A double-Manning approach to compute robust rating curves and hydraulic geometries

Wickert, Andrew D.; Jones, Jabari C.; Ng, Gene-Hua Crystal

doi:https://doi.org/10.5194/egusphere-2023-3118

Preprints

https://doi.org/10.5194/egusphere-2023-3118

Preprints

08 Jan 2024

| 08 Jan 2024

A double-Manning approach to compute robust rating curves and hydraulic geometries

Andrew D. Wickert, Jabari C. Jones, and Gene-Hua Crystal Ng

Abstract. Rating curves describe river discharge as a function of water-surface elevation ("stage"), and 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 equation to address both of these issues. This "double-Manning" approach employs Manning's equation for flow in and above the channel and a Manning-inspired power-law relationship for flows across the floodplain. When applied to data from established stream gauges, we can solve for Manning's n, channel-bank height, and two floodplain-flow parameters. When applied to limited data from a field campaign, constraints from Manning's equation and the surveyed cross section permit a robust fit that matches ground truth. Using these double-Manning fits, we can dynamically adjust the rating curve to account for channel width, depth, and/or slope evolution. Such rating-curve flexibility, combined with a formulation based in flow mechanics, enables predictions amidst coupled hydrologic – geomorphic change, which increasingly occurs as climate warms and humans modify the land surface and subsurface. Open-source software with example implementations is available via GitHub, Zenodo, and PyPI.

Received: 22 Dec 2023 – Discussion started: 08 Jan 2024

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Andrew D. Wickert, Jabari C. Jones, and Gene-Hua Crystal Ng

Status: closed

RC1:
'Comment on egusphere-2023-3118', Anonymous Referee #1, 21 Feb 2024

The main purpose of this research paper is to introduce a novel approach, called double-Manning approach, for computing stage-discharge relations (rating curves) in compound channels, i.e. at river sites with a main channel and a floodplain. The method, based on the addition of two simplified Manning equations, is illustrated through three gaging stations with variable density of data, size and geometry.
Sadly, I have to say that I don’t think that this approach is really a valuable innovation compared to already published rating curve methods, an active field of research in the past decade (cf. the collective paper by Kiang et al. (2018) which compares 7 methods for rating curve determination and uncertainty analysis). Kiang’s paper and the main references they cite should be considered by the Authors. The third site included in their comparison (the Taf River at Clog-y-Fran, Wales, United Kingdom) actually has a compound channel, which several of the compared methods (the 3 Bayesian methods NVE, BaRatin, BayBi, at least) represent using an addition of power-law equations (one for each channel) with explicit parameters derived from the Manning equation. The wide-channel approximation is necessary to get a relation in the form of Q=a(h-h0)^b for each channel equation, as also mentioned in this article, however it does not make a large difference for most natural rivers.
Typically, the SPD model proposed by Manasanarez et al. (2019b) and cited in the article is based on such principles, which the Authors seem to ignore. It allows the management of shifts, flexible rating curves and hydrogeomorphic feedbacks, unlike what is claimed in Sections 6.3 and 6.4. Bayesian methods do provide additional information (through the prior knowledge) unlike what is stated page 2 line 55 and elsewhere throughout the paper. The parameters of the rating curve equation can be interpreted hydraulically and can be assumed to vary over time (including the channel width, roughness, etc.). Also, many hydrometric sites may not be represented by just a combination of two Manning equations, especially when natural or artificial section controls are active (cf. the Mahurangi site in Kiang’s paper). Then, the proposed model will not suit, whereas other software accommodate much more diverse rating curve models.
Therefore, rating curve methods based on the same (or very similar) ‘double-Manning’ approach already exist, especially Bayesian methods. They offer the same opportunity of hydraulic and geomorphic interpretation of the rating curve parameters, through a stochastic framework that accounts for the uncertainty of the stage-discharge data, of the rating curve parameters and of the discharge outputs (cf. eg. Petersen-Overleir and Reitan 2005, Le Coz et al. 2014). This is a huge advantage over the deterministic optimization procedure introduced in this paper, since hydrometric uncertainty quantification is recognized as absolutely necessary (McMillan et al., 2017). The NVE, BaRatin and SPD software are open-source and publicly available.
Therefore, I don’t clearly see what is the added value of the proposed method.
I quickly reproduced the rating curve analyses using the open-source software BaRatinAGE 2.2 (https://github.com/BaRatin-tools/BaRatinAGE), using a ‘double-Manning’ approach and similar prior information as in the paper, and assuming 10% uncertainty for all the discharge measurements. See the obtained rating curves with uncertainty results in the attached supplement. There seems to be some cleaning needed in the datasets, and there are reproducibility issues: there are several data files (.tsv) for the Cannon case. It is unclear which is used in the article. The Minnesota .tsv files is in ft and cfs instead of the SI units of the paper. The stage and discharge measurements at LaDormida look rather uncertain according to the comments.
References
Kiang, J. E., Gazoorian, C., McMillan, H., Coxon, G., Le Coz, J., Westerberg, I. K., et al. (2018). A comparison of methods for streamflow uncertainty estimation. Water Resources Research, 54, 7149–7176. https://doi.org/10.1029/2018WR022708
Le Coz, J., Renard, B., Bonnifait, L., Branger, F., & Le Boursicaud, R. (2014). Combining hydraulic knowledge and uncertain gaugings in the estimation of hydrometric rating curves: A Bayesian approach. Journal of Hydrology, 509, 573–587. https://doi.org/10.1016/j.jhydrol.2013.11.016
McMillan, H., Seibert, J., Petersen-Overleir, A., Lang, M., White, P., Snelder, T., et al. (2017). How uncertainty analysis of streamflow data can reduce costs and promote robust decisions in water management applications. Water Resources Research, 53, 5220–5228. https://doi.org/10.1002/2016WR020328
Petersen-Overleir, A., & Reitan, T. (2005). Objective segmentation in compound rating curves. Journal of Hydrology, 311(1–4), 188–201. https://doi.org/10.1016/j.jhydrol.2005.01.016

Citation: https://doi.org/10.5194/egusphere-2023-3118-RC1
- AC1: 'Reply on RC1', Andy Wickert, 11 Jun 2024
  
  We would first like to thank the first referee for their constructive comments and inclusion of some useful references that we either missed or considered in a different light. These opened some new considerations and a desire to better define our goals in developing the double-Manning approach. We also appreciate that the referee took the time to reproduce some of our analyses using the BaRatinAGE software.
  I (Wickert, here and beyond) think that there is a fundamental difference in purpose between our aim here and those of the papers that the referee cites. Here, we seek to develop simple but physically based stage–discharge rating curves, especially in locations in which we are concerned about morphological change or in which data coverage is sparse. These are conditions that – to my experience – are not as commonly considered in hydrological research: Morphological change is most commonly often spoofed as a bed-elevation-only change (even though, geomorphically, channel width is much more variable). Although many hydrologists work with data-rich sites, there is still a significant need for better methods to constrain river discharge at low-accessibility and (therefore often) data-poor sites. These are places in which statistical methods (e.g., Bayesian) might help a bit, but in which the channel dynamics or data-set limitations are the key problems to overcome. I will respond with these ideas in mind.
  
  First, regarding the Kiang et al. (2018) study, I again thank the referee for sharing this. I had not seen it, and it indeed seems a very operationally useful guidance for different approaches. Its purpose, however, is not in the functional form of the equations, but rather in the statistical approaches. This makes it somewhat orthogonal to our goals here.
  The referee notes that the "addition of power-law equations" approach presented in a few methods by Kiang et al. (2018) is derived from Manning's equation. This seems both reasonable and likely. However, neither my reading nor a CTRL+F-aided search found any mention of the Manning equation. Furthermore, I could not find any listing of the parameters used in the fits – only of the goodnesses of the fits themselves. Therefore, I am unable to assess the referee's note that these parameters are explicitly derived from the Manning equation, even if the functional form is.
  This lack of direct confirmation that the best-fit parameters are reasonable in a Manning-equation framework highlights one of the key benefits of the double-Manning approach: with it, we are able to use ancillary data from channel and floodplain morphology to constrain the parameter sets. Fundamentally, the methods demonstrated in the Kiang et al. paper should be able to be applied as such, but there is no evidence that relevant constraints and/or priors were given. This said, the GesDyn method demonstrated there may be something that could be, in future work, combined with morphological data (including those from remote sensing) to further inform the time-evolving rating curve.
  
  Second, the referee notes that the Mansanarez et al. (2019b) study incorporates an equation with functional form $Q=a(h-h_0)^b$. I heartily agree with this, and Manasanarez et al. (2019b) indeed note this generic form in their paper. However, they take the equation in a different direction than we do for two reasons.
  First, they indicate that channel width is typically expected to be fixed and focus on variations in rating curves due to channel depth. To be fair, rating curves are often developed for engineered reaches of rivers with fixed width. However, this is far from universally the case, and both geomorphic theory and observation are that natural alluvial channels may vary in width as well as depth, with similar response times for both (e.g., Naito and Parker, 2020). Fortunately, when operating their model, they do permit both channel width and bed elevation to change.
  Second: they do not return to the question of parameter realism when compared to the Manning equation. This likewise makes them entirely dependent on paired stage–discharge data, which, as we point out in the submitted manuscript, are relatively difficult to obtain.
  This brings us to your note about our text, which I agree was somewhat unfair, "And although Bayesian methods may help to better understand sparse data sets (Mansanarez et al., 2019b), they do not provide fundamental new information." I should (and will) update this to note that, when the priors are data-informed, they do provide additional information. The Mansanarez et al. (2019b) study is a good example for including data-informed priors.
  Finally, the referee notes that some cross-sections might require more than two power-law curves to describe the rating curve, and holds up the Mahurangi River in New Zealand, as studied by Kiang et al. (2018), as an example. I have a few thoughts in response to this: (a) Yes, some rivers will have channel and/or valley geometries for which a double-Manning approach is not appropriate. Our goal in this paper is to craft an equation that jointly optimizes for simplicity and its ability to abstract and capture the dynamics of most rivers. (b) Including an arbitrary number of power-law equations can easily lead to an exercise in overfitting, which can rapidly lose straightforward comparability with diverse data sets and theory. I think that we all wish to avoid this. Indeed, the core of the double-Manning approach involves fixing or constraining these power-law parameters in order to limit the number of degrees of freedom in the fit. Therefore, for more complex channels and valleys, an approach invoking more realistic hydraulics along complex walls might be more apt (e.g., ray-isovel: Houjou et al., 1990; Kean & Smith, 2004). (c) Upon visual inspection of the stage–discharge data from the Mahurangi River (Fig. 6 in the paper by Kiang et al., 2018 – in case it does not reproduce below successfully or is not readable in its max-500-px size for inclusion here), I believe that a reasonable fit could be achieved using a two-power-law approach like the double-Manning approach here. I believe this because in the semilog plot there is a single roll-over separating straight segments. I apologize for not fully testing this idea: The authors do not provide the data used to generate this plot, and I have chosen for expediency and accuracy not to attempt to locate them, download what I believe to be the appropriate portion of their data, and (with some doubt) work to reproduce what they have done.
  
  Third, the referee notes that, "rating curve methods based on the same (or very similar) ‘double-Manning’ approach already exist, especially Bayesian methods". I would add critical nuance to this statement. Existing methods and associated software tools can be implemented with double-Manning constraints to reproduce the work that we have done here. This is absolutely excellent. However, to my knowledge, these methods were not built with constraints in place to acknowledge that the physical system is a river, or with straightforward guidance on how to take simple geomorphic and sedimentological measurements and use them to help guide parameter estimates.
  This incorporation of physically meaningful and simply obtained ancillary data constraints is the power of the double-Manning approach. The idea behind it is simple, but in these tests that we have performed, it is also powerful. The "doublemanning" software package need not – and ideally should not – be its only implementation. Therefore, in a revised manuscript, I will also include information on these software packages, with further thanks for you running these tests using BaRatinAGE.
  
  Before continuing on these thoughts regarding software implementations of the double-Manning approach, I will reply to the referee's notes towards implementation using BaRatinAGE. (a) It is true that there are multiple data subsets given in the "CannonWelch" data set. Due to rating-curve drift and a desire for transparency, I kept all of these. As described in the article and the "doublemanning" package documentation, the "config.yaml" file specifies that "CannonWelch05355200_RatingCurve_2002-2012.tsv" is used. This should therefore resolve the referee's stated concern that, "It is unclear which is used in the article.". (b) The referee also notes that, "The Minnesota .tsv files is in ft and cfs instead of the SI units of the paper." This is because these are the units of the source data. In this case, I chose to keep these units in order to implement the "doublemanning" package's unit conversions. (c) The referee also notes that "The stage and discharge measurements at LaDormida look rather uncertain according to the comments." Without knowing what "rather uncertain" means, I think I probably agree with the referee: these measurements were made infrequently and under harsh field conditions (rain, cold, and flows that were difficult to stand in while holding the velocimeter). This is far from an established survey at a gauging site with infrastructure. Yet, this is a common scenario, and one for which scientists may take great advantage of a more solidly mechanistically constrained approach (e.g., double-Manning) due to a relative shortage of paired in-situ stage–discharge measurements.
  
  Returning to the software implementations, I want to share two key thoughts to close. The first is that I plan to discuss the BaRatin method and how it can also (naturally) produce similar results when given similar source data as priors. I will also discuss the BaRatinAGE open-source implementation of BaRatin.
  Towards this second point, I did not know that BaRatinAGE was available as an open-source package, despite completing what I felt was a reasonable literature review. If I did, I may have used it instead of creating my own code! Looking at the GitHub repository (thanks for the URL), it has only two stars (three with mine) and "issues" that seem to be raised only by the dev team, indicating little external traffic or awareness. Furthermore, there are no instructions for compilation or running, though some information is found in "CONTRIBUTING.MD". These are significant barriers to discovery or use. "doublemanning", on the other hand, includes full documentation and straightforward installation via "pip": https://github.com/MNiMORPH/doublemanning. Therefore, I would argue that BaRatinAGE is not very accessible, even though the source code is open. If you would like to avoid a repeat of the present scenario, in which another scientist independently creates some code that you consider to not solve a problem as well as existing work has, I would strongly urge those involved with BaRatinAGE (including yourself or colleagues whom you can nudge, I hope?) to (a) document the code to at least a minimum level to permit straightforward usage, (b) publicize it, and (c) make sure that others know about what seems to me to be a just-published companion paper (Le Coz et al., 2024). I just found this paper while replying here – it was just published in April – and I am glad that I did.
  
  To sum up, I will:
  
  (a) change some of the language regarding Baysesian approaches;
  
  (b) better clarify the problem that we seek to solve (incorporating more diverse field data and more tightly constraining the system with more than just stage–discharge data) and that our ideas towards these solutions are, strictly speaking, agnostic from the code used to do so; and
  
  (c) reference one or more alternative (existing) methods to solve the same systems of equations in a Bayesian framework.
  
  I hope that this adequately represents the background and mission of our work, contextualizes the value that we seek to bring, and inspires better publicity and accessibility of existing tools.
  
  Citation: https://doi.org/10.5194/egusphere-2023-3118-AC1
CC1:
'Comment on egusphere-2023-3118', Richard Koehler, 26 Mar 2024

Authors have not completed a basic spell check or grammar check. Multiple misspelled words, grammar errors, and inconsistent phrasing detracts from this article. Article rating curve figures are inconsistent with USGS rating curves for same site. Authors have reversed the axes and have used linear, not log10, scaling. This makes it very difficult to compare figures from this article to official, quality controlled, USGS rating curves. This reviewer does not recommend this article for publication.

Citation: https://doi.org/10.5194/egusphere-2023-3118-CC1
- AC2: 'Reply on CC1', Andy Wickert, 11 Jun 2024
  
  Dear Dr. Koehler;
  It is unfortunate that you have found spelling and grammatical errors. All of the authors revised the article. I went back through it and found a couple of mistakes that will be fixed upon revision and included in the tracked-changes document associated with this. In the future, I would suggest giving targeted comments as to where improvements would be needed, especially in this case, in which wholesale changes seem unnecessary.
  The rating-curve figures in this article were not designed to match those from the US Geological Survey. Because we have published all of the data, replotting these for comparison is possible, if desired.
  
  Citation: https://doi.org/10.5194/egusphere-2023-3118-AC2
RC2:
'Comment on egusphere-2023-3118', Anonymous Referee #2, 16 Apr 2024

In this paper, the authors introduce a compound channel rating curve method/tool which they describe as a “double-Manning approach”. This approach relies on parameterization of two empirical flow equations (Qch and Qfp) to predict discharge at all depths, both channel-contained and at flood stage. While the tool can be used when discharge data are unavailable, the results presented in this manuscript at three test sites use an optimization routine (included in the tool) to select parameter sets that best fit this piecewise Manning’s power function.
While the approach using a piecewise power function to estimate compound channel flows is not necessarily novel, I see this tool as being a good additional resource for individuals that need a quick estimate of discharge, particularly in situations where flow data is limited. The tool could lend itself well to widespread use, however, end-users need a basic understanding of the CLI which may be prohibitive.
It does seem appropriate that the authors submit a manuscript that accompanies the software tool, but for this paper to be publishable, the authors may want to structure it in a way that 1) does a better job acknowledging the other piecewise power function regression tools described in the literature, and 2) improves on the method to account for error and uncertainty.

Citation: https://doi.org/10.5194/egusphere-2023-3118-RC2
- AC3: 'Reply on RC2', Andy Wickert, 11 Jun 2024
  
  We thank Referee 2 for their comments about our work. Following both Referee 2's comments here and those from Referee 1, we plan to incorporate more information and background on piecewise power-function approaches, including the references noted by Referee 1 and in our response to them.
  Regarding managing error and uncertainty, I (Wickert, here and after) suggest the following combination. First, our current method provides the fit RMSE. Second, as pointed out by Referee 1, applying the double-Manning approach as priors and constraints through an existing Bayesian method (BaRatin, with accompanying software BaRatinAGE) is possible. Therefore, my plan here is to highlight how we quantify uncertainty while incorporating prior work in a discussion of Bayesian methods of uncertainty quantification. This, I think, will answer both points without recreating existing effort and software products.
  Third, I want to address your comments on the basic knowledge of the CLI and potential roadblocks that this might cause. In case you did not see it, the README.md for "doublemanning" includes a full description of the yaml configuration file as well as how to run the CLI, and this includes prints of the built-in help functions: https://github.com/MNiMORPH/doublemanning. This documentation should, I think and hope, significantly lower barriers to usage.
  We thank you once more for your time and constructive comments.
  
  Citation: https://doi.org/10.5194/egusphere-2023-3118-AC3

Status: closed

RC1:
'Comment on egusphere-2023-3118', Anonymous Referee #1, 21 Feb 2024

The main purpose of this research paper is to introduce a novel approach, called double-Manning approach, for computing stage-discharge relations (rating curves) in compound channels, i.e. at river sites with a main channel and a floodplain. The method, based on the addition of two simplified Manning equations, is illustrated through three gaging stations with variable density of data, size and geometry.
Sadly, I have to say that I don’t think that this approach is really a valuable innovation compared to already published rating curve methods, an active field of research in the past decade (cf. the collective paper by Kiang et al. (2018) which compares 7 methods for rating curve determination and uncertainty analysis). Kiang’s paper and the main references they cite should be considered by the Authors. The third site included in their comparison (the Taf River at Clog-y-Fran, Wales, United Kingdom) actually has a compound channel, which several of the compared methods (the 3 Bayesian methods NVE, BaRatin, BayBi, at least) represent using an addition of power-law equations (one for each channel) with explicit parameters derived from the Manning equation. The wide-channel approximation is necessary to get a relation in the form of Q=a(h-h0)^b for each channel equation, as also mentioned in this article, however it does not make a large difference for most natural rivers.
Typically, the SPD model proposed by Manasanarez et al. (2019b) and cited in the article is based on such principles, which the Authors seem to ignore. It allows the management of shifts, flexible rating curves and hydrogeomorphic feedbacks, unlike what is claimed in Sections 6.3 and 6.4. Bayesian methods do provide additional information (through the prior knowledge) unlike what is stated page 2 line 55 and elsewhere throughout the paper. The parameters of the rating curve equation can be interpreted hydraulically and can be assumed to vary over time (including the channel width, roughness, etc.). Also, many hydrometric sites may not be represented by just a combination of two Manning equations, especially when natural or artificial section controls are active (cf. the Mahurangi site in Kiang’s paper). Then, the proposed model will not suit, whereas other software accommodate much more diverse rating curve models.
Therefore, rating curve methods based on the same (or very similar) ‘double-Manning’ approach already exist, especially Bayesian methods. They offer the same opportunity of hydraulic and geomorphic interpretation of the rating curve parameters, through a stochastic framework that accounts for the uncertainty of the stage-discharge data, of the rating curve parameters and of the discharge outputs (cf. eg. Petersen-Overleir and Reitan 2005, Le Coz et al. 2014). This is a huge advantage over the deterministic optimization procedure introduced in this paper, since hydrometric uncertainty quantification is recognized as absolutely necessary (McMillan et al., 2017). The NVE, BaRatin and SPD software are open-source and publicly available.
Therefore, I don’t clearly see what is the added value of the proposed method.
I quickly reproduced the rating curve analyses using the open-source software BaRatinAGE 2.2 (https://github.com/BaRatin-tools/BaRatinAGE), using a ‘double-Manning’ approach and similar prior information as in the paper, and assuming 10% uncertainty for all the discharge measurements. See the obtained rating curves with uncertainty results in the attached supplement. There seems to be some cleaning needed in the datasets, and there are reproducibility issues: there are several data files (.tsv) for the Cannon case. It is unclear which is used in the article. The Minnesota .tsv files is in ft and cfs instead of the SI units of the paper. The stage and discharge measurements at LaDormida look rather uncertain according to the comments.
References
Kiang, J. E., Gazoorian, C., McMillan, H., Coxon, G., Le Coz, J., Westerberg, I. K., et al. (2018). A comparison of methods for streamflow uncertainty estimation. Water Resources Research, 54, 7149–7176. https://doi.org/10.1029/2018WR022708
Le Coz, J., Renard, B., Bonnifait, L., Branger, F., & Le Boursicaud, R. (2014). Combining hydraulic knowledge and uncertain gaugings in the estimation of hydrometric rating curves: A Bayesian approach. Journal of Hydrology, 509, 573–587. https://doi.org/10.1016/j.jhydrol.2013.11.016
McMillan, H., Seibert, J., Petersen-Overleir, A., Lang, M., White, P., Snelder, T., et al. (2017). How uncertainty analysis of streamflow data can reduce costs and promote robust decisions in water management applications. Water Resources Research, 53, 5220–5228. https://doi.org/10.1002/2016WR020328
Petersen-Overleir, A., & Reitan, T. (2005). Objective segmentation in compound rating curves. Journal of Hydrology, 311(1–4), 188–201. https://doi.org/10.1016/j.jhydrol.2005.01.016

Citation: https://doi.org/10.5194/egusphere-2023-3118-RC1
- AC1: 'Reply on RC1', Andy Wickert, 11 Jun 2024
  
  We would first like to thank the first referee for their constructive comments and inclusion of some useful references that we either missed or considered in a different light. These opened some new considerations and a desire to better define our goals in developing the double-Manning approach. We also appreciate that the referee took the time to reproduce some of our analyses using the BaRatinAGE software.
  I (Wickert, here and beyond) think that there is a fundamental difference in purpose between our aim here and those of the papers that the referee cites. Here, we seek to develop simple but physically based stage–discharge rating curves, especially in locations in which we are concerned about morphological change or in which data coverage is sparse. These are conditions that – to my experience – are not as commonly considered in hydrological research: Morphological change is most commonly often spoofed as a bed-elevation-only change (even though, geomorphically, channel width is much more variable). Although many hydrologists work with data-rich sites, there is still a significant need for better methods to constrain river discharge at low-accessibility and (therefore often) data-poor sites. These are places in which statistical methods (e.g., Bayesian) might help a bit, but in which the channel dynamics or data-set limitations are the key problems to overcome. I will respond with these ideas in mind.
  
  First, regarding the Kiang et al. (2018) study, I again thank the referee for sharing this. I had not seen it, and it indeed seems a very operationally useful guidance for different approaches. Its purpose, however, is not in the functional form of the equations, but rather in the statistical approaches. This makes it somewhat orthogonal to our goals here.
  The referee notes that the "addition of power-law equations" approach presented in a few methods by Kiang et al. (2018) is derived from Manning's equation. This seems both reasonable and likely. However, neither my reading nor a CTRL+F-aided search found any mention of the Manning equation. Furthermore, I could not find any listing of the parameters used in the fits – only of the goodnesses of the fits themselves. Therefore, I am unable to assess the referee's note that these parameters are explicitly derived from the Manning equation, even if the functional form is.
  This lack of direct confirmation that the best-fit parameters are reasonable in a Manning-equation framework highlights one of the key benefits of the double-Manning approach: with it, we are able to use ancillary data from channel and floodplain morphology to constrain the parameter sets. Fundamentally, the methods demonstrated in the Kiang et al. paper should be able to be applied as such, but there is no evidence that relevant constraints and/or priors were given. This said, the GesDyn method demonstrated there may be something that could be, in future work, combined with morphological data (including those from remote sensing) to further inform the time-evolving rating curve.
  
  Second, the referee notes that the Mansanarez et al. (2019b) study incorporates an equation with functional form $Q=a(h-h_0)^b$. I heartily agree with this, and Manasanarez et al. (2019b) indeed note this generic form in their paper. However, they take the equation in a different direction than we do for two reasons.
  First, they indicate that channel width is typically expected to be fixed and focus on variations in rating curves due to channel depth. To be fair, rating curves are often developed for engineered reaches of rivers with fixed width. However, this is far from universally the case, and both geomorphic theory and observation are that natural alluvial channels may vary in width as well as depth, with similar response times for both (e.g., Naito and Parker, 2020). Fortunately, when operating their model, they do permit both channel width and bed elevation to change.
  Second: they do not return to the question of parameter realism when compared to the Manning equation. This likewise makes them entirely dependent on paired stage–discharge data, which, as we point out in the submitted manuscript, are relatively difficult to obtain.
  This brings us to your note about our text, which I agree was somewhat unfair, "And although Bayesian methods may help to better understand sparse data sets (Mansanarez et al., 2019b), they do not provide fundamental new information." I should (and will) update this to note that, when the priors are data-informed, they do provide additional information. The Mansanarez et al. (2019b) study is a good example for including data-informed priors.
  Finally, the referee notes that some cross-sections might require more than two power-law curves to describe the rating curve, and holds up the Mahurangi River in New Zealand, as studied by Kiang et al. (2018), as an example. I have a few thoughts in response to this: (a) Yes, some rivers will have channel and/or valley geometries for which a double-Manning approach is not appropriate. Our goal in this paper is to craft an equation that jointly optimizes for simplicity and its ability to abstract and capture the dynamics of most rivers. (b) Including an arbitrary number of power-law equations can easily lead to an exercise in overfitting, which can rapidly lose straightforward comparability with diverse data sets and theory. I think that we all wish to avoid this. Indeed, the core of the double-Manning approach involves fixing or constraining these power-law parameters in order to limit the number of degrees of freedom in the fit. Therefore, for more complex channels and valleys, an approach invoking more realistic hydraulics along complex walls might be more apt (e.g., ray-isovel: Houjou et al., 1990; Kean & Smith, 2004). (c) Upon visual inspection of the stage–discharge data from the Mahurangi River (Fig. 6 in the paper by Kiang et al., 2018 – in case it does not reproduce below successfully or is not readable in its max-500-px size for inclusion here), I believe that a reasonable fit could be achieved using a two-power-law approach like the double-Manning approach here. I believe this because in the semilog plot there is a single roll-over separating straight segments. I apologize for not fully testing this idea: The authors do not provide the data used to generate this plot, and I have chosen for expediency and accuracy not to attempt to locate them, download what I believe to be the appropriate portion of their data, and (with some doubt) work to reproduce what they have done.
  
  Third, the referee notes that, "rating curve methods based on the same (or very similar) ‘double-Manning’ approach already exist, especially Bayesian methods". I would add critical nuance to this statement. Existing methods and associated software tools can be implemented with double-Manning constraints to reproduce the work that we have done here. This is absolutely excellent. However, to my knowledge, these methods were not built with constraints in place to acknowledge that the physical system is a river, or with straightforward guidance on how to take simple geomorphic and sedimentological measurements and use them to help guide parameter estimates.
  This incorporation of physically meaningful and simply obtained ancillary data constraints is the power of the double-Manning approach. The idea behind it is simple, but in these tests that we have performed, it is also powerful. The "doublemanning" software package need not – and ideally should not – be its only implementation. Therefore, in a revised manuscript, I will also include information on these software packages, with further thanks for you running these tests using BaRatinAGE.
  
  Before continuing on these thoughts regarding software implementations of the double-Manning approach, I will reply to the referee's notes towards implementation using BaRatinAGE. (a) It is true that there are multiple data subsets given in the "CannonWelch" data set. Due to rating-curve drift and a desire for transparency, I kept all of these. As described in the article and the "doublemanning" package documentation, the "config.yaml" file specifies that "CannonWelch05355200_RatingCurve_2002-2012.tsv" is used. This should therefore resolve the referee's stated concern that, "It is unclear which is used in the article.". (b) The referee also notes that, "The Minnesota .tsv files is in ft and cfs instead of the SI units of the paper." This is because these are the units of the source data. In this case, I chose to keep these units in order to implement the "doublemanning" package's unit conversions. (c) The referee also notes that "The stage and discharge measurements at LaDormida look rather uncertain according to the comments." Without knowing what "rather uncertain" means, I think I probably agree with the referee: these measurements were made infrequently and under harsh field conditions (rain, cold, and flows that were difficult to stand in while holding the velocimeter). This is far from an established survey at a gauging site with infrastructure. Yet, this is a common scenario, and one for which scientists may take great advantage of a more solidly mechanistically constrained approach (e.g., double-Manning) due to a relative shortage of paired in-situ stage–discharge measurements.
  
  Returning to the software implementations, I want to share two key thoughts to close. The first is that I plan to discuss the BaRatin method and how it can also (naturally) produce similar results when given similar source data as priors. I will also discuss the BaRatinAGE open-source implementation of BaRatin.
  Towards this second point, I did not know that BaRatinAGE was available as an open-source package, despite completing what I felt was a reasonable literature review. If I did, I may have used it instead of creating my own code! Looking at the GitHub repository (thanks for the URL), it has only two stars (three with mine) and "issues" that seem to be raised only by the dev team, indicating little external traffic or awareness. Furthermore, there are no instructions for compilation or running, though some information is found in "CONTRIBUTING.MD". These are significant barriers to discovery or use. "doublemanning", on the other hand, includes full documentation and straightforward installation via "pip": https://github.com/MNiMORPH/doublemanning. Therefore, I would argue that BaRatinAGE is not very accessible, even though the source code is open. If you would like to avoid a repeat of the present scenario, in which another scientist independently creates some code that you consider to not solve a problem as well as existing work has, I would strongly urge those involved with BaRatinAGE (including yourself or colleagues whom you can nudge, I hope?) to (a) document the code to at least a minimum level to permit straightforward usage, (b) publicize it, and (c) make sure that others know about what seems to me to be a just-published companion paper (Le Coz et al., 2024). I just found this paper while replying here – it was just published in April – and I am glad that I did.
  
  To sum up, I will:
  
  (a) change some of the language regarding Baysesian approaches;
  
  (b) better clarify the problem that we seek to solve (incorporating more diverse field data and more tightly constraining the system with more than just stage–discharge data) and that our ideas towards these solutions are, strictly speaking, agnostic from the code used to do so; and
  
  (c) reference one or more alternative (existing) methods to solve the same systems of equations in a Bayesian framework.
  
  I hope that this adequately represents the background and mission of our work, contextualizes the value that we seek to bring, and inspires better publicity and accessibility of existing tools.
  
  Citation: https://doi.org/10.5194/egusphere-2023-3118-AC1
CC1:
'Comment on egusphere-2023-3118', Richard Koehler, 26 Mar 2024

Authors have not completed a basic spell check or grammar check. Multiple misspelled words, grammar errors, and inconsistent phrasing detracts from this article. Article rating curve figures are inconsistent with USGS rating curves for same site. Authors have reversed the axes and have used linear, not log10, scaling. This makes it very difficult to compare figures from this article to official, quality controlled, USGS rating curves. This reviewer does not recommend this article for publication.

Citation: https://doi.org/10.5194/egusphere-2023-3118-CC1
- AC2: 'Reply on CC1', Andy Wickert, 11 Jun 2024
  
  Dear Dr. Koehler;
  It is unfortunate that you have found spelling and grammatical errors. All of the authors revised the article. I went back through it and found a couple of mistakes that will be fixed upon revision and included in the tracked-changes document associated with this. In the future, I would suggest giving targeted comments as to where improvements would be needed, especially in this case, in which wholesale changes seem unnecessary.
  The rating-curve figures in this article were not designed to match those from the US Geological Survey. Because we have published all of the data, replotting these for comparison is possible, if desired.
  
  Citation: https://doi.org/10.5194/egusphere-2023-3118-AC2
RC2:
'Comment on egusphere-2023-3118', Anonymous Referee #2, 16 Apr 2024

In this paper, the authors introduce a compound channel rating curve method/tool which they describe as a “double-Manning approach”. This approach relies on parameterization of two empirical flow equations (Qch and Qfp) to predict discharge at all depths, both channel-contained and at flood stage. While the tool can be used when discharge data are unavailable, the results presented in this manuscript at three test sites use an optimization routine (included in the tool) to select parameter sets that best fit this piecewise Manning’s power function.
While the approach using a piecewise power function to estimate compound channel flows is not necessarily novel, I see this tool as being a good additional resource for individuals that need a quick estimate of discharge, particularly in situations where flow data is limited. The tool could lend itself well to widespread use, however, end-users need a basic understanding of the CLI which may be prohibitive.
It does seem appropriate that the authors submit a manuscript that accompanies the software tool, but for this paper to be publishable, the authors may want to structure it in a way that 1) does a better job acknowledging the other piecewise power function regression tools described in the literature, and 2) improves on the method to account for error and uncertainty.

Citation: https://doi.org/10.5194/egusphere-2023-3118-RC2
- AC3: 'Reply on RC2', Andy Wickert, 11 Jun 2024
  
  We thank Referee 2 for their comments about our work. Following both Referee 2's comments here and those from Referee 1, we plan to incorporate more information and background on piecewise power-function approaches, including the references noted by Referee 1 and in our response to them.
  Regarding managing error and uncertainty, I (Wickert, here and after) suggest the following combination. First, our current method provides the fit RMSE. Second, as pointed out by Referee 1, applying the double-Manning approach as priors and constraints through an existing Bayesian method (BaRatin, with accompanying software BaRatinAGE) is possible. Therefore, my plan here is to highlight how we quantify uncertainty while incorporating prior work in a discussion of Bayesian methods of uncertainty quantification. This, I think, will answer both points without recreating existing effort and software products.
  Third, I want to address your comments on the basic knowledge of the CLI and potential roadblocks that this might cause. In case you did not see it, the README.md for "doublemanning" includes a full description of the yaml configuration file as well as how to run the CLI, and this includes prints of the built-in help functions: https://github.com/MNiMORPH/doublemanning. This documentation should, I think and hope, significantly lower barriers to usage.
  We thank you once more for your time and constructive comments.
  
  Citation: https://doi.org/10.5194/egusphere-2023-3118-AC3

Andrew D. Wickert, Jabari C. Jones, and Gene-Hua Crystal Ng

Data sets

Minnesota River near Jordan: stream-gauge data and double-Manning fit J. Jones and A. D. Wickert https://doi.org/10.5281/zenodo.10334289

Cannon River at Welch: stream-gauge data (stage, discharge, and sediment grain size) and double-Manning fit J. Jones et al. https://doi.org/10.5281/zenodo.10334497

Stream-gauging Data: La Dormida Captación G. C. Ng et al. https://doi.org/10.5281/zenodo.10334038

Model code and software

doublemanning A. D. Wickert https://doi.org/10.5281/zenodo.7495274

Andrew D. Wickert, Jabari C. Jones, and Gene-Hua Crystal Ng

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Jan 2024	126	56	3	185
Feb 2024	29	19	3	51
Mar 2024	50	15	3	68
Apr 2024	51	14	5	70
May 2024	33	22	0	55
Jun 2024	85	26	11	122
Jul 2024	33	23	3	59
Aug 2024	14	12	1	27
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Nov 2024	18	5	0	23
Dec 2024	2	5	0	7

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Short summary

For over a century, scientists have used a simple algebraic relationship to estimate the amount of water flowing through a river (its discharge) from the height of the flow (its stage). Here we add physical realism to this approach by explicitly representing both the channel and floodplain, thereby allowing channel and floodplain geometry and roughness to these estimates. Our proposed advance may improve predictions of floods and water resources, even when the river channel itself changes.


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