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With v = = = = 0.226.
The relative root-mean-square error of the average long-term value of the annual river flow for a given period is equal to:
5,65 %
The relative standard error of the coefficient of variability C v when it is determined by the method of moments is:
18,12 %.
The length of the series is considered sufficient to determine Q o and C v if 5-10%, and 10-15%. The value of the average annual runoff under this condition is called the runoff rate. If and (or) is greater than the allowable error, it is necessary to lengthen the series of observations.
3. Determination of the runoff rate in the absence of data by the method of hydrological analogies
The analogue river is selected according to:
– similarity of climatic characteristics;
– synchronism of runoff fluctuations in time;
- homogeneity of the relief, soils, hydrogeological conditions, close degree of coverage of the watershed with forests and swamps;
- the ratio of catchment areas, which should not differ by more than 10 times;
- the absence of factors that distort the runoff (dam construction, withdrawal and discharge of water).
An analogue river must have a long-term period of hydrometric observations to accurately determine the flow rate and at least 6 years of parallel observations with the river under study.
Annual flow modules of the Ucheba river and the analogue river Table 5.
| year | M, l/s*km2 | Man, l/s*km2 |
| 1963 | 5,86 | 6,66 |
| 1964 | 4,72 | 4,55 |
| 1965 | 3,88 | 3,23 |
| 1966 | 3,29 | 4,24 |
| 1967 | 5,15 | 6,22 |
| 1968 | 4,82 | 8,19 |
| 1969 | 6,86 | 7,98 |
| 1970 | 4,71 | 3,74 |
| 1971 | 3,17 | 3,03 |
| 1972 | 3,72 | 5,85 |
| 1973 | 6,29 | 8,16 |
| 1974 | 5,24 | 5,67 |
| 1975 | 4,36 | 3,97 |
| 1976 | 3,61 | 5,15 |
| 1977 | 5,70 | 7,49 |
| 1978 | 5,37 | 7,00 |
Picture 1.
Graph of the relationship between the average annual runoff modules of the Ucheva River and the analogue river
According to the communication schedule, M o is 4.9 l / s.km 2
Q O \u003d M o * F;
Annual runoff variability coefficient:
C v \u003d A C va,
where C v is the coefficient of runoff variability in the design section;
C va - in the alignment of the analogue river;
Моа is the mean annual runoff of the analogous river;
A is the tangent of the slope of the communication graph.
In our case:
With v = 0.226; A=1.72; M oa \u003d 5.7 l / s * km 2;
Finally, we accept M o =4.9; l / s * km 2, Q O \u003d 163.66 m 3 / s, C v \u003d 0.046.
4. Construction and verification of the annual runoff supply curve
In this work, it is required to construct an annual runoff probability curve using a three-parameter gamma distribution curve. To do this, it is necessary to calculate three parameters: Q o - the average long-term value (norm) of the annual runoff, C v and C s of the annual runoff.
Using the results of calculations of the first part of the work for r. Laba, we have Q O = 169.79 m 3 / s, C v \u003d 0.226.
For a given river, we take C s =2С v =0.452 with subsequent verification.
The ordinates of the curve are determined depending on the coefficient C v according to the tables compiled by S.N. Kritsky and M.F. Menkel for C s =2С v .To improve the accuracy of the curve, it is necessary to take into account the hundredths of C v and interpolate between adjacent columns of numbers. Enter the ordinates of the supply curve in the table.
Coordinates of the theoretical endowment curve. Table 6
| Provision, Р% | 0,01 | 0,1 | 1 | 5 | 10 | 25 | 50 | 75 | 90 | 95 | 99 | 99,9 |
| Curve ordinates (Cr) | 2,22 | 1,96 | 1,67 | 1,45 | 1,33 | 1,16 | 0,98 | 0,82 | 0,69 | 0,59 | 0,51 | – |
Construct a security curve on a probability cell and check its actual observational data. (Fig.2)
Table 7
Data to test the theoretical curve
| No. p / p | Modular coefficients descending K | Actual security
P = |
Years corresponding to K |
| 1 | 1,43 | 5,9 | 1969 |
| 2 | 1,31 | 11,8 | 1973 |
| 3 | 1,22 | 17,6 | 1963 |
| 4 | 1,19 | 23,5 | 1977 |
| 5 | 1,12 | 29,4 | 1978 |
| 6 | 1,09 | 35,3 | 1974 |
| 7 | 1,07 | 41,2 | 1967 |
| 8 | 1,00 | 47,1 | 1968 |
| 9 | 0,98 | 52,9 | 1964 |
| 10 | 0,98 | 58,8 | 1970 |
| 11 | 0,91 | 64,7 | 1975 |
| 12 | 0,81 | 70,1 | 1965 |
| 13 | 0,78 | 76,5 | 1972 |
| 14 | 0,75 | 82,4 | 1976 |
| 15 | 0,68 | 88,2 | 1966 |
| 16 | 0,66 | 94,1 | 1971 |
To do this, the modular coefficients of annual costs must be arranged in descending order and for each of them, calculate its actual provision according to the formula Р = , where Р is the provision of a member of the series, located in descending order;
m is the serial number of a member of the series;
n is the number of members of the series.
As can be seen from the last graph, the plotted points average the theoretical curve, which means that the curve is built correctly and the ratio C s =2With v corresponds to reality.
The calculation is divided into two parts:
a) off-season distribution, which is of the greatest importance;
b) intra-seasonal distribution (by months and decades), established with some schematization.
The calculation is carried out according to hydrological years, i.e. for years beginning with a high-water season. The dates of the seasons begin the same for all years of observations, rounded up to a whole month. The duration of the high-water season is assigned so that the high water is placed within the boundaries of the season both in the years with the earliest onset and with the latest end date.
In the assignment, the duration of the season can be taken as follows: spring-April, May, June; summer-autumn - July, August, September, October, November; winter - December and January, February, March of the next year.
The amount of runoff for individual seasons and periods is determined by the sum of average monthly flows. In the last year, expenses for 3 months (I, II, III) of the first year are added to the December expense.
| Calculation of the intra-annual distribution of the runoff of the Ucheba river by the layout method (off-season distribution). Table 8 | |||||||||||||
| № | Year | Water consumption for the winter season (limiting season) | winter runoff | Qm runoff for a low-water low-water period | To | K-1 | (K-1)2 | Water discharges in descending order (total runoff) | p=m/(n+1)*100% | ||||
| XII | I | II | winter | Spring | summer autumn | ||||||||
| 1 | 1963-64 | 74,56 | 40,88 | 73,95 | 189,39 | 883,25 | 1,08 | 0,08 | 0,00565 | 264,14 | 2043,52 | 814,36 | 5,9 |
| 2 | 1964-65 | 93,04 | 47,64 | 70,83 | 211,51 | 790,98 | 0,96 | -0,04 | 0,00138 | 255,06 | 1646,21 | 741,34 | 11,8 |
| 3 | 1965-66 | 68,53 | 40,62 | 75,27 | 184,42 | 679,62 | 0,83 | -0,17 | 0,02982 | 246,72 | 1575,96 | 693,86 | 17,6 |
| 4 | 1966-67 | 61,00 | 75,85 | 59,10 | 195,95 | 667,87 | 0,81 | -0,19 | 0,03497 | 240,35 | 1535,03 | 689,64 | 23,5 |
| 5 | 1967-68 | 39,76 | 40,88 | 51,36 | 132,00 | 730,81 | 0,89 | -0,11 | 0,01218 | 229,04 | 1456,13 | 673,52 | 29,4 |
| 6 | 1968-69 | 125,99 | 40,88 | 42,57 | 209,44 | 862,01 | 1,05 | 0,05 | 0,00243 | 228,15 | 1308,68 | 670,73 | 35,3 |
| 7 | 1969-70 | 83,02 | 65,79 | 91,54 | 240,35 | 869,70 | 1,06 | 0,06 | 0,00345 | 213,65 | 1277,64 | 652,57 | 41,2 |
| 8 | 1970-71 | 106,58 | 75,85 | 72,63 | 255,06 | 793,34 | 0,97 | -0,03 | 0,00117 | 211,51 | 1212,54 | 629,35 | 47,1 |
| 9 | 1971-72 | 99,09 | 61,94 | 52,62 | 213,65 | 631,92 | 0,77 | -0,23 | 0,05325 | 211,46 | 1207,80 | 598,81 | 52,9 |
| 10 | 1972-73 | 122,69 | 47,51 | 58,84 | 229,04 | 902,56 | 1,10 | 0,10 | 0,00974 | 209,63 | 1185,05 | 579,47 | 58,8 |
| 11 | 1973-74 | 82,97 | 49,59 | 78,90 | 211,46 | 1025,82 | 1,25 | 0,25 | 0,06187 | 209,44 | 1057,65 | 564,21 | 64,7 |
| 12 | 1974-75 | 102,30 | 68,10 | 76,32 | 246,72 | 917,45 | 1,12 | 0,12 | 0,01365 | 195,95 | 969,18 | 538,28 | 70,1 |
| 13 | 1975-76 | 77,21 | 70,42 | 80,52 | 228,15 | 792,36 | 0,96 | -0,04 | 0,00126 | 189,39 | 785,60 | 537,44 | 76,5 |
| 14 | 1976-77 | 69,20 | 72,73 | 67,70 | 209,63 | 747,07 | 0,91 | -0,09 | 0,00820 | 184,42 | 727,76 | 495,20 | 82,4 |
| 15 | 1977-78 | 48,28 | 49,04 | 56,55 | 153,87 | 843,51 | 1,03 | 0,03 | 0,00072 | 153,87 | 714,91 | 471,92 | 88,2 |
| 16 | 1978-63 | 140,06 | 77,36 | 46,72 | 264,14 | 1005,48 | 1,22 | 0,22 | 0,05017 | 132,00 | 679,69 | 418,27 | 94,1 |
| sum | 13143,75 | 16,00 | 0,00 | 0,28992 | |||||||||
Work description
During the period of high water (flood), part of the excess water is temporarily retained in the reservoir. In this case, there is a slight increase in the water level above the FSL, due to which a forced volume is formed and the high water (flood) hydrograph is transformed (flattened) into a discharge hydrograph. The formation of a forced volume equal to the accumulating part of the high water runoff makes it possible to reduce the maximum flow of water entering the downstream, and thereby prevent floods in the downstream sections of the river, as well as reduce the size of spillway hydraulic structures.
2. Initial data………………………………………………………………………………….…4
3. Determination of the average long-term value (norm) of the annual runoff in the presence of observational data……………………………………………………………………………..…….8
4. Determination of the coefficient of variability (variation) Сv of the annual flow………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………….10
5. Determination of the running norm with a lack of data by the method of hydrological analogy …………………………………………………………………………………………………………………………………………………………………
6. Construct and check the curve of annual flow availability……………………………………………………………………….…………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………….
7. Calculate the intra-annual distribution of runoff by the layout method for irrigation purposes with the estimated probability of exceeding Р=80%............................................ ................................................. ...................................21
8. Determination of the estimated maximum flow, melt water P = 1% in the absence of hydrometric observation data according to the formula……………….23
9. Construction of the bathygraphic curves of the reservoir…………………………………………………………………………………………………………………………………24
10. Determination of the minimum ULV water level……………………………………………………………………….……..26
28
12. Determination of the operation mode of the reservoir by the balance table-numerical calculation…………………………………………………………………..……………...30
13. Integral (calendar) flow and return curves……………………………………………………………………………………….34
14. Calculation of the reservoir of long-term regulation………………………………………………………………………………...36
15. Bibliographic list…………………………………………………………………………………
Water discharge is the volume of water flowing through the cross section of a river per unit time. Water flow is usually measured in cubic meters per second (m3/s). The average long-term water flow of the largest rivers of the republic, for example, the Irtysh, is 960 m/s, and the Syr Darya - 730 m/s.
The flow of water in rivers in a year is called the annual flow. For example, the annual flow of the Irtysh is 28,000 million m3. Water runoff determines surface water resources. The runoff is unevenly distributed throughout the territory of Kazakhstan, the volume of surface runoff is 59 km3. The amount of annual river flow depends primarily on the climate. In the flat regions of Kazakhstan, the annual runoff mainly depends on the nature of the distribution of snow cover and water reserves before the snow melts. Rainwater is almost completely used to moisten the topsoil and evaporate.
The main factor influencing the flow of mountain rivers is the relief. As the absolute height increases, the amount of annual precipitation increases. The moisture coefficient in the north of Kazakhstan is about one, and the annual flow is high, and there is more water in the river. The amount of runoff per square kilometer on the territory of Kazakhstan is on average 20,000 m3. Our republic is ahead of only Turkmenistan in terms of river flow. The flow of rivers varies with the seasons of the year. Plain rivers during the winter months provide 1% of the annual flow.
Reservoirs are built to regulate river flows. Water resources are equally used both in winter and in summer for the needs of the national economy. There are 168 reservoirs in our country, the largest of them are Bukhtarma and Kapchagai.
All solid material carried by the river is called solid runoff. The turbidity of the water depends on its volume. It is measured in grams of a substance contained in 1 m³ of water. The turbidity of lowland rivers is 100 g/m3, while in the middle and lower reaches it is 200 g/m3. The rivers of Western Kazakhstan carry a large amount of loose rocks, turbidity reaches 500-700 g/m3. The turbidity of mountain rivers increases downstream. Turbidity in the river is 650 g/m3, in the lower reaches of the Chu - 900 g/m3, in the Syr Darya 1200 g/m3.
Nutrition and river regime
Kazakhstani rivers have different nutrition: snow, rain, glacial and groundwater. There are no rivers with the same nutrition. The rivers of the flat part of the republic are divided into two types according to the nature of the supply: snow-rain and predominantly snow supply.
Snow-rain fed rivers include rivers located in the forest-steppe and steppe zones. The main ones of this type - Ishim and Tobol - overflow their banks in spring, 50% of the annual runoff falls in April-July. Rivers are first fed by melt water, then rain. Since the low water level is observed in January, at this time they feed on groundwater.
Rivers of the second type have exclusively spring flow (85-95% of the annual flow). This type of food includes rivers located in the desert and semi-desert zones - these are Nura, Ural, Sagyz, Turgay and Sarysu. The rise of water in these rivers is observed in the first half of spring. The main source of food is snow. The water level rises sharply in the spring when the snow melts. In the CIS countries, such a regime of rivers is called the Kazakhstani type. For example, 98% of its annual flow flows along the Nura River in a short time in spring. The lowest water level occurs in summer. Some rivers dry up completely. After the autumn rains, the water level in the river rises slightly, and in winter it drops again.
In the high-mountainous regions of Kazakhstan, rivers have a mixed type of food, but snow-glacier prevails. These are the Syrdarya, Ili, Karatal and Irtysh rivers. The level in them rises in late spring. The rivers of the Altai Mountains overflow their banks in spring. But the water level in them remains high until mid-summer, due to non-simultaneous snowmelt.
The rivers of the Tien Shan and Zhungarskiy Alatau are full-flowing in the warm season; In spring and summer. This is explained by the fact that in these mountains the melting of snow stretches until autumn. In spring, snowmelt begins from the lower belt, then during the summer, snow of medium height and highland glaciers melt. In the runoff of mountain rivers, the share of rainwater is insignificant (5-15%), and in low mountains it rises to 20-30%.
The flat rivers of Kazakhstan, due to low water and slow flow, quickly freeze with the onset of winter and are covered with ice at the end of November. The ice thickness reaches 70-90 cm. In frosty winters, the ice thickness in the north of the republic reaches 190 cm, and in the southern rivers 110 cm. second half of April.
The glacial regime of high mountain rivers is different. There is no stable ice cover in mountain rivers due to strong currents and groundwater supply. Coastal ice is observed only in some places. Kazakh rivers gradually erode rocks. Rivers flow, deepening their bottom, destroying their banks, rolling small and large stones. In the flat parts of Kazakhstan, the river flow is slow, and it carries solid materials.
The flow of a certain land area is measured by indicators:
- water flow - the volume of water flowing per unit of time through the living section of the river. It is usually expressed in m3/s. The average daily water discharges make it possible to determine the maximum and minimum discharges, as well as the amount of water flow per year from the basin area. Annual flow - 3787 km a - 270 km3;
- drain module. It is called the amount of water in liters, flowing per second from 1 km2 of area. It is calculated by dividing the runoff by the area of the river basin. The tundra and rivers have the largest module;
- runoff coefficient. It shows what proportion of precipitation (in percent) flows into rivers. Rivers of the tundra and forest zones have the highest coefficient (60-80%), while in the rivers of the regions it is very low (-4%).
Loose rocks - products are carried by runoff into rivers. In addition, the (destructive) work of rivers also makes them a supplier of loose . In this case, a solid runoff is formed - a mass of suspended, drawn along the bottom and dissolved substances. Their number depends on the energy of moving water and on the resistance of rocks to erosion. Solid runoff is divided into suspended and bottom runoff, but this concept is arbitrary, since when the flow velocity changes, one category can quickly move into another. At high speed, bottom solid runoff can move in a layer up to several tens of centimeters thick. Their movements are very uneven, since the speed at the bottom changes dramatically. Therefore, sandy and rifts can form at the bottom of the river, hindering navigation. The turbidity of the river depends on the value, which, in turn, characterizes the intensity of erosion activity in the river basin. In large river systems, solid runoff is measured in the tens of millions of tons per year. For example, the runoff of elevated sediments of the Amu Darya is 94 million tons per year, the Volga river is 25 million tons per year, - 15 million tons per year, - 6 million tons per year, - 1500 million tons per year, - 450 million tons per year, Nile - 62 million tons per year.
Flow rate depends on a number of factors:
- first of all from . The more precipitation and less evaporation, the more runoff, and vice versa. The amount of runoff depends on the form of precipitation and their distribution over time. The rains of a hot summer period will give less runoff than a cool autumn period, since evaporation is very large. Winter precipitation in the form of snow will not provide surface runoff during the cold months, but is concentrated in the short spring flood period. With a uniform distribution of precipitation throughout the year, the runoff is uniform, and sharp seasonal changes in the amount of precipitation and evaporation rate cause uneven runoff. During prolonged rains, the infiltration of precipitation into the ground is greater than during heavy rains;
- from the area. As the masses rise up the slopes of the mountains, they cool down, as they meet with colder layers, and water vapor, so here the amount of precipitation increases. Already from insignificant hills, the flow is greater than from adjacent ones. So, on the Valdai Upland, the runoff module is 12, and on the neighboring lowlands - only 6. An even greater volume of runoff in the mountains, the runoff module here is from 25 to 75. The water content of mountain rivers, in addition to an increase in precipitation with height, is also affected by a decrease in evaporation in the mountains due to the lowering and steepness of the slopes. From the elevated and mountainous territories, water flows quickly, and from the plains slowly. For these reasons, lowland rivers have a more uniform regime (see Rivers), while mountainous ones react sensitively and violently to;
- from cover. In areas of excessive moisture, soils are saturated with water for most of the year and give it to rivers. In zones of insufficient moisture during the snowmelt season, the soils are able to absorb all the melt water, so the runoff in these zones is weak;
- from vegetation cover. Studies of recent years, carried out in connection with the planting of forest belts in, indicate their positive effect on runoff, since it is more significant in forest zones than in the steppe;
- from influence. It is different in zones of excessive and insufficient moisture. Bogs are regulators of runoff, and in the zone their influence is negative: they suck in surface and water and evaporate them into the atmosphere, thereby disrupting both surface and underground runoff;
- from large flowing lakes. They are a powerful flow regulator, however, their action is local.
From the above brief review of the factors affecting runoff, it follows that its magnitude is historically variable.
The zone of the most abundant runoff is, the maximum value of its module here is 1500 mm per year, and the minimum is about 500 mm per year. Here, the runoff is evenly distributed over time. The largest annual flow in .
The zone of minimum runoff is the subpolar latitudes of the Northern Hemisphere, covering. The maximum value of the runoff module here is 200 mm per year or less, with the largest amount occurring in spring and summer.
In the polar regions, the runoff is carried out, the thickness of the layer in terms of water is approximately 80 mm in and 180 mm in.
On each continent there are areas from which the flow is carried out not into the ocean, but into inland water bodies - lakes. Such territories are called areas of internal flow or drainless. The formation of these areas is associated with fallout, as well as with the remoteness of inland territories from the ocean. The largest areas of drainless regions fall on (40% of the total territory of the mainland) and (29% of the total territory).
Let us determine the average long-term value (norm) of the annual runoff Kolp River, Upper Dvor point according to the data from 1969 to 1978. (10 years).
The resulting norm in the form of an average long-term water flow must be expressed in terms of other runoff characteristics: modulus, layer, volume, and runoff coefficient.
Calculate the average multi-year runoff module by the ratio:
l/s km 2
where F - catchment area, km2.
Runoff volume - the volume of water flowing from the catchment for any time interval.
Let us calculate the average long-term runoff volume per year:
W 0 \u003d Q 0 xT \u003d 22.14. 31.54 . 10 6 \u003d 698.3 10 6 m 3
where T is the number of seconds in a year, equal to 31.54. 10 6
The average long-term runoff layer is calculated from the dependence:
220.98 mm/year
Average long-term runoff coefficient
where x 0 is the average long-term precipitation per year
The assessment of the representativeness (sufficiency) of a series of observations is determined by the value of the relative root-mean-square error of the average long-term value (norm) of annual runoff, calculated by the formula:
where C V is the coefficient of variability (variation) of the annual runoff; the length of the series is considered sufficient to determine Q o if ε Q ≤10%. The value of the average long-term runoff is called the runoff rate.
Determination of the coefficient of variability Cv of annual runoff
The coefficient of variability C V characterizes the runoff deviations for individual years from the runoff norm; it is equal to:
where σ Q is the root-mean-square deviation of annual discharges from the runoff norm
If the runoff for individual years is expressed in the form of modular coefficients 
the coefficient of variation is determined by the formula
Compiling a table for calculating the annual runoff Kolp River, Verkhny Dvor point (Table 1)
Table 1
Data for calculation FROM v
Let us determine the coefficient of variability C v of the annual runoff:
The relative root-mean-square error of the average long-term value of the annual runoff of the Kolp River, Verkhny Dvor point for the period from 1969 to 1978 (10 years) is equal to:
Relative standard error of coefficient of variability FROM v when it is determined by the method of moments, it is equal to:
Determining the runoff rate in case of insufficient observational data by the method of hydrological analogy
Fig.1 Graph of connection of average annual runoff modules
of the studied basin the Kolp River, Verkhny Dvor point and the basin of the analogue of the river. Obnora, p. Sharna.
According to the graph of the connection of the average annual runoff modules, the Kolp River, the Verkhny Dvor point and the basin of the analogue of the river. Obnora, p. Sharna.M 0 \u003d 5.9 l / s km 2 (removed from the graph by the value of M 0a \u003d 7.9 l / s km 2)
Calculate the annual runoff variability coefficient using the formula
C v is the coefficient of runoff variability in the design section;
FROM V a - in the alignment of the analogous river;
Моа is the mean annual runoff of the analogue river;
BUT is the tangent of the slope of the communication graph.
Finally, to plot the curves, we accept Q o =18.64 m 3 /s, C V =0.336.
Construction of an analytical endowment curve and verification of its accuracy using an empirical endowment curve
The coefficient of asymmetry C s characterizes the asymmetry of the hydrological series and is determined by selection, based on the condition of the best correspondence of the analytical curve with the points of actual observations; for rivers located in flat conditions, when calculating the annual runoff, the best results are given by the ratio C s = 2C V. Therefore, we accept for the Kolp River, point Upper Yard C s \u003d 2С V=0.336 followed by verification.
The ordinates of the curve are determined depending on the coefficient C v according to the tables compiled by S N. Kritsky and M. F. Menkel for C S \u003d 2C V.
Ordinates of the analytical curve of provision of average annual
water discharge Kolp River, Verkhniy Dvor point
The security of a hydrological quantity is the probability of exceeding the considered value of a hydrological quantity among the totality of all its possible values.
We arrange the modular coefficients of annual expenses in descending order (Table 3) and for each of them calculate its actual empirical supply using the formula:
where m is the serial number of a member of the series;
n is the number of members of the series.
P m 1 \u003d 1 / (10 + 1) 100 \u003d 9.1 P m 2 \u003d 2 / (10 + 1) 100 \u003d 18.2, etc.
Figure - Analytical security curve
Plotting points with coordinates on the graph ( Pm , Q m ) and averaging them by eye, we obtain the curve of availability of the considered hydrological characteristic.
As can be seen, the plotted points lie very close to the analytical curve; from which it follows that the curve is constructed correctly and the relation C S = 2 C V corresponds to reality.
Table 3
Data for constructing an empirical endowment curve
Kolp River, Verkhny Dvor point
|
Modular coefficients (K i) descending |
Actual security |
Years corresponding to K i |
|
Figure - Empirical security
