data ܾ: the slope of graph to which records are adjusted ܾ: the slope of graph at time ܲ was observed Significance of an apparent break in the double-mass curve can be determined by an analysis- of-variance test (or covariance). The statistics used in this test is F, the ratio of the among-periods variance to the within periods variance. The larger apparent shift in the relation the larger this ratio becomes. The test consists of computing f,the variance ratio, from the data being tested and comparing the computed value with the tabulated value of the F distribution for the level of significance selected (generally 5%) [12].The null hypothesis is that the break point is not statically significant and doesn’t need to be adjusted, while the alternative hypothesis is that the break is statistically significant and needs to be adjusted. The steps of computing f value is are explained in Searcy and Hardison [12]. If ݂ is smaller than F(α,k-1,N-k), where α is a level of significance (generally 5%), then the null hypothesis is accepted but if ݂ is larger than F(α,k-1,N-k) then the null hypothesis must be rejected and the break is statistically significant and it should be adjusted. 4. TESTING OF HOMOGENEITY BY PETTITT TEST The first step in Pettitt test is ranking the data. After that, the value of each ܸଵis calculated, by using Equation (1). The value of each ܷ is calculated by using Equations (2) and (3)and then the value of ܭேis detected which equals the maximum absolute value of ܷ as stated in Equation (4). The value of ܲை ߙ (ߙ ൌ 0.05ሻ which means that the data are homogenous.Table 1 shows the Final calculations of Pettitt test for Iraqi stations.
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International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online), Volume 5, Issue 5, May (2014), pp. 76-87 © IAEME 81 Table (1): Final calculations of Pettitt test for Iraqi stations Station Sample Size (N) Significance Level (α) ܭே ܲை Mosul 492 0.05 4568 0.700489714 Kirkuk 492 0.05 5405 0.288621138 Khanaqin 492 0.05 4366 0.76702084 Ramadi 360 0.05 3310 0.490704409 Baghdad 492 0.05 2708 1.383267725 Hilla 372 0.05 5278 0.078473797 Karbala 492 0.05 4935 0.587826385 Najaf 492 0.05 3332 1.454581077 Diwaniyah 492 0.05 3108 1.230576698 Samawa 492 0.05 5376 0.467692963 Nasiriyah 492 0.05 3834 0.955127429 Amara 492 0.05 2586 1.428923379 Basrah 492 0.05 4699 0.659013415 5. TESTING OF HOMOGENEITY BY BUISHAND TEST As mentioned in Section (1.2) Buishandtest is used to detect a change in the mean by studying the cumulative deviation from the mean. The null hypothesis is that the observations have no break point, while the alternative hypothesis is that the observations have a significant break point .The first procedure in calculations of Buishand test is finding the mean of data of a station and calculating ܵ כ using Equation (6) and then calculating ܵ ככ and ܦ ଶ ,using Equations (7) and (8) respectively, to find the value of ܳ and comparing the value of ܳ/√ܰwith the critical values which are given by Buishand [1].Table 2 shows the final results of Buishand test for Iraqi stations which are used in the study. Table (2): Final calculations of Buishand test for Iraqi stations Station Sample Size (N) Critical ܳ/√ܰ ܳ Observed ܳ/√ܰ Mosul 492 1.36 25.60554 1.15487 Kirkuk 492 1.36 29.04000 1.30922 Khanaqin 492 1.36 26.38292 1.18943 Ramadi 360 1.36 24.60151 1.29661 Baghdad 492 1.36 26.61862 1.20006 Hilla 372 1.36 16.65578 0.86356 Karbala 492 1.36 32.40925 1.46112 Najaf 492 1.36 28.31888 1.27671 Diwaniyah 492 1.36 21.49021 0.96885 Samawa 492 1.36 24.63553 1.11066 Nasiriyah 492 1.36 14.4932 0.65340 Amara 492 1.36 11.11098 0.50092 Basrah 492 1.36 20.56312 0.92706
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International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online), Volume 5, Issue 5, May (2014), pp. 76-87 © IAEME 82 7. DOUBLE MASS CURVE TEST Double mass curve is used for checking the homogeneity of twelve Iraqi stations. The procedure that used in calculations of double mass curve is explained in section 1.3. Mosul station has been taken as a reference station for all stations because it is homogenous, as proving by Pettitt and Buishand tests and it is also has no missing data. F-test is used for checking the significance of an apparent break in the double mass curve. Figs 2, 3, 4,…, 13 show the double mass curve for the Iraqi stations that used in this study. Table 3 shows Final Calculations of F-test for Iraqi stations. Figure (2): Double mass curve for Kirkuk station Figure (3): Double mass curve for Khanaqin station y = 1.0077x R=.999 y = 0.932x + 946.4 R=0.999 0 2000 4000 6000 8000 10000 12000 14000 16000 0 2000 4000 6000 8000 10000 12000 14000 16000 CumulativeRainfallforKirkukStation Cumulative Rainfall for Mosul Station Apr-98 Series1 Series2 Linear (Series1) Linear (Series2) y = 0.8423x R=0.997 y = 0.7713x + 705.08 R=0.997 0 2000 4000 6000 8000 10000 12000 14000 0 2000 4000 6000 8000 10000 12000 14000 16000 CumulativeRainfallforKhanaqinstation Cumulative Rainfall for Mosul station Apr-98 Series1 Series2 Linear (Series1) Linear (Series2)
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International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online), Volume 5, Issue 5, May (2014), pp. 76-87 © IAEME 83 Figure (4): Double mass curve for Ramadi station Figure (5): Double mass curve for Baghdad station Figure (6): Double mass curve for Hilla station y = 0.3068x R=0.997 y = 0.2918x + 166.39 R=0.993 0 1000 2000 3000 4000 0 2000 4000 6000 8000 10000 12000 CumulativeRainfallforRamadistation Cumulative Rainfall for Mosul station Mar-97 Series1 Series2 Linear (Series1) Linear (Series2) y = 0.3768x R=0.985 y = 0.3159x + 302.23 R=0.988 0 1000 2000 3000 4000 5000 6000 0 2000 4000 6000 8000 10000 12000 14000 16000 CumulativeRainfallforBaghdadstation Cumulative Rainfall for Mosul station Apr-95 Series1 Series2 Linear (Series1) Linear (Series2) y = 0.2696x R=0.997 y = 0.2929x - 158.83 R=0.999 0 1000 2000 3000 4000 0 2000 4000 6000 8000 10000 12000 CumulativeRainfallforHillastation Cumulative Rainfall for Mosul station Mar-98 Series1 Series2 Linear (Series1) Linear (Series2)
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International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online), Volume 5, Issue 5, May (2014), pp. 76-87 © IAEME 84 Figure (7): Double mass curve for Karbala station Figure (8): Double mass curve for Najaf station Figure (9): Double mass curve for Diwaniyah station y = 0.3005x R=0.991 y = 0.2234x + 717.16 R=0.999 0 1000 2000 3000 4000 5000 0 2000 4000 6000 8000 10000 12000 14000 16000 CumulativeRainfallforKarbalastation Cumulative for Mosul station Mar-98 Series1 Series2 Linear (Series1) Linear (Series2) y = 0.2955x R=0.997 y = 0.239x + 536.91 R=0.995 0 1000 2000 3000 4000 5000 0 2000 4000 6000 8000 10000 12000 14000 16000 CumulativeRainfallforNajafstation Cumulative Rainfall for Mosul station Mar-98 Series1 Series2 Linear (Series1) Linear (Series2) y = 0.2977x R=0.995 y = 0.2428x + 887.38 R=0.996 0 1000 2000 3000 4000 5000 0 2000 4000 6000 8000 10000 12000 14000 16000 CumulativeRainfallforDiwaniyahstation Cumulative Rainfall for Mosul station Apr-02 Series1 Series2 Linear (Series1) Linear (Series2)
10.
International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online), Volume 5, Issue 5, May (2014), pp. 76-87 © IAEME 85 Figure (10): Double mass curve for Samawa station Figure (11): Double mass curve for Nasiriya station Figure (12): Double mass curve for Amara station y = 0.2293x R=0.993 y = 0.3502x - 1115.8 R=0.9887 0 1000 2000 3000 4000 5000 0 2000 4000 6000 8000 10000 12000 14000 16000 CumulativeRainfallforSamawastation Cumulative Rainfall of Mosul station Dec-91 Series1 Series2 Linear (Series1) Linear (Series2) y = 0.3308x R=0.998 y = 0.3907x - 630.57 R=0.995 0 1000 2000 3000 4000 5000 6000 0 2000 4000 6000 8000 10000 12000 14000 16000 CumulativeRainfallforNsiriyahstation Cumulative Rainfall for Mosul station Jan-97 Series1 Series2 Linear (Series1) Linear (Series2) y = 0.4533x R=0.995 y = 0.5689x - 1152.5 R=0.990 0 1000 2000 3000 4000 5000 6000 7000 8000 0 2000 4000 6000 8000 10000 12000 14000 16000 CumulativeRainfallforAmarastation Cumulative Rainfall for Mosul station Mar-96 Series1 Series2 Linear (Series1) Linear (Series2)
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International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online), Volume 5, Issue 5, May (2014), pp. 76-87 © IAEME 86 Figure (13): Double mass curve for Basrah station Table 3 shows the final calculations of F-test forstations that are used in this study. Table 3: Final Calculations of F-test for Iraqi stations Station Sample size (N) Date of Break point Critical F Observed f Kirkuk 492 April-1998 3.861 1.491116 Khanaqin 492 April-1998 3.861 1.07195 Ramadi 360 March-1997 3.8742 0.141963 Baghdad 492 April-1995 3.861 1.84374 Hilla 372 March-1998 3.873 0.02773 Karbala 492 March-1998 3.861 4.106744 Najaf 492 March-1998 3.861 2.101180 Diwaniyah 492 April-2002 3.861 3.028565 Samawa 492 December-1991 3.861 7.044241 Nasiriyah 492 January-1997 3.861 0.856929 Amara 492 March-1996 3.861 2.89694 Basrah 492 December-1993 3.861 0.275821 As shown in Table 3 there are two stations that have a significant break, they are Karbala and Samawa stations. The breaks of these stations are corrected using Equation (10).The value of correction factor for Karbala station is 1.345 and for Samawa station is 0.655. The corrected stations are retested by Pettitt test and Buishand and it is found that they have no break. For Pettittt test the value of ܲை for Karbala station is 1.287031 and for Samawa station is 1.168666. For Buishand test the value of ܳ/√ܰ for Karbala station is 0.9409098 and for Samawa station is 1.1719214. 8. CONCLUSIONS Pettitt test indicates that there is no break point at any station of the studied stations. Buishand test indicates that there is a non-homogeneity in Karbala station at March-1998 while, the other stations are homogeneity. Double mass curve indicates that there is a break in each of Karbala station at March-1998 and Samawa station at December-1991 and they are corrected by double mass curve itself. Double mass curve indicates that there is no break in other stations. KarbalaSamawa stations are retested by Buishand and Pettitt tests and it is found that they have no break. y = 0.3853x R=0.998 y = 0.433x - 511.54 R=0.994 0 1000 2000 3000 4000 5000 6000 7000 0 2000 4000 6000 8000 10000 12000 14000 16000 CumulativeRainfallforBasrahstation Cumulative Rainfall for Mosul station Dec-93 Series1 Series2 Linear (Series1) Linear (Series2)
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International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online), Volume 5, Issue 5, May (2014), pp. 76-87 © IAEME 87 9. REFERENCES 1. Buishand T. A., “Some Methods for Testing the Homogeneity of Rainfall Records”, Journal of Hydrology, Elsevier Scientific Publishing Company, Amsterdam, Netherlands, 58, 1982, PP. (11-27). 2. Busuioc A. and Storch H. V., “Changes in the Winter Precipitation in Romania and its Relation to the Large-Scale Circulation”, Munksgraad, Tellus, U. K., Vol. 48A, 1996, PP. (538-552). 3. Tarhule A. and Woo M. K., “Changes in Rainfall Characteristics in Northern Nigeria”, International Journal of Climatology, Royal Meteorological Society, 18, 1998, PP. (1261–1271). 4. Karabork M. C., Kahya E. and Komuscu A. U., “Analysis of Turkish Precipitation Data: Homogeneity and the Southern Oscillation Forcings on Frequency Distributions”, Hydrological Processes, John Wiley & Sons, Ltd., 21, 2007, PP.(3203–3210). 5. Kang H. M. and Yusof F., “Homogeneity Tests on Daily Rainfall Series in Peninsular Malaysia”, Int. J. Contemp. Math. Sciences, Department of Mathematics, Johor, Malaysia Vol. 7, No. 1, 2012, PP. (9 – 22). 6. Arndt D. S. and Redmond K. T., “Toward an Automated Tool for Detecting Relationship Changes within Series of Observations”, Paper, Posted at the internet website https://ams.confex.com/ams/pdfpapers/72826. pdf, 2004. 7. Metzger L. and Sohn S. J., “Validation of APCC Precipitation Forecasts by Using Hydrologic Models”, Report, APEC Climate Center, Busan, Korea, 2011. 8. Frenken K., Irrigation in the Middle East region in figures”, Survay, Food and Agriculture Organization of the United Nations (FAO), Rome, Italy, ISBN 978-92-5-106316-3, 2009. 9. Iraqi Meteorological Office, Baghdad, Iraq. 10. Pettitt A. N., “A Non-Parametric Approach to the Change-Point Problem”, Applied Statistics, Leics, England, 28, No. 2, 1979, PP. (126-135). 11. Costa A. C. M., Negreiros J. and Soares A., “Identification of Inhomogeneities in Precipitation Time Series Using Stochastic Simulation”, Springer Science+Business Media, 2008, PP.(275–282). 12. Searcy J. K. and Hardison C. H., “Double-Mass Curves”, Geological Water Supply Paper 1541-8, Geological Survey, Washington, U.S, 1960.
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