## Assessment of Flow Periodicity of the Danube River and its Major Tributaries in Serbia

Zoran Simić^{1}, Miodrag Milovanović^{1}, Mirko Melentijević^{2}

^{1} Jaroslav Černi Institute for the Development of Water Resources, Jaroslava Černog 80, 11226 Belgrade, Serbia; E-mail:
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^{2} Kneza od Semberije 6a, 11000 Beograd, Serbia

### Abstract

In the present paper an analysis of the flow periodicity for the period from 1926 to 2016 was carried out on nine significant gauging stations on the Danube River and large tributaries of the first-order tributaries through Serbia (Sava, Tisza, Drava, Velika Morava) and large tributaries of the second-order (Drina, Južna Morava and Zapadna Morava). The paper presents a morphological description of the Danube basin and sub-basins on the analyzed rivers and indications of meteorological conditions for the occurrence of discharges in the Danube basin. For the analyzed series of average monthly discharges, the differences between the discharges into the Danube (Bezdan and Orsova) and individual tributaries are shown. Statistical parameters of data series stationary data series, mean monthly discharges distribution, etc are analyzed,. The time dependence of the stochastic component is described by the linear Markov first-order model. The periodic component in a series of average monthly discharges is described by Fourier transforms with a 12-month periodicity and its subperiodicity. For the analysis of monthly and annual discharges, the correlogram and variance spectrum are applied. When analyzing annual discharges, for each profile the "summary Δq curve" is calculated, based on which the alteration of rainy and dry periods is considered. Conclusions and observations are presented at the end of the work.

**Keywords:** flow periodicity, models of discharge transformation, river Danube.

Introduction

The Danube is the most important river in central and southeastern Europe and is the second largest by discharge in Europe. It rises in Germany and empties into the Black Sea. The Danube is 2850 km long and drains a land area of more than 800,000 km^{2}. Given that as many as 19 countries share the Danube River Basin, in that regard it is the top-ranking river on a global scale.

A large part of the Danube's course runs through Serbia (588 km). Its drainage area in Serbia is 81,646 km^{2}, which accounts for 92.2% of Serbia's territory. Along its course in Serbia, the Danube receives several major tributaries: the Drava, the Tisza, the Sava, and the Velika Morava.

The Danube River Basin (DRB) borders on the Alps to the west and the Dinaric Alps and Balkan Mountains to the south. The Carpathian Mountains are also located within the DRB, in its northeastern part.

Due to the elongated shape of the DRB in the west to east direction and diverse topography, the climate conditions in the basin vary. The western parts of the upper DRB are exposed to Atlantic influences, while the climate in the eastern parts is continental. Along the upper and middle course of the Danube, particularly in the drainage areas of the Drava and the Tisza, the climate is affected by the Mediterranean Sea. In general, the climate in the DRB is driven by a morphology ranging from tall ice-capped and medium-high mountains through to rolling hills and spacious lowlands. (Hock and Kovacs, 1987).

Annual precipitation totals in the DRB range from 2300 mm in the high mountainous region to about 400 mm in the Danube Delta. Precipitation levels greater than 2000 mm are recorded in the upper part of the basin – the Alpine region, the central part of the basin and the southern parts (Julian Alps and the Dinaric system), exposed to the influence of humid and warm air from the Mediterranean (Stancik and Jovanovic, 1988).

Figure 1: The Danube River Basin and drainage areas of its tributaries addressed in the paper.

Generally speaking, humidity in the DRB originates from seven regions: North Atlantic, North Africa, Mediterranean, Black Sea, Caspian Sea, DRB, and central and eastern Europe. The main source of precipitation in winter (October through March) is the Mediterranean, and in summer (April to September) the DRB itself (Ćirić et al. 2017).

Figure 2: Schematic representation of precipitation sources in the Danube River Basin (Ćirić et al. 2017).

It is well known that hydrologic and meteorological data traditionally provide inputs (river discharges, precipitation, etc.) for hydrologic studies. Particularly interesting projections are provided by climate change assessments, use of satellite data for long-term forecasting, and systematic studies of the energy spectrum of meteorological data that provide an indication of the causes of cycles, trends and oscillatory tendencies of observations. Analyses of consecutive (periodic and cyclical) river discharges are also very important. Monthly and seasonal flows exhibit a higher level of consistency, reflected in strong river discharge correlations between consecutive time periods (Melentijević, 1970).

The Danube, along with its major tributaries, is suitable for cyclic and periodic assessments because long and reliable discharge time-series are available (Ducić V. et al. 2006; Pekárová P. et al. 2010). It is a known fact that river catchments in Serbia exhibit distinct alternation of wet and dry periods of varying intensity and duration, and there are sums of modular deviation of mean annual flows with a similar pattern at most of the considered gauging stations on the Danube and its tributaries (Isailović and Srna, 2001).

The present research analyzes different transformations of mean monthly river discharges to derive second-order stationarity and apply certain mathematical and statistical models to the datasets. The structure of the time series, particularly of monthly and annual river discharges, is tested. In addition, a brief analysis of the time-series stationarity and periodicity is provided, along with a short discussion of the significance of autocorrelation coefficients and the variance spectrum in structural analysis of time series.

The assessment of monthly and annual discharges of the Danube included data from gauging stations on the Danube (Orșova GS and Bezdan GS), the Sava (Sremska Mitrovica GS), the Tisza (Senta GS), the Drava (Donji Miholjac GS), the Drina (Radalj & Zvornik GS), the Velika Morava (Ljubičevski Most GS), the Zapadna Morava (Jasika GS + Bivolje GS), and the Južna Morava (Mojsinje & Stalać GS), collected during the period from 1926 to 2016. A hydrologic year was assumed to begin on 1 September and end on 30 September of the following year.

The Sava, Drava, Tisza and Velika Morava are first-order tributaries of the Danube, while the Drina, which is a tributary of the Sava and the Velika Morava, originates at the junction of the Zapadna Morava and the Južna Morava.

Table 1 shows that, among the studied rivers, the Drina and the Sava have the highest specific discharge, followed by the Drava (q=14÷21 l/s/km^{2}). The specific discharge of the other five rivers is less than 10 l/s/km^{2}.

Monthly Discharges

The mean monthly discharge time-series of the studied rivers were averaged out by month to define mean monthly and annual discharges for the study period from 1926 to 2016.

The figure in Fig.4 is a graphical representation of mean monthly discharges of the studied rivers over the study period from 1926 to 2016.

Figure 3: Gauging stations on the studied rivers in the Danube River Basin.

Figure 4: Graphical representation of mean monthly discharges of the studied rivers over the study period from 1926 to 2016.

The above table and figure show that during the year, the mean discharges of the Danube (at Orșova) strongly correlate with those of the Tisza (0.985), then with those of the Velika Morava, Zapadna Morava and Južna Morava (0.86-0.89). The correlation is somewhat weaker with the mean discharges of the Sava, the Danube at Bezdan and the Drina (0.78-0.84). The weakest correlation of mean monthly discharges is between the Danube at Orșova and the Drava (0.368). The mean discharges of the Velika Morava, Zapadna Morava and Južna Morava exhibit a strong mutual correlation (0.998). The discharges of the Drina correlate very well with those of the Sava and the Tisza (0.977). There is a significant difference between the discharge pattern of the Drava and those of the other studied rivers. A particularly weak correlation exists between the mean monthly discharges of the Drava and those of the Sava and the Tisza (-0.158).

In statistical analysis of mean monthly discharges the stability (stationarity) of datasets might be questionable, but this is not always a problem in the case of annual discharge time-series. Each monthly data point has its own mean, variance, skewness coefficient, excess coefficient, and the like. For a dataset analysis to be as accurate as possible, the mean of each of these parameters needs to be constant for all the calendar months. Consequently, the transformation of the main time-series must also be such that the desired stability (stationarity) is achieved. When a time series is stationary, statistical analysis is used to determine the structure of the time series and describe it by means of a suitable mathematical model.

Stationarity

As is well known, time series of mean monthly discharges are not stationary because the mean discharges vary from month to month. Likewise, higher-order moments around the mean vary between months during the year. The months that exhibit a higher mean discharge need not also show a higher variance.

The time series of monthly discharges *Q _{t}* are comprised of values from twelve sets (each month has its own distribution function, or its own statistical parameters (mean, standard deviation, skewness coefficient, excess coefficient, etc.)), which explain their non-stationarity.

First-order stationarity is obtained when *Q _{t}* is transformed into

*X*as follows:

_{t}(1)

Here *Q _{τ}* is the monthly mean of month

*τ, n*is the number of years since the beginning of monitoring,

*N*is the number of years of monitoring, and

_{1}*X*is not only a monthly time series whose mean is zero, but also a time series whose total average value is equal to zero. This results in first-order stationarity.

_{t}If the variable *X _{t}* is transformed further, we get a new variable

(2)

where *S _{τ}* is the standard deviation of month

*τ*. Variable

*Q*is thus standardized. The time series

_{t}*q*now has a distribution function whose mean is 0 and standard deviation 1 for all monthly values. Then the time series of standard random components of monthly discharges Є

_{t}*can be derived from*

_{t}(3)

The maximum amplitude of random minimum and maximum monthly discharges (*A _{τ}*), expressed as a percentage of the average discharge, is calculated from:

(4)

Table 4 highlights the months in which there is a maximum amplitude of random monthly discharges, or the sum of absolute minimum and maximum discharges as a percentage of the average discharge. It is apparent that as the size of the river (and of the discharge) increases, the amplitudes of minimum and maximum monthly discharges decrease. Thus, this value of the Danube at Orșova is 137% on average (maximum in October, 167%), whereas in the case of the Južna Morava it is 421% on average (maximum in July, as high as 723%).

Table 4 and Figure 5 show that the largest deviations from average discharges for most of the studied rivers can be expected in October, except for the Tisza in September and the Južna Morava in July. In view of the large share of the random component in total discharges, the discharges of the studied rivers cannot be predicted even one month in advance.

Figure 5: Graphical representation of the maximum amplitude of random minimum and maximum discharges as a percentage of average discharge.

Periodicity

If time series exhibit a relatively high frequency of repeated oscillations, their periodicity needs to be tested. Each periodic movement can be approximated by the main period and its cycles. It is known from experience that the average values (*Q _{τ}*) of time series

*Q*are periodically correlated with the main period of 12 months. The periodic movement of monthly discharges

_{t}*Q*can be described applying Fourier series.

_{t}Mean monthly *Q _{t}* can be represented by the following mathematical expression:

(5)

Likewise, the standard deviation of monthly discharges can be written as:

(6)

The Fourier coefficients in Eqs. (5) and (6) are defined by the following expressions:

(7)

(8)

(9)

(10)

To describe the periodic movement of mean monthly discharges, 12 constants, Ai and Bi, need to be determined for a period of 12 months (i=1), as well as their five cycles (i = 2, 3, 4, 5, 6), evident in Eqs. (5) and (6). In physical terms, it is clear in hydrology that there is a one-year (12-month) cycle. However, observed data very often show a second, 6-month cycle. The number of cycles in the main 12-month period depends on the shape of the periodic function. If a 12-month periodic movement of and St can be properly approximated by a simple sine or cosine function, the 12-month period is sufficient, without any of its cycles. However, if the 12-month periodic movement does not resemble, even approximately, a sine or cosine function, say with sharp maxima and spread-out minima, only a 6-month cycle would not be sufficient. All the other cycles: 4-, 3-, 2.4- and 2-monthly, are needed as well.

Figure 6: Periodicity of the Danube River at Orsova.

Figure 7: Periodicity of the Danube River at Bezdan.

Figure 8: Periodicity of the Sava River at Sremska Mitrovica.

Figure 9: Periodicity of the Tisza River at Senta.

Figure 10: Periodicity of the Drava River at Donji Miholjac.

Figure 11: Periodicity of the Drina River at Radalj.

Figure 12: Periodicity of the Velika Morava River at Ljubičevski Most.

Figure 13: Periodicity of the Zapadna Morava River at Jasika.

Figure 14: Periodicity of the Južna Morava River at Mojsinje.

Based on the monthly discharges of the studied rivers shown in the above tables and figures, it is apparent that 26-80% of the total variance belongs to the 12-month cycle, 4-31% to the 6-month cycle, and 2.4-11% to the 4-month cycle, while the share of each of the other cycles is less than 3.4%. Given this distribution of the variance of monthly discharges, the conclusion is that the impact of yearly seasonal climate variations on the discharges is small. The seasonal variation of the mean monthly discharge time-series of the studied rivers is low. The stochastic component (*q _{t}*) causes the major discharge variations. Keeping in mind the physics of the problem, a small impact of the stochastic component can be expected only in the case of rivers largely fed by reservoirs. Such rivers include glacial streams, outflows from large natural lakes, rivers whose watersheds are capable of storing large amounts of water underground, and rivers backed up by artificial reservoirs that equalize seasonal flows.

Autocorrelation Coefficients

It is also possible to determine the periodicity of time series by analyzing autocorrelation coefficients. The general equation for deriving autocorrelation coefficients is:

(11)

It is well known that in the case of time series that can be represented by

(12)

(where: *C* is the amplitude, *θ* is the frequency of the cyclical component, and *Z _{t}* is the stochastic component), the autocorrelation coefficients of the cyclical component can be expressed as

(13)

where: is the variance of *X _{t}*. This means that if there is a frequency

*θ*, there is also a cyclical component in the correlogram which cannot be suppressed.

Figure 15: Correlogram of monthly discharges of the studied rivers over the study period (1926 to 2016).

The values of the first autocorrelation coefficient of the eight studied rivers range from 0.549 (Drina) to 0.640 (Drava). This value is 0.638 for the Danube, 0.604 for the Sava, 0.627 for the Tisza, 0.627 for the Velika Morava, 0.591 for the Zapadna Morava, and 0.638 for the Južna Morava.

In cases where the time series have more than one cycle, the correlogram is a linear combination of individual periodic durations.

Variance Spectrum

Even though the autocorrelation function is very useful for analysis of collected data, it is sometimes difficult to draw conclusions from it related to physics. However, it is possible to gain further insight into the nature of a time series by Fourier's transformation of the autocorrelation function. This operation of transformation from the time domain into the frequency domain is referred to as the variance spectrum.

The first determination of the variance spectrum is based on the equation

(14)

where: *r* = 0, 1, 2, . . . , *m*; and Δ*ι* = frequency interval.

From a statistical perspective, however, this approach does not yield the best determination of the variance spectrum. Various approaches are followed to obtain a "smooth" function, or, more precisely, to smoothen the curve resulting from Eq. (14). The method known in literature as "hanning" (after Julius von Hann) is used in the present research. It is defined by the following equations:

(15)

(16)

(17)

The dimensions of the spectral quantities are indicated in Eq. (14). Variance spectrum (*V _{r}*) calculations involve autocorrelation coefficients (

*ρ*), instead of autocovariance coefficients (

_{t}*C*), such that (

_{r}*V*) is obtained as a dimensionless number relative to the number of cycles in a month. If the variance spectrum as entered in the diagram is the ordinate, relative to the abscissa that shows the frequency of cycles in a month, the area below the variance spectrum line will be equal to one. If autocovariance coefficients were to be used, the area below this curve would be equal to the square of the standard deviation of the time series. If Δ

_{r}*t*is the interval between data points in the time series and Δ

*ι*the frequency interval for the calculations, then the abscissa is

*m*Δ

*t*Δ

*ι*.

Figure 16: Variance spectrum of autocorrelation coefficients of monthly discharges of the studied rivers over the study period (1926 to 2016).

The variance spectrum line is in effect the total variance distribution line of the time series. Each distinct periodicity of the time series will be identified on the variance spectrum line by peaks, and the area below the curve at those peaks represents the share of the corresponding period in the total variance of the time series. The above variance spectrum clearly shows that the 12-month (1/0.083) and 6-month (1/0.167) cyclical components are the most pronounced, or, in other words, that in addition to a stochastic component they have the largest share in the total monthly discharges.

Spectral analysis can be extremely useful for data assessment where river basins feature various human interventions. It can reveal the changes that have occurred in the time series.

According to all of the above, the conclusion would be that the studied mean monthly discharges exhibit small seasonal changes. It is the stochastic component (*q _{t}*) that causes major discharge variations. Keeping in mind the physics of the problem, a small impact of the stochastic component can only be expected if rivers are largely fed by reservoirs. Such rivers include glacial streams, outflows from large natural lakes, rivers whose watershed are capable of storing large amounts of water underground, and rivers whose seasonal flows are equalized by artificial reservoirs. Based on the results, the existing reservoirs in the Danube River Basin do not have the retention capacity that would have a significant effect on natural discharge variations in Serbia.

Mathematical Models

Following the transformation of a non-stationary time series *Q _{t}* into a stationary (second-order) time series qt, two models can be used to describe the time series

*q*. The two models are: (i) independent variables and (2) linear first-order Markov model.

_{t}(1) Independent variables

If for a certain probability the autocorrelation coefficients are *ρ _{ι}* ≅ 0 for time series qt, then it is comprised of a series of stochastic variables independent of each other. As previously stated, qt is a distribution whose mean is zero and whose standard deviation is one. To determine if the independent time series is a mathematical model, the correlogram of that time series is checked to see whether it substantially differs from zero. Anderson derived Eq. (18) to assess if

*ρ*differed considerably from zero or not. He found

_{ι}*ρ*to have a nearly normal distribution, with a mean of -1/(

_{ι}*N - ι - 1*) and a variance of (

*N - ι - 2*) / (

*N - ι - 1*)

^{2}. The calculated values of ρι are compared with theoretical quantities obtained from:

(18)

where: *t _{α}* is the standard variable that corresponds to probability

*α*, and

*N*is the total number of collected data points. If , for a given probability

*ρ*is significantly different from zero.

_{ι}

Figure 17: Correlogram of the random component of monthly discharges of the studied rivers.

Figure 17 shows the correlogram of the random component (Є* _{t}*) of mean monthly discharges. The confidence intervals according to Eq. (18) were also calculated for different probabilities (70% and 90%). The plots indicate that the values of Є

_{t}are indeed random quantities.

(2) Linear first-order Markov model

If the time-series correlogram follows the values derived from the following equation:

(19)

then the time series *q _{t}* is a first-order Markov process that satisfies the equation:

(20)

where *η* is the random component, independent of *q _{t}, q_{t-1}, q_{t},* ..., and other quantities

*η*.

If *η _{t} = β*Є

_{t}, where 1/

*β*is the standard deviation of variable

*η*, then Є

_{t}*is the standardized independent stochastic variable. Given that the standard deviation of*

_{t}*q*is equal to one for each

_{t}*t*, it follows from Eq. (20) that

(21)

If Eqs. (2) and (20) are equated, then:

(22)

This equality yields the following expression for *Q _{t}*:

(23)

Equation (23) is the first-order Markov model of the variable *Q _{t}*. If we want to know whether

*Q*is really a first-order Markov model, then for the order of Є

_{t}_{t}we derive from

(24)

we need to calculate the autocorrelation coefficients and test them against Eq. (18) to see if it is really a series of mutually independent quantities. If so, the model can be adopted.

For the linear first-order Markov model (Eq. 23), the first autocorrelation coefficient *ρ _{1}* of the monthly discharges of the studied rivers is 0.55-0.64. These coefficients are rather high because large amounts of water are moved from month to month (i.e. because of the capacity of the basin to store certain amounts of water over a short time period). As the time period lengthens, this effect decreases.

Any monthly discharge of a studied river is defined by Eq. (23). The values of the first-order autocorrelation coefficients *ρ _{1}* of the studied rivers are determined in Section 2.3.

It should be noted that these models are not entirely accurate because they only take into account second-order stationarity for *Q _{t}*. If there is a time series

*Q*for which skewness coefficients are high, then calculations need to include ln

_{t}*Q*. This will reduce the effect of outliers (i.e. the value of greater central moments).

_{t}The correlogram of the stochastic component (*q _{t}*) is shown in Fig. 15. It is apparent that the time series

*q*is consistent with the linear first-order Markov model.

_{(t)}

Assessment of Annual River Discharges

Annual Discharge Fluctuations

In cyclical time series, maximum and minimum quantities occur at equal time intervals, with a constant amplitude. The stochastic component, if present, tends to destroy this regularity. In any oscillatory time order, the amplitude and time interval between the maximum and minimum quantities are spread around the mean. Cyclical time series are also oscillatory, but oscillatory time series need not be cyclical. If annual time series of river discharges are to be properly analyzed, the random component needs to be separated from other components (if any), and each should be analyzed on its own.

According to Table 6, there is a strong correlation between mean annual discharges of the Velika Morava, Zapadna Morava and Južna Morava, then of the Danube at Orșova and the Danube at Bezdan, and ultimately a good correlation between the Sava and the Tisza. Mean annual discharges of the Drina show a solid correlation with the Zapadna Morava, the Sava and the Danube at Orșova, which is as expected. The Drava exhibits the weakest correlation with the other rivers, especially the three Moravas and the Tisza.

Figure 18: Mean annual discharges of the studied rivers.

Figure 19: Annual discharge correlograms of the studied rivers.

We will know if the annual discharges are random quantities if we calculate their autocorrelation coefficients according to Eq. (11) and test them against Eq. (18). If not random, then a suitable mathematical model (first-order Markov, second-order Markov or other) should be found to extract the random component.

Figure 20: Variance spectrum of annual discharges of the analyzed rivers.

Based on Fig. 20, Table 7 shows the most significant periods of annual discharges of the studied rivers.

Although it is apparent that for some of the rivers certain periods of occurrence of a higher or lower order (50, 20, 3.7 years, etc.) are dominant, it is very difficult to define the multiyear periodicity of the studied rivers at the selected gauging stations. Rather, each river appears to be specific in terms of multiyear periodicity and some of the rivers together show a similar multiyear pattern, but no distinct periodicity of the Danube and its studied tributaries (the Sava, Tisza, Drava, Drina, Velika Morava, Zapadna Morava and Južna Morava) was established in this research.

Annual Discharges

The annual discharges of the studied rivers were assessed over the monitoring period from 1926 to 2016. This period of 90 years is suitable for analysis of annual discharges.

Figure 21 shows the summary curve of Δ_{q} derived from the equation:

(25)

where:

*Q _{g}* – mean annual discharge;

*Q _{g}* – average discharge of the monitoring period.

Figure 21: Summary Δ*q* curves of the studied rivers (1926-2016).

It is apparent that there is a very good correlation between the alternating dry (decreasing ∑Δ*q*) and wet (increasing ∑Δ*q*) periods at most of the gauging stations of the studied rivers. Exceptions are the Drava and the Danube at Bezdan in the 1970's, attributable to the specific features of their drainage areas (large parts belong to the Alpine region).

Interestingly, for more than 20 years (from 1995 to 2016) most of the studied rivers did not exhibit distinct periods in either direction. The Tisza is an exception, given a rainy period in the first decade of the 21st century and a dry period in the second decade.

Figure 22: Summary Δ*q* curve of the Danube at Orșova (1840-2016).

Figure 22 of the summary Δ*q* curve of the Danube at Orșova shows distinctly alternating wet and dry periods. The above-average periods in terms of water abundance were 1842-1853, 1875-1881, 1911-1917, 1936-1945, 1962-1969 and 1974-1981, and the below-average (drier) periods were 1855-1869, 1883-1897, 1897-1909, 1926-1935, 1946-1954 and 1982-1994.

According to the plot in Fig. 22, the periods of above-average discharges lasted from 7 to 12 years (8.7 on average), and those below average from 9 to 15 years (12.2 on average). It should be noted that in the periods of above-average discharges there have been isolated years with below-average discharges (dry years), and vice-versa. The summary curve suggests that the annual discharges adhere to a certain periodic function, and that this function might be cyclical.

Figure 23: Variance spectrum of the summary Δ*q* curves of the studied rivers.

It is apparent from the variance spectrum that the macro cycles of the summary Δ*q* curves of the Danube have macroperiods of 50 (33.3), 20 and 14.3 years, of the Sava 100 and 20 years, of the Tisza 33.3 and 100 years, of the Drava 50, 25 and 14.3 years, of the Drina 50 and 20 years, of the Velika Morava 50 and 25 years, of the Zapadna Morava 50 and 20 years, and of the Južna Morava 100 and 25 years.

Conclusions

- The study period from 1926 to 2016 is representative for drawing conclusions with regard to the periodicity of mean monthly time series of all the studied rivers (and gauging stations). The assessment of the nine gauging stations of the studied rivers – the Danube, the Sava, the Tisza, the Drava, the Drina, the Velika Morava, the Zapadna Morava and the Južna Morava, could shed theoretical light on the changes that occur in hydrologic time series. It should be noted, however, that the assessment approach differs for each time series, depending on the nature of the problem addressed. Still, a harmonic assessment, as a predecessor of autocorrelation and spectral analyses, is convenient for explaining phenomena associated with river discharges.
- The morphology of the Danube River Basin and the sub-basins of the other studied rivers (the Drava, the Tisza, the Danube up to Bezdan, the Sava, the Drina, the Velika Morava), which affects the discharge regime of these rivers, was described and analyzed. In addition, an analysis of the meteorological conditions in the Danube River Basin was provided, along with which cycles, or dominant directions, contribute to precipitation in the river basin.
- The differences between the discharges of the Danube (at Bezdan and Orșova) and the various tributaries were discussed. The monthly discharge distribution is strongly correlated among seven of the studied gauging stations, while that of the Drava and the Danube at Bezdan reflects specific hydrologic processes in their drainage areas. The time series of monthly discharges contain a periodic component and a highly time-dependent stochastic component. The time dependence of the stochastic component can be described by Markov models. Based on the maximum amplitude of random minimum and maximum monthly discharges, the largest deviations from mean discharges can be expected in October for most of the studied rivers, except for the Tisza and the Danube at Bezdan (September) and the Južna Morava (July). Given the large share of the random component in the total discharges of the studied rivers, it is not possible to predict discharges even one month in advance.
- The periodic component of the monthly discharge time-series can be described by Fourier transforms, with a 12-month cycle and its sub-cycles. Approximations of mean monthly discharges by harmonic functions are difficult because of large variations in the periods of maximum and minimum discharges. Autumn rains and spring snowmelt have the greatest impact on these variations, causing high discharges of many rivers for three or four months during the year. Harmonic analyses of such discharge regimes require a 12-month component plus at least a 6-month component. If other components are also used, an even better fit of the observed and modeled discharges is achieved. In the case of the monthly discharges of the studied rivers, the 12-month cycle accounts for 26-80% of the total variance, the 6-month cycle for 4-31%, the 4-month cycle for 2.4-11%, and the other cycles for less than 3.4%. Given this variance distribution of the monthly discharges, the impact of yearly seasonal climate variations on discharges is small.
- The correlogram and variance spectrum were used in parallel to assess the monthly and annual discharges. The correlogram showed the physical cycles in the monthly discharge time-series and the variance spectrum the number and significance of the periods. Both the correlogram and variance spectrum are good indicators of the degree of dependency of the stochastic component. As shown, some of the rivers exhibited a distinct periodicity of mean annual discharges but the extracted dominant periods were of similar significance so it is difficult to rank them.
- Based on the research presented, one of the conclusions is that water management of the studied rivers can predominantly be improved by creating large reservoirs in the catchments, as well as applying other measures such as afforestation and the like.
- The "summary Δ
*q*curve" plays an exceptional role in assessments of annual discharges. It can be used to draw conclusions about the occurrence of wet and dry periods, or consecutive wet and dry periods. This is especially important for optimal sizing of reservoirs in the river basin. The assessed summary Δ*q*curve of the Danube at Orșova shows periods of above-average discharges that lasted for 8 to 10 years, and periods of below-average discharges of 9 to 13 years.

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