Fourth Study Conference on BALTEX Scala Cinema Gudhjem

Statistic Parameters
Qср, m 3 /sec σ, m 3 Period of
average
estimation
/sec r(1)
max min max min max min
20 years 231 118 49,7 24,1 0,50 -0,43
30 years 219 181 47,5 24,4 0,52 -0,27
35 years 216 183 46,2 26,6 0,46 -0,24
50 years 209 187 44,2 30,3 0,42 -0,12
Table 2. Statistic parameters of moving average estimation
of the Neman’s water flow at Grodno.
As shown in Table 2, the maximum and minimum values
pertaining to different periods of average estimation vary
greatly. This is caused by the period of the 1920s to the
1940s with relatively high water levels in the Neman. Such
differences in the parameter estimation underline the instability
hypothesis in respect of the time period in question.
The verification of the hypothesis of statistic parameter homogeneity
supports the above hypothesis.
At present, 3 statistic models are generally used for the purpose
of describing long-term flow fluctuation: the sequence
of independent contingency values, the simple Markov
chain, and the complex Markov chain. The application of
the statistic conception is based on the principles of stability
of the process and annual completeness of the observed data
(Ismailov G.H, Fedorov V.M., 2001).
If no trend is clearly detectable, it is necessary that selected
Self-Correlation (SCF) and Particular Self-Correlation
(PSCF) Functions be considered. These functions specify
the type and the order of the annual river flow. SCF and
PSCF of the Neman river at Grodno have a conspicuous
variation with τ=1, while all other ordinate values prove to
be statistically insignificant and are characterized by the alternation
of positive and negative values. It follows that the
process of the annual flow may be identified with the following
model (Ismailov G.H, Fedorov V.M., 2001; Box
G.E.P., Jenkins F.M, 1974):
Q( t ) = Qср
+ r(
1 ) ⋅[
Q(
t −1)
− Qср
] + ξ(
t ) , (1)
where ξ ( t ) is the Gaussian “white noise”
2
and σξ = σQ
⋅ 1 − r(
1)
.
According to equation (1), the annual flow of water in the
Neman at Grodno is described with r(1)=0,19, Qср=194 m 3 /sec (Table 1), σξ = 36,19 m 3 /sec and σ Q =36,86
m 3 /sec, resulting in
Q( t ) = 0 , 19⋅
Q(
t −1)
+ 157 + ξ(
t ) .
In this case the variance of the contingent constituent is
quite high.
The results concerning the long-term fluctuations of the annual
flow of water in the Neman river at Grodno indicates
the possible existence of a certain interconnection between
consecutive flow events. For this reason the Markov simple
chain can be applied to describe the annual flow of water,
i.e.
Q( t ) = r(
1 ) ⋅Q(
t −1)
+ ξ(
t ) . (2)
The first item in the right part of equation (2) may be interpreted
as the river flow part caused by accumulated atmospheric
precipitation in the river basin during the previous
year and its discharge into the river this year. In this case the
contingent constituent ξ ( t ) in (2) must obviously include
- 142 -
that part of the annual outflow of the current year. The result
(2) may be transformed into the following equations:
Q( t ) = 0, 059⋅
Q(
t −1)
+ 0,
493⋅Woc(
t ) + 71,
02 + ξ(
r1
)
(3)
Q( t ) = 0, 205⋅Woc(
t ) + 0,
120⋅Woc(
t −1)
− 9.
173 + ξ(
t2
)
(4)
where Woc( t ) and Woc( t − 1)
denote the annual precipitation
of the current and the previous years, respectively.
The coefficient of multiple correlation between the outflow
and the defining factors in the equation (3) is R=0,48>R Т (45, 5
%)=0,29, while the ±5% and ±10% confidence intervals embrace
40,4% and 55,3% of all points, respectively. In equation
(4) it corresponds to R=0,58>R Т (45, 5 %)=0,29, while the
±5% and ±10% confidence intervals embrace 27,7% and
63,8%, respectively.
We have attempted to describe the annual fluctuation of the
Neman’s water outflow by means of the Markov complex
chain with a shift of 50 years. The regressive analysis of correlations
defines the particular presentation of the model:
Q(
t ) = 0,
139⋅
Q(
t −1)
+ 0,
195⋅
Q(
t −10
) −
. (5)
− 0,
199⋅
Q(
t − 37 ) + 168,
983 + ξ(
t )
The coefficient of multiple correlation in the equation (5) is
equal to R=0,33>R Т (144, 5%)=0,155, while the ±5% and ±10%
confidence intervals include 27,1% and 47,9%, respectively,
of all points.
4. Conclusions
This study has shown that there are statistically significant
changes in the dynamics of the annual flow of water in the
Neman river at the station Grodno within the period of
nearly 200 years investigated here. Changes are influenced
by both natural climate and anthropogenic factors in the hydrological
cycle. Stability of long-term fluctuations of the
Neman’s annual flow is observed only within limited time
intervals of the whole period under investigation. Considering
the consistencies of the long-term fluctuation of the river
annual outflow one needs to apply the methods of contingent
processes theory, analyze the genesis of the process,
and take into account natural, economic and first of all climatic
factors defining its shape.
References
Volchak A.A., Automation of hydrologic calculations, waterworks
construction and preservation of the environment,
Brest – Nottingham, pp. 55 – 59, 1998.
Ismailov G.H., Fedorov V.M, Analysis of perennial fluctuation
of annual flow of Volga, Water Resources, Vol. 28.,
No.5, pp. 517-525, 2001.
Box G.E.P., Jenkins F.M., Time Series Analysis: Forecasting
and Control, 1 st edition, Mir, Moscow, 1974