The study, which analysed streamflow records from 721 catchments in the Reference Observatory of Basins for International hydrological climate change detection (ROBIN) dataset, harmonised to a common analysis period from 1976 to 2020, addresses a long-standing gap. Traditional low-flow frequency analysis assumes that observations are independent and identically distributed, meaning the underlying statistics remain constant over time, the generating processes are homogeneous, and serial dependence is absent. Under climate change, and in seasonal climates where annual low flows may arise from fundamentally different summer and winter processes, these assumptions are increasingly violated.

Summer and winter low flows are generated by distinct mechanisms. Summer minima are typically driven by precipitation deficits and high evaporative demand, while winter minima in snow-influenced catchments are often associated with snow accumulation, freezing conditions, or delayed melt. The annual minimum is therefore a mixture of seasonal processes, and droughts may even extend across seasons, introducing dependence between the summer and winter components. Earlier work by the same research group had addressed this heterogeneity with a mixed distribution approach and, later, a mixed copula framework that models seasonal dependence explicitly, but both were limited to stationary conditions.

The new framework, called NSLFAmixC, extends these approaches by allowing the location parameter of three-parameter Weibull distributions fitted to seasonal low-flow series to vary linearly with time, while keeping the scale and shape parameters constant. The dependence between seasons, modelled with a Gumbel-Hougaard copula, is also allowed to change over time through a time-variant dependence parameter. The authors chose this parsimonious formulation deliberately: allowing additional parameters to vary without strong empirical support would substantially increase estimation uncertainty, and scaling analysis indicates that the moments of streamflow series are so tightly related that independent temporal variation in scale and shape would be structurally inconsistent.

The data reveal a strongly seasonal picture. Summer low-flow regimes dominate the study domain, accounting for 82.5 percent of catchments and concentrated in western, central, and southern Europe, while winter regimes account for 14.4 percent and occur mainly in northern Europe and mountainous regions. Seasonal dependence, assessed with Kendall’s rank correlation, is common but generally weak to moderate: significant low to moderate correlations characterise 41.5 percent and 25.9 percent of catchments respectively, while strong correlations are rare at 0.1 percent. Trend patterns differ sharply by season. Significant trends are more common in summer than in winter, occurring in 36.1 percent and 18.1 percent of catchments respectively, and in summer, decreasing trends dominate, accounting for 73.8 percent of significant cases and concentrating in western, central, and southern Europe.

Not every catchment demands the more complex model. Using a likelihood ratio test at the 5 percent significance level, 84.5 percent of summer catchments and 91.3 percent of winter catchments favoured the simpler stationary model, and combining the test with a Mann-Kendall trend analysis reduced the share classified as non-stationary to 12.2 percent in summer and 7.7 percent in winter. Overall, non-stationary behaviour was evident in approximately 10 percent of the near-natural catchments studied. Crucially, the affected catchments were not randomly scattered: summer-only non-stationarity clustered in central and eastern Europe, while catchments requiring non-stationary modelling in both seasons concentrated in northern Europe and other snow-influenced regions, suggesting that temporal change there affects both seasonal components simultaneously.

The practical consequences of ignoring this change are quantified in the study’s model comparisons. In catchments with evident non-stationarity, comparing return-period estimates between the beginning of the observation period in 1976 and its end in 2020 revealed median relative absolute deviations of 41.73 percent for the non-stationary mixture-copula estimator and 41.99 percent for the non-stationary mixture estimator at a 100-year return period, with deviations of roughly 20 to 26 percent even at a 10-year return period. When a fixed stationary estimate was contrasted with the non-stationary estimate at the end of the record, the median deviation at the 100-year level was 29.42 percent, meaning that in catchments where non-stationarity is evident, stationary estimates may differ by around 30 percent or more from non-stationary estimates for rare events. Deviations were systematically smaller across the full sample of catchments, dropping to about 25.2 percent at the 100-year level, because that sample includes catchments where non-stationarity is weak or absent.

A detailed case study illustrates what these numbers mean on the ground. At the Stauf Vöckla catchment in Upper Austria, which covers 99.7 square kilometres and has a mixed seasonal regime with moderate dependence between seasons, trend analysis identified a significant decreasing trend in summer flow but no significant trend in winter. All non-stationary estimates of the 100-year low-flow quantile declined over the observation period, and the seasonal curves crossed over time, marking a gradual shift from winter dominance to summer dominance in the low-flow regime. According to the fitted non-stationary model, an event corresponding to a 100-year return period in 1976 was associated with a return period of about 43 years by 2020, meaning such an event became approximately 2.3 times as frequent on average. The authors are careful to stress that this comparison is a statistical diagnostic of change within the historical record and should not be interpreted as a projection of future event frequency. A contrasting example, the winter-dominated Oksfjordvatn catchment in Norway, showed the opposite pattern, with low-flow magnitudes increasing over time and conditions becoming less severe toward the end of the record.

One of the study’s more subtle findings concerns the hierarchy of model improvements. Once seasonal structure was already represented through the mixture framework, the largest additional change came from allowing the distribution parameters to vary with time rather than from modelling the dependence between seasonal events. The summary statistics for the mixture and mixture-copula estimators differed only slightly throughout the comparisons. The authors conclude that the largest gain comes from explicitly representing seasonal process heterogeneity, followed by temporal non-stationarity, while the added refinement from modelling inter-seasonal dependence is more modest, an ordering that echoes their earlier stationary analyses.

The framework comes with clear caveats. Non-stationarity is introduced only through the location parameter, so temporal change is represented mainly as a shift in the lower bound of the seasonal low-flow distribution, and variability may also change over time in ways the model cannot capture. The non-stationary model is used solely to describe historical change and is not intended to predict future trends, and robust estimation of temporal change generally requires sufficiently long data records, which remain scarce in many catchments. The authors also note that residual human influences, such as small abstractions, local water management, or flow regulation, cannot be fully excluded even in the near-natural ROBIN reference network and may contribute to temporal changes in some catchments. Future work, they suggest, could link the inferred changes to physical drivers such as temperature, precipitation deficits, and snow processes, extend the framework to multivariate settings such as combined low flow and high water temperature, and explore how non-stationary estimates can inform drought management and water-resources planning. For now, the message for practitioners is concrete: in the roughly one in ten catchments where temporal change is evident, the assumption of a constant climate can materially distort the assessment of how rare a given low-flow event really is.

Subject of Research: Non-stationary low-flow frequency analysis across 721 European catchments using mixed Weibull distributions and a copula-based dependence framework.

Article Title: Non-stationary low-flow frequency analysis with mixed Weibull components and Copula-based dependence framework

Article References: Fitriana, F. S., Fischer, S., Weigelhofer, G., Laimighofer, J., & Laaha, G. (2026). Non-stationary low-flow frequency analysis with mixed Weibull components and Copula-based dependence framework. Hydrology and Earth System Sciences, 30(20), 6599-6622. https://doi.org/10.5194/hess-30-6599-2026

Image Credits: AI Generated

DOI: 10.5194/hess-30-6599-2026

Keywords: low flows, non-stationarity, frequency analysis, Weibull distribution, copula, climate change, hydrology, ROBIN dataset, return periods, seasonal regimes, river drought, Europe