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Article

Probability Characteristics of High and Low Flows in Slovakia: A Comprehensive Hydrological Assessment

by
Pavla Pekárová
,
Veronika Bačová Mitková
* and
Dana Halmová
Institute of Hydrology Slovak Academy of Sciences, Dúbravská Cesta 9, 841 04 Bratislava, Slovakia
*
Author to whom correspondence should be addressed.
Hydrology 2025, 12(8), 199; https://doi.org/10.3390/hydrology12080199
Submission received: 30 June 2025 / Revised: 24 July 2025 / Accepted: 29 July 2025 / Published: 31 July 2025
(This article belongs to the Section Water Resources and Risk Management)

Abstract

Frequency analysis is essential for designing hydraulic structures and managing water resources, as it helps assess hydrological extremes. However, changes in river basins can impact their accuracy, complicating the link between discharge and return periods. This study aims to comprehensively assess the probability characteristics of long-term M-day maximum/minimum discharges in the Carpathian region of Slovakia. We analyze the long-term data from 26 gauging stations covering 90 years of observation. Slovak rivers show considerable intra-annual variability, especially between the summer–autumn (SA) and winter–spring (WS) seasons. To allow consistent comparisons, we apply a uniform methodology to estimate T-year daily maximum and minimum specific discharges over durations of 1 and 7 days for both seasons. Our findings indicate that 1-day maximum specific discharges are generally higher during the SA season compared to the WS season. The 7-day minimum specific discharges are lower during the WS season compared to the SA season. Slovakia’s diverse orographic and climatic conditions cause significant spatial variability in extreme discharges. However, the estimated T-year 7-day minimum and 1-day maximum specific discharges, based on the mean specific discharge and the altitude of the water gauge, exhibit certain nonlinear dependences. These relationships could support the indirect estimation of T-year M-day discharges in regions with similar runoff characteristics.

1. Introduction

In the context of contemporary climate change, hydrology faces growing challenges arising from both natural climatic variability and ongoing climate change. Predicting future river flow regimes has become a complex task, particularly due to the interplay of extreme hydrological events, such as droughts and floods, and their potential drivers. A key issue remains the identification of trends in these extremes and understanding the extent to which they are influenced by climate factors versus human interventions. Numerous studies, e.g., [1,2,3,4], have highlighted the increasing frequency and intensity of extreme events worldwide, largely attributed to a changing climate. In recent decades, several climatic anomalies have been observed across Europe, leading to a shift in long-term hydrometeorological patterns and influencing surface and groundwater interactions within river basins. Since the mid-20th century, the anthropogenic impact on the global water cycle has become increasingly evident, with recent research (e.g., [5,6]) documenting the growing influence of socioeconomic development on flow regimes. These trends underscore the importance of long-term observational records and robust statistical analysis to better understand and manage water extremes.
Confirming or refuting this information is only possible through statistically processing the long-term data series of the hydrological cycle. In response to these challenges, the European Union has implemented a legal framework aimed at sustainable water management and effective flood risk mitigation [7]. For instance, Directive 2007/60/EC on the assessment and management of flood risks requires Member States to develop flood hazard and risk maps as well as risk management plans [8]. The directive underscores the need to understand hydrological extremes and their probabilities, which are critical for formulating robust flood risk management strategies. On the other hand, in 2012, the EC issued the “Report on the Review of the European Water Scarcity and Droughts Policy” [9]. The report aimed to develop and describe a policy on water scarcity and droughts in Europe during the period 2014–2020. Furthermore, the revised EU Climate Adaptation Strategy emphasizes the importance of enhancing societal resilience to hydrological extremes, including both floods and droughts, as a core priority in building climate resilience [10].
Addressing the uncertainty in hydrological forecasting requires the use of long-term hydrological datasets. As data records become increasingly comprehensive and accessible, opportunities emerge to perform more detailed and statistically robust analyses. However, extrapolating trends from hydrological data remains highly sensitive to the observation period, the length of the data series, the presence of historical outliers, and the selection of appropriate probability distribution functions. A thorough understanding of the statistical nature of extreme flows (both high and low) is essential for anticipating hydrological extremes such as floods and droughts. These events continue to pose significant risks for water infrastructure and resource management. Probabilistic characterization of daily discharge extremes over extended periods supports the development of resilient flood protection measures, sustainable water use strategies, and hydraulic design standards. The choice of probability models, parameter estimation techniques, and the selected time frame must be carefully aligned with established hydrological practices in each region. Equally important is the quantification of uncertainty, from both measurement errors and methodological limitations. For instance, Ref. [11] emphasized that discharge uncertainty is strongly influenced by local site conditions, underscoring the need for station-specific uncertainty assessment prior to any quantitative analysis. Ref. [12] further explored uncertainty in the estimation of design discharges, demonstrating that hydrological model assumptions and runoff-generating mechanisms significantly impact flood predictions. The geological attributes of catchments can also explain discrepancies in expected flood behavior. Ref. [13] investigated how non-stationarity in extreme events affects flood frequency estimates by comparing multiple models based on fit, reliability, and parameter behavior. In Central Europe, various studies have contributed to regional flood frequency analyses using L-moments and annual maxima series, such as [14] on the Pannonian Basin. The IHP UNESCO framework has supported significant hydrological research in the Danube basin, including work [15] on water balance. Using the same framework, a monography of the flood regime of rivers in the Danube River basin was carried out [16]. Both publications provide a good starting point for further water balance studies and for determining the impact of climate change on the Danube water regime [17]. These fundamental studies, together with broader historical analyses, e.g., [18,19,20,21,22,23,24,25], continue to inform the assessment of climate change impacts on flood risks in European river systems.
In river flow regimes, periods of low flows represent a critical component of hydrological variability. While hydrological frequency analyses often prioritize peak flows due to their association with flooding hazards, the characterization of low flows is equally essential, particularly in the context of water scarcity and ecosystem health. Numerous studies have investigated historical patterns and long-term trends in high or low flows conditions, especially within the European context, e.g., [26,27,28]. Analytical approaches typically use multi-day low-flow averages (such as 7-, 10-, or 30-day minima) to capture meaningful trends in basins to minimize or exclude anthropogenic influences. Various probability distribution models have been applied to describe these low-flow events, with comprehensive overviews provided by [29,30,31,32,33]. Comparative analyses of statistical methods, such as [34], have highlighted the advantages of Bayesian techniques like Markov chain–Monte Carlo simulations, particularly under conditions of limited data availability.
Low-flow dynamics have significant implications for water resource planning, affecting supply reliability, water quality, and aquatic habitats—especially during prolonged dry periods. These phenomena exhibit distinct seasonal patterns and are driven by a combination of climatic and catchment-specific factors. As emphasized in recent literature, e.g., [35,36], it is crucial to distinguish between the influence of climate variability and human-induced landscape changes when analyzing such events. Understanding historical changes in climate-driven seasonal low flows will help researchers to better anticipate future changes. Drought, closely related to persistent low-flow episodes, presents a growing challenge for hydrological and environmental management. Despite its widespread impact, there is no universally accepted definition of drought; disciplinary perspectives categorize it into meteorological, agricultural, hydrological, and socioeconomic types. Climate-induced reductions in streamflow can adversely affect freshwater ecosystems [37], although in some arid regions, such reductions may offer a competitive advantage to native biota if invasive species are less tolerant of flow intermittency [38]. Monitoring and interpreting low-flow variability is particularly relevant in areas experiencing increasing water demand [39], as it supports the development of adaptive management strategies in the face of changing climate conditions.
Seasonal fluctuations in the water regime are pronounced in the Carpathian area of Central Europe. The seasonality of runoff in snow-dominated rivers across Central Europe has declined, characterized by increased discharge in winter and spring and reduced flows during summer and early autumn. This shift is primarily attributed to the construction of hydropower reservoirs in the Alpine region over the past century [40]. The overall water balance regime in these basins is closely linked to the climatic, hydrological, hydrogeological, and physico-geographic characteristics of individual river catchments. Moreover, natural water circulation is increasingly influenced or disrupted by anthropogenic activities within these river basins [41].
Slovakian streams have a variable flow regime, with a relatively frequent occurrence of extreme flows, both in time and space. Their variability is caused by physical, geographical, and climatic conditions. Long periods of drought, which alternate with heavy rainfall causing flash floods, are increasingly common [42]. Climate change is expected to significantly impact flood regimes in Slovakia, with possible increases in winter runoff and summer droughts [43]. Climate and land use changes are projected to increase winter runoff and decrease summer runoff in mountainous basins in Slovakia, with more pronounced changes in the long term due to changing temperature and altered snowmelt timing [44]. Impacts of future climate change on runoff or water balance in catchments of Slovakia have been investigated and analyzed in recent years by [45,46,47,48,49]. A long-term analysis of changes in the seasonal and maximum daily discharges of Slovak rivers during 1931–2020 detected changes in the runoff regime curves for rivers with altitudes below 500 m a.s.l. [50]. The comparative analysis of the effects of selecting different reference periods (1961–2000, 1971–2020, 1981–2020, and 1991–2020) on the assessment of long-term hydrological characteristics, specifically monthly and annual discharges, at 140 water-gauging stations in Slovakia highlighted that the choice of reference period impacts the assessment of possible changes in discharge. This raises the question of how these changes affected the classification of water management rules that were commonly used at some stations during the periods 1991–2020 and 1961–2000 [51,52].
This study aims to analyze and assess the probability characteristics of long-term M-day maximum and minimum discharges in the Carpathian region of Slovakia to propose a uniform methodology for harmonizing and generalizing hydrologically extreme value assessments in the selected region. We used the theoretical Log-Pearson Type III probability distribution as a mathematical tool. The first section of our report focuses on estimating the T-year 1-day maximum specific discharge (q1dmax,T) and T-year 7-day minimum specific discharge (q7dmin,T) using a Log-Pearson III probability distribution function. The skew coefficient measures the asymmetry of that distribution and is sensitive to extreme events. The second section of this study focuses on testing the effects of the relationships between T-year 1-day maximum specific discharges (q1dmax,100) and T-year 7-day minimum specific discharges (q7dmin,100) on the annual mean specific discharges or on the altitude of the water gauge, aiming to indirectly estimate the T-year 1-day maximum specific discharges and T-year 7-day minimum specific discharges.
The findings of this study can help inform adaptive management strategies that ensure sustainable water supply, protect ecosystems, and mitigate flood risks. Moreover, establishing a unified methodology for frequency analysis could contribute to developing frameworks that can be utilized in regions experiencing similar hydrological shifts. This is crucial for policymakers and water resource managers, who need to make informed decisions based on reliable data and predictive models.
This research is particularly relevant given the pressing challenges facing water resource management today, including climate change, population growth, and urbanization. Climate change affects hydrological extremes on multiple scales, leading to varied spatial patterns and mechanisms. To improve monitoring and adaptation to these extremes, decision-makers should incorporate diverse sources of information, particularly focusing on the abrupt transitions between droughts and floods. As regions face increased variability in precipitation and extreme weather events, understanding how river flow patterns have changed becomes essential for effective water management.

2. Materials and Methods

2.1. Study Area and Data

Slovakia, located in Central Europe and spanning 49,035 km2, shares borders with five countries, including Poland, Ukraine, Hungary, Austria, and the Czech Republic. Its physiography is dominated by the Carpathian Mountains in the north and central parts, while the southern parts are formed by lowland areas such as the Danubian and Eastern Slovak Lowlands. The highest elevation is 2655 m (Gerlachovský štít) in the High Tatras. This complex topography substantially influences regional climate and hydrological regimes. The climate is temperate continental, with marked seasonal variability. Winter temperatures often fall below −10 °C at higher altitudes, whereas summer maxima can exceed 35 °C in lowland regions. Precipitation is spatially variable, ranging from approximately 500 mm annually in the lowlands to over 2000 mm in mountainous areas. The highest long-term average precipitation during the period 1991–2015 was recorded at Lomnický štít (2634 m), amounting to P1991–2015 = 1851.24 mm. In contrast, the lowest value for the same period was observed in Senec (125 m), with P1991–2015 = 527.43 mm.
The majority of Slovakia’s land area, approximately 96%, lies within the hydrological catchment of the Black Sea. This includes extensive sections of the Danube River basin, a key fluvial system in Central Europe, along with several of its major Slovak tributaries, specifically the Morava, Váh, Hron, and Ipeľ Rivers. Additional river basins, such as those of the Slaná, Bodva, Hornád, and Bodrog, form part of the Tisa River sub-catchment, which also contributes to the Black Sea outflow through the Danube. In contrast, only about 4% of Slovak territory drains northward toward the Baltic Sea, primarily through the Poprad and Dunajec River basins. Table 1 lists a detailed overview of selected rivers and corresponding gauging stations, including information on catchment size, periods of hydrological monitoring, and basic flow characteristics computed for the designated time intervals. The analysis utilized average daily discharge data as the primary input. The locations of the twenty-six selected rivers and gauge stations in the Slovak region are presented in Figure 1. All daily discharge data were obtained from the Slovak Hydrometeorological Institute (SHMÚ) and processed according to the hydrological year. In Slovakia, the hydrological year is calculated from 1 November of the current year to 31 October of the following year [53,54]. A hydrological year is a 12-month period usually chosen so that precipitation falling during that period can participate in the runoff process in the same period. Figure 1a,b illustrate long-term changes and variability in mean areal annual precipitation (P), air temperature (Ta), and runoff (R) for two representative basins: Belá, Podbanské, with the highest station altitude, and Litava, Plášťovce, with the second-lowest station altitude.
To ensure the consistency and reliability of the daily discharge dataset for long-term hydrological analysis, the data were homogenized using the method of hydrological analogy. This approach involves comparing the target time series with data from nearby reference stations that exhibit similar hydrological behaviors. By identifying and correcting discontinuities caused by non-natural influences, such as changes in measurement procedures, instrumentation, or station relocation, the method helps to maintain the homogeneity of the dataset, which is essential for accurate trend detection and climatological interpretation.
For processing T-year maximum daily discharges (Qdmax,T), sets of 1-day maximum discharges (Q1dmax) were chosen. For a complex analysis of the effects of climate change on the flow regime of the maximum daily discharges, we also divided the hydrological data into two seasons: summer–autumn (SA: 1 May–31 October) and winter–spring (WS: 1 November–30 April).
For processing T-year low flow, the annual minimum discharges from 1-day, 3-day, and 7-day minimums per summer–autumn (SA: 1 May–31 October) and winter–spring (WS: 1 November–30 April) seasons of the hydrological year were computed. The 7-day minimums were derived from the moving averages of the daily discharges calculated for every possible season completely within the hydrological year. These specific durations were chosen to capture short-term, seasonal, and weekly low-flow events, which are critical for understanding varying scales of hydrological extremes and their implications for water resource management under projected climate change scenarios. These time series, consisting of 90 years of data, were used to estimate T-year minimum discharges (Q7dmin,T) and T-year minimum specific discharges (q7dmin,T)

2.2. Methods

2.2.1. Log-Pearson Type III Probability Distribution

To analyze the statistical behavior of the high and low flow series, we selected the Log-Pearson Type III (LPIII) distribution, a widely recognized model for representing extreme hydrological events. Its frequent application across various natural processes has established it as a standard tool in hydrological frequency analysis. Parameter estimation for the LPIII distribution was addressed by the authors of [55], who formulated a log-likelihood function to support more accurate fitting. A broader methodological framework was later introduced in [56], presenting a frequency factor-based technique capable of generating several probabilistic models, including normal, log-normal, extreme value type 1, Pearson Type III, and LPIII distributions. Further validation of the LPIII distribution’s effectiveness in the analysis of extremes was demonstrated in studies by the authors of [16,57,58]. Relying on a single distribution model also enables extrapolation of T-year flood estimates to ungauged river segments by leveraging regional statistics—specifically, the long-term average of annual maximum discharges and distribution parameters from adjacent gauging stations. An application in combination with spatial modeling in the case was implemented in [59].
To estimate the distribution parameters, the method described in the Interagency Committee on Water Data Bulletin 17B [60] was used. Bulletin 17B introduced revised procedures for integrating station skew values with results from generalized skew analyses, including both identified and adjusted outliers, through dual station comparisons and the computation of frequency curve confidence limits. In the United States, flood estimation has traditionally been based on two key methods: using frequency analysis of peak flows for floodplain management and levee design and applying deterministic Probable Maximum Flood estimates for infrastructure like dams and nuclear power plants [61].
The Log-Pearson Type III distribution is a three-parameter gamma distribution with a logarithmic transformation of the variables. It is widely used for flood analyses because the data quite frequently fit the assumed annual maximum discharge series. The probability density function of the Pearson Type III distribution takes the following form:
f x τ , α , β = x τ β α 1 e x p x τ β β Γ α ,
with
x τ β 0 ,
where τ is the location parameter, α is the shape parameter, β is the scale parameter, and Γ(α) is the gamma function, as given by (3):
Γ α = 0 t α 1 e x p t d t .

2.2.2. Conditions of Variable Series

The distribution is fitted by computing the base 10 logarithms of the variable, and it is calculated at a selected probability exceedance using the following equation:
log X =   X ¯ + K σ ,
where X ¯ is the mean, σ is the standard deviation, and K is a factor of the skew coefficient at the selected probability of exceedance. The formulas for these parameters are provided below:
Mean:
X ¯ = 1 n i = 1 n X i .
Standard Deviation:
σ = 1 n 1 i = 1 n X i X ¯ 2 .
Skew Coefficient:
G = n n 1 n 2 σ 3 i = 1 n X i X ¯ 3 .
The Kolmogorov–Smirnov test was performed to test the assumption that the discharge magnitudes followed the theoretical distributions. The p-value (p ≥ 0.05) was used as a criterion for rejecting the proposed distribution hypothesis. The probability estimates were calculated for the [62,63] plotting positions. The plotting position did not affect the calculations for the frequency curve, but only affected the graphical representation of the observed values (Bulletin 17B). A basic plotting position formula for symmetrical distributions is given by [64]
P i = i α n + 1 2 α ,
where i is the rank of the observation ranked from largest (i = 1) to smallest (i = n), n is the sample size, and α is a plotting position parameter (0 ≤ α ≥ 0.5).
The relationship between the probability of exceeding a certain value in any year and its average recurrence period T < 10, according to the Slovak standard [53], is
p = 1 e 1 T ,
while for T ≥ 10, a simplified form is used:
P = 1 T .

2.2.3. Simple Case of Parameter Estimation

The method of moments uses the logarithms of flood discharges to estimate the distribution parameters. The first three sample moments are used to estimate the LPIII parameters: the mean ( μ ^ ), standard deviation ( σ ^ ), and skew coefficient ( γ ^ ). In cases where only systematic data are available with no historical information, the mean, standard deviation, and skew coefficient of the station data may be computed using the following equations:
μ ^ = 1 n i = 1 n X i
σ ^ = 1 n 1 i = 1 n X i μ ^ 2
γ ^ = n n 1 n 2 σ ^ 3 i = 1 n X i μ ^ 3
where n is the sample size and (ˆ) is a sample estimate. The sample standard deviation and skew coefficient include the bias correction factors (n − 1) and (n − 1)·(n − 2), respectively, for small samples.

3. Results

3.1. Direct Estimate of the T-Year 1-Day Maximum Specific Discharges

We selected the data series of the maximum daily discharges with runoff durations of 1 day (Q1dmax) to estimate the coefficients of the Log-Pearson III distribution in selected rivers in the Slovak region during the entire measurement period (see Table 1). The maximum daily discharges (Q1dmax) were recalculated to the maximum daily specific discharges (q1dmax) to enable a comparison of the runoff from different basins and to identify regional patterns. As the first step, we estimated the LPIII distribution function parameters (mean, standard deviation, and skew coefficient) for each station separately (Figure 2).
The estimation of the T-year maximum daily specific discharges with runoff durations of 1 day was carried out. To comprehensively analyze the river runoff’s extreme regime, we divided the dataset of M-day maxima specific discharges into two seasons: summer–autumn (SA: 1 May–31 October) and winter–spring (WS: 1 November–30 April). We proceeded with this experiment based on an analysis of the flow regime of the rivers in the Slovak region. Examples of the graphic results of the LPIII probability of exceedance curve of the 1-day maximum specific discharges for two seasons, summer–autumn (SA) and winter–spring (WS), are illustrated in Figure 3a–c. The presented examples point to various seasonal regimes and differences in the return periods of the specific discharges (q1dmax): the winter–spring season (Figure 3a), the summer–autumn season regime (Figure 3b), and no prevailing specific season (Figure 3c).
The graphical comparison of the differences between the estimated values of the 50-, 100-, and 500-year maximum specific daily runoff for all analyzed rivers is illustrated in Figure 4. The estimations of the 50-, 100-, and 500-year 1-day maximum daily specific discharges for selected rivers in the Slovak region (WS season, SA season) show that most of the rivers analyzed reach their maximum mean specific discharges in the summer–autumn season, especially for return periods of over 100 years (Table 2). The difference in the estimated 1-day maximum mean runoff at T = 100 years for some small rivers can reach more than 100% between seasons (Table 2), especially for rivers with extreme or very high intra-annual variability runoff according to the ratio of the maximum/minimum monthly Pardé coefficient [50].

3.2. Direct Estimate of the T-Year 7-Day Minimum Specific Discharges

In this section, we focus on constructing theoretical probability curves for the 7-day minimum specific discharge (q7dmin) for the selected stations. As the first step, we estimated the LPIII distribution function parameters: mean (q7dmin), standard deviation (S7dmin), and station skew coefficient (G7dmin) for each station individually (Figure 5).
We constructed the curves, paying particular attention to the lower values to ensure that the empirical and theoretical values aligned as closely as possible in the lower part of the graph. This was achieved by adjusting the skew coefficient G. At some stations, such as Ipeľ (Holiša), outliers were identified in the SA season series in 1947 and 1993 (see Figure 6a,b). Including such extreme values in the series significantly affected the estimates of T-year values. After verifying that these outliers were not erroneous data, we decided to include them in the calculations. We constructed the curves, paying attention to the lower values so that the empirical and theoretical values in the lower part of the graph aligned as closely as possible. This was achieved by adjusting the shape parameter α in Equations (1) and (2). The LPIII distribution curves were also constructed for the summer–autumn (SA, example Figure 6c) and winter–spring (WS, example Figure 6d). A comparison of the estimated values of the 20-, 50-, and 100-year minimum specific discharges for 7-day runoff durations for selected rivers in the Slovak region from 1930/1931 to 2019/2020 for WS, SA, and the whole analyzed period is illustrated in Figure 7. The estimated T-year 7-day minimum specific discharges (q7dmin,T) calculated for WS and SA, and for the whole period, are presented in Table 3.

3.3. Indirect Estimation of the T-Year 1-Day Maximum and 7-Day Minimum Specific Discharges

Given the significant orographic, soil, and geological variability in Slovakia, the maximum and minimum specific discharges of rivers/streams in different regions vary greatly, especially in summer–autumn. For example, the 1-day 100-year maximum specific discharges (q1dmax,100) range from 842 L·s−1·km−2 on the Belá River in Podbanské to 51 L·s−1·km−2 on the Morava River in Moravský Svätý Ján during the SA season (see Table 2). Moreover, the 7-day 100-year minimum specific discharges (q7dmin,100) range from 0.04 on the Ipeľ River to 7.56 L·s−1·km−2 on the Belá River in Podbanské during the SA season (see Table 3). Due to the low number of stations processed in this study, detailed regionalization of the streams is not possible. However, it was possible to derive simple relationships for indirectly estimating the T-year specific discharges from the 26 Slovak stations based on the direct estimated T-year specific discharges and the long-term average annual specific discharges from the given stations or a set of the different physical–geographical characteristics of the basins, doing so separately for both the SA and WS seasons. The parameters of the regression curves were estimated using specialized statistical software, which also provided standard errors, as well as the upper and lower bounds of the 95% confidence intervals. For the final equations presented in the study, we selected parameter values from within the estimated confidence intervals that best fit the empirical relationships. This approach ensured both a statistically reasonable basis and a good empirical fit, while avoiding overfitting.
In the case of maximum specific discharges and long-term average annual specific discharges (q), there was no strong relationship for deriving a satisfactory curve to estimate the T-year 1-day specific discharges. Next, we tested the relationship between the q1dmax,100 and a set of the different physical–geographical characteristics of the basins. Satisfactory results were mainly achieved based on the dependence between q1dmax,100 and the altitude of the gauge station. From the 26 Slovak stations, we derived simple relationships for estimating the 100-year 1-day maximum specific discharges for stations without observations. We derived a nonlinear relationship that used only the altitude of the station, H, as the input, as follows:
q 1 d m a x , 100 S A = 4.10 6 H 3 0.0049 H 2 + 2.1211 H ,
q 1 d m a x , 100 W S = 325 1 + e 0.009 H 277 ,
The individual curves (Figure 8a,b) indicate that the 100-year 1-day maximum specific discharges in SA reach higher values and dependencies above an altitude of 800 m a.s.l., which is significantly weaker than the dependency for WS. The results suggest that the 1-day maximum specific discharges are higher in the SA than in the WS season. The correlation between the modeling and measurement data reached a value of R = 0.48 for WS and R = 0.74 for SA. The lower correlations could be caused by the size and diversity of the Slovak region.
In the case of minimum specific discharges, it was possible to derive simple relationships for estimating the 100-year 7-day minimum specific discharges from the 26 Slovak stations based on the long-term average annual specific discharges (q) from the given stations, doing so separately for both SA and WS in the form of an S-curve (16):
q 7 d m i n , 100 S A = 8 1 + e 0.18 q 17 0.2 ,
q 7 d m i n , 100 W S = 4.5 1 + e 0.2 q 13
where q is the long-term average annual specific discharge from 2.94 to 34.23 L·s−1·km−2.
For stations without any observations, we also derived a nonlinear relationship that uses only the altitude of the station, H, as its input [65]:
q 7 d m i n , 100 S A = 8 1 + e 0.007 H 580 ,
q 7 d m i n , 100 W S = 4.5 1 + e 0.0095 H 430
where H is the altitude of the station ranging from 100 to 1000 m above sea level.
The individual curves (Figure 9a,b) indicate that the dependence on altitude is significantly weaker than the dependence on average specific discharge. It is evident from Figure 9b that the value of q7dmin,100 at the Bystrianka (Bystrá) station is overestimated. The likely reason is the high water withdrawal from the Bystrianka stream during the summer period. The dependencies derived for indirectly estimating T-year 7-day minimum specific discharges from the 26 Slovak stations showed a slightly stronger correlation with the long-term average annual specific discharges (q) (R = 0.92) for both WS and SA than for the altitudes of the stations (R = 0.80 for both seasons). The dependencies of the T-year 7-day minimum specific discharges (q7dmin,T) on the long-term average annual specific discharge (q) for T = 1000, 100, 50, and 20 are presented for both SA and WS in Figure 10. The coefficient α—the upper asymptote—values in Equation (1) for T = 1000, 100, 50, and 20 are 6, 8, 9.5, and 10.5 for SA and 3.75, 4.5, 5.25, and 6 for WS. Please note that these are only estimates of the T-year values.
Based on the relationship (16), it is possible to estimate the 100-year minimum specific discharge for SA for Slovakia (Figure 11). This highlights the occurrence of extremely low flows, particularly in the streams of southern Slovakia. Extremely low flows may not occur every year. However, in general, in all Slovak rivers, the values of 7-day discharges were high between 1936 and 1941. In all regions, the minimum 7-day flow rates generally decreased until 1990 and have shown an increasing trend since then. For the entire period from 1931 to 2020, the 7-day minimum flow rates increased by 12.8% and 3.5% in the northern and eastern parts of Slovakia, respectively, while they decreased by 15.7% in the central part and by 23% in the southern part (lowland areas) of Slovakia. This development was influenced by the increase in precipitation levels after 1996, which affected groundwater levels as well as the regulation of river flow during dry periods through reservoirs.

4. Discussion

This study builds upon our previous research [50,66], which focused on long-term changes in the hydrological regime of Slovak rivers over the past 90 years (1931–2020), particularly in terms of intra-annual and seasonal variability, trends, average discharges, low, and maximum discharges. A decrease in average discharges was recorded in three regions during the period 1931–2020, while no significant trends were observed in the series of minimum and maximum discharges at the given 26 stations. While those studies provided important insights into the temporal evolution of river flows, the current manuscript extends this work by offering a comprehensive probabilistic and statistical assessment of high and low flows. This type of frequency analysis has not been addressed in the previously mentioned works. Our study does not directly analyze meteorological drivers, such as precipitation or temperature trends, and thus we do not aim to establish a causal relationship between climate change and the observed hydrological changes. Nevertheless, the variability and shifts detected in extreme discharges across Slovak rivers may be consistent with broader regional changes reported in [46].
Projected changes for Slovak basins indicate that future runoff patterns will shift, with increases in winter flows, decreases in summer flows, and more extreme maximum daily runoff events, driven by anticipated climate change scenarios [67]. The changes in the hydrological extreme’s regime, including the timing of runoff maxima and minima, are intricately linked to the overarching influences of climate change (rising temperatures and the increasing frequency and intensity of summer rainfall events) and human activities (deforestation and urbanization, channel and riverbed management). The comparative analysis in [52] of the hydrological characteristics related to drought in the context of climate change between 2001–2020 and 1961–2000 from 13 selected water gauging stations in Slovakia showed that the incidence of years with subnormal mean annual discharges (less than 90% of the historical average) was higher in 2001–2020 than in 1961–2000, and also revealed that, for most stations, runoff distribution shifted, with spring runoff moving forward into the winter months. However, this shift is less pronounced in the mountainous areas of northern Slovakia.
Our own analyses in the Slovak region reveal that certain rivers exhibit earlier maximum/minimum runoff, whereas others show a delay in these values. This variability is influenced by local climate, terrain, and the specific features of each catchment. For example, catchments located at higher elevations may still utilize snowmelt, while lowland areas respond more promptly to intensive rainfall (Figure 12) [50,66].
For example, ref. [68] identified distinct regional changes in flood discharges across Europe over the past five decades, with both increases and decreases, which the authors suggest could be a consequence of climate changes. Similarly, ref. [69] found that low-flow magnitudes generally declined in European catchments where minimum flows typically occur during the warm season. According to [70], low-flow conditions in the United Kingdom have remained relatively stable over time, despite notable hydrological variability and uncertainty linked to interdecadal climate variability and human influences, whereas [71] observed an increase in low-flow magnitudes in western Ireland. Some studies report a consistent decrease in low-flow values, e.g., in southern France, the Czech Republic, Spain, and Turkey [72,73,74,75], whereas significant upward trends in low-flow magnitudes have been observed in the Alpine region of Europe [76] and in rivers of the Western Balkans [77]. Recent studies have documented significant seasonal shifts and runoff volume changes in low-flow regimes during the dry periods in Serbia [78,79], reporting increasing frequency and intensity of extreme low-flows in the upper Tisza River.
Figure 12. The changes in the seasonality for WS and SA (10th percentile—red point; 50th percentile—blue point; and 90thpercentile—green point), with a comparison across three 30-year periods for selected Slovak rivers: (a) q1dmax discharges and (b) q7dmin discharges [50,78].
Figure 12. The changes in the seasonality for WS and SA (10th percentile—red point; 50th percentile—blue point; and 90thpercentile—green point), with a comparison across three 30-year periods for selected Slovak rivers: (a) q1dmax discharges and (b) q7dmin discharges [50,78].
Hydrology 12 00199 g012
Therefore, monitoring and evaluating extreme hydrological events through diverse methodological and modeling approaches is essential, particularly due to human impacts, which may compromise the robustness of frequency analyses. It is essential to have complete flow regime data in order to draw valid conclusions based on long-term observations from catchments. Estimating the maximum or minimum discharge values with a long return period (once every 100, 500, or 1000 years) remains a complex task associated with considerable uncertainty. For the purpose of this study, the Log-Pearson Type III distribution was selected due to its flexibility and its ability to capture extreme values by incorporating the skewness coefficient (G). Applying a regionalized skewness coefficient enhances the estimation of the T-year discharges in basins with short data records. From the perspective of regionalization, it is more appropriate to apply such an approach to regions with homogenous hydrological and geographical characteristics, or along the course of a river. Successful application of this approach for the Slovak region or the Danube basin has been reported in [80,81]. It is important to note that the limitations of these regression-based models should be acknowledged, particularly in areas with limited data records or significant anthropogenic alterations to the flow regime, as these factors may affect the robustness and reliability of the estimates. The results from Slovak catchments cannot be generalized to the entire Danube River basin. Based on the outcomes of this study, practical recommendations for water management and flood mitigation should focus on enhancing seasonal forecasting, reassessing and potentially redesigning flood defenses and water storage infrastructure, developing flexible water management plans that account for fluctuating annual and seasonal variability, and supporting ecosystem-based solutions. In conclusion, this study contributes to the understanding of long-term seasonal variability in extreme river flows across Slovak catchments. Although our findings align with regional hydrological trends often associated with climate change, further research is needed to link these changes directly to climatic drivers. Future work should integrate meteorological data and climate model outputs to confirm and better understand the role of climate change in shaping hydrological extremes. Today, it is increasingly difficult to identify streams or river systems that are simultaneously not influenced by human activities and have continuous flow observations for more than a 90-year period.

5. Conclusions

The results of our work formulate a unified method for estimating T-year 1-day maximum specific discharges and 7-day minimum specific discharges, a key metric for water resource management, especially in regions where direct measurements are lacking. This methodology has potential applications beyond Slovakia and can be adapted for use in other geographical regions with similar or differing flow regimes.
The results for the first goal of this study, the application of the one theoretical probability distribution function, Log-Pearson III, showed that the 1-day maxima specific discharges are higher in SA compared to WS. With warming temperatures leading to earlier snowmelt in high-altitude regions and more frequent intense rainfall events, flood risks are expected to become more unpredictable. Increased discharge variability, as observed in Slovak rivers, could strain existing flood protection infrastructure and water management practices, especially during maximum flows. For larger catchments, variability in flood hazard trends due to climate influences requires an adaptive, location-specific approach to water resource management.
The second goal of our study was to use the LPIII theoretical probability distribution for estimating T-year minimum 7-day specific discharges, also a key metric for water resource management, especially in regions where direct flow measurements are lacking. Minimum flows are now almost universally affected by the withdrawal of water for irrigation (streams in southern Slovakia), drinking water (the Turiec River), cooling nuclear power plants (e.g., the Hron River), and recreational facilities (the Bystrianka basin). In areas with reservoirs constructed for storing water, it is possible to increase minimum flows in rivers during dry summer periods. For example, in the Turiec River, we identified the impact of the Turček reservoir on minimum flow trends. The construction of the Turček reservoir in 1996 led to a decrease in annual minimum 1-day and 7-day discharges compared to, for example, the Kysuca River. Our results suggest that minimum specific discharge design values are higher in summer–autumn (SA) compared to winter–spring (WS).
However, due to the significant orographic and climatic diversity of Slovakia, the maximum and minimum specific discharge values vary greatly across different regions. Due to the low number of stations processed in this study, detailed regionalization of the streams is not possible. However, it was possible to derive simple relationships for indirectly estimating the T-year specific discharges from the 26 Slovak stations separately for both the SA and WS seasons.
We tested the relationship between specific discharges with a return period of T = 100 years and a set of different physical–geographical characteristics of the basins. In the final part of this study, we derived simple regression relationships and simple regression relationships in the form of an S-curve to estimate the indirect design values of maximum and minimum specific discharges with a 100-year recurrence period based on station elevations and long-term average annual specific discharge.
The relationships of 1-day maxima specific discharges for the whole selected region of Slovakia provided satisfactory results mainly for the dependence between specific discharges with 100-year return periods and the altitude of the gauge stations. The correlation between the modeling and measurement data of the 1-day maxima specific discharges reached 0.48 for WS and 0.74 SA. The lower correlations could be caused by the size and diversity of the Slovak region. The results suggest that the 1-day maxima specific discharges are higher in SA compared to WS.
The relationships for 7-day minima specific discharge for Slovakia provided satisfactory results mainly for the dependence between 7-day specific discharges with 100-year return periods and the altitudes of the gauge stations and the long-term average annual specific discharges (q). The derived dependencies for indirectly estimating T-year 7-day minimum specific discharges from the 26 Slovak stations showed a strong correlation with the long-term average annual specific discharges (q) (R = 0.92) for both WS and SA and the station altitude (H) (R = 0.80). The results (Equations (16)–(19)) suggest that the 7-day minima specific discharges are higher in SA compared to WS.
With respect to the significant orographic and climatic diversity of Slovakia and the results achieved, it would be appropriate to analyze, evaluate, and apply such a method separately for basins with the same hydrological and geographical characteristics or along the length of a river when designing water management measures.

Author Contributions

Conceptualization, V.B.M.; Formal analysis, V.B.M. and P.P.; Methodology (indirect estimations), V.B.M. and P.P.; Investigation, V.B.M., P.P., and D.H.; Writing—original draft, V.B.M.; Writing—review and editing, V.B.M., P.P., D.H. All authors have read and agreed to the published version of the manuscript.

Funding

projects: APVV-20-0374, and VEGA No. 2/0015/23.

Data Availability Statement

The authors obtained the hydrological data used in this study from the Slovak Hydrometeorological Institute (SHMÚ) under a data-sharing agreement for scientific project purposes.

Acknowledgments

This study was supported by the projects APVV-20-0374, “Regional detection, attribution and projection of impacts of climate variability and climate change on runoff regimes in Slovakia”, and VEGA No. 2/0015/23, “Comprehensive analysis of the quantity and quality of water regime development in rivers and their mutual dependence in selected Slovak basins”.

Conflicts of Interest

The authors declare no conflicts of interest.

References

  1. Dai, A. Increased drought under global warming in observations and models. Nat. Clim. Change 2013, 3, 52–58. [Google Scholar] [CrossRef]
  2. Zhao, C.; Brissette, F.; Chen, J.; Martel, J.L. Frequency change of future extreme summer meteorological and hydrological droughts over North America. J. Hydrol. 2020, 584, 124316. [Google Scholar] [CrossRef]
  3. Meresa, H.; Tischbein, B.; Mekonnen, T. Climate change impact on extreme precipitation and peak flood magnitude and frequency: Observations from CMIP6 and hydrological models. Nat. Hazards 2022, 111, 2649–2679. [Google Scholar] [CrossRef]
  4. Xiong, J.; Yang, Y. Climate Change and Hydrological Extremes. Curr. Clim. Change Rep. 2025, 11, 1. [Google Scholar] [CrossRef]
  5. Blöschl, G. Three hypotheses on changing river flood hazards. Hydrol. Earth Syst. Sci. 2022, 26, 5015–5033. [Google Scholar] [CrossRef]
  6. Dey, P.; Mishra, A. Separating the impacts of climate change and human activities on streamflow: A review of methodologies and critical assumptions. J. Hydrol. 2017, 548, 278–290. [Google Scholar] [CrossRef]
  7. WFD. Directive 2000/60/EC of the European Parliament and of the Council of the 23 October 2000, Establishing a Framework for Community Action in the Field of Water Policy. 2000. Available online: https://eur-lex.europa.eu/eli/dir/2000/60/oj/eng (accessed on 28 July 2025).
  8. Cauncil, T.E. Directive 2007/60/EC on the Assessment and Management of Flood Risks. J. Eur. Union Off. 2007, 288, 27–34. Available online: https://eur-lex.europa.eu/eli/dir/2007/60/oj/eng (accessed on 28 July 2025).
  9. European Commission (EU COM). Report on the Review of the European Water Scarcity and Droughts Policy. 2012. Available online: https://environment.ec.europa.eu/topics/water/water-scarcity-and-droughts_en (accessed on 28 July 2025).
  10. European Commission (EU COM). Forging a Climate-Resilient Europe—The New EU Strategy on Adaptation to Climate Change. 2021. Available online: https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=CELEX%3A52021DC0082 (accessed on 28 July 2025).
  11. Coxon, G.; Freer, J.; Westerberg, I.K.; Wagener, T.; Woods, R.; Smith, P.J. A novel framework for discharge uncertainty quantification applied to 500 UK gauging stations. Water Resour. Res. 2015, 51, 5531–5546. [Google Scholar] [CrossRef]
  12. Rogger, M.; Kohl, B.; Pirkl, H.; Viglione, A.; Komma, J.; Kirnbauer, R.; Merz, R.; Blöschl, G. Runoff models and flood frequency statistics for design flood estimation in Austria—Do they tell a consistent story? J. Hydrol. 2012, 456–457, 30–43. [Google Scholar] [CrossRef]
  13. Šraj, M.; Viglione, A.; Parajka, J.; Blöschl, G. The influence of non-stationarity in extreme hydrological events on flood frequency estimation. J. Hydrol. Hydromech. 2016, 64, 426–437. [Google Scholar] [CrossRef]
  14. Leščešen, I.; Dolinaj, D. Regional Flood Frequency Analysis of the Pannonian Basin. Water 2019, 11, 193. [Google Scholar] [CrossRef]
  15. Petrovič, P. Basin-Wide Water Balance in the Danube River Basin. In The Danube and Its Basin—Hydrological Monograph; Follow-up; IH SAS: Bratislava, Slovakia, 2006; Volume VIII. [Google Scholar]
  16. Pekárová, P.; Miklánek, P. (Eds.) Flood Regime of Rivers in the Danube River Basin; Follow-Up Volume IX of the Regional Co-operation of the Danube Countries in IHP UNESCO; IH SAS: Bratislava, Slovakia, 2019; p. 215+527. ISBN 978-80-89139-45-3. (Print Version); Available online: https://ekniznice.cvtisr.sk/view/uuid:0a69b07b-1ab2-4fc6-b1ff-8b4fee53aa53?page=uuid:368e587c-1980-44d8-abe0-8a90f28b2206 (accessed on 28 July 2025).
  17. Pekárová, P.; Bajtek, Z.; Pekár, J.; Výleta, R.; Bonacci, O.; Miklánek, P.; Belz, J.; Gorbachova, L. Monthly stream temperatures along the Danube River: Statistical analysis and predictive modelling with incremental climate change scenarios. J. Hydrol. Hydromech. 2023, 71, 382–398. [Google Scholar] [CrossRef]
  18. Brázdil, R.; Kundzewicz, Z.W.; Benito, G. Historical hydrology for studying flood risk in Europe. Hydrol. Sci. J. 2006, 51, 739–764. [Google Scholar] [CrossRef]
  19. Merz, R.; Blöschl, G. Flood frequency hydrology: 1. Temporal, spatial, and causal expansion of information. Water Resour. Res. 2008, 44, W08432. [Google Scholar] [CrossRef]
  20. Merz, R.; Blöschl, G. Flood frequency hydrology: 2. Combining data evidence. Water Resour. Res. 2008, 44, W08433. [Google Scholar] [CrossRef]
  21. Lugeri, N.; Kundzewicz, Z.W.; Genovese, E.; Hochrainer, S.; Radziejewski, M. River flood risk and adaptation in Europe—Assessment of the present status. Mitig. Adapt. Strateg. Glob. Change 2010, 15, 621–639. [Google Scholar] [CrossRef]
  22. Elleder, L.; Herget, J.; Roggenkamp, T.; Nießen, A. Historic floods in the city of Prague—A reconstruction of peak discharges for 1481–1825 based on documentary sources. Hydrol. Res. 2013, 44, 202–214. [Google Scholar] [CrossRef]
  23. Pekárová, P.; Halmová, D.; Bačová Mitková, V.; Miklánek, P.; Pekár, J.; Skoda, P. Historic flood marks and flood frequency analysis of the Danube River at Bratislava, Slovakia. J. Hydrol. Hydromech. 2013, 61, 326. [Google Scholar] [CrossRef]
  24. Kjeldsen, T.R.; Macdonald, N.; Lang, M.; Mediero, L.; Albuquerque, T.; Bogdanowicz, E.; Brazdil, R.; Castellarin, A.; David, V.; Fleig, A.; et al. Documentary evidence of past floods in Europe and their utility in flood frequency estimation. J. Hydrol. 2014, 517, 963–973. [Google Scholar] [CrossRef]
  25. Paprotny, D.; Sebastian, A.; Morales-Nápoles, O.; Jonkman, S.N. Trends in flood losses in Europe over the past 150 years. Nat. Commun. 2018, 9, 1985. [Google Scholar] [CrossRef]
  26. Cammalleri, C.; Vogt, J.; Salamon, P. Development of an operational low-flow index for hydrological drought monitoring over Europe. Hydrol. Sci. J. 2017, 62, 346–358. [Google Scholar] [CrossRef]
  27. Kay, A.L.; Griffin, A.; Rudd, A.C.; Chapman, R.M.; Bell, V.A.; Arnell, N.W. Climate change effects on indicators of high and low river flow across Great Britain. Adv. Water Resour. 2021, 151, 103909. [Google Scholar] [CrossRef]
  28. Tomaszewski, E.; Kubiak-Wójcicka, K. Low-flows in Polish rivers. In Management of Water Resources in Poland; Springer: Cham, Switzerland, 2021; pp. 205–228. [Google Scholar] [CrossRef]
  29. Tsakiris, G.; Nalbantis, I.; Cavadias, G. Regionalization of low flows based on canonical correlation analysis. Adv. Water Resour. 2011, 34, 865–872. [Google Scholar] [CrossRef]
  30. Gottschalk, L.; Yu, K.X.; Leblois, E.; Xiong, L. Statistics of low flow: Theoretical derivation of the distribution of minimum streamflow series. J. Hydrol. 2013, 481, 204–219. [Google Scholar] [CrossRef]
  31. Sun, P.; Zhang, Q.; Yao, R.; Singh, V.P.; Song, C. Low flow regimes of the Tarim River Basin, China: Probabilistic behaviour, causes and implications. Water 2018, 10, 470. [Google Scholar] [CrossRef]
  32. Bhatti, S.J.; Kroll, C.N.; Vogel, R.M. Revisiting the probability distribution of low streamflow series in the United States. J. Hydrol. Eng. 2019, 24, 04019043. [Google Scholar] [CrossRef]
  33. Langat, P.K.; Kumar, L.; Koech, R. Identification of the most suitable probability distribution models for maximum, minimum, and mean streamflow. Water 2019, 11, 734. [Google Scholar] [CrossRef]
  34. Lee, K.S.; Kim, S.U. Identification of uncertainty in low flow frequency analysis using Bayesian MCMC method. Hydrol. Process. Int. J. 2008, 22, 1949–1964. [Google Scholar] [CrossRef]
  35. Whitfield, P.H.; Kraaijenbrink, P.D.; Shook, K.R.; Pomeroy, J.W. The spatial extent of hydrological and landscape changes across the mountains and prairies of Canada in the Mackenzie and Nelson River basins based on data from a warm-season time window. Hydrol. Earth Syst. Sci. 2021, 25, 2513–2541. [Google Scholar] [CrossRef]
  36. Trenberth, K.E.; Dai, A.; Van Der Schrier, G.; Jones, P.D.; Barichivich, J.; Briffa, K.R.; Sheffield, J. Global warming and changes in drought. Nat. Clim. Change 2014, 4, 17–22. [Google Scholar] [CrossRef]
  37. Arismendi, I.; Johnson, S.L.; Dunham, J.B.; Haggerty, R.O.Y. Descriptors of natural thermal regimes in streams and their responsiveness to change in the Pacific Northwest of North America. Freshw. Biol. 2013, 58, 880–894. [Google Scholar] [CrossRef]
  38. Leprieur, F.; Hickey, M.A.; Arbuckle, C.J.; Closs, G.P.; Brosse, S.; Townsend, C.R. Hydrological disturbance benefits a native fish at the expense of an exotic fish. J. Appl. Ecol. 2006, 43, 930–939. [Google Scholar] [CrossRef]
  39. Assefa, K.; Moges, M. Low Flow Trends and Frequency Analysis in the Blue Nile Basin, Ethiopia. J. Water Resour. Prot. 2018, 10, 182–203. [Google Scholar] [CrossRef]
  40. Rottler, E.; Francke, T.; Bürger, G.; Bronstert, A. Long-term changes in Central European River discharge for 1869–2016: Impact of changing snow covers, reservoir constructions and an intensified hydrological cycle. Hydrol. Earth Syst. Sci. 2020, 24, 1721–1740. [Google Scholar] [CrossRef]
  41. Oki, T.; Valeo, C.; Heal, K. (Eds.) Hydrology 2020: An Integrating Science to Meet World Water Challenges; IAHS Press: Oxfordshire, UK, 2006; Volume 300, p. 190. [Google Scholar]
  42. Poórová, J.; Melová, K.; Lovásová, Ľ.; Blaškovičová, L. The impact of manipulation on selected water reservoirs on flow with regard to the dry season. In Proceedings of the Water Reservoirs Conference, Brno, Czech Republic, 3–4 October 2017; pp. 169–177. Available online: https://www.shmu.sk/sk/?page=2272 (accessed on 28 July 2025).
  43. Rončák, P.; Hlavčová, K.; Kohnová, S.; Szolgay, J. Impacts of Future Climate Change on Runoff in Selected Catchments of Slovakia. In Climate Change Adaptation in Eastern Europe; Climate Change Management; Leal Filho, W., Trbic, G., Filipovic, D., Eds.; Springer: Cham, Switzerland, 2019. [Google Scholar] [CrossRef]
  44. Hlavčová, K.; Rončak, P.; Maliarikova, M.; Latkova, T.; Korbelova, L. Changes in hydrological regime under changed climate and forest conditions in mountainous basins in Slovakia. In EGU General Assembly Conference Abstracts; European Geosciences Union: Munich, Germany, 2016; p. EPSC2016-11995. [Google Scholar]
  45. Halmová, D.; Pekárová, P.; Bačová Mitková, V. Long-term trend changes of monthly and extreme discharges for different time periods. Acta Hydrol. Slovaca 2019, 20, 122–130. [Google Scholar] [CrossRef]
  46. Sabová, Z.; Kohnová, S. On future changes in the long-term seasonal discharges in selected basins of Slovakia. Acta Hydrol. Slovaca 2023, 24, 73–81. [Google Scholar] [CrossRef]
  47. Poórová, J.; Jeneiová, K.; Blaškovičová, L.; Danáčcová, Z.; Kotríková, K.; Melová, K.; Paľušová, Z. Effects of the time period length on the determination of long-term mean annual discharge. Hydrology 2023, 10, 88. [Google Scholar] [CrossRef]
  48. Váš, P.; Bartok, J.; Gaál, L.; Jurašek, M.; Melo, M.; Gera, M. Frequency shifts in thunderstorm patterns as key precursors to flash flood events. J. Hydrol. Hydromech. 2025, 73, 73–83. [Google Scholar] [CrossRef]
  49. Sleziak, P.; Danko, M.; Jančo, M.; Holko, L.; Greimeister-Pfeil, I.; Vreugdenhil, M.; Parajka, J. Accuracy of ASCAT-DIREX Soil Moisture Mapping in a Small Alpine Catchment. Water 2025, 17, 49. [Google Scholar] [CrossRef]
  50. Bačová Mitková, V.; Pekárová, P.; Halmová, D.; Miklánek, P.; Leščešen, I. Long-term analysis of changes in seasonal and maximum discharges of Slovak rivers in the period 1931–2020. J. Hydrol. Hydromech. 2024, 72, 486–498. [Google Scholar] [CrossRef]
  51. Blaškovičová, L.; Jeneiová, K.; Kotríková, K.; Lovásová, Ľ.; Melová, K.; Liová, S. Challenges in selecting the new reference period for long-term hydrological characteristics in Slovakia. Acta Hydrol. Slovaca 2023, 24, 232–241. [Google Scholar] [CrossRef]
  52. Blaškovičová, L.; Melová, K.; Liová, S.; Podolinská, J.; Síčová, B.; Grohoľ, M. The drought characteristics and their changes in selected water-gauging stations in Slovakia in the period 2001–2020 compared to the reference period 1961–2000. Acta Hydrol. Slovaca 2022, 23, 10–20. [Google Scholar] [CrossRef]
  53. OTN ŽP SR 3112-1:03; Surface Water and Subsurface Water Quantity. Hydrological Data of Surface Waters. Quantification of Flood Regime. Part 1: Determination of T-Year Discharges and T-Year Discharge Waves on Larger Streams. Ministry of the Environment: Bratislava, Slovakia, 2003.
  54. OTN ŽP SR 3113-1:04; Surface Water Quantity. Part 1 Determination of Low Water Content in Water Measuring Stations. Ministry of the Environment: Bratislava, Slovakia, 2007; 10p. (In Slovak)
  55. Pilon, P.J.; Adamowski, K. Asymptotic variance of flood quantile in log Pearson type III distribution with historical information. J. Hydrol. 1993, 143, 481–503. [Google Scholar] [CrossRef]
  56. Cheng, K.S.; Chiang, J.L.; Hsu, C.W. Simulation of probability distributions commonly used in hydrological frequency analysis. Hydrol. Process 2007, 21, 51–60. [Google Scholar] [CrossRef]
  57. Griffis, V.W.; Stedinger, J.R. Log-Pearson type 3 distribution and its application in flood frequency analysis, III—Sample skew and weighted skew estimators. J. Hydrol. 2009, 14, 121–130. [Google Scholar] [CrossRef]
  58. Farooq, M.; Shafique, M.; Khattak, M.S. Flood frequency analysis of river swat using Log Pearson type 3, Generalized Extreme Value, Normal, and Gumbel Max distribution methods. Arab. J. Geosci. 2018, 11, 216. [Google Scholar] [CrossRef]
  59. Tian, D.; Wang, L. BLP3-SP: A Bayesian Log-Pearson type III model with spatial priors for reducing uncertainty in flood frequency analyses. Water 2022, 14, 909. [Google Scholar] [CrossRef]
  60. IACWD. Guidelines for Determining Flood Flow Frequency, Bulletin 17-B; Technical report; Interagency Committee on Water Data, Hydrology Subcommittee: Reston, VA, USA, 1982; 194p. [Google Scholar]
  61. Stedinger, J.R.; Griffis, V.W. Flood Frequency Analysis in the United States: Time to Update. J. Hydrol. Eng. 2008, 13, 199–204. [Google Scholar] [CrossRef]
  62. Weibull, W. A Statistical Theory of the Strength of Materials. In Ingeniörs Vetenskaps Akademiens Handlingar; Royal Swedish Institute for Engineering Research: Stockholm, Sweden, 1939; No. 151. [Google Scholar]
  63. Cunnane, C. Unbiased plotting positions—A review. J. Hydrol. 1978, 37, 205–222. [Google Scholar] [CrossRef]
  64. Stedinger, J.R.; Vogel, R.M.; Foufoula-Georgiou, E. Frequency analysis of extreme events. In Handbook of Hydrology, Maidment, D.R., Ed.; McGraw-Hill: New York, NY, USA, 1993; Chapter 18. [Google Scholar]
  65. Obodovskyi, O.; Lukianets, O.; Konovalenko, O.; Mykhaylenko, V. Mapping the Mean Annual River Runoff in the UkrainianCarpathian Region. EREM J. Environ. Res. Eng. Manag. 2020, 76, 22–33. [Google Scholar] [CrossRef]
  66. Pekárová, P.; Halmová, D.; Bačová Mitková, V.; Poórová, J.; Blaškovičová, L.; Pekár, J.; Leščešen, I.; Bajtek, Z. Temporal variability of average and low flows in Slovak rivers: A 90-year perspective. J. Hydrol. Reg. Stud. 2025, 60, 102560. [Google Scholar] [CrossRef]
  67. Sleziak, P.; Výleta, R.; Hlavčová, K.; Danáčová, M.; Aleksić, M.; Szolgay, J.; Kohnová, S. A hydrological modelling approach for assessing the impacts of climate change on runoff regimes in Slovakia. Water 2021, 13, 3358. [Google Scholar] [CrossRef]
  68. Blöschl, G.; Hall, J.; Viglione, A.; Perdigão, R.A.; Parajka, J.; Merz, B.; Lun, D.; Arheimer, B.; Aronica, G.T.; Bilibashi, A.; et al. Changing climate both increases and decreases European river floods. Nature 2019, 573, 108–111. [Google Scholar] [CrossRef] [PubMed]
  69. Stahl, K.; Hisdal, H.; Hannaford, J.; Tallaksen, L.M.; Van Lanen, H.A.J.; Sauquet, E.; Jódar, J. Streamflow trends in Europe: Evidence from a dataset of near-natural catchments. Hydrol. Earth Syst. Sci. 2010, 14, 2367–2382. [Google Scholar] [CrossRef]
  70. Hannaford, J. Climate-driven changes in UK river flows: A review of the evidence. Prog. Phys. Geogr. 2015, 39, 29–48. [Google Scholar] [CrossRef]
  71. Nasr, A.; Bruen, M. Detection of trends in the 7-day sustained low-flow time series of Irish rivers. Hydrol. Sci. J. 2017, 62, 947–959. [Google Scholar] [CrossRef]
  72. Giuntoli, I.; Renard, B.; Vidal, J.-P.; Bard, A. Low flows in France and their relationship to large-scale climate indices. J. Hydrol. 2013, 482, 105–118. [Google Scholar] [CrossRef]
  73. Fiala, T.; Taha, B.M.; Ouarda, J.; Hladný, J. Evolution of low flows in the Czech Republic. J. Hydrol. 2010, 393, 206–218. [Google Scholar] [CrossRef]
  74. Coch, A.; Mediero, L. Trends in low flows in Spain in the period 1949–2009. Hydrol. Sci. J. 2016, 61, 568–584. [Google Scholar] [CrossRef]
  75. Cigizoglu, H.K.; Bayazit, M.; Önöz, B. Trends in the Maximum, Mean, and Low Flows of Turkish Rivers. J. Hydrometeor. 2005, 6, 280–290. [Google Scholar] [CrossRef]
  76. Bard, A.; Renard, B.; Lang, M.; Giuntoli, I.; Korck, J.; Koboltschnig, G.; Janža, M.; d’Amico, M.; Volken, D. Trends in the hydrologic regime of Alpine rivers. J. Hydrol. 2015, 529, 1823–1837. [Google Scholar] [CrossRef]
  77. Leščešen, I.; Gnjato, S.; Vujačić, D.; Petrović, A.M.; Radevski, I. Seasonal variability changes and trends in minimum discharge for Western Balkan rivers. J. Hydrol. Reg. Stud. 2025, 60, 102529. [Google Scholar] [CrossRef]
  78. Blagojević, B.; Mihailović, V.; Bogojević, A.; Plavšić, J. Detecting Annual and Seasonal Hydrological Change Using Marginal Distributions of Daily Flows. Water 2023, 15, 2919. [Google Scholar] [CrossRef]
  79. Gorbachova, L.; Khrystiuk, B. Extreme low flow change analysis on the Tysa River within Ukraine. Acta Hydrol. Slovaca 2021, 22, 200–206. [Google Scholar] [CrossRef]
  80. Mészáros, J.; Miklánek, P.; Pekárová, P. Estimation of the t-year specific discharge using the regionalised skewness coefficient of the log-pearson type III distribution. In Proceedings of the XXVIII Conference of the Danubian Countries on Hydrological Forecasting and Hydrological Bases of Water Management, Kyiv, Ukraine, 6–8 November 2019; Ukrainian Hydrometeorological Institute, Department of Hydrological Research: Kyiv, Ukraine, 2019; pp. 73–85, ISBN 978-966-7067-38-0. [Google Scholar]
  81. Bačová Mitková, V.; Pekárová, P.; Halmová, D.; Miklánek, P. The use of a uniform technique for harmonization and generalization in assessing the flood discharge frequencies of long return period floods in the Danube River Basin. Water 2021, 13, 1337. [Google Scholar] [CrossRef]
Figure 1. Locations of the 26 rivers in the Slovak region and long-term changes and variability in mean areal annual precipitation P, mean areal air temperature Ta, and runoff R for the analyzed basins: (a) the basin with the highest station altitude: Belá, Podbanské; and (b) the basin with the second-lowest station altitude: Litava, Plášťovce.
Figure 1. Locations of the 26 rivers in the Slovak region and long-term changes and variability in mean areal annual precipitation P, mean areal air temperature Ta, and runoff R for the analyzed basins: (a) the basin with the highest station altitude: Belá, Podbanské; and (b) the basin with the second-lowest station altitude: Litava, Plášťovce.
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Figure 2. Map of the estimated skew coefficients, G1dmax, of the 1-day maximum daily specific discharges of the Log-Pearson III distribution in selected rivers in the Slovak region for two seasons, summer–autumn (SA: 1 May–31 October) and winter–spring (WS: 1 November–30 April), during the entire data period of 1930/1931–2019/2020.
Figure 2. Map of the estimated skew coefficients, G1dmax, of the 1-day maximum daily specific discharges of the Log-Pearson III distribution in selected rivers in the Slovak region for two seasons, summer–autumn (SA: 1 May–31 October) and winter–spring (WS: 1 November–30 April), during the entire data period of 1930/1931–2019/2020.
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Figure 3. Examples of the theoretical LPIII probability of exceedance curves (blue line) of the maximum daily specific discharges, q1dmax (green points), with confidence intervals of 5% and 95% (red dashed line) for the (a) Ľubochnianka River (Ľubochňa), (b) Belá River (Podbanské), and (c) Topľa River (Hanušovce) for 1930/1931–2019/2020 (WS—winter–spring season; SA—summer–autumn season).
Figure 3. Examples of the theoretical LPIII probability of exceedance curves (blue line) of the maximum daily specific discharges, q1dmax (green points), with confidence intervals of 5% and 95% (red dashed line) for the (a) Ľubochnianka River (Ľubochňa), (b) Belá River (Podbanské), and (c) Topľa River (Hanušovce) for 1930/1931–2019/2020 (WS—winter–spring season; SA—summer–autumn season).
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Figure 4. A comparison of the estimated values of the 50-, 100-, and 500-year maximum daily specific discharges for runoff durations of 1 day for selected rivers in Slovakia (period 1930/1931–2019/2020: WS—winter–spring season; SA—summer–autumn season).
Figure 4. A comparison of the estimated values of the 50-, 100-, and 500-year maximum daily specific discharges for runoff durations of 1 day for selected rivers in Slovakia (period 1930/1931–2019/2020: WS—winter–spring season; SA—summer–autumn season).
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Figure 5. Map of the estimated skew coefficients (G7dmin) of the 7-day minimum daily specific discharges of Log-Pearson III distribution in selected rivers in the Slovak region for two seasons, summer–autumn (SA: 1 May–31 October) and winter–spring (WS: 1 November–30 April), during 1930/1931–2019/2020.
Figure 5. Map of the estimated skew coefficients (G7dmin) of the 7-day minimum daily specific discharges of Log-Pearson III distribution in selected rivers in the Slovak region for two seasons, summer–autumn (SA: 1 May–31 October) and winter–spring (WS: 1 November–30 April), during 1930/1931–2019/2020.
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Figure 6. T-year 7-day minimum specific discharge (green points) and LP3 theoretical distribution (blue line) with confidence intervals of 5% and 95% (red dashed line), Ipeľ (Holiša) lowland river, (a) with outliers and (b) without outliers for 1930/1931–2019/2020 across the hydrological year. Example distribution curves for the mountainous Belá River (Podbanské) (c) for summer–autumn and (d) winter–spring.
Figure 6. T-year 7-day minimum specific discharge (green points) and LP3 theoretical distribution (blue line) with confidence intervals of 5% and 95% (red dashed line), Ipeľ (Holiša) lowland river, (a) with outliers and (b) without outliers for 1930/1931–2019/2020 across the hydrological year. Example distribution curves for the mountainous Belá River (Podbanské) (c) for summer–autumn and (d) winter–spring.
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Figure 7. A comparison of the estimated values of the 20-, 50-, and 100-year minimum specific discharges for 7-day runoff durations for selected rivers in the Slovak region (1930/1931–2019/2020; WS, winter–spring; SA, summer–autumn).
Figure 7. A comparison of the estimated values of the 20-, 50-, and 100-year minimum specific discharges for 7-day runoff durations for selected rivers in the Slovak region (1930/1931–2019/2020; WS, winter–spring; SA, summer–autumn).
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Figure 8. The dependencies of the 100-year 1-day maximum specific discharges (q1dmax,100) on the altitude of station H in selected water gauges for (a) winter–spring (WS) and (b) summer–autumn (SA) for 1930/1931–2019/2020.
Figure 8. The dependencies of the 100-year 1-day maximum specific discharges (q1dmax,100) on the altitude of station H in selected water gauges for (a) winter–spring (WS) and (b) summer–autumn (SA) for 1930/1931–2019/2020.
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Figure 9. The dependencies of the 100-year 7-day minimum specific discharge (q7dmin,100) (a) on the long-term average annual runoff q and (b) the altitude of the station, H, in the selected water gauges for WS and SA.
Figure 9. The dependencies of the 100-year 7-day minimum specific discharge (q7dmin,100) (a) on the long-term average annual runoff q and (b) the altitude of the station, H, in the selected water gauges for WS and SA.
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Figure 10. The dependencies of the T-year 7-day minimum specific discharges (q7dmin,T) on the long-term average annual specific discharges (q) for T = 1000, 100, 50, and 20 years for WS and SA.
Figure 10. The dependencies of the T-year 7-day minimum specific discharges (q7dmin,T) on the long-term average annual specific discharges (q) for T = 1000, 100, 50, and 20 years for WS and SA.
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Figure 11. A schematic of the 100-year 7-day minimum daily specific discharge for SA in Slovakia.
Figure 11. A schematic of the 100-year 7-day minimum daily specific discharge for SA in Slovakia.
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Table 1. List of 26 selected rivers in the Slovak region and the Danube River and its tributaries, with basic characteristics and long-term mean discharge (Qma), long-term mean runoff (Rma), and station altitude (H).
Table 1. List of 26 selected rivers in the Slovak region and the Danube River and its tributaries, with basic characteristics and long-term mean discharge (Qma), long-term mean runoff (Rma), and station altitude (H).
ID SKGauge
Station
RiverPeriodArea [km2]Forestry [%]Valley Length
[km]
Qma
[m3.s−1]
Rma
[mm]
H
[m a.s.l]
5040Moravský Svätý JánMorava1930–202024,129.330260.4105.1137146
5330VýchodnáBiely Váh1930–2020105.643018.81.58472730.65
5340Kráľova LehotaBoca1930–2020116.68018.32.01544655.08
5400PodbanskéBelá1930–202093.4930163.531191922.7
5550Liptovský MikulášVáh1930–20201107.216058.820.47583568
5740PodsucháRevúca1930–2020217.956021.94.82697558.31
5790ĽubochňaĽubochnianka1930–2020118.4890242.37631442.71
6130MartinTuriec1930–20208276059.610.17388389.88
6200Kysucké Nové MestoKysuca1930–2020955.095057.616.15533346
6300PoluvsieRajčanka1930–2020243.66012.53.53457393.03
6730Nitrianska StredaNitra1930–20202093.715078.714.49218158.3
7045HronecČierny Hron1930–2020239.418023.52.85375480.48
7060BystráBystrianka1930–202036.0180120.92806574.54
7065Mýto pod ĎumbieromŠtiavnička1930–202047.18010.91.05703616.75
7070Dolná LehotaVajskovský potok1930–202053.0270151.36809495.28
7160Banská BystricaHron1930–20201766.4860100.525.78460334
7290BrehyHron1930–20203821.3850181.446.22381195
7440HolišaIpeľ1930–2020685.263056.13.03139172
7580PlášťovceKrupinica1930–2020302.794054.51.76183139
7600PlášťovceLitava1930–2020214.423043.91.09160142.02
7730ŠtítnikŠtítnik1930–2020129.634018.l1.36331284.95
7860Lehota nad RimavicouRimavica1930–2020148.953029.61.5318263.65
8320ChmeľnicaPoprad1930–20201262.414085.215.51387507.44
8870Košické OlšanyTorysa1930–20201298.330116.47.73188185.88
8970Nižný MedzevBodva1940–202090.158013.50.78273310.24
9500HanušovceTopľa1930–20201050.055084.68.04241160.4
Table 2. Estimated values of the 50-, 100-, and 500-year 1-day maximum daily specific discharges, q1dmax,T, in [L·s−1·km−2], for the selected rivers (WS season, SA season).
Table 2. Estimated values of the 50-, 100-, and 500-year 1-day maximum daily specific discharges, q1dmax,T, in [L·s−1·km−2], for the selected rivers (WS season, SA season).
ID SKPeriodq1dmax,50q1dmax,100q1dmax,500q1dmax,50 WSq1dmax,50 SAq1dmax,100 WSq1dmax,100
SA
q1dmax 500 WSq1dmax,500
SA
50401930–2020576689554364518573
53301930–2020277323445237259286302432409
53401930–2020294335434254256288293367381
54001930–202070682711522667152878423271186
55501930–2020235265339163231185263240343
57401930–2020316361478290257335289450362
57901930–2020236272400214169259189396237
61301930–2020249310502185180217238305439
62001930–2020474534686343434375491447623
63001930–2020346398528277324319392424578
67301930–2020142161203127132143177179325
70451930–2020310383600214264248358333698
70601930–2020320370497211304234363287494
70651930–2020352439721217347239450289797
70701930–2020291333442261267309301444383
71601930–2020215256375152190167240199399
72901930–2020207237313171177188216225326
74401930–2020159179219160129186174244323
75801930–2020277313393250235279312341548
76001930–2020300350470250276281380343719
77301930–2020272334513221235266304382521
78601930–20203985008003372994234416691033
83201930–2020330388538150364171445226674
88701930–2020201238339129214148263193394
89701940–2020335428716261304332433540932
95001930–2020251296417204213235252315346
average290342485215262249325334521
Table 3. Estimated values of the 7-day minimum specific discharge (q7dmin,T) for T = 100, 50, and 20 years [L·s1·km−2], hydrological year, SA, and WS for 1930/1931–2019/2020.
Table 3. Estimated values of the 7-day minimum specific discharge (q7dmin,T) for T = 100, 50, and 20 years [L·s1·km−2], hydrological year, SA, and WS for 1930/1931–2019/2020.
ID SKPeriodq7dmin,20q7dmin,50q7dmin,100q7dmin,20 WSq7dmin,20
SA
q7dmin,50 WSq7dmin,50
SA
q7dmin,100 WSq7dmin,100
SA
50401930–20200.600.510.460.890.600.750.510.670.46
53301930–20203.663.222.953.724.503.253.942.963.60
53401930–20202.692.352.152.733.352.412.862.222.57
54001930–20205.304.714.355.2610.014.678.494.327.56
55501930–20204.424.133.954.485.544.205.134.044.89
57401930–20204.083.312.844.255.113.334.442.784.03
57901930–20204.653.763.215.125.484.224.413.673.76
61301930–20203.222.982.843.443.373.163.062.992.87
62001930–20201.181.010.921.471.221.151.040.980.94
63001930–20201.691.471.352.221.731.931.481.761.34
67301930–20201.181.050.971.401.221.131.080.970.99
70451930–20201.871.721.632.211.882.011.701.911.59
70601930–20203.873.513.313.874.723.513.813.313.30
70651930–20204.474.284.194.554.964.274.594.124.40
70701930–20205.605.214.985.776.795.206.344.856.09
71601930–20203.303.092.963.423.573.113.352.923.23
72901930–20202.372.192.082.672.432.432.242.302.13
74401930–20200.12 */
0.18 **
0.07 */
0.13 **
0.04 */
0.11 **
0.360.110.250.060.200.04
75801930–20200.140.100.080.330.170.250.120.210.10
76001930–20200.090.070.060.290.090.210.070.170.06
77301930–20200.990.730.591.601.001.290.711.120.56
78601930–20200.650.430.311.110.670.820.430.660.31
83201930–20202.282.071.932.332.922.122.642.002.47
88701930–20200.660.540.480.810.690.690.560.630.48
89701940–20200.510.400.340.790.510.680.400.620.34
95001930–20200.870.780.731.050.900.920.790.840.73
average2.332.071.912.542.832.232.472.052.26
* With outliers. ** Without outliers.
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Pekárová, P.; Bačová Mitková, V.; Halmová, D. Probability Characteristics of High and Low Flows in Slovakia: A Comprehensive Hydrological Assessment. Hydrology 2025, 12, 199. https://doi.org/10.3390/hydrology12080199

AMA Style

Pekárová P, Bačová Mitková V, Halmová D. Probability Characteristics of High and Low Flows in Slovakia: A Comprehensive Hydrological Assessment. Hydrology. 2025; 12(8):199. https://doi.org/10.3390/hydrology12080199

Chicago/Turabian Style

Pekárová, Pavla, Veronika Bačová Mitková, and Dana Halmová. 2025. "Probability Characteristics of High and Low Flows in Slovakia: A Comprehensive Hydrological Assessment" Hydrology 12, no. 8: 199. https://doi.org/10.3390/hydrology12080199

APA Style

Pekárová, P., Bačová Mitková, V., & Halmová, D. (2025). Probability Characteristics of High and Low Flows in Slovakia: A Comprehensive Hydrological Assessment. Hydrology, 12(8), 199. https://doi.org/10.3390/hydrology12080199

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