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.

Response: In our approach, the uncertainty models are built from the residuals, i.e. the difference between the observed and estimated values. Various uncertainty models are considered (e.g. homoscedastic and heteroscedastic) and adjusted to the distribution of residuals. We therefore assume that all sources and types of uncertainties (stage, flow, model parameters, rating curve uncertainties) are accounted for in residual distributions. It is a straightforward approach where there is no need to identify or separate uncertainty contributions. By doing so we assume that residual distributions are representative of uncertainties on future discharge estimated from the rating curve. This would be inappropriate if some source of uncertainties, for example, more precise flow measuring devices were used at some point in time. Otherwise, we assume that uncertainties remain unchanged over the process of the construction the rating curve.

We therefore disagree with the reviewer when he says that, since our approach didn’t specifically address model parameter, it cannot be used to address uncertainty in predictions. We think that, even though this approach doesn’t explicitly identify and set a statistical framework of the various uncertainty contributions to total uncertainties, it is rigorous and useful.

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.

Response: The proposed approach doesn’t specifically address the uncertainties below the Smallest Gauged Stage (SGS) or above the Largest Gauged Stage (LGS). In fact, the uncertainty models estimated in the interpolated part are used to assess uncertainties in extrapolated parts of the rating curve, and cautionary remarks are made about this issue. For instance, Figure 8 shows that uncertainties for many rating curves increases as stage decreases which can be reasonably related to a progressive change of hydraulic control. Uncertainty models can partly account for such changes but a more detailed analysis of the hydraulic is needed to further assess uncertainties for these parts (below the SGS and above the LGS) of the rating curve. Leaving out 2 or 4 gauged stage-discharge pairs with the largest stages, as suggested by the reviewer, may be view as a possible way to estimate the uncertainties in the extrapolated part of the RC, but may be questionable since the main source of uncertainties is related to changes in hydraulic control or to changes in channel characteristics. Therefore, we didn’t use resampling approaches and preferred to stress the limits of our approach (and of any resampling approaches) and to reiterate the importance of hydraulic analysis.  

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).

Response: Mean Square Relative Error (MSRE) is a normalized measure of the difference between estimated and observed values. Is it not per se based on a particular stochastic model but on the hypothesis that the best adjustment of the power function to the stage-discharge pairs is obtained through the minimization of this metric. Intuitively we can expect that an adequate RC should correspond to the one where relative errors don’t depend on stage (homoscedastic model). Results support this hypothesis since 75% of the RC are adequately represented by a normal or logistic homoscedastic distribution with mean standard deviation of the order of 5-7%, values close to the estimated flow measurements.

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.

Response: Of the 361 values used to build Figure 4, 3 of the c exponents for 1 PF rating curve, and 14 of the c1 exponents for the 2 PF rating curve have values larger than 5, which represent less than 5% of the estimated c values. As explained in the paper, these peculiar cases were not investigated prior to the redaction of the manuscript. They were however further analyzed since the submission of the paper. Our analysis examined all RC with c > 3 and concluded that they occur in four situations:

1. Small number of stage-discharge pairs and small range of gauged stages: Many unusual c values are observed under these conditions. It is incidentally the case for the 3 RC with c < 1. Also in some cases, even if the number of gauging may seem reasonable, the range of gauged flow and stage remains small. The available stage-discharge pairs ‘under sampled’ the stage range and, as suggested by the reviewer, may be explained by sampling errors.
2. Hydrometric stations downstream of dams: Streams downstream of dams often have much smaller flow variability and therefore much narrower range of gauged stage. The largest reported c values correspond to such situation.
3. Control section upstream a natural spillway: In some cases, hydraulic control corresponds to a natural spillway. This type of configuration is often observed but could be problematic if the crest of the natural spillway is close to water surface at low stage. In such case, the adjusted ‘b’ value of equation 2 is unrealistic, ‘a’ is very small, and must be compensated by a large ‘c’ value.
4. Compensating effect between a and c parameters of equation 2: In some cases, under specific conditions, e.g. when small stages are over-represented compared to high stages, adjusting the CT led to very small ‘a’ values and large ‘c’ values. This effect is often observed, but not only, downstream dams as mentioned previously.

As can be seen, these unusual ‘c’ values are related to very peculiar situations suggesting that power function may be tricky to estimate in these cases. We suggest adding the preceding discussion about these cases in the manuscript in response to reviewer’s comment.

RC Technical corrections

Line 11: Please consider changing "usually represented by power functions, a mathematical function" to "usually represented by power functions, mathematical functions".

Response: Done.

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.

Response: Done.

Line 15: Please consider changing "This paper proposed an approach" to "This paper proposes an approach“.

Response: Done.

Line 17: Please consider changing "and assessed if one or two" to "and assesses whether one or two".

Response: Done.

Line 23: Please add a comma in "they increase significantly at low stages, reaching values larger than 20%".

Response: Done.

Line 40: Please change "bias on estimated flows" to "bias in estimated flows".

Response: Done.

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.

Response: Overview of RC has been updated and references added.

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.

Response: The sentence has been rephrased and references added.

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).

Response: These citations have been added.

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).

Response: We didn’t find reference Reitan and Overleir 2004.

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.

Response: Done.

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. ".

Response: Done.

Line 128-130: Maybe it is sufficient to state that "Relative errors on estimated discharge were considered. However, this assumption will be further investigated. "?

Response: We agree.

Line 131: Please explain why six gaugings are set as the minimum to adjust a single RC.

Response: A minimum number of gaugings has obviously to be set to adjust a power law. The choice of six gaugings is subjective and seemed reasonable.

Lines 135-139: Please provide reference/references after "hereafter called the transition level. " in Line 139.

Response: We proposed this expression and therefore no reference can be provided. 

Line 177: Please consider changing "an average of 2.0 RC/station" to "an average of 2.0 RC per station".

Response: Done.

Line 178: Please consider changing "a 5-95% confidence interval" to "a 90% confidence interval".

Response: Done.

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.

Response: Done.

Line 189: Please change "the complexity of Darienzo et al.’s approach" to "the complexity of Darienzo et al. (2021) approach".

Response: Done.

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.

Response: Done.

Line 284: In Figure 5, the box for model M2 uses M3 instead of M2. Please revise Figure 5 accordingly.

Response: Done.

Lines 306-309: The model L-M1d should be referred to as Logistic distribution but not as Normal distribution.

Response: Done.

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

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.

Response: A careful rereading of the paper was conducted.

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.

Response: As shown in Figure 1, most gaugings are realized in spring, during the peak flow associated with snow melt, and during summer/autumn. For operational reasons, gaugings are most likely made under medium to low stage conditions as pointed out by the reviewer. Ideally, as pointed out by the reviewer, we would prefer to have uniformly distributed gaugings across the h-range. However sub-selection, or sub-sampling, eliminates some gaugings and deprive us of valuable information. Also, as pointed out by the reviewer, such strategy could be applyied only to stations with ‘reasonable number of gaugings’ and this ‘reasonable’ number would need to be assessed.