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Ensemble forecasting of spring runoff volumes and hydrograph

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					International conference on innovation advances and implementation of flood forecasting technology



LONG-TERM ENSEMBLE FORECASTING SPRING RUNOFF
VOLUME AND HYDROGRAPH PEAK

L. S. Kuchment (1) and A. N. Gelfan (1)
(1) Water Problems Institute of Russian Academy of Sciences, 119991 Gubkin 3, Moscow, Russia


Abstract
A technique for long-term (i.e. two to three months) ensemble forecasting of the spring runoff
volumes and the peak discharges has been developed. The technique is based on using physically
based model of runoff generation combined with a weather generator. The distributed physically
based model includes a description of the snow accumulation and melt, soil freezing and redistribution
of soil moisture during the autumn and winter period, and the processes of runoff generation after the
beginning of spring snowmelt period. The weather generator, which consists of stochastic models of
daily air temperature and precipitation, is used to simulate the meteorological inputs for lead time
periods. Long series of possible hydrographs are calculated by the physically based model with Monte
Carlo simulated inputs. The probability distributions of the forecast runoff volumes and peak
discharges are estimated on the basis of these hydrographs. In addition, the runoff volume and the
hydrograph peak are forecast using climatological values of air temperature and precipitation for each
chronological day during the lead time period (deterministic forecast). The results of the deterministic
forecast of spring runoff volumes, with the help of the physically based model, are compared with the
results, which have been obtained on the basis of the regression relationships between the spring
runoff volume and the initial river basin indexes under the mean meteorological conditions during the
lead time period (the present day procedure of long-term runoff forecasting). The case study has been
carried out for the Sosna River basin. This basin covers an area of 16,400 km2.

Keywords: long-term ensemble forecast, physically-based model, snowmelt flood, weather generator

INTRODUCTION
Long-term (i.e. seasonal) forecasts of spring-summer flood volumes are widely used in Russia for
water management and for runoff regulation where there are cascades of reservoirs in large river
basins. The traditional techniques for such forecasts are based on the application of nonlinear or linear
multiple regression relationships between flood volume and the main snowmelt runoff factors, which
can include (depending on the physiographic region) the snow water equivalent before melt, the
measured autumn soil moisture content (or its calculated index), and the measured depth of a frozen
soil (or its calculated index). It is assumed that the precipitation and the air temperature during the
lead time period are equal to their mean, climatological values. Opportunities for such forecasts
depend on physiographic, climatic conditions and runoff generation mechanism in the river basin.
However, in most of Russia these forecasts provide an acceptable level of accuracy. Since the applied
regression relationships only account for the integrated characteristics of the hydrometeorological
conditions before snowmelt, these forecasting techniques can lead to significant errors if the
conditions of snowmelt runoff generation are unusual, (for example, the complicated processes of soil
freezing and soil moisture redistribution, the high rate of snow melt, several periods of thaws during a
cold season, large liquid precipitation before melt, etc), or the measurements or the indexes of the
autumn soil moisture content and the depth of a frozen soil are not representative. It seems that
continuous physically based models taking into account snow and soil processes during the winter and
spring period give opportunity to improve the forecast. In this paper, we test this approach together
with an ensemble form of long-term forecasting of meteorological variables. As a result, we obtain a
technique that provides a probabilistic technique of forecasting both snowmelt flood volume and flood
peak discharge. The case study has been carried out for the Sosna River basin, which has a catchment
area of approximately 16,400 km2. This river is a tributary of the Don River and is located in the
forested-steppe zone of the European Russia.


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THE USED PHYSICALLY BASED MODEL OF SNOWMELT FLOOD
GENERATION
The model for runoff generation is based on the finite-element schematization of the river basin shown
in Figure 1. This describes the following main processes: snow cover formation and snowmelt, soil
freezing and thawing, infiltration into frozen and unfrozen soil, vertical water transfer in an unfrozen
soil, evaporation, overland and channel flow. The main aim is to provide a long-term probabilistic
forecast. This needs a long series of runoff simulation. As a result we simplified the description of
some of these processes in comparison with the previous version of the model (see Kuchment et al.,
1986).




The bold lines represent the river network
The thin lines represent the boundaries of the finite elements

Figure 1             Finite-element schematization of the Sosna River watershed

Snow cover formation and snowmelt

To calculate the characteristics of the snow pack, a system of vertically averaged equations of snow
processes has been applied (Kuchment & Gelfan, 1996). The system includes the description of
temporal change of the snow depth, content of ice and liquid water, snow density, snowmelt,
sublimation, re-freezing melt water, snow metamorphism. This is modelled as follows:

           dHs
            dt
                        [                                    ]
                 = ρw X s ρ0 −1 − ( S + E s )( ρi I s ) −1 − V                                                            (1a)

           d
              ( ρ i IH ) = ρ w ( X s − S − E s ) + S i                                                                    (1b)
           dt
           d
              ( ρ w ws H s ) = ρ w ( X l + S − E l − R s ) − S i                                                          (1c)
           dt

where Hs is the snow depth; I s and ws are the volumetric content of ice and liquid water in snow,
respectively; X s and X l are the snowfall rate and the rainfall rate, respectively (it is assumed that


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if the air temperature Ta  ≥ 0 oC only rainfall occurs and if Ta < 0 oC only snowfall occurs); S is the
snowmelt rate; ρ s is the density of snowpack calculated as ρ s = ρ i I S + ρ w w S ; ρw , ρi , and ρ0 are
the density of water, ice, and new snow, respectively; E s is the rate of snow sublimation; E l is the
rate of liquid water evaporation from snow; S i is the rate of re-freezing melt water in snow; Rs is the
meltwater outflow from snowpack; V is the compression rate.

The melting rate S is determined as
           S = βρ sT a                                                                                                     (2)
where β is the empirical constant.

Soil freezing and redistribution of soil moisture during a cold period

The hydrothermal processes in the soil during the cold period are described by the following equations
(Kuchment & Gelfan, 1993):

              ∂T ∂ ⎛ ∂T ⎞
           Cf       = ⎜λ              ⎟, 0 < z < H (t )                                                                    (3)
              ∂t ∂ z ⎝ f ∂z ⎠
             ∂T ∂ ⎛ ∂T ⎞
          C        = ⎜λ            ⎟, H (t ) < z < L                                                                       (4)
             ∂t ∂z ⎝ ∂z ⎠
          T (0, t ) = T0 (t ); T ( H , t ) = 0; T ( L, t ) = TL ; T ( z,0) = T ( z )
          ∂θ       ∂ ⎛ ∂θ             ⎞
               = ⎜D               − K ⎟, H (t ) < z < L                                                                    (5)
           ∂t ∂z ⎝ ∂z                  ⎠
          θ ( L, t ) = θ L ; θ ( H ) = θ 0 ; θ (z,0) = θ (z)
               ∂T                     ∂T                                   dH
          λf          z = H −0 = λ           z = H + 0 + χρ w (θ − − θ 0 )                                                 (6)
               ∂z                     ∂z                                    dt
          H ( 0) = 0

where H ( t ) is the depth of frozen soil at time t ; T ( z , t ) is the soil temperature at the depth z and
time t ; λ f and λ are the thermal conductivities of frozen and unfrozen layers of soil, respectively;
 C f and C are the heat capacities of frozen and unfrozen layers of soil, respectively; χ is the latent
heat of ice fusion; θ ( z , t ) is the volumetric liquid water content in unfrozen layer of soil; θ − is the
liquid water content just above the freezing front; θ 0 is the liquid water content at 0oC (assumed to be
equal to the moisture content at the wilting point); D is the diffusivity of soil moisture; K is the
hydraulic conductivity of soil; L is the depth of the soil where the soil temperature and the volumetric
moisture content can be considered as constants equalled TL and θ L , respectively ( L was taken to be
equal to 2 m).

In this system, the equations (3) and (4) describes heat transfer in the frozen soil layer and in the
unfrozen soil layer, respectively. The equation (5) describes soil moisture transfer from the unfrozen
layer of soil to the freezing front. According to experimental data, this process plays an important role
in the vertical redistribution of soil moisture during the cold period for soils which are typical for
forested-steppe zone.

The upper boundary condition is formulated as




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             Ta − T0        ∂T
           λs        = −λ f          ,                 Hs > 0
               Hs           ∂z z = 0                                                                                       (7)
          T0 = Ta,                                      Hs = 0
where λ s is the thermal conductivity of snow.

The diffusivity, the hydraulic conductivity of unfrozen soil, the heat capacities and the thermal
conductivities of frozen and unfrozen soil were calculated by the formulas suggested by Kuchment et
al. (1983).

Soil thawing

Soil thawing is calculated for snow-free areas of the catchment area from the end of snowmelt. The
movement of the soil thawing front is described by the equations similar to ones used for soil freezing
description excluding equation (5). The description of soil thawing model is presented in (Kuchment,
et al., 2000).

Meltwater infiltration into frozen soil

Saturated hydraulic conductivity of the frozen soil K 0 is calculated as (Kuchment & Gelfan, 1993):
                                                                        f

                                                  4
                    ⎛θ      − I − θ0 ⎞     1
       K 0 f = K 0 ⎜ max
                    ⎜ θ              ⎟
                                     ⎟                                                                (8)
                    ⎝ max − θ 0 ⎠ (1 + 8 I )
                                                2

where K 0 is the saturated hydraulic conductivity of an unfrozen soil; θ max is the porosity; I is the ice
content of the upper 0.1m-layer of soil.

Detention of melt water by basin storage

It was assumed that the spatial distribution of the free basin storage capacity before snow melt can be
described by exponential probability function. In this case, detention Β of water by the basin storage
up to time t after the beginning of melting is determined as (Kuchment et al., 2000):
               ⎡       ⎛ R ⎞⎤
       Β = Β 0 ⎢1 − exp⎜ −
                       ⎜ Β ⎟⎥ ⎟                                                                (9)
               ⎢
               ⎣       ⎝    0 ⎠⎥⎦
where Β 0 is the mean value of the free storage capacity (or the maximum possible detention); R is
the total melt and rainfall water yield on the basin area up to time t.

Vertical water transfer in an unfrozen soil and evaporation

The changes of the unfrozen soil moisture content and infiltration into the soil during the warm period
are calculated by the equation (5).

The evaporation rate E is calculated as (Kuchment & Gelfan, 1993):
           E = k E d aθ u                                                                                                 (10)
where d a is the air humidity deficit; θ u is the soil moisture content at the upper soil layer; k E is the
empirical constant.

Overland and channel flow

Schematization of the river basin is based on presentation of the overland flow areas by 32 rectangular



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strips of different width, length and slope, which are perpendicular to the river channels. The choice
of strips is determined by topography, soil types, vegetation, and meteorological data. To calculate the
water movement along strips and in the river channels, the kinematic wave equations are applied.

The runoff excess is calculated for each strip area taking into account statistical distribution of snow
water equivalent and depth of a frozen soil inside of this area. To approximate the distributions,
gamma distribution is applied (the mean values are calculated by the model and coefficients of
variation are found by the empirical formulas).

Calibration and validation of the physically based model

The hydraulic parameters of soil in equations (5), (8) and the parameters of the heat flow equation (3),
(4) were calculated by the formulas presented in (Kuchment et al., 1983). Soil constants which are
necessary for using these formulas was taken from (Reference book of hydrological processes…,
1975). Five parameters (the factor β in formula (2), the mean value of the free storage capacity in
formula (9), the saturated hydraulic conductivity K 0 in formula (8), Manning’s roughness
coefficients for overland and channel flow) were calibrated against the measured daily discharges at
the Sosna river outlet for 15 years (1952-1966). The model validation was carried out using the
measured discharges for the next 15 years (1967-1981).

THE WEATHER GENERATOR
The weather generator includes the stochastic models of daily precipitation and air temperature for the
period from 1 March to 30 April (for this period we neglect evaporation). The model of daily
precipitation occurrence throughout this period is represented as the first-order Markov chain. Daily
precipitation amount is described as a gamma-distributed random value. The air temperature series
are simulated by the procedure suggested in (Kuchment et al., 2003). Parameters of the stochastic
models were estimated using available meteorological records for 30 year (1952-1981).

FORECASTING TECHNIQUES
Continuous simulation of runoff generation processes for each spring snowmelt flood began on 1 May
of the previous year. Up to 1 March (the date of the forecast issue) the input data for simulation
included the observed daily precipitation amount, air temperature and air humidity. After 1 March, the
input meteorological data for the lead time period were assigned by one from the following two ways:

1.        The daily meteorological data were assigned as the climatological means for each
          chronological day. In this case, it is possible to obtain the deterministic forecast of the spring
          flood hydrograph.
2.        The daily meteorological inputs were simulated using a weather generator and the Monte
          Carlo procedure.

In this case, we have an opportunity to calculate long series of possible hydrographs and determine
probability distribution of the forecasting flood volume and peak discharge (we simulated 1000
hydrographs for each year).

Schematic diagrams of the both forecasting techniques are shown in Figures 2 and 3.




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                                                Deterministic long-term forecast

                                                Characteristics of snow and soil condition
                                                    on the date of the forecast issue

                                    Average temperature and precipitation for the lead-time period
                                                             (2-3 months )
                            10.00




                                                                                            01.03



                                                                                                    08.03



                                                                                                            15.03



                                                                                                                    22.03



                                                                                                                            29.03



                                                                                                                                    05.04



                                                                                                                                            12.04



                                                                                                                                                    19.04
                             5.00                                                   0.000
                                                                                    0.010
                                                                                    0.020
                             0.00                                                   0.030
                                                                                    0.040
                            -5.00                                                   0.050
                                                                                    0.060
                                                                                    0.070
                           -10.00                                                   0.080
                                                                                    0.090
                                28.02   20.03     09.04      29.04                  0.100




                                                          Physically based, distributed
                                                              hydrological model



                                                          Flood volume and peak discharge


Figure 2             Schematic diagram of the deterministic forecast


                                         Probabilistic long-term forecast

                                           Characteristics of snow and soil condition
                                               on the date of the forecast issue


                                                                     Weather generator



                                    Monte Carlo generated series of daily temperature and
                                      precipitation for the lead-time period (2-3 months)



                                                   Physically based, distributed
                                                       hydrological model



                               Probability distributions of flood volume and peak discharge


Figure 3             Schematic diagram of the probabilistic forecast

Comparison of the deterministic and probabilistic forecasts

The results of the deterministic forecast of flood volume and peak discharge for 30 snowmelt floods
(1952-1981) as well as the efficiency criterion R2 of Nash & Sutcliffe (1970), which was adopted to
estimate the goodness of fit of the forecasted and measured flood characteristics, are shown in Figure
4. The comparison of the observed flood characteristics with their mean forecast values obtained by
averaging 1,000 hydrographs calculated for each year using the generated meteorological inputs is
given in Figure 5. As can be seen from these Figures, the forecasting techniques used give satisfactory


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results for both the flood volume and peak discharge. However, the forecast of the flood volume has
appears to be much better.




                     160                                                                    2
                                                                                         R = 0.8149
                     140
                     120
                     100
         Ycalc, mm




                      80
                      60
                      40
                      20
                       0
                           0        20        40        60          80          100          120         140        160
                                                               Yobs, mm




                                                                                                     R2 = 0.6142
                 4000

                 3000
   Qcalc, m /s
  3




                 2000

                 1000

                      0
                           0          1000         2000           3000             4000               5000          6000
                                                                         3
                                                              Qobs, m /s


Figure 4                       Deterministic forecast of the flood volume (top) and flood peak discharge (bottom)




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                              160
                                                                                                   R2 = 0.8351
                              140
                              120
                  Ycalc, mm




                              100
                               80
                               60
                               40
                               20
                                0
                                     0          20          40          60        80       100       120          140         160

                                                                              Yobs, mm




                              4000

                              3000
    Qcalc, m /s
   3




                                                                                                           2
                              2000                                                                       R = 0.6441

                              1000

                                0
                                     0               1000        2000            3000       4000           5000            6000
                                                                                       3
                                                                             Qobs , m /s


Figure 5                                 Forecast of the flood volume (top) and flood peak discharge (bottom) obtained by the
                                         averaging of 1000 simulated hydrographs

Comparison of the suggested techniques of the flood volume forecasting with the
traditional method

For the number of river basins located within the forested-steppe zone of European Russia, the spring
snowmelt flood volume is forecast on the basis of the experimental formula suggested by Komarov
(1974):
                             ⎡       ⎛ SWE + X ⎞⎤ˆ
        Y = ( SWE + X ) − P0 ⎢1 − exp⎜ −
                     ˆ
                                     ⎜             ⎟⎥
                                                   ⎟                                            (11)
                             ⎢
                             ⎣       ⎝      P0     ⎠⎥⎦
        P0 = a exp[− W (bH + c )]
where Y is the flood volume, mm; SWE is the snow water equivalent on the date of the forecast
            ˆ
issue, mm; X is the climatological mean of the precipitation total for the lead time period (equals 27
mm for Sosna basin); P0 is the maximum possible runoff losses depending on the watershed
conditions on the date of the forecast issue, mm; W is the dimensionless index of moisture content of




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the 1 metre layer of soil; is the freezing depth, cm; а, b, с are the empirical coefficients equal 750 mm,
0.051 and 0.11, respectively.

We applied the formula (11) to forecast snowmelt flood volume in the Sosna river for the same 30
years (1952-1981) as used in the previous section. The efficiency criterion of the forecast =0.77, i. e.
the forecast is only slightly worse than one obtained with the help of the physically based model
(Figures 6, 7). However, formula (11) systematically underestimated the flood volume; mean error is
20 mm. To remove the systematic error, we re-calibrated the coefficients а, b, с in comparison with
ones suggested by Komarov (1974) using 15 years of observations (1952-1966). The obtained values
of the coefficients а, b, с are 930 mm, 0.040 and 1.85, respectively. The forecast of the flood volume
for 30 years by formula (11) with the obtained values of the coefficients is shown in Figure 7.
                                                                                             1969                                                                                                                                        1972
                            35                                                                                                                                             30


                            30
                                                                                                                                                                           25

                            25
            Frequency (%)




                                                                                                                                                                           20
                            20


                                                                                                                                                                           15
                            15


                            10                                                                                                                                             10


                                5
                                                                                                                                                                           5

                                0
                                     0                   10       20         30           40        50        60       70          80          90          100
                                                                                                                                                                           0
                                                                                  Flood volume, mm                                                                                   10        20         30         40          50      60        70       80      90     100         110     120



                                                                                              1973                                                                                                                                       1977
                        55                                                                                                                                                 35

                        50
                                                                                                                                                                           30
                        45

                        40                                                                                                                                                 25

                        35
                                                                                                                                                                           20
                        30

                        25
                                                                                                                                                                           15
                        20

                        15                                                                                                                                                 10


                        10
                                                                                                                                                                            5
                            5

                            0                                                                                                                                               0
                                    10                    20           30               40          50          60          70             80               90                   60           70     80        90         100     110        120   130     140   150     160     170     180    190




Figure 6                                                 Histograms of the forecasted flood volumes
                                                                                                          1965                                                                                                                               1966
                                                         40
                                                                                                                                                                                35

                                                         35
                                                                                                                                                                                30
                                                         30
                                         Frequency (%)




                                                                                                                                                                                25
                                                         25


                                                         20                                                                                                                     20


                                                         15
                                                                                                                                                                                15

                                                         10
                                                                                                                                                                                10
                                                          5

                                                                                                                                                                                5
                                                          0
                                                                  0         400         800     1200     1600      2000     2400        2800        3200     3600   4000

                                                                                                                                           3
                                                                                              Flood peak discharge, m /s                                                        0
                                                                                                                                                                                          0         300        600         900        1200     1500      1800    2100    2400     2700       3000

                                                                                                              1969                                                                                                                           1971
                                         40
                                                                                                                                                                                45


                                         35                                                                                                                                     40


                                         30                                                                                                                                     35


                                                                                                                                                                                30
                                         25

                                                                                                                                                                                25
                                         20

                                                                                                                                                                                20
                                         15
                                                                                                                                                                                15

                                         10
                                                                                                                                                                                10

                                            5
                                                                                                                                                                                5

                                            0                                                                                                                                   0
                                                              0       250         500         750      1000     1250      1500      1750            2000     2250   2500                  0         450        900        1350        1800     2250      2700    3150    3600     4050       4500




Figure 7                                                 Histograms of the forecasted flood peak discharge the 1 meter layer of soil; H is the
                                                         freezing depth, cm; а, b, с are the empirical coefficients equal 750 mm, 0.051 and
                                                         0.11, respectively.


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                160
                140                                                                      R2 = 0.8202
                120
                100
    Ycalc, mm




                 80
                 60
                 40
                 20
                  0
                      0       20        40           60           80          100          120         140       160
                                                              Yobs, mm


Figure 8                  Comparison of the flood volumes forecasted by formula (11) with the observed values

On average, the forecast accuracy is very close to one obtained with the help of the physically based
model. However, in some cases the accuracy differs significantly. For example, for a large flood that
occurred in 1970 (the flood volume, 160 mm, was the largest for the period of observations) the
forecast error by formula (11) is 52 mm, whereas the deterministic forecast by our model gives the
error of 15 mm. Such a difference may be explained by the peculiarities of the runoff losses formation
during the spring of 1970, which are accounted by the model but not by formula (11). There were
many thaws during the winter of 1969 to 1970 which, in accordance with an upward migration of soil
moisture during soil freezing, result in significant decreasing permeability of the upper 10 cm to 30 cm
of soil in the beginning of spring melt. The model accounts for such a decreasing, however, the
predictor of formula (11) (the water content of the 1 meter layer of soil) was changed only slightly
during the winter.

CONCLUSION
The suggested technique of ensemble forecasting spring runoff volume and hydrograph can be
efficiently applied for both deterministic and probabilistic long-term flood forecasts. However, it is
necessary to investigate opportunities and reliability of this technique for river basins in different
physiographic zones.

ACKNOWLEDGEMENT
This research was supported by the Russian Foundation of the Basic Research (grant 05-05-64828).

REFERENCES
Apollov, BA, Kalinin, GP, & Komarov, VD 1974, Hydrological Forecasting, Gidrometeoizdat, Leningrad.
(in Russian).
Kuchment, LS & Gelfan, AN 1993, Dynamic-stochastic models of river runoff generation, Nauka,
Moscow. (In Russian).
Kuchment, LS & Gelfan, AN 1996, ‘The determination of the snowmelt rate and meltwater outflow
from a snowpack for modelling river runoff generation’, Journal of Hydrology, vol. 179, no. 1/4, pp.
23–36.
Kuchment, LS, Demidov, VN & Motovilov, YuG 1983, River runoff formation (physically-based
models), Nauka, Moscow. (In Russian)



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Kuchment, LS, Demidov, VN & Motovilov, YuG 1986, ‘A physically based model of the formation of
snowmelt and rainfall runoff’, in EM Morris (ed). Symposium on the Modeling Snowmelt-Induced
Processes, IAHS Publications, no. 155, pp. 27–36.
Kuchment, LS, Gelfan, AN & Demidov, VN 2000, ‘A distributed model of runoff generation in the
permafrost regions’, Journal of Hydrology, vol. 240, no. 1/2, pp. 1-22
Kuchment, LS, Gelfan, AN & Demidov, VN 2003, ‘Application of dynamic-stochastic models of
runoff generation for estimating extreme flood frequency distribution’, in G Blöschl, S Franks, M
Kumagai, K Musiake & D Rosbjerg (eds). Water Resources Systems—Hydrological Risk, Management
and Development, IAHS Publications, no. 281, pp. 107-114.
Nash, JE & Sutcliffe, JV 1970, ‘River flow forecasting through conceptual models’, Journal of
Hydrology, vol. 10, no. 3, pp. 282-290.
Reference book of the hydrological properties of soils in the Central-Chernozem Region 1975,
Hydrometeorological Survey of the Central-Chernozem Regions, Kursk. (In Russian).




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