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Section 2. Geography
Isakova Aziza Yadgarovna, Trafimov Gennadiy Nikolaevich, Prof. Dr., National University of Uzbekistan named after Mirzo Ulugbek, teacher, of the Faculty of geology and geography, Tashkent city, Uzbekistan E-mail: isakovaziza@mail.ru
ESTIMATION EMPIRIC SUPPLY OF MAXIMUM FLOWS OF MUD FLOODS ON SMALL RIVERS IN UZBEKISTAN
Abstract: The paper deals with the estimation of empirical supply of maximum mudflows recorded on small rivers in Uzbekistan.
Keywords: small rivers, intense rains, mudslides, the maximum cost nanosovodnye mudslides, debris flows repeatability.
Introduction. Calculations of maximum costs are one of the most difficult tasks in the design and construction of various kinds of structures on rivers. Underestimation of maximum costs leads to the destruction of structures, the flooding of coastal areas and often to human casualties. In turn, the overesti-mation of these highs increases the cost of facilities and reduces their economic efficiency. The problem of calculating maxima is complicated many times, when it comes to mudflows. Let us briefly dwell on the reasons that determine this problem:
1. Litter flows, characterized by large, or better to say, outstanding maximum costs - an objective phenomenon, one can say the necessary in the life of the river;
2. There are no data on the composition of the mud-flow mass of "classical" mudflows;
3. In Uzbekistan, so-called "nano-driven mudslides" are formed on small, especially low-mountain rivers. The main reason for the formation of such mudflows is intense rainfall;
4. In the overwhelming majority of cases, the maximum flow of a large flood is characterized as a mudflow, especially if its passage was accompanied by some destruction;
5. Information on the maximum expenditure of mudflows hydrologists receive on the basis of a survey of "traces of mudflows", which are often not expressed throughout the course of the riverbed, but only in some of its places;
6. There are no information on mudflows for 33.7% of the rivers in Uzbekistan (29.3% for the rivers of the Amu Darya basin and 42.7% for the rivers of the Syr Darya basin). For more than 70% of cases of registered mudflows, it is impossible to determine the place (range) of estimating maxima [9];
7. It is difficult to give a probabilistic estimate of such maxima, since firstly, the series of observations are very short, and secondly, initially, for example, because of paragraphs 4 and 6, the rows are not homogeneous.
Nevertheless, the needs of practice require at least an approximate probabilistic evaluation of such maxima. As can be seen from what has been said above, the only possibility for statistical calculations remains the use of annual maxima recorded at the water station.
Results and its discussion. The maxima of the mudflows, given in the reference materials [10], should be used as a definite reference point, or "background" of the general picture of the mudflow hazard of the watercourse.
For very small rivers with catchment areas 10-20 km2, it is possible to include the data of the mudflow catalog and other reference books on maxima, since due to the small extent of the watercourses, the change in the location of the measurement range of this maximum will not have a significant effect on its size.
As is known, the probability of excess (security) of the maximum flow is determined by the type (class) of the structure, its cost, operating conditions, etc. Usually, according to construction standards, the security of maximum costs varies from 0.01 to 1% and, if it is an annual maximum, a series of observations of 100 years or more are required for such an assessment. In the absence of such data it is recommended to use those or other theoretical laws of distribution of variables. In hydrology, the distributions of K. Pearson, S. N. Kritsky, and MF Menkel are used most often, and the distribution of E. Gambel is less often [2, 6, 7].
A large number of papers have been devoted to the conditions for the applicability of a particular theoretical distribution in hydrology. Here I would like to note that very short, according to the concepts of mathematical statistics, series of observations of the maximum annual water discharge and the statistical features of these series in most cases do not give a positive result when using one or another theoretical distribution law. So, SN Kritsky and MF Menkel note that "the points depicting the observed values of the maximum runoff on the availability graph are sometimes deviating significantly from the upper section of the theoretical curve ..." [4].
The formation of outstanding maximum costs is an objective phenomenon caused by a combination of random processes occurring in the river basin. This can be a rain of exceptional intensity, breakthroughs in the dams of high-mountain lakes, sudden movements of glaciers, the descent of landslides and swamps, partition walls of rivers, etc. Therefore, before carrying out statistical calculations, it is necessary to make sure that such situations are possible in the future. For example, on the Isfayramsy River in 1963, with the breakthrough of the dam of the high-altitude lake Yashinkul, an extraordinary maximum of 1770 m3/s was formed. Note that the second maximum in the ranked series was "only" 286 m3/s. In the absence of other such objects in the river basin, the repetition of such maxima is excluded [10].
The basis of statistical calculations and their results is a series of observations. To describe the empirical distribution function and extrapolate these calculations to the region of rare occurrence in hydrology, a number of formulas are used. Most often these are the formulas: m - 0,5
- A. Hazena p =
•100% ;
- N. N. Chegodaeva p = ™ ^ ^ ■100% ;
n + 0,4
- S. N. Kritsky-M.F. Menckel p =
(1) (2)
m n +1
- E. G. Blokhinova p -
- D. Cowden p =
1
m - 0,4
n + 0,2
f
m
•100% ;
1
vn +1 \y[n 2^
■100%; (3)
(4)
(5)
Note that the formula of Kritsky-Menckel completely corresponds to the previously proposed formula of E. Gambel, and the formula of D. Cowden is often called the formula "min / max". All the above formulas do not take into account either the absolute variations of the original variables or their statistical characteristics. The main arguments in them are the ordinal number of the variable m and the total length of the series n. A detailed analysis of such formulas was made by Yu. B. Vinogradov and MA Mamedov [1, 5].
Unfortunately, in mathematical statistics, methods of probabilistic estimation of "sharply allocated quantities", which are often called "emissions", are practically not developed [3]. Mathematicians, mainly interested in the question of excluding such "emissions", distorting, in their opinion, the statistical characteristics of the series. The procedure for excluding such anomalous values is elaborated in detail. To hydrologists, and especially to practical hydrologists, it is precisely such sharply differing members of the series.
As is known, mathematical statistics use those or other criteria for estimating both the laws of distributions and the characteristics of aggregates. Among these criteria, Irwin's criterion [8] is intended to give an answer to the question of whether the extreme terms of a series of a given set belong:
(6)
A = xL_xi±I
o„
where x . and x.+1 are adjacent terms of the ranked series To calculate empirical security, we have proposed the formula
n
P* =
mn-A2
(7)
i(n + X1)
The verification of this formula on highly asymmetric series of maximum annual water discharge for a number of rivers showed that:
- it is the least biased in comparison with the formulas (1-5);
- with a small difference between the neighboring
terms of the series, we practically get an unbiased esti-^ m
mate ot the security p* = — ;
- for sharply asymmetric series, formula (7) gives the smallest variance of estimated p *;
- we add that with increasing n the probability of exceeding p * tends to 1, which indicates its consistency.
To demonstrate the advantages of formula (7) in calculating the empirical abundances of the maxima, we
give the observation data on the Shaugazsai River (left tributary of the Akhangaran River). The catchment area of the river to the water point (Karata§ tract) is 65.8 km2, the length of the watercourse from its source to the water point is 15 km, the total length of the river is 22 km, the average height of the catchment area is 1.66 km.
There are a number of observations from 1951 to 2004, i.e. somewhat more than 50 years. The maximum water discharge measured at the station was 172 m3/s (July 27, 1964), the smallest annual maximum of 0.55 m3/s, which was observed twice in 1962 and 1977. The average of the maximum costs is 7.46 m3/s. The coefficient of variation of this series is 3.20, and the asymmetry coefficient is 6.61. The results of calculating the empirical security according to the above formulas for the upper part of the curve (the first 10 members of the ranked series) are given in (table 1).
Table 1. - Empirical assurances of the maximum water flow of the river Shaugazsay (without a mudflow maximum)
Empirical assurance of maximum water flow
№ Qmax m3/S m m m-0,3 m-0,5 m-0,4 1 , m s rr^ + 0.5) n +1 n m ,2 mn-A
n n+1 n+0,4 n n+0,2 n+1+kmZ 2 n(n+A )
1. 172 0.0189 0.0185 0.0131 0.0094 0.0113 0.0771 0.0017 0.0042
2. 33.5 0.0377 0.0370 0.0318 0.0283 0.0301 0.0937 0.0270 0.0376
3. 24.0 0.0566 0.0556 0.0506 0.0472 0.0489 0.1104 0.0466 0.0565
4. 19.3 0.0755 0.0741 0.0693 0.0660 0.0677 0.127 0.0659 0.0755
5. 17.2 0.0943 0.0926 0.0880 0.0849 0.0865 0.1436 0.0893 0.0941
6. 8.79 0.1132 0.1111 0.1067 0.1038 0.1053 0.1602 0.1083 0.1132
7. 7.58 0.1321 0.1296 0.1255 0.1226 0.1241 0.1768 0.1272 0.1321
8. 6.23 0.1509 0.1481 0.1442 0.1415 0.1429 0.1935 0.1463 0.1509
9. 5.29 0.1698 0.1667 0.1629 0.1604 0.1617 0.2101 0.1651 0.1698
10. 5.09 0.1887 0.1852 0.1816 0.1792 0.1805 0.2267 0.1836 0.1887
Table 2.- Empirical assurances of the maximum water flow of the river Shaugazsay (with a maximum of a maximum)
№ Empirical assurance of maximum water flow
Qmax m3/s m n m n+1 m-0,3 m-0,5 m-0,4 1 , m s m ,2 mn-A
n+0,4 n n+0,2 i- A r 1 °.5) n +1 n n+1+kmZ 2 n(n+A )
1 2 3 4 5 6 7 8 9 10
1. 274 0.0189 0.0185 0.0131 0.0094 0.0113 0.0771 0.0011 0.0024
2. 33.5 0.0377 0.0370 0.0318 0.0283 0.0301 0.0937 0.0300 0.0377
3. 24.0 0.0566 0.0556 0.0506 0.0472 0.0489 0.1104 0.0495 0.0566
4. 19.3 0.0755 0.0741 0.0693 0.0660 0.0677 0.127 0.0687 0.0755
5. 17.2 0.0943 0.0926 0.0880 0.0849 0.0865 0.1436 0.0872 0.0942
6. 8.79 0.1132 0.1111 0.1067 0.1038 0.1053 0.1602 0.1093 0.1132
7. 7.58 0.1321 0.1296 0.1255 0.1226 0.1241 0.1768 0.1281 0.1321
1 2 3 4 5 6 7 8 9 10
8. 6.23 0.1509 0.1481 0.1442 0.1415 0.1429 0.1935 0.1469 0.1509
9. 5.29 0.1698 0.1667 0.1629 0.1604 0.1617 0.2101 0.1657 0.1698
10. 5.09 0.1887 0.1852 0.1816 0.1792 0.1805 0.2267 0.1842 0.1887
Based on a study of the traces of the catastrophic flood in 1964, the maximum flow was estimated at 274 m3/s for a track located slightly below the water point. Naturally, the statistical characteristics have changed: the average value of the flow is 9,38 m3/s, the coefficients ofvariation and asymmetry are 4.00 and 7.00, respectively (table 2).
When analyzing the data (tables 1 and 2), it is noteworthy that if the empirical supply is equal to 0,0042 a for the flow rate of 172 m3/s, naturally, the supply security of 274 m3/s is much less and according to formula (7) it is equal to 0,0024, i.e. almost 2 times less. Obviously, formulas (1-5) of such an adjustment can not fulfill the estimation of empirical security.
Obviously, with the above statistical characteristics of the series, the use of the three-parameter distribution of Kritsky-Menckel, traditionally used in hydrology, is impossible and, to compare the empirical assumptions
calculated from formulas (1-7) with the theoretical, we used the distribution of K. Pearson. For illustration, a maximum of 1% of the provision is selected, the results of the calculations are presented in (table 3).
When comparing the maxima calculated from the empirical formulas with their magnitude, obtained from the distribution of K. Pearson, it is clear that the amount of expenditure obtained by the formula "min / max" is unreasonably overestimated. It is understandable to overestimate the highs calculated by the formulas (1-5), because with a decrease in the number of points, the security in the region of its small values inevitably increases. A different picture of the change in the maxima of1% of security, calculated by the formula (7). We add that on average, the smallest deviation of the empirical peaks from their Pearson value was obtained by the formula (7) - 22.0%.
Table 3.- The maximum water expenditure of 1% of provision, calculated by different formulas (without taking into account the mudflow maximum)
Number of points Maximum water consumption 1% of supply, m3/s
According m m m-0,3 m-0,5 m-0,4 1 „ m , -i—ri-T + 0.5) n +1 n m mn-X2
to Pearson n n+1 n+0,4 n n+0,2 n+1+kmZ 2 n(n+X )
53. 120 222 220 180 152 166 740 91,2 120
26. 161 310 306 251 212 232 764 121 138
18. 187 355 348 287 244 266 765 142 143
10. 240 430 418 348 297 323 770 174 120
Table 4.- The maximum water expenditure of 1% of the provision, calculated by different formulas (taking into account the mudflow maximum)
Number of points Maximum water consumption 1% of supply, m3/s
According m m m-0,3 m-0,5 m-0,4 1 , m „ -i—ri-T + 0.5) n +1 n m ,2 mn-X
to Pearson n n+1 n+0,4 n n+0,2 n+1+kmZ 2 nin+X )
53. 186 354 351 351 238 261 938 127 159
26. 255 496 488 399 337 369 1230 178 188
18. 300 568 557 459 389 425 1230 212 199
10. 381 687 668 556 475 516 1230 271 174
Returning to the problem of including, or not including, mudslides in the general long-term series of observations, we refer to the opinion of I. F. Gorosh-kov [2].
Guided by these considerations, we performed calculations of a maximum of 1% of the availability of the Shaugasai River, taking into account the maximum of 1964, obtained on the basis of hydraulic calculations for
the cross-section of the channel located below the hydrostatic, by about 1.5-2 km. This maximum, as noted above, is estimated by the harvesters at a rate of 274 m3/s [9]. Using a new series of maxima, we repeated the calculation of 1% of the water flow for a different number of terms in the series (table 4).
As can be seen from (tables 3 and 4), the mudslides are higher than the peaks observed at the water station by about 1.4-1.6 times, which is to some extent due to the mismatching of the calculated alignments. The smallest average deviation of the expenditure of1% of the security compared with the Pearson curve was obtained from the formula (5) - 28.7%. However, according to formula (7), the analogous deviation is 32.3%.
Apparently, we should add that the ratio of the largest 1% maximum (with a small number of terms of the series), to the smallest (at the maximum length of the series), in calculations by formulas (1-4) is approximately 1.9-2.0. This ratio is substantially smaller for formulas (5 and 8), but, as already noted, the calculation by the formula "min / max" greatly overestimates, in comparison with the others, the value of 1% of the maximum, which raises doubts about the expediency of its use in
practical calculations. This conclusion, in our opinion, is important when using short series and it is impossible to supplement these series with any data, for example, using observation materials on the analog rivers.
Conclusions. So, you can summarize the findings:
- an estimate of the empirical supply of maximum costs by formula (7) is the least biased compared to the rest of the biased estimates;
- its small variability speaks of its solvency;
- estimating the maximums in the region of rare frequency of occurrence, as Yu. B. Vinogradov points out, "... it should not be embarrassing that the minimal term of the variational series for any n will have the estimate p * = 1" [1];
- the empirical security calculated by formula (7) is least dependent on the number of terms in the series;
- in practical calculations for small rivers, it is expedient to take into account the so-called mudflow peaks, and the use of formula (7) ensures the smallest increase in design values (about 1.3-1.4 times), which, to some extent, prevents justified overestima-tion of highs and little justified rise in the cost of the designed facilities.
References:
Vinogradov Yu. B. Mathematical modeling offlow formation processes.-L.: Gidrometeoizdat,- 1988.- P. 203-251. Goroshkov I. F. Hydrological calculations.- L .: Gidrometeoizdat,- 1979.- P. 214-231. Dlin A. M. Mathematical statistics in engineering.- M.: Science,- 1958.- P. 49-51. Kritsky S. N., Menckel M. F. Hydrological basis of river flow management.- M.: Science,- 1981.- P. 27-77. Mammadov Magbet Adil ogli. Analysis of the conditions of formation and methods for calculating the maximum expenditure of mountain rivers (on the example of the rivers of Transcaucasia and Dagestan).- Author's abstract. Diss. Doct. geogr. sciences.-L., GGI,- 1984.- 39 p.
International guidelines on methods for calculating basic hydrological characteristics. -L.: Gidrometeoizdat,-1984.- P. 23-63.
Manual on the calculation of hydrological characteristics.-L.: Gidrometeoizdat,- 1984.- P. 41-70.
8. Smirnov N. V., Dunin-Barkovsky I. V. Course of the theory of probability and mathematical statistics (for technical applications).- Moscow: Nauka,- 1965.- 510 p.
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