ANALYZING CHANGES AND FREQUENCY DISTRIBUTION IN MAXIMUM RUNOFF VOLUMES WITH DIFFERENT DURATION OF THE DANUBE RIVER AT BRATISLAVA

The volume of the flood waves and its importance is evaluated rarely. But, without these data, it would not be possible to imagine the extreme nature of the distribution system. Several hypotheses claim that more extremes in climatic and hydrologic phenomena are anticipated. In the present study, the annual maximum runoff volumes with t-day durations were calculated for a 144-year series of mean daily discharge of the Danube River at Bratislava gauge (Slovakia). The statistical methods were used to estimate T-year annual maximum runoff volumes and clarify how the annual maximum runoff volumes of the Danube River at Bratislava changed over period 1876–2019 and over dry and wet periods. The conclusion is that the runoff volume regime during floods has not changed significantly during the last 144 years. The annual maximum runoff volumes of the wet period have a greater impact on changes in LPIII exceedance curves at volumes with a time duration more than 20-days.


Introduction
In studying the flood wave parameters, the attention is often focused on the culmination or maximum water level. However, these basic data on extreme floods are not enough for modern water management needs. Apart from the peak runoff another very important streamflow characteristic of a river is its runoff volume. Some issues in water management and engineering hydrology require that peak discharge (Qmax) and the shape of a flood wave or a flood runoff volume (Vmax) are known. In applied hydrology, it is often difficult to assign exact values of a flood wave volume to a particular probability of exceedance and hence to its corresponding T-year discharges. Such relationships are very irregular in nature, so a flood wave hydrograph of a given exceedance probability must be a priory known. Pekárová and Miklánek (2019) described statistical processing of the maximum discharges and flood volumes based on a set of distribution function. The team of authors Hladný et al. (1970) dealt with the processing of volumes of flood waves from stations throughout the territory of the Czechoslovakia. To define the volumes of individual waves, the authors introduced the parameter tthe duration of the flood wave in days. In this way, they determined maximum runoff volumes lasting 2, 5, 10 and 30 days. Čermak (1956) studied the flood wave volumes of flood events with peaking above the long-term annual mean discharge Qa, on the regional scale. The determination of maximum volumes was, for example, addressed by Beard (1956). Beard (1956) also used theoretical exceedance probability curves and the parameter tfor flood wave duration, to determine the annual extreme runoff volumes. The values of tparameter depend on the catchment characteristics of the river being studied. However, the author selected only one value for the extreme runoff volumes in every year of the data sets. Knowledge of flood-wave volumesas an important hydrological characteristicwas apparent during a flood on the Danube in 1965(Zatkalik, 1970Hladný et al., 1970). During this flood event, the river dikes broke under the pressure exerted by the long duration of high water stages, but not because of the extremely high water stages themselves. Recently in Slovakia a few studies, analyses and estimations of flood wave volumes corresponding to the maximum design discharge with a return period of T-year on the River Danube at Bratislava have been carried out by e.g. Mitková et al. (2002) and Halmova et al. (2008), but more studies have focused on the joint probability of total volume and peak discharges, e.g. Bačová Mitková and Halmová (2014), Gaal et al. (2010), Szolgay et al. (2012), Szolgay et al. (2016), and Pekárová et al. (2018). The influence of human activity occurs in the Danube River to change the transformation properties of the channel, to change the travel time of the waves as well as to a different outflow behaviour at the passage of the flow wave through the riverbed compared to the past. Case study of large flood events and hydrological simulation of flood transformations in the upper Danube River were investigated in Bačová Mitková et al. (2016). Authors concluded that the travel times of high floods have not significantly changed at the Kienstock-Devín section. On the other hand, the peak water levels for recent river conditions are higher at the same discharges. In assessment of the climate change impacts on the river discharge regime (extremes, flood hydrographs and drought periods), it is expected that an increase in air temperature may cause (or already has caused) an increase in extreme discharges and flood volumes (Blöschl et al., 2017;. Since a 143-year series of the mean daily discharge of the Danube at Bratislava gauging station is available. Therefore, we could calculate the 144-year series of the highest (annually) 2-, 5-, 10-, 20-, 30-and 60consecutive days' wave volumes. Than their probability distribution functions and trends were analysed. These series were subsequently divided into dry and wet periods and changes in their probability distribution functions and trends were analysed. The aim of this study is:  assess the maximum annual runoff volumes Vtmax lasting 2-, 5-, 10-, 20-, 30-and 60-days of the wave belongs to annual maximum discharges of the Danube River at Bratislava (1876-2019);  determine the theoretical exceedance probability curves;  estimate the T-year annual maximum runoff volumes with t-day durations belongs to annual maximum discharges;  analyze changes in the maximum annual runoff volumes Vtmax of the Danube River at Bratislava during the period (1876-2019);  analyze the changes in the maximum annual runoff volumes Vtmax in wet and dry periods during the period 1876-2019.

Methodology
In Czechoslovakia, Bratránek (1937) was the first who investigated the issue of runoff volumes. He used direct and indirect methods of peak vs. flood volume assessment.
The direct method was based on compiling runoff volumes higher than a chosen discharge threshold with assistance of the probability of exceedance (related to the T-year return period). The T-year discharges were then determined by extrapolating probability curves to the domain of low exceedance probability. In the latter method the author used generalized results of empirical relationships among several flood hydrograph characteristics.
In order to determine the runoff volume W20,Q1 (Equation 1), the author used not only the highest discharge Q reduced to (Q20 -Q1), but also values of the flood wave duration t20, related to the catchment area. extreme 20-year discharge; Q1 extreme 1-year discharge; t20 the average flood wave duration of the reduced discharge (Q20-Q1) [hrs].
The main disadvantage of this method is that it is not applicable for flood-wave volumes with smaller exceedance probabilities than once in 20 years. The treatment of the calculation of the maximum volumes of the Danube was dealt with by Zatkalík (1970). When calculating the maximum volumes of flood waves, he chose as a basis a procedure taking into account the duration of the flood wave in dayst.
The introduction of this parameter allows:  a clear assessment of the probability of exceeding the volume of a given flood wave,  creation of a basis for solving the problem of design flood, which would consist in allocating such a flood wave volume that would be maximum for a given T-year peak flow.

Log-Pearson III distribution
For the estimation of the maximum annual runoff volume Vtmax series distribution function we used Log-Pearson type III distribution. The Log-Pearson distribution type III. is used to estimate extremes in many natural processes and it is one of the most commonly used probability distribution in hydrology (Bobee, 1975;Pilon and Adamowski, 1993;Griffis and Stedinger, 2007;Pawar and Hire, 2018). In some previous works (Pekárová et al., 2018; Miklánek, 2019) we compared LPIII distribution with theoretical probability distributions, which were and still are also among the most used in Slovak hydrological practice. The Log-Pearson Type III distribution is a three-parameter gamma distribution with a logarithmic transformation of the variable. The cumulative distribution function and probability distribution function according Hosking and Wallis (1997) are defined as: If ≠ 0 let α=4/ 2 and ξ=μ-2σ/ If ˃0 then: where: ξlocation parameter; αshape parameter; βscale parameter Γ -Gamma function.
If <0 then The Kolmogorov-Smirnov test was performed to test the assumption that the discharge magnitudes follow the theoretical distributions. The p-value (p≥0.05) was used as a criterion for rejection of the proposed distribution hypothesis.

Mann-Kendal nonparametric test
The Mann-Kendall nonparametric test (M-K test) was used for determining the significant trends detection in time series. The nonparametric tests are more suitable for the detection of trends in hydrological time series, which are usually irregular, with many extremes (Hamed, 2008;Yue, et al. 2002;Gilbert, 1987). By M-K test, we want to test the null hypothesis H0 of no trend, i.e. the observations xi is randomly ordered in time, against the alternative hypothesis H1, where there is an increasing or decreasing monotonic trend.
For n (number of tested values) ≥ 10, the statistic S is approximately normally distributed with the mean and variance as follows where: qis the number of tied groups, tpthe number of data values in the p group.
The standard test statistic Z is computed as follows The presence of a statistically significant trend is evaluated using Z value. A positive (negative) value of Z indicates an upward (downward) trend. The statistic Z has a normal distribution. To test for either an upward or downward monotone trend (a two-tailed test) at α level of significance, hypothesis H0 (no trend) is rejected if the absolute value of |Z| is greater than Z1-α/2, where Z1-α/2 is obtained from the standard normal cumulative distribution tables. The M-K test detects trends at four levels of significance: α=0.001, 0.01, 0.05 and α=0.1.
Significance level of 0.001 means that there is a 0.1% probability that the value of xi is from a random distribution and are likely to make a mistake if we reject the hypothesis H0; Significance level of 0.1 means that there is a 10% probability that we make a mistake if we reject the hypothesis H0. If the absolute value of Z is less than the level of significance, there is no trend. For the four tested significance levels the following symbols are used in the template: *** if trend at α=0.001 level of significance -H0 seems to be impossible, ** if trend at α=0.01 level of significance, * if trend at α=0.05 level of significance -5%, mistake if we reject the H0, + if trend at α=0.1 level of significance.
Blank: the significance level is greater than 0.1, cannot be excluded that the H0 is true. The most significant trend is assigned three stars (***), with a gradual decrease in importance, the number of stars also decreases.

The Danube River and Input data
The Danube River is the second greatest river in Europe, after the Volga. The basin covers an area of 817 000 km 2 . The river originates from the Black Forest in Germany at the confluence of the Briga and the Breg streams. The Danube then discharges southeast for 2 872 km (1 785 mi), passing through four Central European capitals before emptying into the Black Sea via the Danube Delta in Romania and Ukraine. The Danube River Basin landscape geomorphology is characterized by a diversity of morphological patterns. The territory of the Danube River Basin is also one of the most floodendangered regions in Europe.
The mean daily discharges ( Fig. 1) and maximum annual discharges of the Danube River at Bratislava from the period 1876-2019 were used as input data. The course of annual peak discharges, long-term linear trend and 5-year moving trend are illustrated on the Fig. 2a. The annual peak discharges of the Danube River at Bratislava show increasing long-term linear trend during the selected period of 1876-2019. There were also occurred some extreme floods in 1899,1954,2002 or 2013 (Fig. 2a). The Fig. 2b shows the alternation of dry and wet years at intervals of approximately every three to four years. The minimum annual discharge was occurred in 1934 (3000 m 3 s -1 ). Since year 2014 to year 2019 we also record dry years. The maximum number of the events with annual maximum flows occurs in month of July it can be cause by summer rainfall especially in upper part of the Danube basin. The second peak of the number in occurrence of the annual maximum flows is in month of March when snow melts in higher parts of the basin and rainfall occur in lower part of the basin. The Fig. 3a illustrates the distribution of the annual maximum flows occurrence in individual months during the period of 1876-2019 and in dry and wet years. The Fig. 3a shows, that distribution of the annual maximum occurrence during the period 1876-2019 has maximum in the month of July. The dry and wet years has annual maximum flows occurrence in July. The annual maximum flows occurrence during the last nineteen years (2001-2019) was in January (Fig. 3b).

Determination of maximum volumes of Danube flow waves for the period 1876-2019
In the present paper for determining the maximum annual runoff volumes Vtmax, we used the procedure published in Zatkalík (1970). The mean daily discharges and maximum annual discharges of the Danube River at Bratislava from the period 1876-2019 were used as input data. We introduced the parameter trunoff duration in days to define the t-days maximum volumes of individual waves. In this way, we determined the maximum runoff volumes of t=2-, 5-, 10-, 20-, 30-and 60-days. The series of mean daily discharges were used to determine the annual maximum runoff volume Vtmax lasting 2-, 5-, 10-, 20-, 30-and 60-days. In each year separately, a flood event with an annual maximum flow was selected for the set of volumes. Than the 2-, 5-, 10-, 20-, 30-and 60days moving averages of the volume around the peak flow were calculated and the maximum volume for each t-days were selected. Fig. 4 presents an example of the determination of maximum volumes with a given runoff duration. The runoff volume series are shown in Fig. 5. Considering the 2-day and 5-day maximum runoff volumes, the flood of 1899 was the highest one and the lowest one was in 1934, within the period 1876-2019. But considering the 10-to 60-day runoff volumes, the highest flood was that of 1965.   1905 1920 1935 1950 1965 1980 1995 2010 2025 V20max [mil  1905 1920 1935 1950 1965 1980 1995 2010 2025 V30max [mil  1905 1920 1935 1950 1965 1980 1995 2010 2025 V60max [mil m 3 ] year Danube: Bratislava 1876-2019

Calculation of the theoretical probability curves of the maximum annual runoff volumes Vtmax for various flood durations t
A Log-Pearson III distribution was selected to calculate the T-year maximum runoff volume with the given runoff time duration (t). The examples of the theoretical (LPIII) exceedance probability curves of the maximum annual runoff volumes with durations t equal to 2-, 5-, 10-, and 60-days are demonstrated in Fig. 6

Trend analysis of annual maximum runoff volumes Vtmax on the Danube River at Bratislava (1876-2019)
The Mann-Kendall nonparametric test (M-K test) was used for detection of the significance in long-term trends of annual maximum runoff belong to annual maximum flows of the Danube River at Bratislava for period 1876-2019. The M-K trend test did not show significant longterm trends in annual maximum runoff volumes with duration t belong to annual maximum flows of the Danube River at Bratislava for selected period. After dividing the whole time series1876-2019 into shorter periods, about 40 years the M-K trend analysis indicates an increasing long-term trends in annual maximum runoff volumes Vtmax with t=2 days and t=5 days and we can reject the hypothesis H0 at significance level α=0. 05 and α=0.1, for period 1921-1960 (Table 3). The long-term trend of the M-K test for annual maximum runoff volumes with duration t=2 days and t=5 day of the Danube River at Bratislava (1921Bratislava ( -1960 are illustrated in Fig. 7.

Analysis of annual maximum runoff volumes Vtmax on the Danube River at Bratislava (1876-2019) in dry and wet periods
In this part of the work, we divided the sets of volumes with t-day into two sub-sets based on dry and wet multiannual periods. The dry and wet periods we determined on the basis of double 5-year moving averages of the Danube River flows at Bratislava for the period 1876-2019 (Fig. 8a). The wetness of individual years is different and more or less independent of each other. It is to be understood that the various physical causes also distort the action of the decisive factors to such an extent that we can speak of randomness. According to this, but also from experience, we can say that years of a similar nature usually group together, (Dub, 1957). The limit value for determining the dry and wet periods was the value of the long-term average annual flow Qa=2049 m 3 s -1 . Due to the fact that we took the period as a result of the moving average, dry and wet years can also occur in it. Table 1. The number of 71 years were included in the dry period and number of 73 years in the wet period (Fig. 8b). Fig.  8b also shows that since 2005 to 2019 the dry period is recorded. The M-K trend analysis indicates an increasing long-term trend in annual maximum runoff volumes Vtmax with t=2-days and we can reject the hypothesis H0 at significance level α=0.05 for dry period (Fig. 9). A Log-Pearson III distribution was used to calculate the T-year maximum runoff volume with the given runoff time duration (t) for dry and wet periods. The T-year maximum annual runoff volumes of duration t=2-days, 5-days and 10-days has no significant changes in estimation. In the wet period (except for volumes with t=2-and 5-days), with the same probabilities of exceeding, higher values of maximum volumes may occur compared to the dry period. Dividing of the period 1876-2019 into dry and wet periods had a greater impact on changes in LPIII exceedance curves of the maximum runoff at higher values of volumes with time duration t=20-, 30-, and 60-days. The examples of differences between estimated T-year maximum annual runoff volumes for dry and wet period are illustrated in Fig. 10. Bratislava (1921Bratislava ( -1960.

Conclusion
In the present paper we analyzed, the occurrence of annual maximum runoff volumes with t-day durations for a 144-year series of mean daily discharge of the Danube River at Bratislava gauge (Slovakia). The statistical methods were used to clarify how the maximum runoff volumes of the Danube River at Bratislava changed over period 1876-2019 and over dry and wet periods. On the Danube River is usually the maximum annual flow occur simultaneously with the annual maximum runoff volume of waves with a given time duration t. However, the corresponding values in terms of significance are not equivalent. Based on the exceeding probability curves of the annual maximum runoff volumes, it is possible to determine to the selected volume V for different t the probability of its exceeding and return period. The M-K tests showed increasing trend of the annual maximum runoff volume at significance level α=0. 05 and α=0.1, for period 1921-1960. Based on M-K test we can conclude that the runoff volume regime during floods has not changed substantially during the last 144 years, which is of importance to water management. This conclusion pertains not only to the short-term flood runoff episodes (V2max), but also the long-term ones (V60max). Based on the division of the given interest period 1876-2019 into dry and wet ones, we also plotted and compared the probabilities of exceedance the annual maximum runoff volumes on the Danube at Bratislava for selected runoff durations t = 2, 5, 10, 30, 60 days in given periods. The M-K trend analysis indicates an increasing long-term trend in annual maximum runoff volumes Vtmax with t=2 days and we can reject the hypothesis H0 at significance level α=0.05 for dry period. The results suggest that the T-year maximum annual runoff volumes of duration t=2-days, 5-days and 10-days has no significant changes in estimation. In the wet period (except for volumes with t=2-and 5-days), with the same probabilities of exceeding, higher values of maximum volumes may occur compared to the dry period. Dividing the period 1876-2019 into dry and wet periods had a greater impact on changes in LPIII exceedance curves of the maximum runoff at higher values of volumes with time duration t=20-, 30-, and 60-days. The results are useful in the water planning and flood protection and can help mapping flood risk areas and developing river management plans in Danube River basin. In the future, it would be desirable to confirm the conclusions also on other rivers in Slovakia with satisfactory long runoff data series, or on the basis of reconstructed discharge values by indirect methods (analogy, mathematical runoff modeling etc.).