the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 11 Jun 2024
Assessment of uncertainties on stage-discharge rating curves: A large scale application to Québec hydrometric network
Abstract. Rating curves (RC), which establish a relationship between stage and discharge at a given cross-section of a river, are largely used by national agencies to measure flow. RC are constructed from gauging measurements and are usually represented by power functions, a mathematical function frequently used to represent stage-discharge relationship of standard hydraulic structures. Uncertainties on estimated flows based on rating curves can be significant, especially for high and low flow regimes. It is therefore important to report these uncertainties as accurately as possible. Many approaches estimating the sources of uncertainties on flows have been proposed but are generally too complex for large scale application to hydrometric networks. This paper proposed an approach to develop rating curves and assess the corresponding uncertainties on estimated flow that can be readily applied to large-scale hydrometric networks. This approach takes into consideration possible changes in RC over time due to hydraulic or geomorphologic modifications and assessed if one or two power functions are needed to adequately represent the stage-discharge relationship over the available range of gauged stages. RC at Quebec hydrometric stations have been constructed. Relative differences between flows estimated from the RC and gauged flows are used to assess uncertainties on estimated flow. They were adjusted to normal or logistic distributions with constant (stage-independent uncertainties) or stage-dependent scale parameters (stage-dependent uncertainties). Mean standard deviation on estimated flows for RC with stage-independent uncertainties (75.0 % of the RC) is 6.5 %, while for RC with stage-dependent uncertainties, they increase significantly at low stages reaching values larger than 20 % for some RC at the lowest gauged stage.
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RC1: 'Comment on egusphere-2024-1389', Anonymous Referee #1, 30 Jul 2024
RC General comments
This paper presents a robust method to fit rating curves to hydrometric networks, that is, to fit a large number of rating curves over a long period in an automatic manner. The method accounts for possible temporal changes in RCs over the observed period of each station and determines whether one or two power-law rating curves are required to represent the gauged stage-discharge relationship. The paper also proposes models for representing a stage-dependent scale parameter. The proposed method is applied to the Québec hydrometric network, which includes 173 hydrometric stations mainly located in southern Québec.
RC Specific comments
1)The uncertainty of model parameters is not addressed in this paper. Thus, the uncertainty in the prediction of discharge for a particular value of the stage, h, corresponding to the rating curve only, can‘t be accessed. This is true for both stage h within the observed stage-discharge pairs and stage h outside the range of the observed stage-discharge pairs. If the uncertainty of future discharge observation is needed, then both the uncertainty of the model parameters and the uncertainty due to the error term (representing the measurement error and potential model error) are needed. The uncertainty of the error terms is addressed in this paper but that alone is not enough to address uncertainty in predictions. Please evaluate the uncertainty of the model parameters and report this uncertainty.
2)Following Comment 1), it should be described how the uncertainty of the model parameters enters predictions of discharges for h above the range of the observed stage-discharge pairs. Furthermore, the prediction performance of the method should evaluated by leaving out 2-4 observed stage-discharge pairs holding the largest stage values. As the left-out observed data pairs include measurement error then the prediction should take into account the uncertainty in both the model parameters and the error term.
3)Please provide a reference for the function in Eq. (2) and explain what it is based on. Is it based on the likelihood function of a particular stochastic model? There should be a discussion about other modeling approaches and other inference schemes for discharge rating curves found in the literature. The approach based Eq. (2) only focuses on finding point estimates for the unknown parameters but there is no mention of uncertainty in the parameters due to the lack of information in the data (as mentioned in Comment 1).
4)In Section 4, it is mentioned that values of c larger than 8/3 should not be a priori discarded. This is a reasonable claim for values of c close to 8/3, however, values of c greater than 5 are questionable for natural open channels. For example, the inverse parabolic cross section has c between 3.67 and 4.33 (Hrafnkelsson et al., 2022), and it is unlikely to appear in nature. Values of c greater than 5 correspond to cross sections more extreme than the inverse parabolic cross section, that is, narrow width close to the bottom and a large increase in width with the stage moving further from the bottom. Estimates of c greater than 5 are more likely to be observed due to sampling error than extreme cross sections. Furthermore, estimates of c that are less than 1 are most likely observed due to sampling error. Please consider tackling this issue by adding a penalty term to Eq. (2) (or to a function other than Eq. (2)) that penalizes values of c that are too large (c > 5) and too small (c < 1). This can, for example, be approached by using a parabolic term resembling the logarithmic transformation of a Gaussian prior density. Please consider also adding a penalty term for the parameters b (in the 1 PF model) and b_1 (in the 2 PF model) that ensures that b and b_1 are less than the smallest measured water level and penalizes for too small values of b and b_1.
RC Technical corrections
Line 11: Please consider changing "usually represented by power functions, a mathematical function" to "usually represented by power functions, mathematical functions".
Line 10-12: Please mention that a power function in the context of discharge rating curve models is also referred to as the power law in the literature.
Line 15: Please consider changing "This paper proposed an approach" to "This paper proposes an approach“.
Line 17: Please consider changing "and assessed if one or two" to "and assesses whether one or two".
Line 23: Please add a comma in "they increase significantly at low stages, reaching values larger than 20%".
Line 40: Please change "bias on estimated flows" to "bias in estimated flows".
Lines 60-74: Here the main representations of RC are reviewed. However, only the power-law representation and the approaches based on cubic splines and Chebyshev polynomials are mentioned while multi-segment discharge rating curves (e.g. Petersen-Øverleir, A., and Reitan, T., 2005; Reitan, T., and Petersen-Øverleir, A., 2009; Hodson et al. (2024)) and the generalized power-law rating curve (Hrafnkelsson et al., 2022) are not mentioned. Please consider revising the overview of RC.
Lines 73-74: It is stated that „Despite these legitimate criticisms, wide application of the power function has shown that stage-discharge curves are surprisingly well-represented by such functions, which is the case in the actual application.“ It is true that the power function works well in actual applications but that is only true for a certain proportion of rivers as it has been shown that for some rivers the power function is inadequate (e.g., Petersen-Øverleir and Reitan, 2005; Reitan and Petersen-Øverleir, 2009; Hrafnkelsson et al., 2022; Hrafnkelsson et al., 2023). Please consider rephrasing the above statement.
Lines 85-86: After mentioning that more than one power function must be used it would be appropriate to cite papers on multi-segment rating curves, e.g., Petersen-Øverleir and Reitan (2005), Reitan and Petersen-Øverleir (2009), Hodson et al. (2024).
Lines 92-95: The main objective of the paper is mentioned here, that is, modeling of error terms (or the residuals) of the models. Here previous work on models for the error terms should be cited (e.g., Reitan and Overleir 2004, Hrafnkelsson et al. 2012; Hrafnkelsson et al. 2022).
Line 116: Section 3 might be better described with a title including the word "method" as opposed to the word "approach". Please consider revising the title of Section 3.
Line 120: Please consider changing "The following sections further details each of these steps. " to "The following sections provide further details for each of these steps. ".
Line 128-130: Maybe it is sufficient to state that "Relative errors on estimated discharge were considered. However, this assumption will be further investigated. "?
Line 131: Please explain why six gaugings are set as the minimum to adjust a single RC.
Lines 135-139: Please provide reference/references after "hereafter called the transition level. " in Line 139.
Line 177: Please consider changing "an average of 2.0 RC/station" to "an average of 2.0 RC per station".
Line 178: Please consider changing "a 5-95% confidence interval" to "a 90% confidence interval".
Lines 177-178: It is reported that the RMSRE value is 7.3% with a 90% confidence interval equal to (2.3%,18.7%). It would help to note that this estimate is based on the relative residuals from all the stations.
Line 189: Please change "the complexity of Darienzo et al.’s approach" to "the complexity of Darienzo et al. (2021) approach".
Lines 254-257: Adding 95% confidence intervals to the sample mean (blue dots) and sample standard deviation (black squares) of the RR would improve Figure 5.
Line 284: In Figure 5, the box for model M2 uses M3 instead of M2. Please revise Figure 5 accordingly.
Lines 306-309: The model L-M1d should be referred to as Logistic distribution but not as Normal distribution.
RC References
Hodson, T. O., Doore, K. J., Kenney, T. A., Over, T. M., and Yeheyis, M. B. (2024). Ratingcurve: A python package for fitting streamflow rating curves. Hydrology, 11(2).
Hrafnkelsson, B., Ingimarsson, K.M., Gardarsson, S.M., Snorrason, A. (2012). Modeling discharge rating curves with Bayesian B-splines. Stoch. Env. Res. Risk A. 26 (1), 1-20.
Hrafnkelsson, B., Sigurdarson, H., Rögnvaldsson, S., Jansson, A. Ö., Vias, R. D., and Gardarsson, S. M. (2022). Generalization of the power-law rating curve using hydrodynamic theory and Bayesian hierarchical modeling. Environmetrics, 33(2):e2711
Hrafnkelsson, B., Vias, R. D., Rögnvaldsson, S., Jansson, A. Ö., and Gardarsson, S. M. (2023). Bayesian discharge rating curves based on the generalized power law. In Hrafnkelsson, B., editor, Statistical Modeling Using Bayesian Latent Gaussian Models : With Applications in Geophysics and Environmental Sciences, pages 109–127. Springer International Publishing, Cham.
Petersen-Øverleir A (2004). Accounting for heteroscedasticity in rating curve estimates. Journal of Hydrology 292:173–181.
Petersen-Øverleir, A., and Reitan, T. (2005). Objective segmentation in compound rating curves. Journal of Hydrology, 311(1–4), 188–201.
Reitan, T., and Petersen-Øverleir, A. (2009). Bayesian methods for estimating multi-segment discharge rating curves. Stochastic Environmental Research and Risk Assessment, 23(5), 627– 642.
Citation: https://doi.org/10.5194/egusphere-2024-1389-RC1 -
RC2: 'Comment on egusphere-2024-1389', Anonymous Referee #2, 24 Aug 2024
Review of egusphere-2024-1389: Assessment of uncertainties on stage-discharge rating curves: A large scale application to Québec hydrometric network Alain Mailhot, Guillaume Talbot, Samuel Bolduc, and Claudine Fortier
Based on stage discharge data from the Quebec hydrometric network the authors present an analysis of the uncertainty of stage-discharge rating curves. In their manuscript detailed description of i.) the selection of hydrometric sides, ii.) the analysis of the temporal stability/instability of the gauging sides, iii.) the fitting of either one or two rating curves to the gaugings as well as iv.) the fitting of different models to quantify the uncertainty of the established rating curves is given. In general, the paper is scientifically sound and well designed and written. However, there are quite some careless errors (e.g. different naming of the uc-models in fig 6 and in the text, etc – see the technical points raised by my fellow reviewer colleague) which complicates the reading of the paper quite a bit.
One point I would like to be tackled is the fact that generally the highest density in available gaugings is for medium stage levels, whereas it seems that there are less dense recordings at more extreme (low and high) stages – at least that is what I take from the two examples depicted in fig 7. Would the result of the study (e.g. the selected uc-models) change if - at least for the gauges with reasonable number of gaugings – the records would be somehow randomly (and repeatedly) sub-selected, so that the data points are somehow uniformly distributed (e.g. one record every 10 cm) over the h-range the rating curve is valid for. This maybe could add a further aspect towards the robustness of the selected/assigned uc-models.
Citation: https://doi.org/10.5194/egusphere-2024-1389-RC2
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