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

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 (q1d_max,T) and T-year 7-day minimum specific discharge (q7d_min,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 (q1d_max,100) and T-year 7-day minimum specific discharges (q7d_min,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 P_1991–2015 = 1851.24 mm. In contrast, the lowest value for the same period was observed in Senec (125 m), with P_1991–2015 = 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. 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 SK	Gauge Station	River	Period	Area [km2]	Forestry [%]	Valley Length [km]	Qma [m3.s−1]	Rma [mm]	H [m a.s.l]
5040	Moravský Svätý Ján	Morava	1930–2020	24,129.3	30	260.4	105.1	137	146
5330	Východná	Biely Váh	1930–2020	105.64	30	18.8	1.58	472	730.65
5340	Kráľova Lehota	Boca	1930–2020	116.6	80	18.3	2.01	544	655.08
5400	Podbanské	Belá	1930–2020	93.49	30	16	3.53	1191	922.7
5550	Liptovský Mikuláš	Váh	1930–2020	1107.21	60	58.8	20.47	583	568
5740	Podsuchá	Revúca	1930–2020	217.95	60	21.9	4.82	697	558.31
5790	Ľubochňa	Ľubochnianka	1930–2020	118.48	90	24	2.37	631	442.71
6130	Martin	Turiec	1930–2020	827	60	59.6	10.17	388	389.88
6200	Kysucké Nové Mesto	Kysuca	1930–2020	955.09	50	57.6	16.15	533	346
6300	Poluvsie	Rajčanka	1930–2020	243.6	60	12.5	3.53	457	393.03
6730	Nitrianska Streda	Nitra	1930–2020	2093.71	50	78.7	14.49	218	158.3
7045	Hronec	Čierny Hron	1930–2020	239.41	80	23.5	2.85	375	480.48
7060	Bystrá	Bystrianka	1930–2020	36.01	80	12	0.92	806	574.54
7065	Mýto pod Ďumbierom	Štiavnička	1930–2020	47.1	80	10.9	1.05	703	616.75
7070	Dolná Lehota	Vajskovský potok	1930–2020	53.02	70	15	1.36	809	495.28
7160	Banská Bystrica	Hron	1930–2020	1766.48	60	100.5	25.78	460	334
7290	Brehy	Hron	1930–2020	3821.38	50	181.4	46.22	381	195
7440	Holiša	Ipeľ	1930–2020	685.26	30	56.1	3.03	139	172
7580	Plášťovce	Krupinica	1930–2020	302.79	40	54.5	1.76	183	139
7600	Plášťovce	Litava	1930–2020	214.42	30	43.9	1.09	160	142.02
7730	Štítnik	Štítnik	1930–2020	129.63	40	18.l	1.36	331	284.95
7860	Lehota nad Rimavicou	Rimavica	1930–2020	148.95	30	29.6	1.5	318	263.65
8320	Chmeľnica	Poprad	1930–2020	1262.41	40	85.2	15.51	387	507.44
8870	Košické Olšany	Torysa	1930–2020	1298.3	30	116.4	7.73	188	185.88
8970	Nižný Medzev	Bodva	1940–2020	90.15	80	13.5	0.78	273	310.24
9500	Hanušovce	Topľa	1930–2020	1050.05	50	84.6	8.04	241	160.4

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 (T_a), 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 (Qd_max,T), sets of 1-day maximum discharges (Q1d_max) 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 (Q7d_min,T) and T-year minimum specific discharges (q7d_min,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:

$$f\left(x|\tau ,\alpha ,\beta \right)={\displaystyle \frac{{\left(\frac{x-\tau }{\beta }\right)}^{\alpha -1}exp\left(-\frac{x-\tau }{\beta }\right)}{\left|\beta \right|\Gamma \left(\alpha \right)}},$$

(1)

with

$${\displaystyle \frac{x-\tau }{\beta }}\ge 0,$$

(2)

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):

$$\Gamma \left(\alpha \right)={\int }_0^{\infty }{t}^{\alpha -1}exp\left(-t\right)dt.$$

(3)

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:

$$\mathit{log}X=\overline{X}+K\sigma ,$$

(4)

where $\overline{X}$ 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:

$$\overline{X}={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{X}_i.$$

(5)

Standard Deviation:

$$\sigma =\sqrt{{\displaystyle \frac{1}{n-1}}{\sum }_{i=1}^n{\left(X_i-\overline{X}\right)}^2}.$$

(6)

Skew Coefficient:

$$G={\displaystyle \frac{n}{\left(n-1\right)\left(n-2\right){\sigma }^3}}{\sum }_{i=1}^n{\left(X_i-\overline{X}\right)}^3.$$

(7)

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]

$$P_i={\displaystyle \frac{i-\alpha }{n+1-2\alpha }},$$

(8)

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

$$p=1-e^{{\displaystyle \frac{-1}{T}}},$$

(9)

while for T ≥ 10, a simplified form is used:

$$P={\displaystyle \frac{1}{T}}.$$

(10)

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 ($\widehat{\mu }$), standard deviation ($\widehat{\sigma }$), and skew coefficient ($\widehat{\gamma }$). 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:

$$\widehat{\mu }={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{X}_i$$

(11)

$$\widehat{\sigma }=\sqrt{{\displaystyle \frac{1}{n-1}}{\sum }_{i=1}^n{\left(X_i-\widehat{\mu }\right)}^2}$$

(12)

$$\widehat{\gamma }={\displaystyle \frac{n}{\left(n-1\right)\left(n-2\right){\widehat{\sigma }}^3}}{\sum }_{i=1}^n{\left(X_i-\widehat{\mu }\right)}^3$$

(13)

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 (Q1d_max) 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 (Q1d_max) were recalculated to the maximum daily specific discharges (q1d_max) 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 (q1d_max): 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. Estimated values of the 50-, 100-, and 500-year 1-day maximum daily specific discharges, q1d_max,T, in [L·s⁻¹·km⁻²], for the selected rivers (WS season, SA season).

ID SK	Period	q1dmax,50	q1dmax,100	q1dmax,500	q1dmax,50 WS	q1dmax,50 SA	q1dmax,100 WS	q1dmax,100 SA	q1dmax 500 WS	q1dmax,500 SA
5040	1930–2020	57	66	89	55	43	64	51	85	73
5330	1930–2020	277	323	445	237	259	286	302	432	409
5340	1930–2020	294	335	434	254	256	288	293	367	381
5400	1930–2020	706	827	1152	266	715	287	842	327	1186
5550	1930–2020	235	265	339	163	231	185	263	240	343
5740	1930–2020	316	361	478	290	257	335	289	450	362
5790	1930–2020	236	272	400	214	169	259	189	396	237
6130	1930–2020	249	310	502	185	180	217	238	305	439
6200	1930–2020	474	534	686	343	434	375	491	447	623
6300	1930–2020	346	398	528	277	324	319	392	424	578
6730	1930–2020	142	161	203	127	132	143	177	179	325
7045	1930–2020	310	383	600	214	264	248	358	333	698
7060	1930–2020	320	370	497	211	304	234	363	287	494
7065	1930–2020	352	439	721	217	347	239	450	289	797
7070	1930–2020	291	333	442	261	267	309	301	444	383
7160	1930–2020	215	256	375	152	190	167	240	199	399
7290	1930–2020	207	237	313	171	177	188	216	225	326
7440	1930–2020	159	179	219	160	129	186	174	244	323
7580	1930–2020	277	313	393	250	235	279	312	341	548
7600	1930–2020	300	350	470	250	276	281	380	343	719
7730	1930–2020	272	334	513	221	235	266	304	382	521
7860	1930–2020	398	500	800	337	299	423	441	669	1033
8320	1930–2020	330	388	538	150	364	171	445	226	674
8870	1930–2020	201	238	339	129	214	148	263	193	394
8970	1940–2020	335	428	716	261	304	332	433	540	932
9500	1930–2020	251	296	417	204	213	235	252	315	346
average		290	342	485	215	262	249	325	334	521

). 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 (q7d_min) for the selected stations. As the first step, we estimated the LPIII distribution function parameters: mean (q7dmin), standard deviation (S7dmin), and station skew coefficient (G7d_min) 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 (q7d_min,T) calculated for WS and SA, and for the whole period, are presented in

Table 3. Estimated values of the 7-day minimum specific discharge (q7d_min,T) for T = 100, 50, and 20 years [L·s⁻¹·km⁻²], hydrological year, SA, and WS for 1930/1931–2019/2020.

ID SK	Period	q7dmin,20	q7dmin,50	q7dmin,100	q7dmin,20 WS	q7dmin,20 SA	q7dmin,50 WS	q7dmin,50 SA	q7dmin,100 WS	q7dmin,100 SA
5040	1930–2020	0.60	0.51	0.46	0.89	0.60	0.75	0.51	0.67	0.46
5330	1930–2020	3.66	3.22	2.95	3.72	4.50	3.25	3.94	2.96	3.60
5340	1930–2020	2.69	2.35	2.15	2.73	3.35	2.41	2.86	2.22	2.57
5400	1930–2020	5.30	4.71	4.35	5.26	10.01	4.67	8.49	4.32	7.56
5550	1930–2020	4.42	4.13	3.95	4.48	5.54	4.20	5.13	4.04	4.89
5740	1930–2020	4.08	3.31	2.84	4.25	5.11	3.33	4.44	2.78	4.03
5790	1930–2020	4.65	3.76	3.21	5.12	5.48	4.22	4.41	3.67	3.76
6130	1930–2020	3.22	2.98	2.84	3.44	3.37	3.16	3.06	2.99	2.87
6200	1930–2020	1.18	1.01	0.92	1.47	1.22	1.15	1.04	0.98	0.94
6300	1930–2020	1.69	1.47	1.35	2.22	1.73	1.93	1.48	1.76	1.34
6730	1930–2020	1.18	1.05	0.97	1.40	1.22	1.13	1.08	0.97	0.99
7045	1930–2020	1.87	1.72	1.63	2.21	1.88	2.01	1.70	1.91	1.59
7060	1930–2020	3.87	3.51	3.31	3.87	4.72	3.51	3.81	3.31	3.30
7065	1930–2020	4.47	4.28	4.19	4.55	4.96	4.27	4.59	4.12	4.40
7070	1930–2020	5.60	5.21	4.98	5.77	6.79	5.20	6.34	4.85	6.09
7160	1930–2020	3.30	3.09	2.96	3.42	3.57	3.11	3.35	2.92	3.23
7290	1930–2020	2.37	2.19	2.08	2.67	2.43	2.43	2.24	2.30	2.13
7440	1930–2020	0.12 */ 0.18 **	0.07 */ 0.13 **	0.04 */ 0.11 **	0.36	0.11	0.25	0.06	0.20	0.04
7580	1930–2020	0.14	0.10	0.08	0.33	0.17	0.25	0.12	0.21	0.10
7600	1930–2020	0.09	0.07	0.06	0.29	0.09	0.21	0.07	0.17	0.06
7730	1930–2020	0.99	0.73	0.59	1.60	1.00	1.29	0.71	1.12	0.56
7860	1930–2020	0.65	0.43	0.31	1.11	0.67	0.82	0.43	0.66	0.31
8320	1930–2020	2.28	2.07	1.93	2.33	2.92	2.12	2.64	2.00	2.47
8870	1930–2020	0.66	0.54	0.48	0.81	0.69	0.69	0.56	0.63	0.48
8970	1940–2020	0.51	0.40	0.34	0.79	0.51	0.68	0.40	0.62	0.34
9500	1930–2020	0.87	0.78	0.73	1.05	0.90	0.92	0.79	0.84	0.73
average		2.33	2.07	1.91	2.54	2.83	2.23	2.47	2.05	2.26

* With outliers. ** Without outliers.

.

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 (q1d_max,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 (q7d_min,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 q1d_max,100 and a set of the different physical–geographical characteristics of the basins. Satisfactory results were mainly achieved based on the dependence between q1d_max,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:

$${q1d}_{max,100}{SA=4.10}^{-6}{H}^3-0.0049H^2+2.1211H,$$

(14)

$${q1d}_{max,100}WS={\displaystyle \frac{325}{1+e^{-0.009\left(H-277\right)}}},$$

(15)

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):

$$q7d_{min,100}SA={\displaystyle \frac{8}{1+e^{-0.18\left(q-17\right)}}}-0.2,$$

(16)

$$q7d_{min,100}WS={\displaystyle \frac{4.5}{1+e^{-0.2\left(q-13\right)}}}$$

(17)

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]:

$$q7d_{min,100}SA={\displaystyle \frac{8}{1+e^{-0.007\left(H-580\right)}}},$$

(18)

$$q7d_{min,100}WS={\displaystyle \frac{4.5}{1+e^{-0.0095\left(H-430\right)}}}$$

(19)

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 q7_dmin,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 (q7d_min,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) q1d_max discharges and (b) q7d_min 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) q1d_max discharges and (b) q7d_min discharges [50,78].

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.