TARIM BILIMLERI DERGISI 2002, 8 (3) 256-260
Frequency Analysis for Kelkit Stream's the Daily Extreme Flows
Kadri YÜREKLİ/ Hüseyin ŞIMŞEK°
Geliş Tarihi: 07.02.2002
Abstract: The main purpose of this study is to determine the best suitable probability distribution for the daily extreme flows of Kelkit stream. In this study, the daily flow values measured from 1938 to 1988 in the flow station numbered 1401 which is operated by General Directorate of Electric Power Research Survey and Development Administration (EIE) were used. The daily extreme flows among daily flows for each month and each year were selected. Since reserving water by Kılıçkaya dam built on Kelkit stream had been started after 1988, the daily flows, that came after this date were not taken into account for homogeneity. To determine a suitable probability distribution for daily extreme flows, normal, log normal, three parameter log normal, two parameter gamma, Pearson type III, log Pearson type III, extreme value type I, three parameter Weibull distributions were used. The selected maximum daily flows of each month were fitted better to, Pearson type III for the second, fifth and tenth months; log Pearson type III for the first, eleventh and twelveth months; log normal for the third, seventh and eighth months; three parameters log normal for the fourth and nineth months and extreme value type I for the sixth month. The minimum daily flows of each month were fitted better to, three parameter Weibull for the first, second, nineth and tenth months; three parameter log normal for the third, fourth and eleventh months; Pearson type III for the fifth and sixth months; extreme value type 1 for the seventh month; normal for the eighth month and log normal for the twelveth month. The daily extreme flows for each year were also fitted better to Pearson type III distribution.
Key Words : frequency analysis, extreme flow, probability distribution, Kelkit stream
Kelkit Çay
ı
Günlük Ekstrem Ak
ı
mlar
ı
n
ı
n Frekans Analizi
Özet: Bu çalışmanın amacı, Kelkit çayının günlük ekstrem akımları için en uygun olasılık dağılım biçimini saptamaktır. Çalışmada, Elektrik İşleri Etüt idaresi Genel Müdürlüğü (EİE) tarafından işletilen 1401 numaralı akım gözlem istasyonunda 1938-1988 yıllarında ölçülmüş olan akım miktarları kullanılmıştır. Günlük ekstrem akım miktarları her ay ve yıl için günlük akım miktarları arasından seçilmiştir. Kelkit çayı üzerine inşa edilen Kılıçkaya barajı 1988 yılından sonra su tutmaya başladığından, bu yıldan sonraki akımlar homojenlik için dikkate alınmamıştır. Günlük ekstrem akımlar için uygun bir olasılık dağılımı belirlemek amacıyla, normal, log normal, üç parametreli log normal, iki parametreli gama, Pearson tip ııı, log Pearson tip ııı, uç-ı ve Weibull ııı dağılımları kullanılmıştır. Her ay için seçilen günlük en büyük akımlar, 2., 5. ve 10. aylar için Pearson tip ııı dağılımına; 1., 11. ve 12 aylar için log Pearson tip ııı dağılımına; 3., 7. ve 8. aylar için log normal dağılıma; 4. ve 9. aylar için üç parametreli log normal dağılıma; 6. ay için uç-ı dağılımına en iyi uyumu sağlamıştır. Her ay için seçilen günlük en küçük akımlar, 1., 2., 9. ve 10. aylar için Weibull nı dağılımına; 3., 4. ve 7. aylar için üç parametreli log normal dağılıma; 5. ve 6. aylar için Pearson ııı dağılımına; 7. ay için uç-ı dağılımına; 8. ay için normal dağılıma ve 12. ay için log normal dağılıma en iyi uyumu sağlamıştır. Her yıl için seçilen günlük ekstrem akımlar ise Pearson ııı dağılımına en iyi uyumu sağlamıştır. Anahtar Kelimeler: frekans analizi, ekstrem akım, olasılık dağılım, Kelkit çayı
Introduction
Streams are one of the most important sources for drink, use and irrigation water, and their flows vary with precipitation. Distribution and return periods of the precipitation fairly differentiate over an area by depending on many factors.
Flow of a stream increases with increasing precipitation that fall in the stream basin, sometimes, these flows extremly increase and cause damage around of it. However, during the dry periods, these flows decrease and as a result, it will be difficult to supply our needs. In general, the flows of the streams do not accord with water demand in Turkey. This is the case especially during the irrigation season. When the flows are not sufficient, reservoirs are built on streams. Thus, the excess flows are collected for the dry periods that is not enough for water demand. Reservoirs are built to ensure drinking, use and irrigation water, to generate electric
power, to avoid floods. The volume and flow of a stream vary with physical characteristics of the watersheds and stream systems.
The amounts of hydrologic phenomena in the future should be known to develop water supplies and to provide optimum benefit of these supplies. As the hydrologic phenomena occur under the effects of many factors, these phenomena exhibit differences. Therefore, statistical methods are used to predict hydrologic phenomena in the future time.
This study was conducted to determine problems that will appear due to flood and drought in Kelkit stream watershed, and to obtain information that deal with quality and quantity of water that is necessary for planning, projecting, building and management of hydraulic structures on Kelkit stream.
Material and Methods
In this study, the daily flow values measured from 1938 to 1988 in the flow station numbered 1401 that is operated by General Directorate of Electric Power Research Survey and Development dministration (EIE) were used. The flow station was situated nearby Fatlı bridge in Tokat.
Kelkit stream is came into existance by joining together small streams that spring up Spikör, Pulur, Otlukbeli, Sarhan and Balaban mountains located in Erzincan, near to Kelkit district. Kelkit stream passes through Suşehri, Niksar and Erbaa plains and then, combines with Yeşilırmak river in the north of Erbaa plain. The watershed area of Kelkit stream is 11455 km 2 with 245.5 km length (Anonymous 1970).
Determination of daily extreme flows: The daily flow data were used to obtain the frequency of the daily extreme flows. Although the flow records have been gone on the mentioned flow station, the measurements after 1988 were disregarded, because of Kılıçkaya dam. Okman (1975) expressed that the flows between 1938 and 1988 are homogenous.
The daily extreme flows for every month and year were taken from the daily flows recorded from 1938 to 1988, and the remaining data were disregarded (Luthin 1973). Thus, the daily extreme flows that cover the daily flows remained out of treatment were supposed to be random and continuous variable (Linsley et al. 1958).
The return periods determination for the daily extreme flows: To obtain the frequency lines of the daily extreme flows, normal, log normal, three parameter log normal, two parameter gamma, Pearson type III (three parameter gamma), log Pearson type III (three parameter log gamma), extreme value type I (Fisher-Tippet type I) and three parameter Weibull (Fisher-Tippet type III) probability distributions were used. The frequency lines were determined according to main relationships of the mentioned distributions. The parameters of these distributions were predicted by the method of moments.
The probability levels based on normal probability distribution for the frequency lines of the daily extreme flows were calculated by Equation 1 (Chow et al. 1988). The parameters in the equation were predicted by. Okman (1994).
_(x—Iix
F(X 20. x) — 1 le x2 dx x (1) ag2n —GoTo obtain the frequency lines of the daily extreme flows based on log normal probability distribution, logaritmic transformation of the observed flows (y=lnx) were made, and then the probability levels of the transformed flows were calculated by Equation 1 (Tülücü
1990).
Beyazıt (1981) stated that three parameter log normal distribution can be used if the least value of the variable was wanted to be bigger than zero. Therefore, Sangal and Biswas(1970) expressed that a parameter as xo should be subtracted from the observed flows. The x o parameter giyen in Equation 2 may be positive, zero or negative.
2 ax xo =
2(Px
To obtain the frequency lines of the daily extreme flows according to three parameter log normal probability distribution, the xo parameter was subtracted from the observed flows (x-xo ), and then logaritmic transformation of these values y =[In(x-xo )] was made. The probability levels of the transformed values were calculated by Equation 1 (Sangal and Biswas 1970).
The probability levels based on two parameter gamma and Pearson type III probability distributions for the frequency lines of the daily extreme flows were calculated by using Equation 3 (Diler 1982).
paP x (a p -1) —pp (x—x o )
F(X — jx e dx (3)
1-(a ) X p O
xo , base limit, is bigger than zero for three parameter gamma (Pearson type III) probability distribution, but, xo is zero for two parameter gamma probability distribution. Gamma distribution's a p , 13 p , and xo parameters were predicted by the following equations (Chow et al. 1988). Px x o a pPp 2 r, 2 ax = apPp 2 C S — (6) ap
Logarithmic transformation of the observed flows (y=lnx) were made to obtain the frequency lines of Kelkit stream's daily extreme flows for log Pearson type III probability distribution, and then the probability levels of the transformed flows were calculated by Equation 3 (Haan 1977).
The probability levels of Kelkit stream's daily extreme flows were obtained from the following equations to form the frequency lines based on extreme value type I probability distribution (Beyazıt 1981).
F(X x) = e [—eFY) ( 7 ) y = ai(x -131) (8) (2) (4) (5)
R Fa (x. )= —
N
(15)
258 TARIM BILIMLERI DERGISI 2002, Cilt 8, Sayı 3
The formulas for al and (3, parameters can be
= max(i)14xi )– Fa (xi
presented as follows (Chow et al. 1988). (16)
0.5772 Px =
P
ı
(9)
a
i
1.2825 cix (10)a
l
The probability levels of the lowest flows of Kelkit stream were determined by the following equations to obtain the frequency lines based on three parameter Weibull probability distribution (Haan 1977). Beyazıt (1981) stressed that Weibull probability distribution were used for the frequency lines of the lowest values.
F(X s x) = 1– e–Y
(11)
x – xa aw Y
[3 – x w o
xs and 13, parameters can be predicted by using Equation 13 and 14 (Haan 1977).
Pw =Px +axA(aw) (13)
xo = (3w – ax13(aw) (14)
A(aw) and B(aw) taken from the table prepared by Haan (1977) based on skewness coefficient.
Determination of probability distribution fitted to the daily extreme flows: Goodness of fit to the mentioned distributions in this study for Kelkit stream's daily extreme flows was obtained from Simirnov-Kolmogorov test (Beyazıt, 1981). For this reason, Kelkit stream's daily extreme flows were arranged in an ascending order from the smallest to the largest value (x, x2S x3s xN) and then the rank of each observation was determined. The frequency of each observation was obtained from Equation 15 by dividing rank of each observation
with,
the total numbers of observations.After calculating the frequencies of the observations, the probability level [F(xi)] of each flow amount that is equal ith observation or lower than ith observation was determined for each distribution taken into account. Then, the differences between the frequencies [Fs(x,)] and
probability levels [F(x,)] of the observations were calculated. Finally, the highest difference for each distribution was determined by Equation 16, and the distribution that had the least value among the highest differences from the distributions was accepted for Kelkit stream's the daily extreme flows.
Notation:
NX : average of the sample
ax : standard deviation of the sample X0 : base limit (bigger than zero)
: median of the sample
ap : shape parameter for two parameter gama and Pearson ııı distribution : scale parameter for two parameter
gama and Pearson ııı distribution Rap): gamma function
CS : skewness coefficient of the sample al : parameter for extreme value type I
distribution
: parameter for extreme value type I distribution
aw : parameter for Weibull distribution
p
w
:
parameter for Weibull distribution Fa(xı): the frequency for ith observation F(xı): the probability level for theobservation (equal to ith observation or lower than that)
R : the rank of ith observation N : total observation
Results and Discussion
For Kelkit stream's daily extreme flows, Simirnov-Kolmogorov test results were giyen in Table 1 and Table 2. The selected maximum daily flows of each month have been fitted better to, Pearson type III for the second, fifth and tenth months; log Pearson type III for the first, eleventh and twelveth months; log normal for the third, seventh and eighth months; three parameters log normal for the fourth and nineth months and extrema value type I for the sixth month (Table 1). The minimum daily flows of each month have been fitted better to three parameter Weibull for the first, second, nineth and tenth months; three parameter log normal for the third, fourth and eleventh months; Pearson type III for the fifth and sixth months; extreme value type I for the seventh month; normal for the eighth month and log normal for the twelveth month (Table 2). The daily extreme flows of each year have been fitted beter to Pearson type III distribution (Table 1 and Table 2).
Kelkit stream's daily extreme flows for every month and year were skewed to the right according to skewness coefficients, but the fifth month's daily maximum flows were skewed to the left.
The return periods of the flows that are equal to the averages of the daily extreme flows or lower than their averages varied from 1.37 to 1.99 year for each month's daily maximum flows, and from 1.62 to 2.00 year for each month's daily minimum flows respectively, 1.93 year for each year's daily maximum flows and 1.90 year for each year's daily minimum flows (Table 3 and Table 4). aw >O, 13w >O, )c..x. (12)
The daily extreme flows taken from the frequency lines for the probability levels of 1 %, 2 %, 20 %, 50 %, 80 % and 99 % are shown in Table 3 and Table 4. The daily extreme flows obtained from the frequency lines for 50 % probability level are smaller than the averages of the daily extreme flows. But the flows taken the frequency lines are close to the averages (Table 3 and Table 4). This is the result of Kelkit stream's daily extreme flows which have a skewed distribution. In other words, the daily extreme flows have scattered around average, median and mod by not being homogenous. Therefore, the standard deviations of the daily extreme flows were high. This is very clear for the daily maximum flows.
The duration curves based on 20 %, 50 % and 80 % probability levels are shown in Figure 1 and Figure 2. This is to obviously show the variation of the daily
extreme flows selected for every month. As it can be seen from these "figures, the difference between the average of the daily minimum flows selected for every month and the daily minimum flows for 50 % probability level were less than the daily maximum flows. There was no difference for the daily minimum flows of the eighth month as the eight month's daily minimum flows were fitted to normal probability distribution (Table 4).
As a result, it is very important to perform the frequency analysis of Kelkit stream flows to correctly determine the system capacities and management of hydraulic structures which have to be built on Kelkit stream. Therefore, if the design flow of the hydraulic structures is correctly predicted, the building cost and the flood damages will be decreased, and the demand of water will be supplied.
Table 1. Smirnov-Kolmogorov test results for the daily maximum flows (Amin)
Months
Probability distributions Goodness of
fit
N LN LN III . G II P III LP III EV I
10 0.2108 0.1203 0.1982 0.4753 0.0829 0.1004 0.1761 0.0829 11 0.2445 0.1667 0.2247 0.4771 0.1995 0.1051 0.2598 0.1051 12 0.2282 0.0979 0.1309 0.3076 0.0948 0.0518 0.1740 0.0518 1 0.3251 0.1625 0.2915 0.6796 0.2694 0.0829 0.2878 0.0829 2 0.1681 0.1094 0.1067 0.2806 0.0787 0.0975 0.1651 0.0787 3 0.1463 0.0832 0.1129 0.7044 0.1153 0.2037 0.0935 0.0832 4 0.0664 0.0729 0.0515 0.7897 0.0614 0.1040 0.1276 0.0515 5 0.0668 0.1180 1.0000 0.0578 0.2069 0.0976 0.0578 6 0.1073 0.0912 0.0634 0.4254 0.0687 0.5362 0.0525 0.0525 7 0.1344 0.0678 0.0918 0.1257 0.0681 ... 0.0776 0.0678 8 0.2040 0.0913 0.1167 0.4137 0.1254 0.3824 0.1349 0.0913 9 0.2064 0.1529 0.0930 0.4767 0.1371 0.1618 0.1500 0.0930 Annual 0.0558 0.0976 ** 0.9584 0.0494 0.2258 0.0897 0.0494
** Negative values appeared
"** Some values were not fitted to the distribution due to the a, parameter very high and the p parameter very small
Table 2. Smirnov-Kolmogorov test results for the daily minimum flows (Amin)
Months
Probability distributions Goodness
of fit
N LN LN III G II P III LP III EVI W III
10 0.1132 0.1021 0.1332 0.2258 0.0849 0.1999 0.0981 0.0802 0.0802 11 0.1826 0.1204 0.0887 0.1625 0.1440 0.2264 0.1325 0.1386 0.0887 12 0.1747 0.0901 0.0973 0.1335 0.1205 0.2235 0.1158 0.1129 0.0901 1 0.1669 0.1053 0.1178 0.4596 0.0966 0.1154 0.1274 0.0956 0.0956 2 0.2201 0.2259 0.1678 0.4388 0.1449 0.1511 0.1823 0.1448 0.1448 3 0.1690 0.0692 0.0566 0.3066 0.0730 0.1182 0.1142 0.1115 0.0566 4 0.1917 0.1194 0.0835 0.1671 0.1322 *M* 0.1391 0.1105 0.0835 5 0.1137 0.1506 ** 0.9694 0.1091 0.7104 0.1643 0.1172 0.1091 6 0.1279 0.0868 0.0736 0.2245 0.0485 *** 0.0718 0.0595 0.0485 7 0.1495 0.1670 0.3659 0.6620 0.4323 0.1860 0.1207 0.4905 0.1207 8 0.0544 0.1170 fr. 0.7745 0.0602 0.1155 0.0881 0.0587 0.0544 9 0.1179 0.1050 0.2383 0.0872 0.6234 0.0906 0.0790 0.0790 Annual 0.0744 0.1059 0.0927 0.7525 0.0548 0.1534 0.0863 0.0624 0.0548
** Negative values appeared
*** Some values were not fitted to the distribution due to the et parameter very high and the [3 parameter very small
Table 3. The daily maximum flows (m3/s)
P (X5x) 0/0 10 11 12 1 2 3 Months 4 5 6 7 8 9 Annual 01 14.1 12.4 13.3 17.3 14.1 61.2 116.3 96.6 62.7 12.7 6.3 7.5 202.6 02 14.2 13.3 14.5 17.7 14.7 70.8 156.2 128.2 70.8 14.7 7.1 8.1 232.6 20 17.3 20.9 24.1 22.5 25.8 135.0 335.6 272.3 115.2 27.5 11.9 12.1 376.2 50 28.5 34.5 38.2 32.3 51.3 211.5 462.1 376.1 157.4 42.5 17.1 17.2 485.9 80 55.1 68.2 67.0 57.2 100.6 331.2 590.4 483.0 214.1 65.7 24.6 25.8 604.2 99 153.4 373.3 235.4 289.3 261.1 730.7 820.8 679.2 369.3 141.7 46.5 57.3 834.3 Aver. 37.4 54.7 50.7 47.8 67.8 241.0 468.2 384.2 167.7 49.1 19.0 19.9 491.6 T (Year) 1.57 1.38 1.49 1.37 1.58 1.68 1.99 1.97 1.75 1.67 1.71 1.63 1.93
700 XII II IV VI VIII Months xıı ıı IV Months VI VIII --a-- 0,2 0,5 --&- 0,8 - A ver.
260 TARIM BILIMLERI DERGISI 2002, Cilt 8, Sayı 3
Table 4. The daily minimum flows (m 3/s)
P (XSx) % 10 11 12 1 2 3 Months 4 5 6 7 8 9 Annual 01 6.6 10.6 8.2 9.7 9.7 7.1 47.6 37.6 12.1 4.9 2.4 4.7 3.7 02 7.0 10.8 9.1 10.0 10.0 9.0 52.2 47.8 13.5 5.7 3.4 5.0 4.3 20 9.7 13.0 13.9 13.2 14.0 20.9 82.9 96.8 24.7 10.0 7.8 7.6 7.3 50 12.5 16.2 18.6 18.3 20.7 33.8 118.9 134.4 39.0 14.1 10.9 10.4 9.6 80 15.7 22.2 25.0 25.8 31.2 52.5 174.6 175.1 60.0 19.6 14.0 13.7 12.3 99 21.8 48.3 42.0 44.9 59.0 108.3 357.7 254.4 115.8 34.6 19.5 20.3 17.6 Avar. 12.8 17.9 19.8 20.0 23.3 38.3 132.3 136.6 43.6 15.1 10.9 10.8 9.8 T (Yaar) 1.86 1.62 1.75 1.70 1.68 1.69 1.68 1.93 1.71 1.75 2.00 1.84 1.90
Figure 1. Duration curves for Kelkit Stream's the daily maximum flows
References
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Yöntemler Rehberi. Enerji ve Tabii Kaynaklar Bakanlığı Devlet Su İşleri Genel Müdürlüğü, Bursa, 82s.
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Figure 2. Duration curves for Kelkit Stream's the daily minimum flows
Luthin, J. N. 1973. Drainage Engineering. Robert E. Kreiger Publishing Company, New York, 250s.
Okman, C. 1975. Tekrarlanma Analizlerinde Hidrolojik Verilerin Seçimi. Topraksu Teknik Dergisi, 40-41, 54-58.
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