Statistical Analysis and Trend Detection of the Hydrological Extremes of the Danube River at Bratislava

The territory of the Danube River Basin is one of the most flood-endangered regions in Europe. The flow regime conditions of the Danube River are continually changing. These changes are the result of natural processes and anthropogenic activities. In the present study, we focused on the statistical analysis and trend detection of the hydrological extremes of the Danube River at Bratislava. This paper firstly analyses the changes in correlation between water levels of the Danube River at Bratislava and Kienstock. Studied period of 1991-2013 included one or three hour measured water levels of the Danube River at Bratislava and Kienstock and shorter periods (1991–1995, 1999–2002, and 2004—2013) were selected for identification of the water level changes at Bratislava. One of the factors that recall the necessity to establish empirical - regression relationships was increasing of water levels of the Danube River at Bratislava (due to sediments accumulation at Bratislava). The results of the analysis indicated an increasing of water levels corresponding to the same flood discharges observed in the past. We also can say that travel time of the Danube floods between Kienstock and Bratislava did not change significantly during the analysed period. In the second part of the paper, we have identified changes in commonly used hydrological characteristics of annual maximum discharges, annual discharges and daily discharges of the Danube River at Bratislava during the period of 1876–2019. We examined whether there is a significant trend in discharges of the Danube River at Bratislava.


Introduction
Three types of hydrological changes may to result from regulation of rivers: increased frequency of high flows; redistribution of water from periods of base flow to periods of storm flow, and increased daily variation in stream flow. These changes do not necessarily occur in all regulated rivers, but they are common and need to be addressed as part of any comprehensive effort to rehabilitate regulated rivers.
The estimation of hydrological characteristics cannot be considered as closed and computing methods (models) have to be constantly updated on the latest conditions and current situation in the river. We focused on the Danube River with respect to the Danube River basin as an area with high economical and water management importance. These anthropogenic activities have a direct or indirect effect on river regime. Then, we can say, that the water level in the Danube is a result of the discharge, depth and shape of the riverbed, including the flood-plain area, which is restricted to the area between flood protection dikes since the last century. Construction of dams in the upper section of the Danube in the 20th century gradually reduced the volume of transported sediments to the middle section of the Danube. For example, Rákoczi [1] analysed regime of the sediments of the Danube River during the period of . The effect of the river bed sedimentation on water level changes of the Danube at Bratislava was also analysed in work of [2] and [3]. Author [3] calculated that the water level with the peak of Q100 at Bratislava would be 20-25 cm higher in 1998 due to the changed geometry of the river bed compared to 1996. The analysis of the changes in the cross profile (bed), flow velocities changes and changes of rating curves of the Danube at Bratislava profile showed that the river barrages causes characteristic changes in the grain-size distribution of both, the suspended sediments and the riverbed materials can cause changes in hydrological regime of the river [4]. The detailed description of the Danube River basin for understanding the occurrence of floods in the basin with: a historical overview of individual floods, analysis of homogeneity, cyclicality and long-term trends, seasonality, extreme discharges of selected water stations, coincidence of flood waves in the main river basin and major side tributaries, least and not last theoretical design hydrographs and regionalization of river basin flood regimes you can find in monography of Pekárová [5]. General summary of floods on tributaries of the Upper Danube and their impact on society and the population is published in [6].
The objective of this paper is to analyse and present changes of the water levels of the Danube River at Bratislava profile. The simple empirical regression relationships were used to illustrate changes of the water levels on the Danube at Bratislava profile. This method allows assigning the water levels from one or more upper stations to water level of the lower station. The period of study included the measured three-hour or one-hour water levels of the Danube River at Kienstock and at Bratislava over the period of 1991-2013. These relationships are necessary to be updated after each major flood situation. One of the factors that recall the necessity to establish empirical -regression relationships was increasing of water levels of the Danube River at Bratislava (due to sediments accumulation at Bratislava). The second part of the paper is focused on the identification of the changes in commonly used hydrological characteristics (annual maximum discharges Qmax, annual discharges Qa and daily discharges Qd) of the Danube River at Bratislava during the period of 1876-2019. We examined whether there is a significant trend in discharges of the Danube River at Bratislava.
The results of the statistical analysis and trend detection in the hydrological characteristics of the Danube River at Bratislava are illustrated in figures, listed in tables and summarized in the conclusions of the paper.

Study area
The Danube River is the second largest river in Europe after the Volga. The basin covers an area of 817 000 km 2 . The length of the river is 2 872 km. The river originates from the Black Forest in Germany at the confluence of the Brigach and the Breg streams. It discharges into the Black Sea via the Danube delta, which lies in Romania and Ukraine (figure 1). The Danube River is an important hydrologic and hydrographic system, which is formed by several significant tributaries. The maximum runoff occurs in June in the upper Danube basin. Over 120 major rivers directly confluence with the Danube and a majority of them has its own significant tributaries. The Slovak part of the Danube River is situated from river-km 1 708.2 to river-km 1 880.2.

Methods
In our study we used the method of corresponding water levels. This method is based on the calculation of the movement of a flood wave through a riverbed and allows the water level in one or more upper stations to be assigned a water level in the lower station. The values of the water level in the lower profile can then be written in the form: , where: Hd,t+τ -water level in lower station in t+τ; Hh,t -vater level in upper station in time t and ΔHtincrease of the water level between upper and lower stations, which is achieved in time τ.
The increase in the water level ΔHt is influenced by several factors: water from smaller tributaries; runoff of groundwater; water in the basin; transformation of the flood wave during the progress in the riverbed. These factors can be neglected on short sections and after adjustment equation (1) takes form: (2) where: β -is the coefficient of water levels, and we assume ΔHt/Hh,t const=β .
The determination of the function of water levels depending on the travel time τ is based on the measured values of water levels in the upper and lower profile and the determined travel times. , where: -time of occurrence of the characteristics points of the water level in upper station, time of occurrence of the characteristics points of the water level in lower station.
The Mann-Kendall nonparametric test (M-K test) was used as tool to identify significant trends in hydrological characteristics. The Mann-Kendall nonparametric test (M-K test) is one of the most widely used non-parametric tests for 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 ( [7]; [8] and [9]). By M-K test, we test the null hypothesis H0 of no trend, i.e. the observation xi is randomly ordered in time, against the alternative hypothesis H1, where there is an increasing or decreasing monotonic trend. 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.

Changes in correlation between water levels of the Danube River at Bratislava and Kienstock
We analysed changes in correlation between water levels of the Danube River at Bratislava and Kienstock. Studied period of 1991-2013 included one-hour or three-hour measured water levels of the Danube River at Bratislava and Kienstock. Station Kienstock is situated in sufficient distance from Bratislava and already captured significant Alpine tributaries of the Danube River (Ybbs, Enns, Traun). Number of 68 maximum water levels over 300 cm (simultaneously at Kienstock and Bratislava) was  (table  2). Table 3 presents comparison between measured and simulated maximum water levels of the Danube River at Bratislava based on water levels of the Danube River at Kienstock.

Trend analysis of the of annual maximum discharges, annual discharge and daily discharges of the Danube River at Bratislava
The second part of present paper, is focused on the statistical analysis of changes in some selected hydrological characteristics like annual maximum discharges, annual discharges and daily discharges of the Danube River at Bratislava during the period of 1876-2019. The time series of annual discharges were calculated from daily mean discharges. Course of the annual discharges and decadal averages, the Danube River at Bratislava are illustrated in figure 5.       According to the IPCC's report of 2001, the consequences of climate change may result in altered distribution curves of climate elements (air temperature, precipitations, dis-charge rates, etc.), thus time series of other periods will likely change also. Therefore, we analysed the long-term average annual regime (365 days in a year) of the average daily discharges for two periods 1876-1947 and 1948-2019. The long-term average daily discharges for two periods 1876-1947 and 1948-2019 are illustrated in figure 7a). The red circles point to the most significant differences between the periods in question. Table 5 summarizes some basic statistical characteristics of the average daily flows for the Danube River taking into account both periods.
The basic statistical characteristics of the two data sets do not indicate any significant changes; the long-term average discharge differs only by 35 m 3 s -1 , which is negligible. The analysis of long-term average daily discharges showed their decrease in the months of May-November in the period 1948-2019 in comparison with the period 1876-947.
In the middle of September, the discharges were

Conclusions
The intensive gravel excavation below Bratislava was stopped after 1980. The Danube dam Gabčíkovo near Cunovo caused increasing the Danube water level at Bratislava after November 1992. Decrease of flow velocities and stream turbulence caused the sedimentation of bed load (originally transported by the stream into lower profiles) in the Danube channel in Bratislava. Such interventions caused acceleration of the travel time of flood waves or decreasing of transformation capacity of the channel and floodplains. It can also cause increasing of water levels corresponding to the same flood flow observed in the past. In conclusion we can say that travel time of the Danube floods between Kienstock and Bratislava did not change significantly during the period, but water levels tend to increase their culmination level at the same flow rate. In conclusion, we can also conclude that the derived regressions simulate water levels of the flood waves of the Danube at Bratislava sufficiently accurate. Underestimation of the water levels above 900 cm according to regression I. (1991)(1992)(1993)(1994)(1995) may be due to the fact that in this period there was no flood wave above 9500 m 3 s -1 . There still applies quote "Let the flood be however large, there always comes up even larger one in the future", as it follows from the theory of extremes and experience confirms this.
In the present paper we also analyzed changes in the characteristics of annual maximum discharges, annual discharge and daily discharges of the Danube River at Bratislava during the period of 1876-2019. Generally, in Bratislava in Danube discharge series there doesn't exist (from the statistical point of view), significant trend of discharges within the used 144-years period 1876-2019. The analysis of the two shorter periods (1876-1947 and 1948-2019 did not confirm significant changes in total distribution of the annual discharges or maximum annual discharges. On the other hand, a time change in the long-term average daily discharges was identified at the end of March and end of December, when these increased over the period 1948-2019. This increase can be attributed to the onset of snow-melt in the Danube basin due to the increased atmospheric temperature. In contrast, the long-term daily discharges dropped in the summer months comparison with the period of 1876-1947. It seems that the amount of water discharged in March is missing in the river outflow in September.