To support a hydrologically meaningful regional interpretation of seasonal low-flow regimes, homogeneous low-flow regimes were delineated using seasonality histograms (SH) and classified using partitive clustering based on the partitioning around medoids (PAM), which helps identify dominant hydrological controls, reduce heterogeneity, and improve model choice . SH describe the seasonal distribution of low flows using 12 variables that represent monthly frequencies of low flows. Low-flow days were identified from the M7Q series, obtained by smoothing the daily discharge series with a 7 d moving average, using a catchment-specific threshold Q95, corresponding to the discharge exceeded on 95 % of days, which is a commonly used threshold in low-flow studies. A low-flow day is when its M7Q value is below Q95 .

For each catchment, monthly occurrence frequencies were calculated as the ratio of low-flow days to total observations in each calendar month, yielding a 12-dimensional SH vector that captures the timing and shape of low-flow seasonality. These SH vectors were then grouped using PAM clustering with the number of clusters selected from the silhouette plot, which orders the silhouette width of each SH and reflects its similarity to the assigned cluster relative to the nearest alternative cluster. PAM clustering of seasonality histograms identifies whether low-flow events occur predominantly in summer, winter or both seasons, thereby providing regional evidence that annual low-flow minima may originate from different seasonal generating processes and supporting the need for a seasonal mixture framework. Clusters were interpreted based on the seasonal peaks of their medoid histograms to identify dominant seasonal low-flow regimes, whereas catchments with silhouette widths below 0.2 were considered to have weak cluster membership and were therefore classified as having mixed seasonality .

The annual low-flow series (AM) consists of the minimum 7 d average streamflow observed in each hydrological year (April to March), obtained from the M7Q series. The moving average filter reduces short-term variability and is widely applied in low-flow studies . Let X1,…,Xn denote a sequence of independent and identically distributed (i.i.d.) random variables with common distribution function F . Under the conventional annual-minima approach, the annual low-flow series is defined as: 1AM=min⁡{X1,…,Xn}. which implicitly assumes that annual low flows are generated by a homogeneous underlying process. However, in seasonal climates, low-flow events may arise from distinct hydrological processes operating in different seasons . In such cases, the annual minimum can be expressed as the minimum of the annual summer minima (AM_S) and winter minima (AM_W) , 2AM=min⁡{AMS,AMW}.

For each catchment, the dependence between the annual summer and winter low-flow series was assessed using Kendall's rank correlation coefficient (τ) . Statistical significance was evaluated at the 5 % level, and correlations with p>0.05 were classified as non-significant. Only significant correlations were subsequently grouped, for interpretation purposes, into weak (0.10≤τ<0.30), moderate (0.30≤τ<0.55), and strong (τ≥0.55) dependence classes. In addition, a mixture rate was calculated as the proportion of years in which the summer minimum 7 d flow was lower than the winter minimum 7 d flow. Based on this rate, catchments were classified as summer-dominated (>0.55), winter-dominated (<0.45), or mixed (0.45–0.55). This provides a gradual measure of seasonal dominance at the individual-catchment scale, whereas PAM clustering yields a discrete classification into mutually exclusive regime groups .

In this study, non-stationarity is assessed by separately detecting trends in the summer (AM_S) and winter (AM_W) low-flow series using the non-parametric modified Mann-Kendall test with block bootstrapping (BBSMK) with block size determined from the number of contiguous significant serial correlations, plus an additional η=1 . Compared to the original Mann-Kendall test, BBSMK preserves the original data structure, accounts for autocorrelation in the time series, and reduces the risk of false trend detection. Due to its resampling technique, the method is robust even for relatively short records . A total of 2000 bootstrap replicates is generated to construct the empirical distribution of the Mann-Kendall test statistic. Statistical significance is evaluated using the empirical bootstrap confidence interval of the standardised test statistic (Z). A trend is considered significant at the 95 % confidence level when the observed Z value lies outside the corresponding 95 % empirical bootstrap confidence interval. The magnitude of the trend is quantified using the Sen slope estimator, a robust non-parametric estimate of the monotonic rate of change based on the median of all pairwise slopes. Positive and negative values indicate increasing and decreasing trends, respectively .

The AM_S and AM_W are samples from two random variables S (annual summer low flow) and W (annual winter low flow). Under the assumption that summer and winter events are independent, the probability that the low-flow event is equal to or smaller than magnitude q can be obtained from its seasonal non-exceedance probabilities: 3FS(q)=P(S≤q),FW(q)=P(W≤q), Given that the same discharge magnitude q is inserted into both marginal distributions, the mixed probability estimator can be written as 4Fmix(q)=1-(1-FS(q))(1-FW(q)). This estimator provides a valid generalisation of the annual-frequency estimator and yields improved probability estimates when the summer and winter distributions differ . Following the Fisher–Tippett–Gnedenko theorem, seasonal low-flow minima are modelled using three-parameter Weibull (WEI3) distributions for both summer and winter, 5Fi(q)=1-exp⁡-q-μiσiξi,i∈{S,W}, defined for q>μi, where μi denotes the location parameter, σi>0 the scale parameter, and ξi>0 the shape parameter of the WEI3 distribution. The parameters are estimated separately for each season by maximum likelihood (ML) which is adopted for its flexibility in accommodating changes in model structure, which is important for extending from stationary to non-stationary settings .

The multiplication rule assumes that the events in the two seasons are independent. However, this is not always the case, leading to a more general bivariate frequency problem in which the joint occurrence probability of events in two variables is required. This can be addressed using bivariate frequency analysis, in which a joint probability model is used to estimate the probability that both S and W do not exceed given magnitudes s and w, 6P(S≤s,W≤w)=FS,W(s,w).

To model the joint distribution FS,W(s,w), a copula-based approach is applied. A copula is a function that fully characterises the dependence structure between random variables independently of their marginal distributions . Following Sklar’s theorem, the joint cumulative distribution function can be written as 7FS,W(s,w)=CFS(s),FW(w), where C(⋅) denotes a bivariate copula.

In this study, the Gumbel-Hougaard copula was chosen, motivated by the pan-European comparative assessment of copula-based mixed estimators presented by . Although alternative families (e.g. Frank and Plackett) performed well for individual catchments, the Gumbel copula emerged as a robust and station-wide suitable model for capturing positive tail dependence in seasonal low-flow extremes and was therefore adopted for continental-scale applications . Following this established evidence, and given that the focus of the present study is on extending the mixed copula framework to non-stationary conditions, alternative copula families are not re-evaluated here. The Gumbel copula is defined as 8Cθ(u,v)=exp⁡-(-ln⁡u)θ+(-ln⁡v)θ1/θ,θ∈[1,∞), where u and v denote the pseudo-observations corresponding to the marginal probabilities of S and W, respectively. The parameter θ is directly related to Kendall’s rank correlation coefficient τ.

The copula-based mixed probability estimator for annual low-flow occurrences is obtained by inserting the same discharge magnitude q into both marginal distributions and applying the AND-operator for minima. This yields 9Fmix,C(q)=FS(q)+FW(q)-CθFS(q),FW(q).

For the special case of independence (θ=1), the copula reduces to the independence copula and CFS(q),FW(q)=FS(q)FW(q), such that Eq. () simplifies to the mixed distribution estimator in Eq. (). The mixed copula estimator therefore represents a valid generalisation of the mixed distribution approach for situations where summer and winter low flows are not independent .

Under non-stationary conditions, both seasonal low-flow behaviour and the dependence between summer and winter low flows may change over time. To account for such changes, the mixed copula framework is extended to allow temporal variation in both the marginal distributions and the dependence structure.

Non-stationarity in seasonal low-flow behaviour is represented by allowing the location parameter of the marginal Weibull distributions to vary linearly with time, while keeping the scale and shape parameters constant. Allowing additional parameters to vary without strong empirical support substantially increases estimation uncertainty. Moreover, scaling analysis indicates that the moments of streamflow series (mean, variance, and skewness) are nearly perfectly related , suggesting that independent temporal variation in scale and shape parameters would be inconsistent with these structural dependencies . Let t=1,…,n denote the (annual) time index of the observation record. The time-dependent location parameter is then defined as 10μ(t)=β0+β1t, where β1 quantifies the temporal trend in low-flow magnitude. Accordingly, the seasonal marginal distributions are expressed as time-dependent cumulative distribution functions, 11FS(q∣t)=P(S≤q∣t),FW(q∣t)=P(W≤q∣t), with non-stationarity in both summer and winter low-flow series introduced through the time-varying location parameter of the WEI3 distribution. Assuming seasonal independence, the non-stationary mixed distribution for annual low flows is obtained by inserting FS(q∣t) and FW(q∣t) into Eq. ().

To explicitly account for inter-seasonal dependence that may itself vary over time, a non-stationary copula-based estimator is adopted. As in the stationary case, a Gumbel-Hougaard copula is used to model positive dependence between seasonal low flows, but now with a time-variant parameter θ. The non-stationary copula is defined as 12Cθ(t)(u,v)=exp⁡-(-ln⁡u)θ(t)+(-ln⁡v)θ(t)1/θ(t),θ(t)∈[1,∞), where u=FS(q∣t) and v=FW(q∣t) denote time-dependent pseudo-observations and 13θ(t)=1+exp⁡γ0+γ1zt, where zt∈[0,1] is a standardized time covariate. This formulation ensures valid parameter values and allows for gradual changes in dependence strength over time, and assuming the observations are independent, the likelihood functions for the stationary and non-stationary WEI3 models are provided in Appendix .

The non-stationary copula-based mixed probability estimator for annual low-flow occurrences is then obtained analogously to Eq. (). The non-stationary mixed copula estimator therefore represents a generalisation of both the stationary mixed copula approach and the non-stationary mixed distribution approach, allowing simultaneous representation of temporal changes in marginal low-flow behaviour and inter-seasonal dependence.

Design low-flow quantiles for a given return period T are finally obtained by numerical inversion of the time-dependent form of Eq. () at selected time points, yielding time-dependent return levels that reflect evolving seasonal processes and dependence structures.

Stationary and non-stationary marginal models are compared using the likelihood ratio (LR) test within a nested modelling framework, where M1 denotes the simpler stationary model and M2 the more complex non-stationary model . The LR deviance statistic is defined as 14D=2ℓ(M2)-ℓ(M1), where ℓ(M1) and ℓ(M2) denote the maximized log-likelihoods of the stationary and non-stationary models, respectively.

Model selection follows a parsimony principle, whereby a non-stationary model is adopted only when the LR test indicates a statistically significant improvement over the stationary formulation.

The first step in applying the NSLFA_mixC framework is to fit the WEI3 distribution to the low-flow time series. The use of the WEI3 distribution is theoretically motivated by the Fisher-Tippett-Gnedenko theorem, as the Weibull distribution represents a limiting distribution for lower extremes with a finite lower bound and is therefore well established in low-flow frequency analysis. Nevertheless, departures from the fitted distribution may occur at individual catchments. Such departures are interpreted not only as a distributional issue, but also as a diagnostic indication of sample properties, particularly process heterogeneity between summer and winter low-flow generation. Alternative low-flow distributions may also provide suitable representations at individual catchments. However, the present study retains WEI3 because of its theoretical basis for lower extremes and its consistency with the established LFA_mixC framework.

In this study, WEI3 with ML estimation converged to feasible parameter estimates in 76.1 % of catchments, indicating that the procedure is suitable for modelling low-flow behaviour in most cases. The remaining optimisation failures are likely related to the constrained parameter space, which may yield infeasible parameter combinations. The unsuccessful fits were distributed throughout the study domain rather than being concentrated in a single geographical region, while the successfully fitted catchments retained broad spatial coverage across Europe. Unsuccessfully fitted catchments tended to be smaller and generally had lower low-flow values relative to catchment area in the annual, summer, and winter series, with less year-to-year variability. Mean catchment elevation was similar between the two groups. The unsuccessful catchments also showed consistently higher positive L-skewness, indicating differences in the distributional shape of their low-flow series. These characteristics may contribute to the greater difficulty in estimating the WEI3 parameters within the constrained parameter space. Similar convergence issues have been reported in earlier low-flow studies , suggesting that they arise from the estimation procedure rather than a general limitation of the WEI3 distribution itself. We also provide the corresponding spatial comparison and detailed catchment, low-flow, and distributional characteristics in Appendix .

After fitting the stationary and non-stationary WEI3 distributions to annual and seasonal low-flow data, the evidence supporting the more complex non-stationary formulation must be evaluated for each catchment to avoid overfitting and increased estimation bias. Based on the likelihood ratio test with 5 % significance level, comparing stationary and non-stationary WEI3 models, 84.5 % of summer catchments and 91.3 % of winter catchments favour the simpler stationary model, whereas 15.5 % and 8.7 %, respectively, show a statistically significant improvement when non-stationarity is introduced.

When combining the results of the likelihood ratio test and the Mann-Kendall trend test, the proportion of catchments classified as non-stationary in both cases decreases slightly to 12.2 % in summer and 7.7 % in winter, while the stationary model remains preferred in 87.8 % and 92.3 % of catchments, respectively. The reduction is expected, as the combined criterion is more restrictive and requires both improved statistical fit and evidence of temporal change. In addition, the Type-I error of statistical tests has to be considered. The results indicate that stationary formulations remain adequate for most catchments, while a non-negligible subset exhibits evidence of temporal change that cannot be ignored and justifies the more complex non-stationary representation, with a clearer signal in summer than in winter, which is consistent with the trend analysis presented in Sect. .

The spatial distribution of the model selection (Fig. d) further shows that non-stationarity is spatially structured rather than randomly distributed across the study domain. Catchments classified as non-stationary in summer-only are more frequent and occur mainly in central and eastern Europe, whereas winter-only non-stationary catchments are less common and more scattered. Catchments requiring non-stationary modelling in both seasons are concentrated particularly in northern Europe and other snow-influenced regions, suggesting that temporal change there affects both seasonal components simultaneously. By contrast, stationary models dominate across much of western and central Europe, indicating that for most catchments the simpler stationary model remains sufficient.

The observed spatial structure therefore suggests that the detected non-stationarity may be linked to differences in low- flow generating processes across regions and catchment types. Although the ROBIN Level-1 network minimises direct human influence, low-flow regimes are strongly controlled by catchment storage and release processes, including soil and aquifer characteristics, recharge, evapotranspiration, vegetation, topography, and climate (Smakhtin, 2001). Therefore, residual effects from small abstractions, groundwater interactions, land- use change, local water management, reservoirs, or flow regulation cannot be fully excluded and may contribute to temporal changes in some catchments.

Overall, these results suggest that the proposed non-stationary framework is particularly relevant for catchments with evident non-stationarity, even though it is not required for all catchments. Its value lies not in universally replacing stationary models, but in identifying and representing catchments where seasonal low-flow regimes exhibit temporal change. In this case, the stationarity assumption may be violated and should not be ignored in the frequency analysis. To keep the framework parsimonious, non-stationarity was introduced as a linear trend in the location parameter and was evaluated formally by comparing the non-stationary and stationary models using a likelihood-ratio test. The linear formulation was chosen as a simple and parsimonious way to represent gradual temporal change. Given the limited record length, more flexible non-linear formulations or time-varying scale and shape parameters would require stronger empirical support and could substantially increase estimation uncertainty and the risk of overfitting. More complex temporal dynamics, including regime shifts, accelerations, thresholds, or oscillatory behaviour linked to climatic variability, may also occur. Nonlinear formulations, change-point approaches, or climatic covariates could therefore be considered where sufficiently long records and supporting evidence are available. Therefore, NSLFA_mixC provides an extension of the stationary framework, preserving its conceptual basis while improving the representation of evolving low-flow conditions where non-stationarity is evident. Given that the stationary model remains a special case of the more complex non-stationary model, the non-stationary model provides a robust framework that can cover both sides.

Finally, intermittent-flow conditions relevant to the applicability of the WEI3-based framework were identified by comparing the ROBIN catchments used in this study with the intermittent-flow classifications of . There are 3.2 % of catchments that were classified as intermittent and were retained in the analysis. Because the WEI3 distribution is continuous and cannot explicitly represent a probability mass at zero, catchments with repeated zero-flow occurrences would require a modelling framework that accounts for zero flows, for example by treating zeros as censored observations within a mixed discrete-continuous L-moment framework . However, explicitly accounting for these data conditions is beyond the scope of the current study..

To illustrate the practical application of the NSLFA_mixC framework, the catchment Stauf Vöckla at Traun in Upper Austria is used as a representative example. The catchment covers an area of 99.7 km² and has a mean elevation of 503 m a.s.l. Figure a shows the time series of observed summer and winter minimum flow. Trend analysis using BBMSK identified a significant decreasing trend in summer flow, whereas no significant trend was detected in winter. The catchment is further characterised by moderate seasonal dependence (Kendall's τ=0.48) and a nearly balanced mixture rate (0.52), indicating a mixed seasonal low-flow regime. Stationary and non-stationary WEI3 distributions were then fitted separately to the annual series, used here as a reference model, and to the seasonal series. Based on the likelihood ratio and trend test, the non-stationary model was preferred for summer, whereas the stationary model was not rejected for winter. Notably, this seasonal contrast is already visible in Fig. a, where temporal change is more evident in summer than in winter.

Under stationary conditions, five low-flow frequency models were fitted using WEI3 distribution and Gumbel-Hougaard copula following the framework of : the annual, summer, winter, mixture, and mixture-copula models. Figure b shows the resulting stationary probability plot across different return periods (T). The practical interpretation focuses mainly on the 100-year return period, which can be used as a stress-test or worst-case scenario in water management and infrastructure planning . In this study, the 100-year return period is used to compare the sensitivity of low-flow estimates to seasonality, dependence, and non-stationarity, while higher return periods are shown only for illustrative comparison of lower-tail behaviour.

For all models, estimated minimum flows decrease with increasing return period. Importantly, the annual curve lies above the mixture-based estimators, indicating that explicit representation of seasonal structure yields lower, and therefore more extreme, design low-flow estimates. The mixture and mixture-copula curves are very close, consistent with the catchment's moderate inter-seasonal dependence. Their differences are more visible at short return periods (T<10 years), which are less relevant for design purposes. In this respect, this behaviour is consistent with showing that under moderate seasonal dependence, the copula-based correction has only a limited influence, whereas larger differences would be expected for increased seasonal dependence between summer and winter low-flow components.

When non-stationarity is explicitly taken into account, the seasonal WEI3 location parameter is allowed to vary over time, while the scale and shape parameters are kept constant. For the copula-based formulation, the dependence parameter is also time-varying (Sect.). The effect of this formulation is illustrated by the non-stationary probability plot in Fig. c. Because non-stationarity was detected in summer, but not in winter, the non-stationarity comparison is shown only for the annual, summer, mixture, and mixture-copula models. The probabilities are evaluated at two reference points: the midpoint in 1998 (dashed lines) and the end of the observation period in 2020 (solid lines) to capture more recent conditions. The separation between the dashed and solid lines indicates a downward shift in estimated low-flow magnitudes over time across all models. Notably, summer low flows were lower in 2020 than in 1998, confirming a shift in the summer low flow regime. At the same time, the mixture-copula model accounting for time-varying dependence yields similar behaviour to the stationary models, where the difference between the mixture and the copula-based mixture remains small. Overall, this Fig. c indicates that the temporal behaviour of the low-flow regime cannot be fully described under a stationary assumption.

This temporal change becomes even clearer when examining the evolution of extreme low-flow Q100 over the full observation period. Figure d shows that all non-stationary Q100 estimates decline over time, indicating a general intensification of low-flow conditions. At the beginning of the record, the summer curve lies above the winter curve, implying that winter low flows are initially more severe than summer low flows. Over time, however, the two seasonal curves then converge around the middle of the period, and towards the end of the period, the summer curve falls below the winter curve. This clearly shows a gradual shift from winter dominance to summer dominance. The annual, mixture, and mixture-copula estimates closely track this transition and remain very similar throughout, suggesting that for this catchment the effect of seasonal dependence on the long-term evolution of Q100 is relatively small compared with the overall temporal decline. Comparing return periods from the NSLFA_mixC (Sect. ) between the start of the observation period in 1976 and its end in 2020 suggests a substantial increase in the frequency of extreme low flows within the observed record. In particular, an event corresponding to a 100-year return period in 1976 is associated with a return period of about 43 years by 2020, indicating that such an event became approximately on average 2.3 times as frequent according to the fitted non-stationary model. This comparison is a statistical diagnostic of change within the historical observation period and should not be interpreted as a projection of future event frequency.

To further illustrate the framework under contrasting hydro-climatic conditions, we analysed the catchment Oksfjordvatn in Norway as a second example. Unlike Stauf Vöckla, which represents a mixed seasonal regime (mixture rate = 0.52), Oksfjordvatn is winter-dominated (mixture rate = 0.40) and exhibits weaker inter-seasonal dependence (Kendall's τ=0.26, compared with 0.48 at Stauf Vöckla). The temporal behaviour also differs between the two catchments. At Stauf Vöckla, non-stationarity is mainly associated with declining summer low flows, whereas at Oksfjordvatn both summer and winter components were identified as non-stationary and low-flow magnitudes increase over time. Consequently, the non-stationary Q100 estimates decrease through time at Stauf Vöckla, indicating increasingly severe low-flow conditions, but increase at Oksfjordvatn, indicating less severe low-flow conditions toward the end of the observation period. The corresponding four-panel analysis for Oksfjordvatn is provided in Appendix .

Overall, both examples demonstrate how the NSLFA_mixC framework captures seasonal contrasts, temporal shifts, and dependence effects within a single coherent analysis. It also illustrates that non-stationarity may alter not only the magnitude of design low flows, but also the relative importance of summer and winter contributions to the annual low-flow regime.