Uncertainty and non-stationarity of empirical streamflow sensitivities

Gnann, Sebastian; Anderson, Bailey J.; Weiler, Markus

doi:10.5194/egusphere-2025-4527

Preprints

https://doi.org/10.5194/egusphere-2025-4527

Preprints

17 Oct 2025

| 17 Oct 2025

Uncertainty and non-stationarity of empirical streamflow sensitivities

Sebastian Gnann, Bailey J. Anderson, and Markus Weiler

Abstract. The sensitivity of streamflow to changes in driving variables such as precipitation and potential evaporation is a key signature of catchment behaviour. Due to increasing interest in climate change impacts, streamflow sensitivities derived from observations have become a widely used metric for catchment characterization, model evaluation, and observation-constrained projections. However, there remain open questions regarding the robustness and stationarity of empirically-derived sensitivities. In this paper, we revisit theoretical and empirical approaches to estimate streamflow sensitivities to precipitation and potential evaporation. First, we compare different estimation methods – primarily based on linear regression – using a synthetic dataset for which the sensitivities are known. Second, we extend this comparison and use two methods selected based on the previous analysis to estimate sensitivities for >1000 near-natural catchments. Third, we investigate how sensitivities change over time due to changes in the ratio between potential evaporation and precipitation (i.e., aridity index). Our results confirm that multiple regression is preferable to single regression, but that in presence of noise and correlation between precipitation and potential evaporation, even multiple regression methods can lead to high uncertainty, especially for potential evaporation. When analysing real catchments, sensitivity to precipitation is estimated consistently across methods, while sensitivity to potential evaporation is highly uncertain and often yields unrealistic values. Further, as the aridity index increases over time – a trend found in observational data – sensitivities decrease (by 22–70 % over 50 years) and are thus non-stationary. These results should urge caution in the use of empirical streamflow sensitivities and call for further investigation.

Received: 15 Sep 2025 – Discussion started: 17 Oct 2025

Competing interests: MW is a member of the editorial board of the journal Hydrology and Earth System Sciences.

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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Sebastian Gnann, Bailey J. Anderson, and Markus Weiler

Status: closed

RC1:
'Congratulations on egusphere-2025-4527', Anonymous Referee #1, 09 Nov 2025
SYNTHESIS
This paper deals with the precipitation and potential evaporation sensitivity of streamflow. It presents a theoretical study on the impact of different uncertainty sources which is very original, and allows to discard definitively one of the classical methods to identify elasticity (never seen anywhere in the literature... would be worth a technical note in itself). Then the paper goes on to show that the ongoing climatic change has already changed the empirical precipitation elasticity of streamflow in Germany, a very interesting and original result in itself.

OVERALL COMMENT
This is a very good paper: excellent substance, excellent analysis, excellent form.
I would like in particular to congratulate the authors for using the sensitivities / absolute elasticities which are easily and logically interpretable (and have easily identifiable physical limits) instead of the relative ones (‘true’ elasticities). The plots showing the dependency of the relative elasticities (derived from the Turc-Mezentsev formula) to aridity, published elsewhere in the literature may be mathematically right but is useless in hydrological terms (the behavior with aridity makes no sense: we, as hydrologists, are not interested to know that a theoretical ratio of two terms that tend towards zero has a mathematical limit, we are interested to know that the two terms tend towards zero).
As a reviewer, my only recommendation is “don’t change a word and publish as it is”.
But since I am not only a reviewer but also a hydrologist interested in the topic, I could not help to comment your paper below. Feel free to consider or not my suggestions. I realize that there is enough matter to publish several very interesting papers, and I am definitely not requesting you to turn this paper into a very long undigestible paper.
Honestly, my only regret is your title, which is a little vague and not at the level of your work. The fact is that there are several very interesting points in your paper, it may be difficult to choose one over the others. Also, I guess that a strict statistician would argue that the term non-stationarity is not well-chosen, and would prefer you to talk about changing behavior, I remember a discussion with Prof. Koutsoyannis 10 years ago on this topic (see e.g. Efstratiadis et al., 2015).

DETAILED COMMENTS
The introduction is excellent

Fig1: why did you choose to plot 1-Q/P and not Q/P? Also, I have a problem with your physical limits: when you use catchment data the water limit should correspond to Q=P and the energy limit to Q=P-EP. Your water limit only corresponds to the “physical limit” (Q=0). I agree that it will be strange to have a “water limit” at 0, this is why I would personally use prefer to plot Q/P and not 1-Q/P. See e.g. Fig. 1 in the paper by Andréassian et al. (2025).

Table 1: please adjust your notations (PET -> E_P) for homogeneity with the rest of the paper

Table 1: I understand what you mean by “same as 𝑄 = 𝑠_𝑃𝑃 + 𝑠_PET𝑃𝐸𝑇 + 𝑐” even if I do not completely agree: when expressed in deltas, the formula offers other opportunities, such as the pooled regression which you mention, and is interesting to reduce the uncertainties and yield hydrologically coherent values

Table 1: I do not understand why you introduced eq #1... a very (too?) simplistic choice, unless you want to show us that when the elasticity coefficients try to adjust to represent at the same time the intercept of the regression and the slope, strange things happen (do you really need to add this option in this paper?)

l130: does it make sense to assume positive correlation between delta Q et delta E? 0 and negative could be enough? Unless there are places with that type of correlation in Australia perhaps.

Also in the generation procedure, I note that all the catchments are considered conservative. However, in many datasets (made of mostly small catchments) catchments are significantly contributing to regional aquifers, they “leak”. I am not sure it’s worth to include this aspect in your theoretical experiment, but it could be worth to discuss potential specificities of “leaking” catchments in the discussion part (the fact is that the leaked quantity itself can be sensitive to Precipitation).

l170 Temporal trend: it is not very clear which type of trends you apply (I understood later that you just use the observed trends: it would be good to add a sentence here on this).

Fig3: I understand now that the nonparametric method of Sankarasubramanian et al (2001) gives wrong (i.e. positive) sensitivities of streamflow to E_P. Did anybody in the literature ever mentioned that?

l240, Fig 4: it would be interesting to note that one of the methods respects (almost) the physical limits, while the other does not. But I still do not understand why you introduced the multiple Regression #1 with intercept set to 0. It was obviously a bad choice... the model uses the degree of freedom of s_P (which is often not even statistically significant) to compensate for the lack of intercept in the regression: you end up fitting Q = aP +b and not Q = aP + bE_P

Fig6: very interesting graph

l265: when you look at the change of sensitivities with aridity defined as a variable it is like using a second-order assumption of “space-for-time trading”. You should mention it.

l274: this is a very interesting and very original result of your paper. I would be particularly interested to understand whether it is linked to a change in seasonality (not accounted for in the regressions), to the incapacity of the E_P model to represent the true evolution of the evaporative demand of the atmosphere, or to some other factor... A suggestion (for another paper) would be to test a monthly or daily rainfall-runoff model. If it is able to represent the change of precipitation sensitivity, then the problem is due to the seasonality that the annual anomaly model cannot account for. Otherwise, it means some other hydrological process is unaccounted for (or the E_P model is inappropriate).

Table 4: I am not sure to understand the sign of the relative values (and the necessity for them, if it is complicated to understand). Is this table really useful? Fig 7 and 8 are already extremely clear.

Figure 7 and Figure 8 are extremely clear and interesting. Without requesting too much additional analysis (because it would turn your paper into a book...), I was wondering whether the Australian dataset would allow for producing 2 subsets one with P and E_P out of phase (the Mediterranean part of Australia), and another with P and E_P in phase: I believe this could help to interpret the trend observed in Germany. Another solution would be to compute an index characterizing the P-E_P phase-shift.

l336: you write “More than half of the values based on method #2 are larger than zero”. But I guess that anyway the p-values from a Student t-test would consider these values as non-significantly different from 0. Perhaps mention it?

REFERENCES
Andréassian, V., Guimarães, G.M., de Lavenne, A., and Lerat, J.: Time shift between precipitation and evaporation has more impact on annual streamflow variability than the elasticity of potential evaporation, Hydrol. Earth Syst. Sci., 29, 5477–5491, https://doi.org/10.5194/hess-29-5477-2025, 2025.
Efstratiadis, A., Nalbantis, I., and Koutsoyiannis, D., 2015. Hydrological modelling of temporally-varying catchments: facets of change and the value of information. Hydrological Sciences Journal, 60 (7–8). doi:10.1080/02626667.2014.982123
Citation: https://doi.org/10.5194/egusphere-2025-4527-RC1
- AC2: 'Reply on RC1', Sebastian Gnann, 12 Dec 2025
  
  Please find our replies in the attached PDF.
  
  Citation: https://doi.org/10.5194/egusphere-2025-4527-AC2
RC2:
'Comment on egusphere-2025-4527', Anonymous Referee #2, 15 Nov 2025
Gnann et al. focus on the robustness and stationarity of streamflow sensitivities to P and Ep because the concepts of sensitivity and the methods used to estimate it are not fully clear in the literature. They approach this by: (1) generating six combinations of synthetic data from the Turc-Mezentsev model to identify methods that perform reliably across conditions; and (2) applying the selected methods to catchments with long-term observations to explore how sensitivities evolve over time and the sources of uncertainty. The manuscript is well structured and several of the results are very insightful. My detailed comments are below.

Major comments:
before going into the specific results, I think it would help if the manuscript clarified how the theoretical sensitivities relate to the empirical ones. The analytical sensitivies come directly from the Turc-Mezentsev curve, whereas the empirical values are estimated from interannual variability using regression. Because the Budyko curve is nonlinear, a regression slope over many years does not necessarily match the local derivative at the long-term mean. This difference might explain part of the mismatch between the analytical lines and the observations in several regions.

related to the point above, using a single Budyko parameter n across all catchments can also affect the comparison. Since n controls the curvature of the Turc-Mezentsev relationship, regional differences in n would naturally show up as differences in the “expected’’ sensitivities, even though the manuscript notes that the exact value of n is not the focus. A fixed n can still influence the shape of the theoretical trends, so it would help to check how sensitive the analytical results are to this choice. This may be particularly relevant for Figs. 7 & 8, where the theoretical line captures the trend over Australia but not Germany. Labeling points by country in Fig. S1 might reveal if this mismatch is regionally systematic. The strong bias in German trends also make it difficult to interpret the degree of non-stationarity, even though this general pattern is consistent. Maybe consider to use boxplots or similar summaries to describe the catchment-level trends.

for the data used, the national forcing of P is very useful, but more information is needed on how Ep is calculated in each region. Why are different formulations used in these datasets? If these formulations were chosen because they best represent local conditions, it would be good to explicitly clarify. If not, part of the apparent non-stationarity in sensitivities might come from the way Ep is estimated. This might also help explain the positive s_Ep values in Fig. 4b. A brief comparison with an alternative Ep method (such as PET_Yang2019) might be benificial, although I think it may be somewhat beyond the main scope.

for the unexplained variation in sensitivities, it might be useful to discuss the role of vegetation. The vegetation cover influences the rainfall-runoff relationship, water storage; and the effective Budyko parameter n. Long-term changes in vegetation traits could shift catchments relative to the theoretical curve and influence sensitivities to both P and Ep. Nijzink and Schymanski (2022) provide an interesting example of how adjustments in vegetation influence the Budyko parameter n, and connecting this to your results might strengthen the interpretation.

Minor comments:
for Line 114 & 117, could you provide these results in supplementary materials?

for Table 1, (a) Log-log regression should be log-linear regression and e_EETshould be e_Ep; (b) why do you use PET and Ep together?

for the Turc-Mezentsev model, how is Ep calculated? It directly influence s_Ep, s_P and Ep/P.

When I first saw Table 4, I misunderstood the relative trend, i.e. positive s_Ep with a negative relative change. I think this table is unnecessary.

Reference:
Yang, Y., Roderick, M. L., Zhang, S., McVicar, T. R. & Donohue, R. J. Hydrologic implications of vegetation response to elevated CO2 in climate projections. Nature Clim Change 9, 44–48 (2019).

Nijzink, R. C. & Schymanski, S. J. Vegetation optimality explains the convergence of catchments on the Budyko curve. Hydrology and Earth System Sciences 26, 6289–6309 (2022).
Citation: https://doi.org/10.5194/egusphere-2025-4527-RC2
- AC1: 'Reply on RC2', Sebastian Gnann, 12 Dec 2025
  
  Please find our replies in the attached PDF.
  
  Citation: https://doi.org/10.5194/egusphere-2025-4527-AC1
- AC3: 'Reply on RC2', Sebastian Gnann, 12 Dec 2025
  
  Apologies, but I uploaded the wrong response and did not find a way to edit it. Attached now the replies to RC2.
  
  Citation: https://doi.org/10.5194/egusphere-2025-4527-AC3

Status: closed

RC1:
'Congratulations on egusphere-2025-4527', Anonymous Referee #1, 09 Nov 2025
SYNTHESIS
This paper deals with the precipitation and potential evaporation sensitivity of streamflow. It presents a theoretical study on the impact of different uncertainty sources which is very original, and allows to discard definitively one of the classical methods to identify elasticity (never seen anywhere in the literature... would be worth a technical note in itself). Then the paper goes on to show that the ongoing climatic change has already changed the empirical precipitation elasticity of streamflow in Germany, a very interesting and original result in itself.

OVERALL COMMENT
This is a very good paper: excellent substance, excellent analysis, excellent form.
I would like in particular to congratulate the authors for using the sensitivities / absolute elasticities which are easily and logically interpretable (and have easily identifiable physical limits) instead of the relative ones (‘true’ elasticities). The plots showing the dependency of the relative elasticities (derived from the Turc-Mezentsev formula) to aridity, published elsewhere in the literature may be mathematically right but is useless in hydrological terms (the behavior with aridity makes no sense: we, as hydrologists, are not interested to know that a theoretical ratio of two terms that tend towards zero has a mathematical limit, we are interested to know that the two terms tend towards zero).
As a reviewer, my only recommendation is “don’t change a word and publish as it is”.
But since I am not only a reviewer but also a hydrologist interested in the topic, I could not help to comment your paper below. Feel free to consider or not my suggestions. I realize that there is enough matter to publish several very interesting papers, and I am definitely not requesting you to turn this paper into a very long undigestible paper.
Honestly, my only regret is your title, which is a little vague and not at the level of your work. The fact is that there are several very interesting points in your paper, it may be difficult to choose one over the others. Also, I guess that a strict statistician would argue that the term non-stationarity is not well-chosen, and would prefer you to talk about changing behavior, I remember a discussion with Prof. Koutsoyannis 10 years ago on this topic (see e.g. Efstratiadis et al., 2015).

DETAILED COMMENTS
The introduction is excellent

Fig1: why did you choose to plot 1-Q/P and not Q/P? Also, I have a problem with your physical limits: when you use catchment data the water limit should correspond to Q=P and the energy limit to Q=P-EP. Your water limit only corresponds to the “physical limit” (Q=0). I agree that it will be strange to have a “water limit” at 0, this is why I would personally use prefer to plot Q/P and not 1-Q/P. See e.g. Fig. 1 in the paper by Andréassian et al. (2025).

Table 1: please adjust your notations (PET -> E_P) for homogeneity with the rest of the paper

Table 1: I understand what you mean by “same as 𝑄 = 𝑠_𝑃𝑃 + 𝑠_PET𝑃𝐸𝑇 + 𝑐” even if I do not completely agree: when expressed in deltas, the formula offers other opportunities, such as the pooled regression which you mention, and is interesting to reduce the uncertainties and yield hydrologically coherent values

Table 1: I do not understand why you introduced eq #1... a very (too?) simplistic choice, unless you want to show us that when the elasticity coefficients try to adjust to represent at the same time the intercept of the regression and the slope, strange things happen (do you really need to add this option in this paper?)

l130: does it make sense to assume positive correlation between delta Q et delta E? 0 and negative could be enough? Unless there are places with that type of correlation in Australia perhaps.

Also in the generation procedure, I note that all the catchments are considered conservative. However, in many datasets (made of mostly small catchments) catchments are significantly contributing to regional aquifers, they “leak”. I am not sure it’s worth to include this aspect in your theoretical experiment, but it could be worth to discuss potential specificities of “leaking” catchments in the discussion part (the fact is that the leaked quantity itself can be sensitive to Precipitation).

l170 Temporal trend: it is not very clear which type of trends you apply (I understood later that you just use the observed trends: it would be good to add a sentence here on this).

Fig3: I understand now that the nonparametric method of Sankarasubramanian et al (2001) gives wrong (i.e. positive) sensitivities of streamflow to E_P. Did anybody in the literature ever mentioned that?

l240, Fig 4: it would be interesting to note that one of the methods respects (almost) the physical limits, while the other does not. But I still do not understand why you introduced the multiple Regression #1 with intercept set to 0. It was obviously a bad choice... the model uses the degree of freedom of s_P (which is often not even statistically significant) to compensate for the lack of intercept in the regression: you end up fitting Q = aP +b and not Q = aP + bE_P

Fig6: very interesting graph

l265: when you look at the change of sensitivities with aridity defined as a variable it is like using a second-order assumption of “space-for-time trading”. You should mention it.

l274: this is a very interesting and very original result of your paper. I would be particularly interested to understand whether it is linked to a change in seasonality (not accounted for in the regressions), to the incapacity of the E_P model to represent the true evolution of the evaporative demand of the atmosphere, or to some other factor... A suggestion (for another paper) would be to test a monthly or daily rainfall-runoff model. If it is able to represent the change of precipitation sensitivity, then the problem is due to the seasonality that the annual anomaly model cannot account for. Otherwise, it means some other hydrological process is unaccounted for (or the E_P model is inappropriate).

Table 4: I am not sure to understand the sign of the relative values (and the necessity for them, if it is complicated to understand). Is this table really useful? Fig 7 and 8 are already extremely clear.

Figure 7 and Figure 8 are extremely clear and interesting. Without requesting too much additional analysis (because it would turn your paper into a book...), I was wondering whether the Australian dataset would allow for producing 2 subsets one with P and E_P out of phase (the Mediterranean part of Australia), and another with P and E_P in phase: I believe this could help to interpret the trend observed in Germany. Another solution would be to compute an index characterizing the P-E_P phase-shift.

l336: you write “More than half of the values based on method #2 are larger than zero”. But I guess that anyway the p-values from a Student t-test would consider these values as non-significantly different from 0. Perhaps mention it?

REFERENCES
Andréassian, V., Guimarães, G.M., de Lavenne, A., and Lerat, J.: Time shift between precipitation and evaporation has more impact on annual streamflow variability than the elasticity of potential evaporation, Hydrol. Earth Syst. Sci., 29, 5477–5491, https://doi.org/10.5194/hess-29-5477-2025, 2025.
Efstratiadis, A., Nalbantis, I., and Koutsoyiannis, D., 2015. Hydrological modelling of temporally-varying catchments: facets of change and the value of information. Hydrological Sciences Journal, 60 (7–8). doi:10.1080/02626667.2014.982123
Citation: https://doi.org/10.5194/egusphere-2025-4527-RC1
- AC2: 'Reply on RC1', Sebastian Gnann, 12 Dec 2025
  
  Please find our replies in the attached PDF.
  
  Citation: https://doi.org/10.5194/egusphere-2025-4527-AC2
RC2:
'Comment on egusphere-2025-4527', Anonymous Referee #2, 15 Nov 2025
Gnann et al. focus on the robustness and stationarity of streamflow sensitivities to P and Ep because the concepts of sensitivity and the methods used to estimate it are not fully clear in the literature. They approach this by: (1) generating six combinations of synthetic data from the Turc-Mezentsev model to identify methods that perform reliably across conditions; and (2) applying the selected methods to catchments with long-term observations to explore how sensitivities evolve over time and the sources of uncertainty. The manuscript is well structured and several of the results are very insightful. My detailed comments are below.

Major comments:
before going into the specific results, I think it would help if the manuscript clarified how the theoretical sensitivities relate to the empirical ones. The analytical sensitivies come directly from the Turc-Mezentsev curve, whereas the empirical values are estimated from interannual variability using regression. Because the Budyko curve is nonlinear, a regression slope over many years does not necessarily match the local derivative at the long-term mean. This difference might explain part of the mismatch between the analytical lines and the observations in several regions.

related to the point above, using a single Budyko parameter n across all catchments can also affect the comparison. Since n controls the curvature of the Turc-Mezentsev relationship, regional differences in n would naturally show up as differences in the “expected’’ sensitivities, even though the manuscript notes that the exact value of n is not the focus. A fixed n can still influence the shape of the theoretical trends, so it would help to check how sensitive the analytical results are to this choice. This may be particularly relevant for Figs. 7 & 8, where the theoretical line captures the trend over Australia but not Germany. Labeling points by country in Fig. S1 might reveal if this mismatch is regionally systematic. The strong bias in German trends also make it difficult to interpret the degree of non-stationarity, even though this general pattern is consistent. Maybe consider to use boxplots or similar summaries to describe the catchment-level trends.

for the data used, the national forcing of P is very useful, but more information is needed on how Ep is calculated in each region. Why are different formulations used in these datasets? If these formulations were chosen because they best represent local conditions, it would be good to explicitly clarify. If not, part of the apparent non-stationarity in sensitivities might come from the way Ep is estimated. This might also help explain the positive s_Ep values in Fig. 4b. A brief comparison with an alternative Ep method (such as PET_Yang2019) might be benificial, although I think it may be somewhat beyond the main scope.

for the unexplained variation in sensitivities, it might be useful to discuss the role of vegetation. The vegetation cover influences the rainfall-runoff relationship, water storage; and the effective Budyko parameter n. Long-term changes in vegetation traits could shift catchments relative to the theoretical curve and influence sensitivities to both P and Ep. Nijzink and Schymanski (2022) provide an interesting example of how adjustments in vegetation influence the Budyko parameter n, and connecting this to your results might strengthen the interpretation.

Minor comments:
for Line 114 & 117, could you provide these results in supplementary materials?

for Table 1, (a) Log-log regression should be log-linear regression and e_EETshould be e_Ep; (b) why do you use PET and Ep together?

for the Turc-Mezentsev model, how is Ep calculated? It directly influence s_Ep, s_P and Ep/P.

When I first saw Table 4, I misunderstood the relative trend, i.e. positive s_Ep with a negative relative change. I think this table is unnecessary.

Reference:
Yang, Y., Roderick, M. L., Zhang, S., McVicar, T. R. & Donohue, R. J. Hydrologic implications of vegetation response to elevated CO2 in climate projections. Nature Clim Change 9, 44–48 (2019).

Nijzink, R. C. & Schymanski, S. J. Vegetation optimality explains the convergence of catchments on the Budyko curve. Hydrology and Earth System Sciences 26, 6289–6309 (2022).
Citation: https://doi.org/10.5194/egusphere-2025-4527-RC2
- AC1: 'Reply on RC2', Sebastian Gnann, 12 Dec 2025
  
  Please find our replies in the attached PDF.
  
  Citation: https://doi.org/10.5194/egusphere-2025-4527-AC1
- AC3: 'Reply on RC2', Sebastian Gnann, 12 Dec 2025
  
  Apologies, but I uploaded the wrong response and did not find a way to edit it. Attached now the replies to RC2.
  
  Citation: https://doi.org/10.5194/egusphere-2025-4527-AC3

Sebastian Gnann, Bailey J. Anderson, and Markus Weiler

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Sebastian Gnann, Bailey J. Anderson, and Markus Weiler

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Short summary

The extent to which streamflow varies in response to variability in precipitation and potential evaporation is essential for understanding climate change impacts on water resources. This so-called streamflow sensitivity is often estimated directly from observational data, but the robustness of these estimates remains unclear. Through systematic examination of existing approaches, we highlight uncertainties inherent in all approaches, discuss their origins, and propose potential alternatives.


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