the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 13 May 2025
Technical Note: A double-Manning approach to compute robust rating curves and hydraulic geometries
Abstract. Rating curves describe river discharge as a function of water-surface elevation ("stage"). They are applied globally for stream monitoring, flood-hazard prediction, and water-resources assessment. Because most rating curves are empirical, they typically require years of data collection and are easily affected by changes in channel hydraulic geometry. Here we present a straightforward strategy based on Manning's classic equation to address both of these issues. This "double-Manning" approach employs Manning's equation for flow in and above the channel. Flow across the floodplain can follow either a Manning-inspired power-law relationship or, in the common case of a rectangular floodplain and valley-wall geometry, a second application of Manning's equation analogous to that applied within the channel. When applied to ample data from established stream gauges, we can solve for Manning's n for in-channel flow, channel-bank height, and two floodplain-flow variables. When applied to limited discharge data from a field campaign, additional constraints from the surveyed floodplain cross section permit a fit to the double-Manning formulation that matches ground truth. Using these double-Manning fits, we can dynamically adjust the rating curve to account for evolution in channel width, depth, and/or slope, as well as in channel and floodplain roughness. Such rating-curve flexibility, combined with a formulation based in flow mechanics, enables predictions during times of coupled hydrologic–geomorphic change. Open-source software with example implementations is available via GitHub, Zenodo, and PyPI.
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Status: final response (author comments only)
- RC1: 'Comment on egusphere-2025-1409', Anonymous Referee #1, 24 Jun 2025
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RC2: 'Comment on egusphere-2025-1409', Anonymous Referee #2, 25 Jun 2025
I was Anonymous Referee #1 for the previous submission of this paper (https://egusphere.copernicus.org/preprints/2024/egusphere-2023-3118/). The paper has been reworked so that the proposed DoubleManning approach (unchanged) is better positioned among other rating curve methods reported in the literature. Another reason for commending the authors is that the paper writing and presentation are excellent. Data and software codes are made available.
The rationale of the proposed method is meaningful. However I’m still sceptical about its novelty and practicality: the Divided Channel approach is classic (cf. the plentiful literature on compound channel flows), as well as segmented rating curves (eg. Rantz et al. 1982 for USGS procedures). As now stated in the article, Bayesian methods allow for the prior specification of the physical parameters of channel controls, similar to what is done here (arguably in a more natural and simple way), to constrain and compare the results. And they account for data uncertainty and provide result uncertainty, a big difference not really stressed in this paper…
In brief, in spite of the lack of novelty and practical usefulness, the offered method and tool are possibly worth a technical note, at least for the discussion of interesting ideas. I provide the following comments that may call for some improvements of the text and arguments.
The addition of two Manning equations does not account for head losses due to friction between main channel flow and floodplain flow. This effect is included in most 1D hydraulic models for instance. At least a discussion of this approximation and its possible consequences would be interesting.
It is unclear how (obvious) correlations in parameter estimates are handled: depending on the studied case, some parameters are fixed and others are let free to be calibrated. This appears to require some significant expertise from the user.
The 3 examples are useful but they also show that the use of the method is not straightforward, as it requires adapting the modelling strategy to the amount of information available in the data and in the prior knowledge on channel and floodplain. Summarizing a general workflow applicable to any situation would be useful, in the end of the paper.
A decisive advantage of modelling the channel and floodplain as separate terms in the rating curve equation (i.e. the divided channel approach, as opposed to modelling the whole river section as a single term) is that the calibration of the main channel parameters still holds for overbank flows, which reduces the discharge uncertainty even when few or no discharge measurement exist for overbank flows. I have not seen such an argument in the paper.
Citation: https://doi.org/10.5194/egusphere-2025-1409-RC2 - RC3: 'Comment on egusphere-2025-1409', Federico Gómez-Delgado, 29 Jul 2025
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Peer Review: Technical Note: A Double-Manning approach to compute robust rating curves and hydraulic geometries
1. Summary of the Paper
The authors introduce the Double-Manning methodology for developing rating curves (Q = k (zs−zb)P), which utilizes knowledge of the underlying physics of flow in open channels to minimize the need for ad-hoc parameters when regression models are used to fit observations.
The double-Manning approach is closely related to a suite of modern efforts aimed at developing more flexible, physically grounded rating curves. The authors aim to provide a middle ground between purely empirical fits and full hydrodynamic models.
The authors argue that, compared to other recent methods, their developments are innovative in coupling two Manning equations to reflect channel and floodplain contributions to flow – a concept simple in formulation yet powerful in practice. The concept of double manning emphasizes practical adaptability (via open-source implementation and easily interpretable parameters), whereas some other state-of-the-art methods emphasize comprehensive uncertainty quantification or hydrodynamic completeness. Each approach has its strengths: double-Manning excels in simplicity and physical interpretability, Bayesian methods in statistical rigor, and dynamic models in capturing transient behavior. The existence of these parallel developments underlines a converging theme in hydrology: the need for rating curve models that can handle non-standard conditions (evolving channels, limited data, unsteady flows) more robustly than the old static empiricism. In this context, Wickert et al.’s contribution stands out as a practically minded yet scientifically sound method that complements recent advances. It pushes the field toward rating curves that are mechanistically informed and update-ready, which is an important step for improving flood forecasting, stream monitoring, and water resources management under changing environmental conditions.
2. Relevance and Coverage of Citations
The authors of this technical note demonstrate a strong awareness of both the foundational and the latest literature in stage–discharge rating curve development and open-channel hydraulic modeling. They explicitly cite classical, seminal works such as Manning’s original formulation for flow resistance (Manning, 1891) and Leopold & Maddock’s landmark study on hydraulic geometry (1953). The paper also covers recent advances (within ~10 years) in rating curve methodology and uncertainty quantification. For example, it cites Kiang et al. (2018), a comprehensive comparison of streamflow uncertainty estimation methods (which includes modern rating curve techniques), as well as Hrafnkelsson et al. (2022), who introduced a generalized power-law rating curve using hydrodynamic theory and Bayesian hierarchical modeling. They also reference Le Coz et al. (2014), an influential study that combined hydraulic knowledge with uncertain gaugings in a Bayesian framework (the “BaRatin” method). The authors even refer to Quintero et al. (2021), which describes “synthetic rating curves” generated via hydrologic/hydraulic models for stage-only gauges, illustrating that they have surveyed contemporary innovations in establishing rating curves when direct measurements are limited.
There do not appear to be obvious omissions of critical recent work.
3. Originality and Publication History
This article is an original contribution. We find no evidence that the core ideas or results have been previously published in any journal or formal conference proceedings by these same authors. The methodology appears to be an original synthesis rather than a repackaging of the authors’ earlier works.
4. Comparison to Recent Methods and Tools
The double-Manning approach enters a landscape of active research on improving rating curves, and it shares goals with several recent methods and tools.
The most closely related developments from the last decade include:
Bayesian/Physical Hybrid Rating Curves (e.g. BaRatin and RUHM): Compared to these, the double-Manning approach is less computationally intensive and forgoes an explicit Bayesian treatment of uncertainty in its current form. Its innovation lies in using two applications of Manning’s equation (for channel and floodplain zones) as a constrained form of a piecewise rating curve, rather than relying on generic power-law segments or full hydrodynamic simulations. However, it currently does not inherently provide probabilistic uncertainty estimates as RUHM or BaRatin do. The trade-off is between ease-of-use and statistical rigor: double-Manning favors a straightforward, deterministic calibration with physically plausible parameters, while methods like RUHM prioritize a full accounting of uncertainties and leverage advanced computation (MCMC or other Bayesian algorithms) to fuse models and data.
Generalized Power-Law and Theoretical Extensions: A notable recent contribution is Hrafnkelsson et al. (2022), who generalized the traditional rating curve by deriving the power-law exponent and coefficient from hydrodynamic considerations and fitting a Bayesian hierarchical model. Their approach maintains the familiar power-law form but links parameters to physical quantities (like channel shape and flow regimes) and pools information across sites via a hierarchical Bayesian structure. The double-Manning approach shares a similar spirit of physically-informed modeling but implements it more directly: instead of modifying the power-law exponent abstractly, it literally employs Manning’s equation in two flow domains. This makes double-Manning somewhat more prescriptive – it assumes a rectangular channel cross-section and, optionally, a rectangular floodplain – whereas Hrafnkelsson’s framework is more flexible in form (adapting the power-law curve shape through theory). In terms of innovation, double-Manning’s two-tier Manning equation is a fresh idea that effectively creates a compound rating curve without an arbitrary breakpoint; its method of using one Manning relation for in-bank flows and another (or a Manning-like power law) for overbank flows is an innovative yet intuitive extension of classical uniform flow theory.
Dynamic and Non-Stationary Rating Methods: Another related thread is the development of rating curve methods that account for non-stationary conditions and flow dynamics (beyond the static stage–discharge assumption). For instance, researchers at the USGS have devised a “dynamic rating” approach to capture hysteresis effects during unsteady flows. Domanski et al. (2022) introduced DYNMOD and DYNPOUND, simplified hydrodynamic models derived from the Saint-Venant equations that can compute discharge from stage while accounting for changing energy slope and storage in the channel/floodplain (hysteresis). These methods effectively produce time-varying rating relationships that adjust during a flood wave, which a single static curve cannot do. The double-Manning method is complementary to such approaches: it addresses spatial complexity (channel vs floodplain flow regimes) and long-term morphological changes, rather than short-term unsteady flow dynamics. Double-Manning assumes quasi-steady uniform flow for given stages, so it will not capture hysteresis loops during events (as DYNMOD/DYNPOUND do).
5. Strengths and Limitations
Strengths of the paper: The proposed methodology would reduce the need for the multiple measurements required in a purely empirical fit of a rating curve. However, this only seems to be the case when all the hypotheses of the double-manning methodology hold, and the authors do not present evidence that this situation is the most common in cross-sections with rating curves around the world.
Weaknesses and limitations: The title suggests a level of generality of the application that is not supported by the results and analyses. The title could more explicitly reflect the methodological context and applicable site conditions—specifically, that it is intended for locations with available stage -discharge measurements and supporting field data. Additionally, emphasizing that the approach is a hybrid hydraulic–empirical model for rating curve fitting would enhance clarity and precision.
The evidence that the methodology of double manning rating curves works is very minimal, and there isn’t a formal comparison of errors with existing methodologies.
The paper does not provide direct evidence of the methodology’s accuracy, as it is applied to two sites with substantial stage–discharge measurements but without quantitative comparison to a reference or “true” rating curve—such as that provided by the USGS. In the third case, where only a few measurements are available, it is not possible to verify the accuracy of the resulting fit, particularly in the floodplain region where no observational data are available.
We cannot find proof that there is an “economy” of data using this approach. We would have expected that the authors would show that a minimum set of observations is needed to obtain the same or less error than a traditional fit of the data.
6. Figures
All the figures should be improved.
Figure 1:
Figures 2, 3, and 5:
Figure 4:
6. Specific Recommendations
Line 11: While the abstract notes that the method “matches ground truth” and “enables predictions,” it does not summarize any specific performance metrics or case study outcomes. Include a brief reference to a specific result or performance.
Line 97: The variability of the coefficients kfp and Pfp, which are influenced by changes in floodplain width and roughness, is not fully addressed in the paper. In real-world settings, floodplains are often heterogeneous and exhibit substantial spatial variability along the river reach. However, the method appears to assume spatial homogeneity within the floodplain zone, which may oversimplify real-world conditions. In practice, floodplain heterogeneity introduces uncertainty that could affect the accuracy of the fitted rating curve, particularly in the overbank flow regime. How does the proposed method account for this heterogeneity, and how is the resulting uncertainty represented in the fitted rating curve? Given that these parameters directly influence the overbank component of the discharge, a discussion of how spatial variations and their associated uncertainties affect the reliability of Qfp would strengthen the analysis.
Line 125: Section 3 (Data Constraints): The text mixing parameter estimation difficulty, data source types, and model sensitivity can be dense. Reformat Table 1 to include a column for “Parameter Sensitivity” (if known), and break Section 3 into clearer subsections for: Measurable parameters (e.g., b, S, hβ) - Estimated parameters (e.g., kfp, Pfp) -Data-sparse strategies
Line 247: To better demonstrate the applicability of the methodology to data-limited sites, it would be helpful to conduct a set of controlled experiments at a single site using progressively reduced subsets of data. For example, the authors could evaluate model performance using only channel-stage measurements (excluding overbank flow), then with a few measurements spanning both channel and floodplain stages and compare the resulting rating curves to the full dataset fit. Each case could also be compared against a reference curve (e.g., the USGS rating curve) to assess the sensitivity and robustness of the approach under constrained data conditions (calculate some metrics). This would provide valuable insights into the model’s behavior and reliability when applied to real-world scenarios with sparse (or none) observations.
Line 369: Replace “nch = 0.38, which is virtually identical to the nch = 0.37” by “nch = 0.038, which is virtually identical to the nch = 0.0
Line 372: It could be valuable to include a fourth case study where the channel geometry deviates from the rectangular assumption—for example, a compound channel. This would allow the authors to explore the applicability and limitations of the double-Manning methodology under more complex geometric conditions, which are common in natural river systems. Such an example would also help assess the method’s flexibility and the potential need for adjustments when applied to non-idealized cross sections. Including this type of case would further strengthen the practical relevance of the approach.
Line 442: The authors should consider expanding the conclusions section, which currently consists of a single paragraph. In addition to summarizing the strengths of the double-Manning approach, the conclusions should also acknowledge the method’s limitations. For example, potential sources of uncertainty—such as assumptions of floodplain homogeneity, sensitivity to field-estimated parameters, and the challenges of validating results in data-sparse settings—deserve mention. Including both the advantages and constraints would provide a more balanced and complete perspective and help guide future applications and developments of the method.
7. Recommendations
Reject