the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 14 Feb 2025
Quantifying controls on rapid and delayed runoff response in double-peak hydrographs using Ensemble Rainfall-Runoff Analysis (ERRA)
Abstract. Double-peak hydrographs are widely observed in diverse hydrological settings, but their implications for our understanding of runoff generation remain unclear. Previous studies of double-peak hydrographs in the extensively instrumented Weierbach catchment have linked he first peak to event water and the second, delayed and broader peak to pre-event water. Here we use Ensemble Rainfall-Runoff Analysis (ERRA) to quantify how precipitation intensity and antecedent wetness influence groundwater recharge and double-peak runoff generation at the Weierbach catchment (Luxembourg). The spiky first peak can be attributed to a rapid response directly linking precipitation to streamflow via near-surface flowpaths. Relative to this first peak, the second peak is delayed (peaking ~1.5 days after rain falls), lower (~1/3 the height of the first peak), and broader (declining to nearly zero in ~10 days), and can be attributed to a groundwater-mediated pathway that links precipitation, groundwater recharge, and streamflow. The sum of these two runoff responses quantitatively approximates the whole-catchment runoff response. Under wet conditions, the first peak increases nonlinearly (particularly above precipitation intensity of 2 mm h-1) and the second peak becomes higher, narrower, and earlier with increasing precipitation intensity. Under dry conditions, the first peak increases nonlinearly with precipitation intensity (particularly above 4 mm h-1), and groundwater recharge also responds to precipitation, but no clear second peak occurs regardless of precipitation intensity. The lack of a second peak under dry conditions plausibly arises from groundwater loss to evapotranspiration and from limited connectivity between groundwater and the stream, rather than from a lack of groundwater recharge. Almost no runoff response occurs at precipitation intensities below ~0.8 mm h‑1 under wet conditions and ~1.5 mm h‑1 under dry conditions. After a precipitation-related threshold that initiates the first peak and a catchment wetness threshold that initiates the second peak, higher precipitation intensities amplify the first peak nonlinearly and trigger a larger and quicker second peak.
Competing interests: Some authors are members of the editorial board of HESS.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.-
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RC1: 'Comment on egusphere-2025-613', Anonymous Referee #1, 27 Apr 2025
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CC1: 'Reply to RC1', James Kirchner, 06 May 2025
We thank Reviewer #1 for these helpful and insightful comments. Our point-by-point response below shows reviewer’s comments in plain text and our responses in bold.
The presented study applies the Ensemble Rainfall Runoff Analysis (ERRA) to explore the mechanisms of double-peak hydrograph emergence in the Weierbach catchment, Luxembourg. The authors test the hypothesis that the double-peak hydrograph is generated by two pathways – a near-surface runoff in direct response to precipitation and a pathway through vadose zone influenced by groundwater recharge. They further explore how double-peak runoff generation varies with precipitation intensities and antecedent catchment wetness. The controls of the first and second hydrograph peak are identified with their specific thresholds. The manuscript thus presents novel insights into the mechanistic functioning of runoff generation inferred with ERRA from multiple observational data in the Weierbach catchment. This is a very interesting study which rigorously works out the controls on the double peak hydrograph in the study catchment and clearly presents the results and in-depth discussion.
Thank you!
I have a few remarks that may further enhance the presentation and potentially increase the generalization of the case study results.
- L65 – 69: The cited studies with specific numbers of catchment storage, precipitation volume and intensities are site-specific and very much depend on the catchment characteristics. It would be good to mention this in this paragraph.
Thank you for the suggestion. We will specify that in our revised manuscript.
- With regards to NRF(P) /NRF (GR) the authors define it as a response function of discharge to amount of precipitation/Groundwater recharge in each increment, and this function in its turn depends on precipitation intensity/GR rate at time t. Should this function also depend on the antecedent moisture state as it controls the responsiveness of discharge to P/GR? In case only overland flow as response to P is considered, this is not of large importance, but for GR it should matter, shouldn’t it? Or is average soil moisture state implicitly considered in NRF analogously to the rational formula? The runoff coefficient in the rational formula is however an integral over an event duration of a longer time period, but NRF is a function resolved in time depending on precipitation intensity. I somehow miss soil moisture in this resolved representation. Please, explain.
Thank you for the question. We assume that the reviewer refers to Figure 8 as “NRF(P)” and Figure 9 as “NRF (GR)”.
Figure 8 shows the nonlinear runoff response to precipitation, where NRF is the product between precipitation-intensity-dependent runoff response function (RRD) and the precipitation rate (as defined in Eq. (4)). It quantifies how streamflow responds to one time step of precipitation at a given intensity. Antecedent wetness is treated in ERRA as a category variable, in which the NRF is estimated for two or more different ranges of soil moisture, groundwater levels, antecedent discharge, or antecedent precipitation. The contrast between wet and dry antecedent moisture conditions, as inferred from water table depth, is illustrated by comparing the "wet" and "dry" curves in the figure. In Figure 10, we show that one obtains similar results, whether one infers antecedent wetness from antecedent soil moisture or from antecedent water table depth.
The “NRF(GR)” in Figure 9 is not defined as “a response function of discharge to amount of Groundwater recharge in each increment”. Figure 9 shows the nonlinear groundwater recharge response to precipitation, where NRF is the product between precipitation-intensity-dependent groundwater recharge response and the precipitation rate. It should be interpreted as the rate of groundwater recharge expected to result at a given time lag from precipitation falling at a specific rate. This coupling between precipitation and groundwater recharge will be sensitive to antecedent wetness, as shown by the two curves in Figure 9, but they do not reflect the influence of soil moisture status on streamflow per se. Again, Figure 10 shows that antecedent soil moisture and antecedent water table depth are nearly equivalent as indicators of antecedent wetness.
- The results presented in section 3 rigorously demonstrate that the formulated hypothesis that the double-peak hydrograph is generated by the combination of near-surface and groundwater-mediated pathway cannot be rejected. An alternative hypothesis that only e.g., groundwater-mediated pathway would be responsible must be rejected. I think the paper would benefit if the authors more clearly articulate the results in Karl Popper’s sense that the formulated hypothesis could not be rejected.
Thank you for the suggestion. Two alternative hypotheses are actually rejected in Section 3.3.2: the disconnect between the two curves in Figure 6b rejects the hypothesis that only near-surface pathway can explain the total rainfall-runoff response (because it doesn't explain the second peak), and the disconnect between the two curves in Figure 6c rejects the hypothesis that only groundwater-mediated pathway would be responsible (because it doesn't explain the first peak). We will make this clear in our discussion of Figure 6 in the revised paper.
- L461-465: Do I understand correctly that in order to analyze the effect of catchment wetness and precipitation intensity on hydrograph generation you need to separate different periods when e.g., WTD falls into three designated classes, and then you estimate NRFs for these periods? Please, explain your methodological steps for clarity.
Yes for WTD. For nonstationary analysis (i.e., runoff may respond differently depending on catchment wetness status), we need to separate the precipitation time series into different categories according to the antecedent WTD when rain falls.
In Section 4.1 (L431-459), we estimate RRD for each antecedent wetness category. Comparing RRDs between different categories only examines nonstationary runoff response.
In Section 4.2 (L461-506), we estimate NRFs for each antecedent wetness category. Comparing NRFs within each category only looks at nonlinear runoff response, and comparing NRFs between different categories jointly looks at both nonlinear and nonstationary runoff response.
We will further clarify this in the revised manuscript.
- Is this also done analogously for the precipitation intensity? Precipitation intensity is however, much more volatile than catchment average WTD. How does this affect the results if one maybe need to pick just one or two hours of intensive precipitation out of the entire event. Does ERRA then clearly separates NRFs for one-two hours of intensive precipitation framed by a few hours of less intensive precipitation belonging to another class (and resulting in a different NRF) prior and after a major downpour?
No for precipitation intensity. For nonlinear runoff response analysis (i.e., runoff may respond more-than-proportionally to changes in precipitation intensity), NRF equals the precipitation-intensity-dependent RRD times the precipitation rate (Eq. 4), which is “approximated in ERRA by a continuous piecewise-linear broken-stick function of precipitation intensity” (L195).
Precipitation is divided into segments between specified precipitation intensity values ("knots"), and the NRF_k in Eq.(4) (the nonlinear response of streamflow to precipitation that falls at a rate P_j-k and lasts for a time step of ∆t) is the sum of these segments, each multiplied by the slopes of the corresponding broken-stick segments (Kirchner 2024a). That is, NRF is an estimate of the ensemble average of the responses to many rainfall events instead of only over an individual hour or two at a given precipitation intensity.
Thank you for comments 4 and 5. We will add a figure illustrating the calculation of the Nonlinear Response Function (NRF) in Section 2.3.3 in our revised manuscript for clearer clarification.
- The analysis of nonstationarity and nonlinearity in section 4 is very interesting, but sensitivities expressed in absolute numbers remain pertinent to the study catchment. I am wondering if one could derive some dimensionless or relative measures that can be generalized when investigations from a large set of catchments would be available. Would it be helpful for example to look not at the catchment average WTD but in relation to the mean annual precipitation?
Thank you. We are indeed conducting a large-sample study of nonlinearity and nonstationarity, but that would be a completely different analysis. An example of large set of catchments study looking at WTD’s influence on runoff response can be found here for your reference:
Eslami, Z., Seybold, H., and Kirchner, J. W.: Climatic, topographic, and groundwater controls on runoff response to precipitation: evidence from a large-sample data set, EGUsphere [preprint], https://doi.org/10.5194/egusphere-2025-35, 2025.
Looking at the relationship to mean annual precipitation may work in inter-catchment comparisons, but not here in the nonstationary analysis of our manuscript because the single value of mean annual precipitation in a catchment cannot be used to split the precipitation time series into different categories in the nonstationary response analysis or to divide different precipitation intensity ranges in the nonlinear response analysis.
- I understand that results presented in section 4 are catchment specific and much of the discussion links to previous studies and mechanisms of runoff generation in this very catchment, but still maybe some ideas for generalizations may enrich the discussion part from L539ff.
Thank you for the point. We presented a site-specific discussion because we are reluctant to generalize beyond the data that we actually have, and have actually analyzed. We will consider this in the revised manuscript but would be cautious about it.
- Finally, the potential artifact with near-zero spike mentioned in section 3.1 seems to remain unclear. It says in L234 it will be explored in the next section, but I somehow missed a detailed analysis. Do you mean that the exploration is given in L274-277 which basically concludes that you do not have enough data to pinpoint the origin of this spike? Or did I miss further elaboration on this issue in the manuscript? If there is not enough data to further explore this artifact, I suggest not to raise expectation in L234.
The exploration of the potential artifact with near-zero spike was indeed given in L274-277. This artifact first appears in Figure 3c, where we inferred that it could potentially be caused by the distortion of the strong short-lag relationship between groundwater recharge and precipitation (observed in Figure 3b) or the rapid runoff effects of groundwater recharge in the near-stream zone. We then explored how the spike can be reduced by jointly analyzing the mixing effects of correlated precipitation and groundwater recharge on streamflow (Figure 4). To resolve the source of the remaining spike would require data that unfortunately don't exist.
Thank you for the suggestion. We will clarify this in our revised manuscript.
Citation: https://doi.org/10.5194/egusphere-2025-613-CC1 -
AC2: 'Reply on RC1', Huibin Gao, 27 May 2025
We thank Reviewer #1 for these helpful and insightful comments. Our point-by-point response below shows reviewer’s comments in plain text and our responses in bold.
The presented study applies the Ensemble Rainfall Runoff Analysis (ERRA) to explore the mechanisms of double-peak hydrograph emergence in the Weierbach catchment, Luxembourg. The authors test the hypothesis that the double-peak hydrograph is generated by two pathways – a near-surface runoff in direct response to precipitation and a pathway through vadose zone influenced by groundwater recharge. They further explore how double-peak runoff generation varies with precipitation intensities and antecedent catchment wetness. The controls of the first and second hydrograph peak are identified with their specific thresholds. The manuscript thus presents novel insights into the mechanistic functioning of runoff generation inferred with ERRA from multiple observational data in the Weierbach catchment. This is a very interesting study which rigorously works out the controls on the double peak hydrograph in the study catchment and clearly presents the results and in-depth discussion.
Thank you!
I have a few remarks that may further enhance the presentation and potentially increase the generalization of the case study results.
- L65 – 69: The cited studies with specific numbers of catchment storage, precipitation volume and intensities are site-specific and very much depend on the catchment characteristics. It would be good to mention this in this paragraph.
Thank you for the suggestion. We will specify that in our revised manuscript.
- With regards to NRF(P) /NRF (GR) the authors define it as a response function of discharge to amount of precipitation/Groundwater recharge in each increment, and this function in its turn depends on precipitation intensity/GR rate at time t. Should this function also depend on the antecedent moisture state as it controls the responsiveness of discharge to P/GR? In case only overland flow as response to P is considered, this is not of large importance, but for GR it should matter, shouldn’t it? Or is average soil moisture state implicitly considered in NRF analogously to the rational formula? The runoff coefficient in the rational formula is however an integral over an event duration of a longer time period, but NRF is a function resolved in time depending on precipitation intensity. I somehow miss soil moisture in this resolved representation. Please, explain.
Thank you for the question. We assume that the reviewer refers to Figure 8 as “NRF(P)” and Figure 9 as “NRF (GR)”.
Figure 8 shows the nonlinear runoff response to precipitation, where NRF is the product between precipitation-intensity-dependent runoff response function (RRD) and the precipitation rate (as defined in Eq. (4)). It quantifies how streamflow responds to one time step of precipitation at a given intensity. Antecedent wetness is treated in ERRA as a category variable, in which the NRF is estimated for two or more different ranges of soil moisture, groundwater levels, antecedent discharge, or antecedent precipitation. The contrast between wet and dry antecedent moisture conditions, as inferred from water table depth, is illustrated by comparing the "wet" and "dry" curves in the figure. In Figure 10, we show that one obtains similar results, whether one infers antecedent wetness from antecedent soil moisture or from antecedent water table depth.
The “NRF(GR)” in Figure 9 is not defined as “a response function of discharge to amount of Groundwater recharge in each increment”. Figure 9 shows the nonlinear groundwater recharge response to precipitation, where NRF is the product between precipitation-intensity-dependent groundwater recharge response and the precipitation rate. It should be interpreted as the rate of groundwater recharge expected to result at a given time lag from precipitation falling at a specific rate. This coupling between precipitation and groundwater recharge will be sensitive to antecedent wetness, as shown by the two curves in Figure 9, but they do not reflect the influence of soil moisture status on streamflow per se. Again, Figure 10 shows that antecedent soil moisture and antecedent water table depth are nearly equivalent as indicators of antecedent wetness.
- The results presented in section 3 rigorously demonstrate that the formulated hypothesis that the double-peak hydrograph is generated by the combination of near-surface and groundwater-mediated pathway cannot be rejected. An alternative hypothesis that only e.g., groundwater-mediated pathway would be responsible must be rejected. I think the paper would benefit if the authors more clearly articulate the results in Karl Popper’s sense that the formulated hypothesis could not be rejected.
Thank you for the suggestion. Two alternative hypotheses are actually rejected in Section 3.3.2: the disconnect between the two curves in Figure 6b rejects the hypothesis that only near-surface pathway can explain the total rainfall-runoff response (because it doesn't explain the second peak), and the disconnect between the two curves in Figure 6c rejects the hypothesis that only groundwater-mediated pathway would be responsible (because it doesn't explain the first peak). We will make this clear in our discussion of Figure 6 in the revised paper.
- L461-465: Do I understand correctly that in order to analyze the effect of catchment wetness and precipitation intensity on hydrograph generation you need to separate different periods when e.g., WTD falls into three designated classes, and then you estimate NRFs for these periods? Please, explain your methodological steps for clarity.
Yes for WTD. For nonstationary analysis (i.e., runoff may respond differently depending on catchment wetness status), we need to separate the precipitation time series into different categories according to the antecedent WTD when rain falls.
In Section 4.1 (L431-459), we estimate RRD for each antecedent wetness category. Comparing RRDs between different categories only examines nonstationary runoff response.
In Section 4.2 (L461-506), we estimate NRFs for each antecedent wetness category. Comparing NRFs within each category only looks at nonlinear runoff response, and comparing NRFs between different categories jointly looks at both nonlinear and nonstationary runoff response.
We will further clarify this in the revised manuscript.
- Is this also done analogously for the precipitation intensity? Precipitation intensity is however, much more volatile than catchment average WTD. How does this affect the results if one maybe need to pick just one or two hours of intensive precipitation out of the entire event. Does ERRA then clearly separates NRFs for one-two hours of intensive precipitation framed by a few hours of less intensive precipitation belonging to another class (and resulting in a different NRF) prior and after a major downpour?
No for precipitation intensity. For nonlinear runoff response analysis (i.e., runoff may respond more-than-proportionally to changes in precipitation intensity), NRF equals the precipitation-intensity-dependent RRD times the precipitation rate (Eq. 4), which is “approximated in ERRA by a continuous piecewise-linear broken-stick function of precipitation intensity” (L195).
Precipitation is divided into segments between specified precipitation intensity values ("knots"), and the NRF_k in Eq.(4) (the nonlinear response of streamflow to precipitation that falls at a rate P_j-k and lasts for a time step of ∆t) is the sum of these segments, each multiplied by the slopes of the corresponding broken-stick segments (Kirchner 2024a). That is, NRF is an estimate of the ensemble average of the responses to many rainfall events instead of only over an individual hour or two at a given precipitation intensity.
Thank you for comments 4 and 5. We will add a figure illustrating the calculation of the Nonlinear Response Function (NRF) in Section 2.3.3 in our revised manuscript for clearer clarification.
- The analysis of nonstationarity and nonlinearity in section 4 is very interesting, but sensitivities expressed in absolute numbers remain pertinent to the study catchment. I am wondering if one could derive some dimensionless or relative measures that can be generalized when investigations from a large set of catchments would be available. Would it be helpful for example to look not at the catchment average WTD but in relation to the mean annual precipitation?
Thank you. We are indeed conducting a large-sample study of nonlinearity and nonstationarity, but that would be a completely different analysis. An example of large set of catchments study looking at WTD’s influence on runoff response can be found here for your reference:
Eslami, Z., Seybold, H., and Kirchner, J. W.: Climatic, topographic, and groundwater controls on runoff response to precipitation: evidence from a large-sample data set, EGUsphere [preprint], https://doi.org/10.5194/egusphere-2025-35, 2025.
Looking at the relationship to mean annual precipitation may work in inter-catchment comparisons, but not here in the nonstationary analysis of our manuscript because the single value of mean annual precipitation in a catchment cannot be used to split the precipitation time series into different categories in the nonstationary response analysis or to divide different precipitation intensity ranges in the nonlinear response analysis.
- I understand that results presented in section 4 are catchment specific and much of the discussion links to previous studies and mechanisms of runoff generation in this very catchment, but still maybe some ideas for generalizations may enrich the discussion part from L539ff.
Thank you for the point. We presented a site-specific discussion because we are reluctant to generalize beyond the data that we actually have, and have actually analyzed. We will consider this in the revised manuscript but would be cautious about it.
- Finally, the potential artifact with near-zero spike mentioned in section 3.1 seems to remain unclear. It says in L234 it will be explored in the next section, but I somehow missed a detailed analysis. Do you mean that the exploration is given in L274-277 which basically concludes that you do not have enough data to pinpoint the origin of this spike? Or did I miss further elaboration on this issue in the manuscript? If there is not enough data to further explore this artifact, I suggest not to raise expectation in L234.
The exploration of the potential artifact with near-zero spike was indeed given in L274-277. This artifact first appears in Figure 3c, where we inferred that it could potentially be caused by the distortion of the strong short-lag relationship between groundwater recharge and precipitation (observed in Figure 3b) or the rapid runoff effects of groundwater recharge in the near-stream zone. We then explored how the spike can be reduced by jointly analyzing the mixing effects of correlated precipitation and groundwater recharge on streamflow (Figure 4). To resolve the source of the remaining spike would require data that unfortunately don't exist.
Thank you for the suggestion. We will clarify this in our revised manuscript.
Citation: https://doi.org/10.5194/egusphere-2025-613-AC2
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CC1: 'Reply to RC1', James Kirchner, 06 May 2025
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RC2: 'Comment on egusphere-2025-613', Anonymous Referee #2, 16 May 2025
The manuscript is focusing on the streamflow analysis with Kirchner’s ERRA- method. The catchment reacts depending on the antecedent conditions with or without a double peaked runoff response. The authors describe the method and the applicability of it. The influence of precipitation intensity is shown and the effect of different antecedent wetness measures are shown. The manuscript delivers a simple analysis method, with which these complex runoff responses could be estimated and is therefore an important contribution for the scientific community for runoff response in small to medium sized hydrological catchments for quantitative but as well for qualitative perspectives. Because in the second peak could be substances transported which were remobilised like plant protection products, fertilisers and their metabolites, etc. depending of the pH values and the solutability.
There are several parts which could be moved to an appendix. And the important topic of measures which could be used as antecedent conditions measures is missing in the introduction. The structure is confusing. The rainfall intensity is quite dominant. The antecedent condition measures have only a small part of the manuscript but are quite important. In chapter 5 it is not clear how they were considered no equation is presented for soil moisture, and antecedent runoff.
The mathematical description of the approach is separated into two parts and should be presented in one block. Figure 3 and 4 are results and should be moved to 3.3.2 or to the appendix.
I would suggest a complete reorganisation of the manuscript and more clear formulation of the hypotheses.
Abstract:
Add the different used antecedent conditions measures and which gave the best results.
Introduction:
Add which antecedent wetness conditions could be used which is a crucial measure to detect the double peak phenomena (antecedent precipitation index, antecedent soil moisture index, antecedent groundwater, and pre- event runoff)
Comparison of different proxies for catchment antecedent wetness conditions
I was expecting that the authors present a threshold value at which double peaks occur.
The authors should explain why they have selected antecedent groundwater table and what speaks against soil moisture and pre- runoff conditions. Explain why the lowest soil moisture probe was selected and not a mean value of the probes.
If soil moisture would be equivalent wouldn’t it be the better proxy because installing probes is easier to install and less cost intensive?
Specific comments:
Page 4 Figure1: which GW gauge was used for the analysis?
Page 5 Line 127 which type of device was used
Page 6 Line 145 is the presented VWC the mean value for all soil moisture monitoring points? Or is it a specific point in figure 1?
Page 9 Line 207: RRDP is defined in eq. 2
Page 10 Line 219 equation is missing for GRRDP - it is presented at page 14 Line: 341
Page 14 Line 350- 352: equation 10 or 11 could be removed
Page 25 Lines: 566-571: Is it important to know the percentiles?
Page 27 Line 622 This is exactly the same conclusion like in the Schaefertal catchment (Graeff et al. 2009) where the double peak events could not be any more observed after mining activities below the catchment started and groundwater was completely disturbed
Citation: https://doi.org/10.5194/egusphere-2025-613-RC2 -
AC1: 'Reply on RC2', Huibin Gao, 27 May 2025
We thank Reviewer #2 for these helpful and insightful comments.
Our point-by-point response below shows reviewer’s comments in plain text and our responses in bold.
The manuscript is focusing on the streamflow analysis with Kirchner’s ERRA- method. The catchment reacts depending on the antecedent conditions with or without a double peaked runoff response. The authors describe the method and the applicability of it. The influence of precipitation intensity is shown and the effect of different antecedent wetness measures are shown. The manuscript delivers a simple analysis method, with which these complex runoff responses could be estimated and is therefore an important contribution for the scientific community for runoff response in small to medium sized hydrological catchments for quantitative but as well for qualitative perspectives.
Thank you for your support.
Because in the second peak could be substances transported which were remobilised like plant protection products, fertilisers and their metabolites, etc. depending of the pH values and the solutability.
We appreciate this point, but we would interpret it more generally: understanding the processes/flowpaths underlying both peaks, not just the second one, may be important because different flowpaths may carry different potential contaminants.
We will add this perspective to the introduction.
There are several parts which could be moved to an appendix. And the important topic of measures which could be used as antecedent conditions measures is missing in the introduction. The structure is confusing. The rainfall intensity is quite dominant. The antecedent condition measures have only a small part of the manuscript but are quite important. In chapter 5 it is not clear how they were considered no equation is presented for soil moisture, and antecedent runoff.
We appreciate the reviewer’s enthusiasm concerning measures of antecedent conditions, but that is not the focus of our paper. ERRA is a method for data analysis, not a simulation model, so it is not straightforward to declare one or another measure of antecedent conditions as the “best”, even at this one site, and any such result would not be transferable to other sites. In Section 5.2, we show that all antecedent water table depth, antecedent soil moisture, and antecedent discharge all yield broadly similar results at this site. We also point out that these different proxies for antecedent conditions have different response times, so that the choice among them (in cases where one is lucky enough to have any such choice at all) will depend on the question one is trying to answer.
We did not present an equation for soil moisture or antecedent runoff because these are just lagged values from the soil moisture and discharge time series. We will see if we can more explicitly point this out in the manuscript.
The mathematical description of the approach is separated into two parts and should be presented in one block. Figure 3 and 4 are results and should be moved to 3.3.2 or to the appendix.
It appears that our use of "Result" as the heading for Section 3.3.2 has created the impression that all of the results are there, or should be there, whereas this is only the result of the hypothesis test posed in Section 3.3.1. In fact, Sections 3 and 4 (and Figures 3-8) are all results, with Section 5 and Figures 9-10 extending the results (which is the point of a discussion section).
Thank you for pointing this out. We will consider changing the heading for Section 3.3.2 (and potentially other sections) in our revised manuscript.
What the reviewer refers to as “the mathematical description of the approach” is actually two different things, which would be very confusing if they were presented “in one block”. The first, which is presented in Section 2.3, is an overview of Ensemble Rainfall-Runoff Analysis, which is essential background for everything that follows. Readers then need to see the results from this analysis (Figures 3 and 4) before they are in a position to understand the motivation behind the hypothesized two pathways underlying the double-peak hydrograph, and the convolution model for formally testing that hypothesis (Section 3.3 and Figure 5). It would not be an effective communication strategy to present this convolution model before readers had any idea what it was needed for.
I would suggest a complete reorganisation of the manuscript and more clear formulation of the hypotheses.
The organizational structure of the manuscript grew out of careful consideration of what readers need to know at each point in the paper (as illustrated, for example, by our response immediately above). We will consider whether this can be further improved, but many "quick fixes" that seem superficially attractive would be counterproductive in practice (like, as discussed above, presenting all the math "in one block").
Regarding the hypothesis in Section 3.3, we will state more clearly that Figure 6 also tests two alternative hypotheses. The disconnect between the two curves in Figure 6b rejects the hypothesis that only the near-surface pathway can explain the total rainfall-runoff response (because it doesn't explain the second peak), and the disconnect between the two curves in Figure 6c rejects the hypothesis that only the groundwater-mediated pathway would be responsible (because it doesn't explain the first peak).
We will make this clear in our discussion of Figure 6 in the revised paper.
Abstract:
Add the different used antecedent conditions measures and which gave the best results.
As indicated above, the purpose of the paper is not to compare different measures of antecedent wetness, and there is no clear standard to assess one or another as being "best".
We will change Line 21 to "Under wet conditions (as inferred from antecedent water table depth ≤ 1.66 m), the first peak increases nonlinearly..." in our revised manuscript to make it more explicit.
Introduction:
Add which antecedent wetness conditions could be used which is a crucial measure to detect the double peak phenomena (antecedent precipitation index, antecedent soil moisture index, antecedent groundwater, and pre- event runoff)
Antecedent wetness proxies are not, in fact "crucial... to detect the double peak phenomena". For example, Figures 3-6 clearly illustrate the double-peak phenomenon without any information about antecedent wetness.
Comparison of different proxies for catchment antecedent wetness conditions
We are not sure what is meant here. Figure 10 compares the results obtained with different antecedent wetness proxies.
I was expecting that the authors present a threshold value at which double peaks occur.
Thank you for the point. We are reluctant to specify a particular threshold for the emergence of double peaks, for several reasons. The occurrence of double peaks depends on both antecedent wetness and precipitation intensity. The general tendencies in these relationships are clear; as Figure 10 shows, the second peak is obvious when water table depth is between 1.66 and 1.30 m, but negligible when water table depth is below 1.66 m. And it is more obvious at higher precipitation intensities. But it is difficult to define a specific threshold at which the second peak emerges (vs. at which it is present but very small). And even if we arbitrarily define a size threshold for the second peak (at which it would be declared to be "present" rather than "absent"), any corresponding threshold of antecedent wetness would be specific to Weierbach and not amenable to generalization.
The authors should explain why they have selected antecedent groundwater table and what speaks against soil moisture and pre- runoff conditions.
We used antecedent water table depth (a) because of the evident role of groundwater in generating the second peak, (b) because the available soil moisture measurements only reflect the wetness state of the upper 60 cm of the subsurface, and (c) because the interpretation of antecedent streamflow necessarily depends on whether that streamflow results from the first peak (which is driven primarily by precipitation intensity) or the second peak (which is driven primarily by catchment wetness, and more specifically groundwater).
Thank you for the suggestion. We will explain this in Section 5.2 in our revised manuscript.
Explain why the lowest soil moisture probe was selected and not a mean value of the probes.
The assumption made here is incorrect. Our analysis used all the available probes. Line 582 says explicitly that antecedent VWC reflects soils ≤ 60 cm (i.e., the entire depth range of all the soil moisture probes, but we will revise Section 5.2 to make this more explicit.
If soil moisture would be equivalent wouldn’t it be the better proxy because installing probes is easier to install and less cost intensive?
We are not prepared to make such categorical value judgments. Soil moisture probes will be a better proxy if the most mechanistically relevant antecedent wetness is soil moisture rather than groundwater storage. Whether soil moisture probes are easier to install and less cost intensive will depend on how many of them are needed to adequately capture the spatial heterogeneity in soil moisture (which will not be known in advance). Conversely, the effort and expense of monitoring groundwater will depend on how heterogeneous the substrate is and how difficult it is to drill, and on how large the effective footprint of each well is (which will also not be known in advance).
In practice, researchers are likely to use whatever antecedent wetness proxies are available, which is why we undertook to compare them in Section 5.2, taking advantage of the fact that all three of them are available at Weierbach.
Specific comments:
Page 4 Figure1: which GW gauge was used for the analysis?
The manuscript already explains that we used the average of all three wells. Lines 124-126 say that we used the three GW wells with the most complete records (the three shown in Figure 1). Line 137 says that we averaged the changes in WTD across all three wells to infer groundwater recharge. And line 433 says that we used the catchment-averaged water table depth as a proxy for antecedent wetness. We will look for ways to make these points even more explicit.
Page 5 Line 127 which type of device was used
Soil moisture is recorded on CR800 loggers using CS650 water content reflectometers (Campbell Scientific, Logan, Utah, USA) (Hissler et al., 2021).
Detailed descriptions of equipment and data collection can be found in Hissler (2021) (Line 118). We will note this in the revised manuscript.
Page 6 Line 145 is the presented VWC the mean value for all soil moisture monitoring points? Or is it a specific point in figure 1?
At each depth it is the average of all available probes.
Thank you for the question. We will add this in our revised version.
Page 9 Line 207: RRDP is defined in eq. 2
Not quite. Equation 2 defines the partial RRD, partialRRDP, discussed in Section 3.2.
RRDP and GRRDP are two different versions of the RRD defined in Equation 1. If the input is precipitation and the output is discharge, the RRD is denoted RRDP (the distribution of runoff response to precipitation). Alternatively, if the input is precipitation and the output is groundwater recharge, the RRD is denoted GRRDP (the distribution of groundwater recharge's response to precipitation).
Although this is already explained at the beginning of Sections 3.1 and 3.2, we do see how readers could miss it, so we will add explicit definitions of RRDP, GRRDP, and RRDGR in Section 2 (along with equations for them).
Page 10 Line 219 equation is missing for GRRDP - it is presented at page 14 Line: 341
Agreed. As noted above, we will add explicit definitions of RRDP, GRRDP, and RRDGR in Section 2.
Page 14 Line 350- 352: equation 10 or 11 could be removed
We disagree. Equation 10 follows from Equations 8 and 9, and without explicitly stating that convolution is associative one cannot get from Equation 10 to Equation 11. Equation 11, in turn, is necessary as a component of Equation 13.
Page 25 Lines: 566-571: Is it important to know the percentiles?
We think so. Otherwise it is difficult for readers to understand the ranges of catchment conditions corresponding to the plots in Figure 10.
Page 27 Line 622 This is exactly the same conclusion like in the Schaefertal catchment (Graeff et al. 2009) where the double peak events could not be any more observed after mining activities below the catchment started and groundwater was completely disturbed
We disagree that this is "exactly the same conclusion", since draining a catchment from below by mining is different from depleting groundwater by evapotranspiration and stream discharge. It is also not correct to say that double-peak events "could not be any more observed after mining activities" at Schaefertal, because Graeff et al. (p. 705) note that double-peak events were still observed "in response to very strong precipitation events".
However, we appreciate the pointer to this interesting study, which we were previously unaware of, and will mention it in the revised manuscript.
Citation: https://doi.org/10.5194/egusphere-2025-613-AC1
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AC1: 'Reply on RC2', Huibin Gao, 27 May 2025
Interactive discussion
Status: closed
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RC1: 'Comment on egusphere-2025-613', Anonymous Referee #1, 27 Apr 2025
The presented study applies the Ensemble Rainfall Runoff Analysis (ERRA) to explore the mechanisms of double-peak hydrograph emergence in the Weierbach catchment, Luxembourg. The authors test the hypothesis that the double-peak hydrograph is generated by two pathways – a near-surface runoff in direct response to precipitation and a pathway through vadose zone influenced by groundwater recharge. They further explore how double-peak runoff generation varies with precipitation intensities and antecedent catchment wetness. The controls of the first and second hydrograph peak are identified with their specific thresholds. The manuscript thus presents novel insights into the mechanistic functioning of runoff generation inferred with ERRA from multiple observational data in the Weierbach catchment. This is a very interesting study which rigorously works out the controls on the double peak hydrograph in the study catchment and clearly presents the results and in-depth discussion.
I have a few remarks that may further enhance the presentation and potentially increase the generalization of the case study results.
L65 – 69: The cited studies with specific numbers of catchment storage, precipitation volume and intensities are site-specific and very much depend on the catchment characteristics. It would be good to mention this in this paragraph.
With regards to NRF(P) /NRF (GR) the authors define it as a response function of discharge to amount of precipitation/Groundwater recharge in each increment, and this function in its turn depends on precipitation intensity/GR rate at time t. Should this function also depend on the antecedent moisture state as it controls the responsiveness of discharge to P/GR? In case only overland flow as response to P is considered, this is not of large importance, but for GR it should matter, shouldn’t it? Or is average soil moisture state implicitly considered in NRF analogously to the rational formula? The runoff coefficient in the rational formula is however an integral over an event duration of a longer time period, but NRF is a function resolved in time depending on precipitation intensity. I somehow miss soil moisture in this resolved representation. Please, explain.
The results presented in section 3 rigorously demonstrate that the formulated hypothesis that the double-peak hydrograph is generated by the combination of near-surface and groundwater-mediated pathway cannot be rejected. An alternative hypothesis that only e.g., groundwater-mediated pathway would be responsible must be rejected. I think the paper would benefit if the authors more clearly articulate the results in Karl Popper’s sense that the formulated hypothesis could not be rejected.
L461-465: Do I understand correctly that in order to analyze the effect of catchment wetness and precipitation intensity on hydrograph generation you need to separate different periods when e.g., WTD falls into three designated classes, and then you estimate NRFs for these periods? Please, explain your methodological steps for clarity.
Is this also done analogously for the precipitation intensity? Precipitation intensity is however, much more volatile than catchment average WTD. How does this affect the results if one maybe need to pick just one or two hours of intensive precipitation out of the entire event. Does ERRA then clearly separates NRFs for one-two hours of intensive precipitation framed by a few hours of less intensive precipitation belonging to another class (and resulting in a different NRF) prior and after a major downpour?
The analysis of nonstationarity and nonlinearity in section 4 is very interesting, but sensitivities expressed in absolute numbers remain pertinent to the study catchment. I am wondering if one could derive some dimensionless or relative measures that can be generalized when investigations from a large set of catchments would be available. Would it be helpful for example to look not at the catchment average WTD but in relation to the mean annual precipitation?
I understand that results presented in section 4 are catchment specific and much of the discussion links to previous studies and mechanisms of runoff generation in this very catchment, but still maybe some ideas for generalizations may enrich the discussion part from L539ff.
Finally, the potential artifact with near-zero spike mentioned in section 3.1 seems to remain unclear. It says in L234 it will be explored in the next section, but I somehow missed a detailed analysis. Do you mean that the exploration is given in L274-277 which basically concludes that you do not have enough data to pinpoint the origin of this spike? Or did I miss further elaboration on this issue in the manuscript? If there is not enough data to further explore this artifact, I suggest not to raise expectation in L234.
Citation: https://doi.org/10.5194/egusphere-2025-613-RC1 -
CC1: 'Reply to RC1', James Kirchner, 06 May 2025
We thank Reviewer #1 for these helpful and insightful comments. Our point-by-point response below shows reviewer’s comments in plain text and our responses in bold.
The presented study applies the Ensemble Rainfall Runoff Analysis (ERRA) to explore the mechanisms of double-peak hydrograph emergence in the Weierbach catchment, Luxembourg. The authors test the hypothesis that the double-peak hydrograph is generated by two pathways – a near-surface runoff in direct response to precipitation and a pathway through vadose zone influenced by groundwater recharge. They further explore how double-peak runoff generation varies with precipitation intensities and antecedent catchment wetness. The controls of the first and second hydrograph peak are identified with their specific thresholds. The manuscript thus presents novel insights into the mechanistic functioning of runoff generation inferred with ERRA from multiple observational data in the Weierbach catchment. This is a very interesting study which rigorously works out the controls on the double peak hydrograph in the study catchment and clearly presents the results and in-depth discussion.
Thank you!
I have a few remarks that may further enhance the presentation and potentially increase the generalization of the case study results.
- L65 – 69: The cited studies with specific numbers of catchment storage, precipitation volume and intensities are site-specific and very much depend on the catchment characteristics. It would be good to mention this in this paragraph.
Thank you for the suggestion. We will specify that in our revised manuscript.
- With regards to NRF(P) /NRF (GR) the authors define it as a response function of discharge to amount of precipitation/Groundwater recharge in each increment, and this function in its turn depends on precipitation intensity/GR rate at time t. Should this function also depend on the antecedent moisture state as it controls the responsiveness of discharge to P/GR? In case only overland flow as response to P is considered, this is not of large importance, but for GR it should matter, shouldn’t it? Or is average soil moisture state implicitly considered in NRF analogously to the rational formula? The runoff coefficient in the rational formula is however an integral over an event duration of a longer time period, but NRF is a function resolved in time depending on precipitation intensity. I somehow miss soil moisture in this resolved representation. Please, explain.
Thank you for the question. We assume that the reviewer refers to Figure 8 as “NRF(P)” and Figure 9 as “NRF (GR)”.
Figure 8 shows the nonlinear runoff response to precipitation, where NRF is the product between precipitation-intensity-dependent runoff response function (RRD) and the precipitation rate (as defined in Eq. (4)). It quantifies how streamflow responds to one time step of precipitation at a given intensity. Antecedent wetness is treated in ERRA as a category variable, in which the NRF is estimated for two or more different ranges of soil moisture, groundwater levels, antecedent discharge, or antecedent precipitation. The contrast between wet and dry antecedent moisture conditions, as inferred from water table depth, is illustrated by comparing the "wet" and "dry" curves in the figure. In Figure 10, we show that one obtains similar results, whether one infers antecedent wetness from antecedent soil moisture or from antecedent water table depth.
The “NRF(GR)” in Figure 9 is not defined as “a response function of discharge to amount of Groundwater recharge in each increment”. Figure 9 shows the nonlinear groundwater recharge response to precipitation, where NRF is the product between precipitation-intensity-dependent groundwater recharge response and the precipitation rate. It should be interpreted as the rate of groundwater recharge expected to result at a given time lag from precipitation falling at a specific rate. This coupling between precipitation and groundwater recharge will be sensitive to antecedent wetness, as shown by the two curves in Figure 9, but they do not reflect the influence of soil moisture status on streamflow per se. Again, Figure 10 shows that antecedent soil moisture and antecedent water table depth are nearly equivalent as indicators of antecedent wetness.
- The results presented in section 3 rigorously demonstrate that the formulated hypothesis that the double-peak hydrograph is generated by the combination of near-surface and groundwater-mediated pathway cannot be rejected. An alternative hypothesis that only e.g., groundwater-mediated pathway would be responsible must be rejected. I think the paper would benefit if the authors more clearly articulate the results in Karl Popper’s sense that the formulated hypothesis could not be rejected.
Thank you for the suggestion. Two alternative hypotheses are actually rejected in Section 3.3.2: the disconnect between the two curves in Figure 6b rejects the hypothesis that only near-surface pathway can explain the total rainfall-runoff response (because it doesn't explain the second peak), and the disconnect between the two curves in Figure 6c rejects the hypothesis that only groundwater-mediated pathway would be responsible (because it doesn't explain the first peak). We will make this clear in our discussion of Figure 6 in the revised paper.
- L461-465: Do I understand correctly that in order to analyze the effect of catchment wetness and precipitation intensity on hydrograph generation you need to separate different periods when e.g., WTD falls into three designated classes, and then you estimate NRFs for these periods? Please, explain your methodological steps for clarity.
Yes for WTD. For nonstationary analysis (i.e., runoff may respond differently depending on catchment wetness status), we need to separate the precipitation time series into different categories according to the antecedent WTD when rain falls.
In Section 4.1 (L431-459), we estimate RRD for each antecedent wetness category. Comparing RRDs between different categories only examines nonstationary runoff response.
In Section 4.2 (L461-506), we estimate NRFs for each antecedent wetness category. Comparing NRFs within each category only looks at nonlinear runoff response, and comparing NRFs between different categories jointly looks at both nonlinear and nonstationary runoff response.
We will further clarify this in the revised manuscript.
- Is this also done analogously for the precipitation intensity? Precipitation intensity is however, much more volatile than catchment average WTD. How does this affect the results if one maybe need to pick just one or two hours of intensive precipitation out of the entire event. Does ERRA then clearly separates NRFs for one-two hours of intensive precipitation framed by a few hours of less intensive precipitation belonging to another class (and resulting in a different NRF) prior and after a major downpour?
No for precipitation intensity. For nonlinear runoff response analysis (i.e., runoff may respond more-than-proportionally to changes in precipitation intensity), NRF equals the precipitation-intensity-dependent RRD times the precipitation rate (Eq. 4), which is “approximated in ERRA by a continuous piecewise-linear broken-stick function of precipitation intensity” (L195).
Precipitation is divided into segments between specified precipitation intensity values ("knots"), and the NRF_k in Eq.(4) (the nonlinear response of streamflow to precipitation that falls at a rate P_j-k and lasts for a time step of ∆t) is the sum of these segments, each multiplied by the slopes of the corresponding broken-stick segments (Kirchner 2024a). That is, NRF is an estimate of the ensemble average of the responses to many rainfall events instead of only over an individual hour or two at a given precipitation intensity.
Thank you for comments 4 and 5. We will add a figure illustrating the calculation of the Nonlinear Response Function (NRF) in Section 2.3.3 in our revised manuscript for clearer clarification.
- The analysis of nonstationarity and nonlinearity in section 4 is very interesting, but sensitivities expressed in absolute numbers remain pertinent to the study catchment. I am wondering if one could derive some dimensionless or relative measures that can be generalized when investigations from a large set of catchments would be available. Would it be helpful for example to look not at the catchment average WTD but in relation to the mean annual precipitation?
Thank you. We are indeed conducting a large-sample study of nonlinearity and nonstationarity, but that would be a completely different analysis. An example of large set of catchments study looking at WTD’s influence on runoff response can be found here for your reference:
Eslami, Z., Seybold, H., and Kirchner, J. W.: Climatic, topographic, and groundwater controls on runoff response to precipitation: evidence from a large-sample data set, EGUsphere [preprint], https://doi.org/10.5194/egusphere-2025-35, 2025.
Looking at the relationship to mean annual precipitation may work in inter-catchment comparisons, but not here in the nonstationary analysis of our manuscript because the single value of mean annual precipitation in a catchment cannot be used to split the precipitation time series into different categories in the nonstationary response analysis or to divide different precipitation intensity ranges in the nonlinear response analysis.
- I understand that results presented in section 4 are catchment specific and much of the discussion links to previous studies and mechanisms of runoff generation in this very catchment, but still maybe some ideas for generalizations may enrich the discussion part from L539ff.
Thank you for the point. We presented a site-specific discussion because we are reluctant to generalize beyond the data that we actually have, and have actually analyzed. We will consider this in the revised manuscript but would be cautious about it.
- Finally, the potential artifact with near-zero spike mentioned in section 3.1 seems to remain unclear. It says in L234 it will be explored in the next section, but I somehow missed a detailed analysis. Do you mean that the exploration is given in L274-277 which basically concludes that you do not have enough data to pinpoint the origin of this spike? Or did I miss further elaboration on this issue in the manuscript? If there is not enough data to further explore this artifact, I suggest not to raise expectation in L234.
The exploration of the potential artifact with near-zero spike was indeed given in L274-277. This artifact first appears in Figure 3c, where we inferred that it could potentially be caused by the distortion of the strong short-lag relationship between groundwater recharge and precipitation (observed in Figure 3b) or the rapid runoff effects of groundwater recharge in the near-stream zone. We then explored how the spike can be reduced by jointly analyzing the mixing effects of correlated precipitation and groundwater recharge on streamflow (Figure 4). To resolve the source of the remaining spike would require data that unfortunately don't exist.
Thank you for the suggestion. We will clarify this in our revised manuscript.
Citation: https://doi.org/10.5194/egusphere-2025-613-CC1 -
AC2: 'Reply on RC1', Huibin Gao, 27 May 2025
We thank Reviewer #1 for these helpful and insightful comments. Our point-by-point response below shows reviewer’s comments in plain text and our responses in bold.
The presented study applies the Ensemble Rainfall Runoff Analysis (ERRA) to explore the mechanisms of double-peak hydrograph emergence in the Weierbach catchment, Luxembourg. The authors test the hypothesis that the double-peak hydrograph is generated by two pathways – a near-surface runoff in direct response to precipitation and a pathway through vadose zone influenced by groundwater recharge. They further explore how double-peak runoff generation varies with precipitation intensities and antecedent catchment wetness. The controls of the first and second hydrograph peak are identified with their specific thresholds. The manuscript thus presents novel insights into the mechanistic functioning of runoff generation inferred with ERRA from multiple observational data in the Weierbach catchment. This is a very interesting study which rigorously works out the controls on the double peak hydrograph in the study catchment and clearly presents the results and in-depth discussion.
Thank you!
I have a few remarks that may further enhance the presentation and potentially increase the generalization of the case study results.
- L65 – 69: The cited studies with specific numbers of catchment storage, precipitation volume and intensities are site-specific and very much depend on the catchment characteristics. It would be good to mention this in this paragraph.
Thank you for the suggestion. We will specify that in our revised manuscript.
- With regards to NRF(P) /NRF (GR) the authors define it as a response function of discharge to amount of precipitation/Groundwater recharge in each increment, and this function in its turn depends on precipitation intensity/GR rate at time t. Should this function also depend on the antecedent moisture state as it controls the responsiveness of discharge to P/GR? In case only overland flow as response to P is considered, this is not of large importance, but for GR it should matter, shouldn’t it? Or is average soil moisture state implicitly considered in NRF analogously to the rational formula? The runoff coefficient in the rational formula is however an integral over an event duration of a longer time period, but NRF is a function resolved in time depending on precipitation intensity. I somehow miss soil moisture in this resolved representation. Please, explain.
Thank you for the question. We assume that the reviewer refers to Figure 8 as “NRF(P)” and Figure 9 as “NRF (GR)”.
Figure 8 shows the nonlinear runoff response to precipitation, where NRF is the product between precipitation-intensity-dependent runoff response function (RRD) and the precipitation rate (as defined in Eq. (4)). It quantifies how streamflow responds to one time step of precipitation at a given intensity. Antecedent wetness is treated in ERRA as a category variable, in which the NRF is estimated for two or more different ranges of soil moisture, groundwater levels, antecedent discharge, or antecedent precipitation. The contrast between wet and dry antecedent moisture conditions, as inferred from water table depth, is illustrated by comparing the "wet" and "dry" curves in the figure. In Figure 10, we show that one obtains similar results, whether one infers antecedent wetness from antecedent soil moisture or from antecedent water table depth.
The “NRF(GR)” in Figure 9 is not defined as “a response function of discharge to amount of Groundwater recharge in each increment”. Figure 9 shows the nonlinear groundwater recharge response to precipitation, where NRF is the product between precipitation-intensity-dependent groundwater recharge response and the precipitation rate. It should be interpreted as the rate of groundwater recharge expected to result at a given time lag from precipitation falling at a specific rate. This coupling between precipitation and groundwater recharge will be sensitive to antecedent wetness, as shown by the two curves in Figure 9, but they do not reflect the influence of soil moisture status on streamflow per se. Again, Figure 10 shows that antecedent soil moisture and antecedent water table depth are nearly equivalent as indicators of antecedent wetness.
- The results presented in section 3 rigorously demonstrate that the formulated hypothesis that the double-peak hydrograph is generated by the combination of near-surface and groundwater-mediated pathway cannot be rejected. An alternative hypothesis that only e.g., groundwater-mediated pathway would be responsible must be rejected. I think the paper would benefit if the authors more clearly articulate the results in Karl Popper’s sense that the formulated hypothesis could not be rejected.
Thank you for the suggestion. Two alternative hypotheses are actually rejected in Section 3.3.2: the disconnect between the two curves in Figure 6b rejects the hypothesis that only near-surface pathway can explain the total rainfall-runoff response (because it doesn't explain the second peak), and the disconnect between the two curves in Figure 6c rejects the hypothesis that only groundwater-mediated pathway would be responsible (because it doesn't explain the first peak). We will make this clear in our discussion of Figure 6 in the revised paper.
- L461-465: Do I understand correctly that in order to analyze the effect of catchment wetness and precipitation intensity on hydrograph generation you need to separate different periods when e.g., WTD falls into three designated classes, and then you estimate NRFs for these periods? Please, explain your methodological steps for clarity.
Yes for WTD. For nonstationary analysis (i.e., runoff may respond differently depending on catchment wetness status), we need to separate the precipitation time series into different categories according to the antecedent WTD when rain falls.
In Section 4.1 (L431-459), we estimate RRD for each antecedent wetness category. Comparing RRDs between different categories only examines nonstationary runoff response.
In Section 4.2 (L461-506), we estimate NRFs for each antecedent wetness category. Comparing NRFs within each category only looks at nonlinear runoff response, and comparing NRFs between different categories jointly looks at both nonlinear and nonstationary runoff response.
We will further clarify this in the revised manuscript.
- Is this also done analogously for the precipitation intensity? Precipitation intensity is however, much more volatile than catchment average WTD. How does this affect the results if one maybe need to pick just one or two hours of intensive precipitation out of the entire event. Does ERRA then clearly separates NRFs for one-two hours of intensive precipitation framed by a few hours of less intensive precipitation belonging to another class (and resulting in a different NRF) prior and after a major downpour?
No for precipitation intensity. For nonlinear runoff response analysis (i.e., runoff may respond more-than-proportionally to changes in precipitation intensity), NRF equals the precipitation-intensity-dependent RRD times the precipitation rate (Eq. 4), which is “approximated in ERRA by a continuous piecewise-linear broken-stick function of precipitation intensity” (L195).
Precipitation is divided into segments between specified precipitation intensity values ("knots"), and the NRF_k in Eq.(4) (the nonlinear response of streamflow to precipitation that falls at a rate P_j-k and lasts for a time step of ∆t) is the sum of these segments, each multiplied by the slopes of the corresponding broken-stick segments (Kirchner 2024a). That is, NRF is an estimate of the ensemble average of the responses to many rainfall events instead of only over an individual hour or two at a given precipitation intensity.
Thank you for comments 4 and 5. We will add a figure illustrating the calculation of the Nonlinear Response Function (NRF) in Section 2.3.3 in our revised manuscript for clearer clarification.
- The analysis of nonstationarity and nonlinearity in section 4 is very interesting, but sensitivities expressed in absolute numbers remain pertinent to the study catchment. I am wondering if one could derive some dimensionless or relative measures that can be generalized when investigations from a large set of catchments would be available. Would it be helpful for example to look not at the catchment average WTD but in relation to the mean annual precipitation?
Thank you. We are indeed conducting a large-sample study of nonlinearity and nonstationarity, but that would be a completely different analysis. An example of large set of catchments study looking at WTD’s influence on runoff response can be found here for your reference:
Eslami, Z., Seybold, H., and Kirchner, J. W.: Climatic, topographic, and groundwater controls on runoff response to precipitation: evidence from a large-sample data set, EGUsphere [preprint], https://doi.org/10.5194/egusphere-2025-35, 2025.
Looking at the relationship to mean annual precipitation may work in inter-catchment comparisons, but not here in the nonstationary analysis of our manuscript because the single value of mean annual precipitation in a catchment cannot be used to split the precipitation time series into different categories in the nonstationary response analysis or to divide different precipitation intensity ranges in the nonlinear response analysis.
- I understand that results presented in section 4 are catchment specific and much of the discussion links to previous studies and mechanisms of runoff generation in this very catchment, but still maybe some ideas for generalizations may enrich the discussion part from L539ff.
Thank you for the point. We presented a site-specific discussion because we are reluctant to generalize beyond the data that we actually have, and have actually analyzed. We will consider this in the revised manuscript but would be cautious about it.
- Finally, the potential artifact with near-zero spike mentioned in section 3.1 seems to remain unclear. It says in L234 it will be explored in the next section, but I somehow missed a detailed analysis. Do you mean that the exploration is given in L274-277 which basically concludes that you do not have enough data to pinpoint the origin of this spike? Or did I miss further elaboration on this issue in the manuscript? If there is not enough data to further explore this artifact, I suggest not to raise expectation in L234.
The exploration of the potential artifact with near-zero spike was indeed given in L274-277. This artifact first appears in Figure 3c, where we inferred that it could potentially be caused by the distortion of the strong short-lag relationship between groundwater recharge and precipitation (observed in Figure 3b) or the rapid runoff effects of groundwater recharge in the near-stream zone. We then explored how the spike can be reduced by jointly analyzing the mixing effects of correlated precipitation and groundwater recharge on streamflow (Figure 4). To resolve the source of the remaining spike would require data that unfortunately don't exist.
Thank you for the suggestion. We will clarify this in our revised manuscript.
Citation: https://doi.org/10.5194/egusphere-2025-613-AC2
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CC1: 'Reply to RC1', James Kirchner, 06 May 2025
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RC2: 'Comment on egusphere-2025-613', Anonymous Referee #2, 16 May 2025
The manuscript is focusing on the streamflow analysis with Kirchner’s ERRA- method. The catchment reacts depending on the antecedent conditions with or without a double peaked runoff response. The authors describe the method and the applicability of it. The influence of precipitation intensity is shown and the effect of different antecedent wetness measures are shown. The manuscript delivers a simple analysis method, with which these complex runoff responses could be estimated and is therefore an important contribution for the scientific community for runoff response in small to medium sized hydrological catchments for quantitative but as well for qualitative perspectives. Because in the second peak could be substances transported which were remobilised like plant protection products, fertilisers and their metabolites, etc. depending of the pH values and the solutability.
There are several parts which could be moved to an appendix. And the important topic of measures which could be used as antecedent conditions measures is missing in the introduction. The structure is confusing. The rainfall intensity is quite dominant. The antecedent condition measures have only a small part of the manuscript but are quite important. In chapter 5 it is not clear how they were considered no equation is presented for soil moisture, and antecedent runoff.
The mathematical description of the approach is separated into two parts and should be presented in one block. Figure 3 and 4 are results and should be moved to 3.3.2 or to the appendix.
I would suggest a complete reorganisation of the manuscript and more clear formulation of the hypotheses.
Abstract:
Add the different used antecedent conditions measures and which gave the best results.
Introduction:
Add which antecedent wetness conditions could be used which is a crucial measure to detect the double peak phenomena (antecedent precipitation index, antecedent soil moisture index, antecedent groundwater, and pre- event runoff)
Comparison of different proxies for catchment antecedent wetness conditions
I was expecting that the authors present a threshold value at which double peaks occur.
The authors should explain why they have selected antecedent groundwater table and what speaks against soil moisture and pre- runoff conditions. Explain why the lowest soil moisture probe was selected and not a mean value of the probes.
If soil moisture would be equivalent wouldn’t it be the better proxy because installing probes is easier to install and less cost intensive?
Specific comments:
Page 4 Figure1: which GW gauge was used for the analysis?
Page 5 Line 127 which type of device was used
Page 6 Line 145 is the presented VWC the mean value for all soil moisture monitoring points? Or is it a specific point in figure 1?
Page 9 Line 207: RRDP is defined in eq. 2
Page 10 Line 219 equation is missing for GRRDP - it is presented at page 14 Line: 341
Page 14 Line 350- 352: equation 10 or 11 could be removed
Page 25 Lines: 566-571: Is it important to know the percentiles?
Page 27 Line 622 This is exactly the same conclusion like in the Schaefertal catchment (Graeff et al. 2009) where the double peak events could not be any more observed after mining activities below the catchment started and groundwater was completely disturbed
Citation: https://doi.org/10.5194/egusphere-2025-613-RC2 -
AC1: 'Reply on RC2', Huibin Gao, 27 May 2025
We thank Reviewer #2 for these helpful and insightful comments.
Our point-by-point response below shows reviewer’s comments in plain text and our responses in bold.
The manuscript is focusing on the streamflow analysis with Kirchner’s ERRA- method. The catchment reacts depending on the antecedent conditions with or without a double peaked runoff response. The authors describe the method and the applicability of it. The influence of precipitation intensity is shown and the effect of different antecedent wetness measures are shown. The manuscript delivers a simple analysis method, with which these complex runoff responses could be estimated and is therefore an important contribution for the scientific community for runoff response in small to medium sized hydrological catchments for quantitative but as well for qualitative perspectives.
Thank you for your support.
Because in the second peak could be substances transported which were remobilised like plant protection products, fertilisers and their metabolites, etc. depending of the pH values and the solutability.
We appreciate this point, but we would interpret it more generally: understanding the processes/flowpaths underlying both peaks, not just the second one, may be important because different flowpaths may carry different potential contaminants.
We will add this perspective to the introduction.
There are several parts which could be moved to an appendix. And the important topic of measures which could be used as antecedent conditions measures is missing in the introduction. The structure is confusing. The rainfall intensity is quite dominant. The antecedent condition measures have only a small part of the manuscript but are quite important. In chapter 5 it is not clear how they were considered no equation is presented for soil moisture, and antecedent runoff.
We appreciate the reviewer’s enthusiasm concerning measures of antecedent conditions, but that is not the focus of our paper. ERRA is a method for data analysis, not a simulation model, so it is not straightforward to declare one or another measure of antecedent conditions as the “best”, even at this one site, and any such result would not be transferable to other sites. In Section 5.2, we show that all antecedent water table depth, antecedent soil moisture, and antecedent discharge all yield broadly similar results at this site. We also point out that these different proxies for antecedent conditions have different response times, so that the choice among them (in cases where one is lucky enough to have any such choice at all) will depend on the question one is trying to answer.
We did not present an equation for soil moisture or antecedent runoff because these are just lagged values from the soil moisture and discharge time series. We will see if we can more explicitly point this out in the manuscript.
The mathematical description of the approach is separated into two parts and should be presented in one block. Figure 3 and 4 are results and should be moved to 3.3.2 or to the appendix.
It appears that our use of "Result" as the heading for Section 3.3.2 has created the impression that all of the results are there, or should be there, whereas this is only the result of the hypothesis test posed in Section 3.3.1. In fact, Sections 3 and 4 (and Figures 3-8) are all results, with Section 5 and Figures 9-10 extending the results (which is the point of a discussion section).
Thank you for pointing this out. We will consider changing the heading for Section 3.3.2 (and potentially other sections) in our revised manuscript.
What the reviewer refers to as “the mathematical description of the approach” is actually two different things, which would be very confusing if they were presented “in one block”. The first, which is presented in Section 2.3, is an overview of Ensemble Rainfall-Runoff Analysis, which is essential background for everything that follows. Readers then need to see the results from this analysis (Figures 3 and 4) before they are in a position to understand the motivation behind the hypothesized two pathways underlying the double-peak hydrograph, and the convolution model for formally testing that hypothesis (Section 3.3 and Figure 5). It would not be an effective communication strategy to present this convolution model before readers had any idea what it was needed for.
I would suggest a complete reorganisation of the manuscript and more clear formulation of the hypotheses.
The organizational structure of the manuscript grew out of careful consideration of what readers need to know at each point in the paper (as illustrated, for example, by our response immediately above). We will consider whether this can be further improved, but many "quick fixes" that seem superficially attractive would be counterproductive in practice (like, as discussed above, presenting all the math "in one block").
Regarding the hypothesis in Section 3.3, we will state more clearly that Figure 6 also tests two alternative hypotheses. The disconnect between the two curves in Figure 6b rejects the hypothesis that only the near-surface pathway can explain the total rainfall-runoff response (because it doesn't explain the second peak), and the disconnect between the two curves in Figure 6c rejects the hypothesis that only the groundwater-mediated pathway would be responsible (because it doesn't explain the first peak).
We will make this clear in our discussion of Figure 6 in the revised paper.
Abstract:
Add the different used antecedent conditions measures and which gave the best results.
As indicated above, the purpose of the paper is not to compare different measures of antecedent wetness, and there is no clear standard to assess one or another as being "best".
We will change Line 21 to "Under wet conditions (as inferred from antecedent water table depth ≤ 1.66 m), the first peak increases nonlinearly..." in our revised manuscript to make it more explicit.
Introduction:
Add which antecedent wetness conditions could be used which is a crucial measure to detect the double peak phenomena (antecedent precipitation index, antecedent soil moisture index, antecedent groundwater, and pre- event runoff)
Antecedent wetness proxies are not, in fact "crucial... to detect the double peak phenomena". For example, Figures 3-6 clearly illustrate the double-peak phenomenon without any information about antecedent wetness.
Comparison of different proxies for catchment antecedent wetness conditions
We are not sure what is meant here. Figure 10 compares the results obtained with different antecedent wetness proxies.
I was expecting that the authors present a threshold value at which double peaks occur.
Thank you for the point. We are reluctant to specify a particular threshold for the emergence of double peaks, for several reasons. The occurrence of double peaks depends on both antecedent wetness and precipitation intensity. The general tendencies in these relationships are clear; as Figure 10 shows, the second peak is obvious when water table depth is between 1.66 and 1.30 m, but negligible when water table depth is below 1.66 m. And it is more obvious at higher precipitation intensities. But it is difficult to define a specific threshold at which the second peak emerges (vs. at which it is present but very small). And even if we arbitrarily define a size threshold for the second peak (at which it would be declared to be "present" rather than "absent"), any corresponding threshold of antecedent wetness would be specific to Weierbach and not amenable to generalization.
The authors should explain why they have selected antecedent groundwater table and what speaks against soil moisture and pre- runoff conditions.
We used antecedent water table depth (a) because of the evident role of groundwater in generating the second peak, (b) because the available soil moisture measurements only reflect the wetness state of the upper 60 cm of the subsurface, and (c) because the interpretation of antecedent streamflow necessarily depends on whether that streamflow results from the first peak (which is driven primarily by precipitation intensity) or the second peak (which is driven primarily by catchment wetness, and more specifically groundwater).
Thank you for the suggestion. We will explain this in Section 5.2 in our revised manuscript.
Explain why the lowest soil moisture probe was selected and not a mean value of the probes.
The assumption made here is incorrect. Our analysis used all the available probes. Line 582 says explicitly that antecedent VWC reflects soils ≤ 60 cm (i.e., the entire depth range of all the soil moisture probes, but we will revise Section 5.2 to make this more explicit.
If soil moisture would be equivalent wouldn’t it be the better proxy because installing probes is easier to install and less cost intensive?
We are not prepared to make such categorical value judgments. Soil moisture probes will be a better proxy if the most mechanistically relevant antecedent wetness is soil moisture rather than groundwater storage. Whether soil moisture probes are easier to install and less cost intensive will depend on how many of them are needed to adequately capture the spatial heterogeneity in soil moisture (which will not be known in advance). Conversely, the effort and expense of monitoring groundwater will depend on how heterogeneous the substrate is and how difficult it is to drill, and on how large the effective footprint of each well is (which will also not be known in advance).
In practice, researchers are likely to use whatever antecedent wetness proxies are available, which is why we undertook to compare them in Section 5.2, taking advantage of the fact that all three of them are available at Weierbach.
Specific comments:
Page 4 Figure1: which GW gauge was used for the analysis?
The manuscript already explains that we used the average of all three wells. Lines 124-126 say that we used the three GW wells with the most complete records (the three shown in Figure 1). Line 137 says that we averaged the changes in WTD across all three wells to infer groundwater recharge. And line 433 says that we used the catchment-averaged water table depth as a proxy for antecedent wetness. We will look for ways to make these points even more explicit.
Page 5 Line 127 which type of device was used
Soil moisture is recorded on CR800 loggers using CS650 water content reflectometers (Campbell Scientific, Logan, Utah, USA) (Hissler et al., 2021).
Detailed descriptions of equipment and data collection can be found in Hissler (2021) (Line 118). We will note this in the revised manuscript.
Page 6 Line 145 is the presented VWC the mean value for all soil moisture monitoring points? Or is it a specific point in figure 1?
At each depth it is the average of all available probes.
Thank you for the question. We will add this in our revised version.
Page 9 Line 207: RRDP is defined in eq. 2
Not quite. Equation 2 defines the partial RRD, partialRRDP, discussed in Section 3.2.
RRDP and GRRDP are two different versions of the RRD defined in Equation 1. If the input is precipitation and the output is discharge, the RRD is denoted RRDP (the distribution of runoff response to precipitation). Alternatively, if the input is precipitation and the output is groundwater recharge, the RRD is denoted GRRDP (the distribution of groundwater recharge's response to precipitation).
Although this is already explained at the beginning of Sections 3.1 and 3.2, we do see how readers could miss it, so we will add explicit definitions of RRDP, GRRDP, and RRDGR in Section 2 (along with equations for them).
Page 10 Line 219 equation is missing for GRRDP - it is presented at page 14 Line: 341
Agreed. As noted above, we will add explicit definitions of RRDP, GRRDP, and RRDGR in Section 2.
Page 14 Line 350- 352: equation 10 or 11 could be removed
We disagree. Equation 10 follows from Equations 8 and 9, and without explicitly stating that convolution is associative one cannot get from Equation 10 to Equation 11. Equation 11, in turn, is necessary as a component of Equation 13.
Page 25 Lines: 566-571: Is it important to know the percentiles?
We think so. Otherwise it is difficult for readers to understand the ranges of catchment conditions corresponding to the plots in Figure 10.
Page 27 Line 622 This is exactly the same conclusion like in the Schaefertal catchment (Graeff et al. 2009) where the double peak events could not be any more observed after mining activities below the catchment started and groundwater was completely disturbed
We disagree that this is "exactly the same conclusion", since draining a catchment from below by mining is different from depleting groundwater by evapotranspiration and stream discharge. It is also not correct to say that double-peak events "could not be any more observed after mining activities" at Schaefertal, because Graeff et al. (p. 705) note that double-peak events were still observed "in response to very strong precipitation events".
However, we appreciate the pointer to this interesting study, which we were previously unaware of, and will mention it in the revised manuscript.
Citation: https://doi.org/10.5194/egusphere-2025-613-AC1
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AC1: 'Reply on RC2', Huibin Gao, 27 May 2025
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Huibin Gao
Laurent Pfister
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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The presented study applies the Ensemble Rainfall Runoff Analysis (ERRA) to explore the mechanisms of double-peak hydrograph emergence in the Weierbach catchment, Luxembourg. The authors test the hypothesis that the double-peak hydrograph is generated by two pathways – a near-surface runoff in direct response to precipitation and a pathway through vadose zone influenced by groundwater recharge. They further explore how double-peak runoff generation varies with precipitation intensities and antecedent catchment wetness. The controls of the first and second hydrograph peak are identified with their specific thresholds. The manuscript thus presents novel insights into the mechanistic functioning of runoff generation inferred with ERRA from multiple observational data in the Weierbach catchment. This is a very interesting study which rigorously works out the controls on the double peak hydrograph in the study catchment and clearly presents the results and in-depth discussion.
I have a few remarks that may further enhance the presentation and potentially increase the generalization of the case study results.
L65 – 69: The cited studies with specific numbers of catchment storage, precipitation volume and intensities are site-specific and very much depend on the catchment characteristics. It would be good to mention this in this paragraph.
With regards to NRF(P) /NRF (GR) the authors define it as a response function of discharge to amount of precipitation/Groundwater recharge in each increment, and this function in its turn depends on precipitation intensity/GR rate at time t. Should this function also depend on the antecedent moisture state as it controls the responsiveness of discharge to P/GR? In case only overland flow as response to P is considered, this is not of large importance, but for GR it should matter, shouldn’t it? Or is average soil moisture state implicitly considered in NRF analogously to the rational formula? The runoff coefficient in the rational formula is however an integral over an event duration of a longer time period, but NRF is a function resolved in time depending on precipitation intensity. I somehow miss soil moisture in this resolved representation. Please, explain.
The results presented in section 3 rigorously demonstrate that the formulated hypothesis that the double-peak hydrograph is generated by the combination of near-surface and groundwater-mediated pathway cannot be rejected. An alternative hypothesis that only e.g., groundwater-mediated pathway would be responsible must be rejected. I think the paper would benefit if the authors more clearly articulate the results in Karl Popper’s sense that the formulated hypothesis could not be rejected.
L461-465: Do I understand correctly that in order to analyze the effect of catchment wetness and precipitation intensity on hydrograph generation you need to separate different periods when e.g., WTD falls into three designated classes, and then you estimate NRFs for these periods? Please, explain your methodological steps for clarity.
Is this also done analogously for the precipitation intensity? Precipitation intensity is however, much more volatile than catchment average WTD. How does this affect the results if one maybe need to pick just one or two hours of intensive precipitation out of the entire event. Does ERRA then clearly separates NRFs for one-two hours of intensive precipitation framed by a few hours of less intensive precipitation belonging to another class (and resulting in a different NRF) prior and after a major downpour?
The analysis of nonstationarity and nonlinearity in section 4 is very interesting, but sensitivities expressed in absolute numbers remain pertinent to the study catchment. I am wondering if one could derive some dimensionless or relative measures that can be generalized when investigations from a large set of catchments would be available. Would it be helpful for example to look not at the catchment average WTD but in relation to the mean annual precipitation?
I understand that results presented in section 4 are catchment specific and much of the discussion links to previous studies and mechanisms of runoff generation in this very catchment, but still maybe some ideas for generalizations may enrich the discussion part from L539ff.
Finally, the potential artifact with near-zero spike mentioned in section 3.1 seems to remain unclear. It says in L234 it will be explored in the next section, but I somehow missed a detailed analysis. Do you mean that the exploration is given in L274-277 which basically concludes that you do not have enough data to pinpoint the origin of this spike? Or did I miss further elaboration on this issue in the manuscript? If there is not enough data to further explore this artifact, I suggest not to raise expectation in L234.