Document Type : Research Paper

Authors

• Aboalhasan Fathabadi ¹
• Seyed Morteza Seydian ¹
• Hossein Emami ²

¹ Associate Professor, Rangeland and Watershed Management Department, Faculty of Agriculture and Natural Resources, Gonbad Kavous University, Gonbad Kavous, Iran

² M.Sc., Department of Reclamation of Arid and Mountainous Regions, Faculty of Natural Resources, College of Agriculture and Natural Resources, University of Tehran, Karaj, Iran

Abstract

Introduction
Any modeling and decision-making process in watershed management fundamentally depends on accurate discharge data at the basin outlet. However, direct measurement of discharge is both costly and time-consuming. Therefore, at hydrometric stations, water stage is routinely recorded, and discharge is estimated using the rating-curve equation corresponding to the observed stage. Nevertheless, several sources of uncertainty, including errors in measuring flow velocity, cross-sectional area, and stage height, as well as model limitations in estimating extreme flows and temporal variations in channel morphology caused by erosion, sediment deposition, and vegetation growth, result in inaccuracies in rating-curve-based discharge estimations. Such uncertainties propagate through hydrological model outputs and management decisions, potentially leading to considerable economic and environmental losses. Consequently, it is essential to quantify the uncertainty bounds of rating-curve-derived discharge estimates in order to mitigate risks associated with measurement and estimation errors. To date, both classical and Bayesian approaches have been employed for uncertainty estimation in rating curves. The Bayesian framework, in particular, offers significant advantages: in addition to incorporating observational data through the likelihood function, it allows prior hydraulic knowledge of the station to be embedded in the model through prior distributions. With recent advances in computational power and the widespread application of Markov Chain Monte Carlo (MCMC) sampling techniques, Bayesian methods have become a powerful and flexible tool for rating-curve uncertainty estimation, and various model structures have been introduced in recent years. Accordingly, the present study estimates rating-curve uncertainty using a Bayesian approach for three hydrometric stations located in Golestan Province.
 
Material and methods
The study area consists of three hydrometric stations: two mountainous stations, Galikesh and Lazoreh, characterized by coarse, gravel-bed channels, and one lowland station, Araz Kooseh, with a fine-grained sedimentary bed. After preliminary evaluation, stage–discharge data corresponding to the 2022–2023 hydrological year were utilized for analysis. Following the development of the rating curve for each station, uncertainty estimation was performed using the GLUE method and three Bayesian approaches: Python Package Ratingcurve (PPR), Generalized Power Law Model (GPLM), and the Norwegian Water Resources and Energy Directorate method (NVE). For the GLUE and GPLM approaches, only the single-segment model structure was applied. In contrast, for the PPR and NVE methods, one-, two-, and three-segment structures were evaluated, and the optimal structure was selected based on statistical performance criteria. Subsequently, uncertainty bounds of the rating curve were estimated accordingly. To assess and compare the performance of the applied methods, the uncertainty ranges were quantified using Mean Prediction Interval (MPI) and Prediction Interval Coverage Probability (PICP) metrics, accompanied by graphical visualization of the estimated uncertainty interval.
 
Result and discussion
The findings revealed that within the GPLM framework, the simplest structural configuration consistently demonstrated superior performance across all three hydrometric stations. Furthermore, at the Lazoreh and Araz Kooseh stations, the variability of uncertainty intervals exhibited a non-uniform pattern. In several cases, the uncertainty width estimated for lower stage values exceeded that of higher stages, indicating sensitivity of the model to flow regime transitions. Additionally, at stage levels for which direct discharge measurements were available, considerably narrower uncertainty bounds were obtained, highlighting the constraining effect of empirical observations on posterior uncertainty. Application of the NVE method showed that the single-segment model exhibited the best performance at the Galikesh and Lazoreh stations, while the two-segment structure provided a more adequate representation of rating-curve behavior at the Araz Kooseh station. For all three stations, however, this method yielded extremely narrow uncertainty bands at lower stage values, resulting in the exclusion of a substantial portion of observed stage–discharge pairs from the estimated uncertainty domain. This systematic deviation suggests a consistent under-representation of uncertainty in low-flow conditions by the NVE approach. In contrast, the PPR method identified the two-segment rating curve as the most appropriate structural formulation for all stations examined. Under this optimal configuration, uncertainty estimates were initially minimal at low stage values but expanded markedly with increasing stage. The observed inflation in uncertainty intervals at higher flows is likely attributable to sparse high-discharge measurements and increased hydraulic variability under such conditions. Overall comparative analysis using MPI and PICP performance indicators demonstrated that the GLUE method exhibited the most robust and reliable behavior among all models evaluated. Among Bayesian approaches, the PPR methodology outperformed both GPLM and NVE, yielding superior uncertainty characterization and predictive coverage.
 
Conclusion
In the present study, rating-curve uncertainty was evaluated using the GLUE method in conjunction with three Bayesian approaches—GPLM, PPR, and NVE. The findings revealed that all Bayesian frameworks yielded notably large uncertainty intervals. This outcome can plausibly be attributed to the reliance on weakly informative priors, stemming from limited hydrometric knowledge and the absence of river-specific hydraulic characteristics within the prior structure. Given the scarcity of empirical datasets and the lack of comprehensive, nationwide analyses of stage–discharge relationships in Iranian rivers, the formulation of richer and more informative prior distributions was not practically achievable in this investigation. This limitation underscores a critical research gap, indicating the necessity for extended field measurements, systematic data collection, and the development of regional hydraulic databases. Such advancements would provide the basis for constructing hydrologically informed priors and consequently enhance the reliability and precision of Bayesian uncertainty quantification in future studies.

Keywords

• GPLM
• NVE
• River cross-section
• Stage–discharge
• Uncertainty