H.Vogtmann
and N. Dobretsov (eds.), Transboundary Water Resources: Strategic
for Regional Security and Ecological Stability, 2735, 2005 Springer.
ON THE PROBLEM OF THE CASPIAN SEA LEVEL FORECASTING
©
2005 Bolgov M.V., Trubetskova M.D., Filimonova M.K.
Water Problems Institute, Russian Academy of Sciences
GSP  1, 3, Gubkin St., 117971, Moscow, Russia
Stochastic fluctuations of climate
and hydrological regime caused by both natural and anthropogenic
factors are the main reason for the big uncertainty of longterm
hydrological forecasts. Consequently, they cause the necessity
to reconsider the risk of economic activities at inland sea
coasts towards its increase. To estimate such a risk some sources
of uncertainty arising under the sea hydrological regime forecasting
are considered in the paper. By use of digital models of a region,
some features of morphometric characteristics (depending on
the sea level) are revealed, and their contribution to the level
regime variability is appreciated.
Introduction
The forecast of longterm fluctuations of an inland water body level
such as the Caspian Sea is the most complicated geophysical problem.
It demands knowledge both of features of a hydrometeorological regime
of the region, and of mechanisms for the occurrence of longterm
climate and runoff fluctuations. Their definition is associated
with different types of mistakes arising because of unreliable measurements,
imperfection of the modeling representations, the approximate character
of hypotheses, limitation of access to observation materials of
recent years, etc.
Model
Fluctuations of an inland (closed) water body level represent
poorly predicted natural phenomenon, which nevertheless can be described
on the basis of stochastic models of hydrometeorological processes
and representations about its water balance.
Fluctuations of a water level h
of a closed water body can be described with the help of the known
water balance equation [4,5]:

(1) 
Where
v(t) – is water inflow per unit time, e(t)
– is a amount of effective evaporation (evaporation minus precipitation),
F(t)
 is the water surface area.
To solve the equation (1), the linearized analogue is offered in
[5]:

(2) 
where a
and
b are the coefficients of linear dependence
of water surface area on the level. Assuming the coefficient constant
under h,
we receive Langeven linear differential equation [11]:

(3) 
Assume that at the initial moment , where and
are water inflow and evaporation expectations.
The solution of the equation (3) can be presented as [11,5]:
Assume that at the initial moment
t = 0 the level equals h_{0}
to relative to the socalled equilibrium value
,
where
and
are water inflow and evaporation expectations.
The solution of the equation (3) can be presented
as [11,5]:

(4) 
Averaging the right and the left parts in (4),
we receive

(5) 
as 

The solution (4) of the equations (3) being considered
as a linear operator transforming the random function
g(t), the expression for the correlation function
of the process h
is received in [5,11]
assuming the entrance process 

to be the Markov one. 
This expression being too bulky, we limit ourselves
here only by the formula for the dispersion of level fluctuations:

(6) 
At the
following expression is true for the correlation function

(7) 
For the dispersion: 


(8) 
The data
From the beginning of regular observations over
the Caspian Sea regime during about one century, its levels were
insignificantly fluctuating near the mark of 26 m. In 30th years
of 20th century a catastrophic decrease of the level on 1.7m occurred.
Further, the decrease of the level continued but much more slowly
and in 1979, the level has reached 29m. The increase begun after
that was observed until 1995, annual average levels exceeding 
27 m.
One of the most important problems of the data
analysis is the representativity of the existing observation series
of the Caspian Sea level and the reliability of the statistical
conclusions received on the limited data.
Researches of fluctuations of river runoff and
evaporation from water body surface and the precipitation analysis
have shown that the socalled simple Markov chain can be accepted
as the mathematical model of these processes [6,8,9]. Simulation
of the Caspian Sea level series executed on the basis of the corresponding
numerical algorithms [7], allows to make a conclusion, that probability
distribution in an interval from 0,1 up to 99,9 % is well approximated
by the normal distribution law. At the same time, it is necessary
to note, that the histogram of the observed sea levels sequence
has the twomodal form.
Hypothesis of the Stationarity of Climatic Conditions
Along with other hypotheses, one can find the explanation
of the Caspian Sea abnormal behavior in a context of a climatic
change problem. Climatic conditions are known to be essentially
nonstationary on long time intervals (more than centuries). For
example, according to some estimations, during last postglacial
period the World Ocean level has grown on 130 m. Instrumental measurements
has demonstrated ocean level growth approximately on 15  20 cm
for 100 years as well. However, this figure lies within the limits
of measurement accuracy and can hardly serve as the evidence of
essential modern climatic changes (or their indicator).
The indirect characteristic allowing estimating
of the "stationary" hypothesis acceptability is the average
duration of time when the sea level is above or below the set level
(occurrence probability of series of years with extreme level values).
It is found from the outlier probability distribution for the prescribed
stochastic model. The solution of this problem for the Caspian Sea
has shown [6] that the recurrence of long series (up to 50 years
and over) in relation to the gravitation level is essential. The
existing stochastic runoff fluctuation model seems to be advanced
in the framework of some quasistationary theory by the account
of longterm tendencies in the process of the Caspian Sea basin
humidation, but such models have not been offered yet.
Recently, the trends discovered in wind speed
on the coastal stations called the hypothesis of the climatic condition
stationarity in question. The evaporation value is known to depend
on wind speed as well as on air temperature and humidity. The lowered
evaporation observed in last decades would be logically associated
with fluctuations of these climatic characteristics. The researches
carried out by a number of scientists have not revealed any of significant
tendencies in air and water temperature and air absolute humidity.
As for the module of a wind speed, the conclusion about the presence
of a negative linear trend for the period 19601990 has been made
[2].
On the basis of the observations on meteorological
stations in the Caspian region, average monthly values of air temperature
and surface wind speed for the period of 19611999 has been received
for Izberg, Makhachkala and Tuleniy. These stations are located
at the western coast of the Caspian Sea closely to each other. Thus,
if any tendencies in change of climatic characteristics take place
all over Caspian region, they should be shown equally at these stations.
On fig.1 graphs of surface wind speed module for
these three meteorological stations for January, April, July and
October are presented.
Fig.1
Longterm changes of wind speed module on stations: Makhachkala
(1),Izberg (2), Tuleniy (3) for January (a), April (b), July(c),
October(d) 19601999
At Tuleniy station, one can see the negative trend
in surface wind speed in January; in April and to a lesser degree
in October the positive one is observed, while in July no essential
change is revealed. At Izberg station, on the contrary, in January
and October no any essential change is observed, in April and in
July obviously expressed positive trend is observed. At Makhachkala
station in January and October wind speed is practically the same
for the given period, in April and  to a lesser degree  in July
wind speed decreases. Thus, at three closely located stations wind
speed turns to behave differently. It is necessary to notice as
well that during the considered period differences in meteorological
observations carrying out on the stations took place. So, at station
Makhachkala during the period up to 1968 observations were carried
out 4 times a day, then up to 1986  3 times a day (9, 15, 21h.),
then again 4 times a day. Up to 1993, the observations were carried
out at 3, 9, 15, 21h, from 1994  at 0, 6, 12, 18h. This could lead
to fluctuations of average wind speed within the limits of 0.5 m/s
that corresponds to approximately 10 % of average norm.
Thus, the analysis of change for last 40 years
does not give us an opportunity to draw a conclusion about the existence
of strongly pronounced tendencies of the meteorological characteristics
that would determine evaporation from the Caspian Sea and changes
of its level. On the contrary, the researches carried out can be
regarded as the support of the hypothesis of climatic conditions
stationarity.
Morphometric Dependences
Some other ideas explaining the "anomaly"
of the Caspian Sea are connected with mistakes of the accepted modelling
representations. So, in linearized differential water balance equations
linear dependence of the water surface area upon the sea level is
used (the socalled morphometric dependence). Modern level of computer
facilities allows not limiting oneself by opportunities of standard
topographical maps when searching morphometric dependences. To solve
such problems, it is necessary to automatize the access to elevation
data. With this purpose, the relief digital model (a matrix of altitudes
with a geographical fixation) was used with the grid step of 30
seconds, with smooth approximation along height. Calculations of
the Caspian Sea morphometry are represented further without taking
into account the Kara Bogas Gol gulf water area. The Caspian Sea
is divided into three parts: Northern (to the north of 44^{î}30’ N),
Middle (from 40^{î}N up to 44^{î}30’ N) and Southern (from 40^{î} N to the
south).
Let us consider the distribution of areas occupied
by various bathymetric steps. The most significant part of the area
 66.6 %  has the depths less than 100 meters, 42.4 % of it (28.2
% of the total sea area) located mainly (70 %) in the Northern part
of the Caspian Sea having the depth less than 10 m. Depths more
than 900 m occupy about 1 % of the area. The reminder area is distributed
rather regularly between 200 and 800 m depths approximately by 4
 5 % per 100m of depth. The general character of depths distribution
is well seen on the bathygraphic curve of the sea (fig.2). One can
see two smooth breaks at the depths of 500 and 100 meters and various
inclinations: very flat one in the upper part, very steep one in
the middle and less steep one in the lower part of the curve.
If to consider the part of the bathygraphic curve
corresponding to the heights of 40  20m abs in more details (Fig.
2B), a presence is obvious of a bathygraphic curve excess at about
28 m abs. It is the Northern part of the sea that is responsible
for the excess, while curves of the Middle and the Southern parts
of the Caspian Sea have no peculiarities within these altitudes.
Such behavior of a curve is explained by high flatness of the relief
in the coastal zone and of the Northern Caspian Sea coast. These
relief features do not allow using linear interpolation in the field
of a coastal zone area for the problems of forecasting of the sea
level change and the coastal zone flooding.
Figure 2. Dependence of the
water surface area (thousand km2) of The Caspian Sea on the
level (m abs.). A – for level from –1000 to 0 m abs. Â – for
level from –100 to –20 m abs.
The account of "new" morphometry of the
Caspian Sea in problems of the longterm level forecast results
in the underestimated values (quantile) with small excess probability
compared with the "linear" problem.
The Forecast
With the modern level of scientific knowledge,
it is impossible to make the forecast (longterm) of water balance
hydrometeorological components for concrete calendar date. Hence,
the method of the longterm calendar forecasting of a sea level
is impossible as well. Only probability forecasts are possible,
for example, the one of the average sea level position and the deviation
from it of the position of the given probability (quantile).
In Table 1, the quantils of the conditional distributions
of level probability for the nearest decades are presented. As follows
from this table, the range of possible level values is wide enough.
The mark  26 m should be taken into account as one having the exceedence
of 1 % when designing engineering protection actions. Low sea levels
on marks  28,29 m are also probable.
Table 1. Probability forecast
of the Caspian Sea level (irrevocable withdrawals = 25 km3/year;
the initial level=27.0m)
Probability
of exceedence, % 
Year 
5 
10 
20 
30 
40 
50 
0.1 
25.87 
25.59 
25.48 
25.48 
25.54 
25.63 
1.0 
26.20 
26.04 
26.02 
26.06 
26.13 
26.21 
5.0 
26.50 
26.44 
26.51 
26.58 
26.66 
26.73 
50.0 
27.22 
27.40 
27.67 
27.83 
27.92 
27.98 
95.0 
27.93 
28.37 
28.84 
29.07 
29.19 
29.23 
99.0 
28.23 
28.76 
29.33 
29.59 
29.71 
29.75 
For the periods until 2005, 2010, 2020, 2030 etc.
the average forecasting level (with 50 % probability of exceedence)
is from 27.05 m (that practically corresponds to the modern coastal
line position) up to 27.98 m for 2050 (actually up to the level
which is considered safe). The most adverse forecast with 0.1 %
probability of exceedence for the same periods is 25.48 m, and
the most adverse forecast with 1 % probability of exceedence is
26.02 m.
Bayes Forecast Estimations
The risk in economic development of coastal territories
arises both as a consequence of stochastic character of influences
as midannual or extreme levels and owing to a wide set of uncertainties,
that should be taken into account when accepting some design (or
organizational) decisions. The models presented above take into
account the basic kind of uncertainty  probability character of
inducing hydrometeorological processes. At that, parameter errors
resulting from estimating by short samples are not considered.
Let us consider the influence of sample properties
of model parameters estimations. Being functions of time, expressions
for sea level expectation and dispersion enable to predict future
fluctuations of the sea in the form of confidence intervals or the
given probability values (the conditional density being approximated
by the Gaussian law). The latest form of the forecast is used at
a substantiation of actions and designing of constructions of coastal
territories engineering protection. We remind that expressions (5)
and (6) has been received in the assumption that estimations of
stochastic models parameters are known exactly and the received
conditional distributions reflect only stochastic character of hydrometeorological
processes variability. However, those parameters estimations are
known to possess the socalled sample properties and to be characterized
by errors in the simplest case.
The simple approach leading to results easy to
be interpreted is based on Bayes ideology supposing the construction
of the socalled forecast density of required value
x
as the conditional distribution
with the given observations of
y.
In accordance with the terminology [9], let us
introduce the probability model for x
as
dependent on some parameter
determined by available values of
y. Further, assuming that posterior distribution
density of this parameter
is known and x
and y
are independent, the forecast probability density can be received
from the following expression:

( 9) 
Calculations according to the equation (9) are
carried out by numerical integration with either sample distribution
of a parameter (estimation), or the distribution of an estimation
on homogeneous objects group (water bodies, lakes, meteorological
stations, etc.) used as the distribution density p
. As it was mentioned above, as g
function it is possible to use the normal distribution
law with parameters determined by formulas (5) or (6) depending
on parameters estimations of stochastic models of river inflow and
water body evaporation. The results of the calculations are presented
in Tables 2 and 3 for cases when the sample dispersion of estimations
of inflow and evaporation average values and the autocorrelation
coefficient estimation are accounted.
Table 2. Probability forecast
of the Caspian Sea level with the account of sample properties
(error) of the inflow expectation estimation (irrevocable
withdrawals = 25 êm3/year; an initial level = 27.0m)
Probability
of exceedence, % 
Year 
5 
10 
20 
30 
40 
50 
0.1 
25.86 
25.56 
25.40 
25.36 
25.40 
25.48 
1.0 
26.20 
26.01 
25.97 
25.97 
26.02 
26.10 
5.0 
26.50 
26.42 
26.47 
26.51 
26.58 
26.65 
50.0 
27.22 
27.40 
27.68 
27.83 
27.92 
27.98 
95.0 
27.94 
28.39 
28.89 
29.14 
29.27 
29.31 
99.0 
28.24 
28.79 
29.39 
29.68 
29.82 
29.86 
Table 3. Probability forecast
of the Caspian Sea level with the account of the sample properties
(errors) of the inflow and evaporation expectation estimations
(irrevocable withdrawals = 25 êm3/year; an initial level =
27.0m)
Probability
of exceedence, % 
Year 
5 
10 
20 
30 
40 
50 
1.0 
26.19 
26.00 
25.94 
25.94 
25.96 
26.01 
99.0 
28.24 
28.80 
29.40 
29.70 
29.86 
29.92 
Conclusions
The problem of the Caspian Sea level forecasting
is closely connected both with the research of natural hydrometeorological
processes variations and with transboundary character of this water
object. The changed status of the sea has led to essential degradation
of the observation network and, correspondingly, to the growth of
hydrological forecasts uncertainty and zones of risk.
As the result, the conclusion is obvious about the necessity of
close international cooperation of scientists in the Caspian region
with participation and under the support of UNESCO, UNEP, etc.
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