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Developing the Actual Precipitation Probability Distribution Based on the Complete Daily Series

Sustainability 2023, 15(17), 13136; https://doi.org/10.3390/su151713136

by Wangyuyang Zhai¹, Zhoufeng Wang^2,3,*, Youcan Feng^1,*

, Lijun Xue⁴, Zhenjie Ma⁵, Lin Tian⁵ and Hongliang Sun⁴

Reviewer 1:

Vojto Rušin

Reviewer 2:

Panayiotis G. Dimitriadis

Reviewer 3:

Sebastian Zabłocki

Sustainability 2023, 15(17), 13136; https://doi.org/10.3390/su151713136

Submission received: 15 June 2023 / Revised: 7 August 2023 / Accepted: 14 August 2023 / Published: 31 August 2023

Round 1

Reviewer 1 Report

The aim of the work was to find out the actual distribution of precipitation in Northeast China based on the analysis of precipitation observations at two stations in the period 1960-2020 using three existing models/M-K test and to predict precipitation in the future based on this analysis. I assume that they were led to solve this task by the unpredictability of precipitation in the framework of global warming and the protection of the population and property from destructive precipitation. The analysis clearly shows that the current methods do not allow a reliable forecast (each method has certain advantages according to the registered rainfall) and it will be necessary to look for other models in the future. If they will be found in this area at all due to dynamic behaviour of weather.

In principle, one can agree with the work as such, but its shortcoming is the missing amount of citations, which are both listed in the text of the article, but also missing citations, which are included in the text but are not included in the references used, e.g. p.34.

Due to the lack of literature, I do not recommend publishing the article in this form

Author Response

Dear Reviewer #1:

Thank you very much for your valuable comments and suggestions on our research paper titled " Developing the Actual Precipitation Probability Distribution based on the Complete Daily Series". We truly appreciate your expertise and comprehensive feedback. We have thoroughly considered your comments and made the necessary revisions and improvements. Here are our specific responses to each of your points:

Comment: its shortcoming is the missing amount of citations, which are both listed in the text of the article, but also missing citations, which are included in the text but are not included in the references used, e.g. p.34.

Response to R1C1: Thanks for your suggestion. The missing references have been added and the format of the references has been rectified.

Once again, we sincerely appreciate the time and effort you have dedicated to the review process. Your feedback is highly regarded, and we believe that these revisions will contribute to the overall quality of our research. If you have any further suggestions or requirements, please don't hesitate to let us know, and we will make every effort to accommodate them.

Thank you once again

Best regards,

Wangyuyang Zhai

Author

E-mail: [email protected]

Reviewer 2 Report

In this work, the authors aim to develop a method to fit a continuous (including wet-dry days) precipitation probability distribution function to a complete daily precipitation timeseries. In my opinion, the manuscript needs multiple major and minor revisions before it can be evaluated for publication, while after being strongly revised, it may serve as a discussion/opinion review letter/paper that compares different probability-fitting models as well as investigates the impact of gaps in the precipitation records. Also, in order to fit the journal's scope, an additional application could be added that highlights how the lack of data affects the sustainability design of, for example, the water resources in a location.

1) Please consider including for the precipitation probability fitting, the so-called Pareto-Burr-Feller probability distribution (which is a generalized form of the Pareto IV or Burr XII distribution by including a method to fit a complete/continuous precipitation timeseries without separating wet and dry values; see section 4.2 in Dimitriadis and Koutsoyiannis, 2018).

Dimitriadis, P., and D. Koutsoyiannis, Stochastic synthesis approximating any process dependence and distribution, Stochastic Environmental Research & Risk Assessment, 32 (6), 1493–1515, doi:10.1007/s00477-018-1540-2, 2018.

2) The presented historical review of precipitation fitting needs major changes, since many mentioned works by the authors include mistakes or older methods of precipitation probability fitting. Please consider looking the review (and mentioned references) presented in a recent extended analysis in the book of Koutsoyiannis (2022). Also, please note that the first global-analyses on the rainfall-extremes probability-fitting was performed by Koutsoyiannis (2004a,b).

Koutsoyiannis, D., Statistics of extremes and estimation of extreme rainfall, 1, Theoretical investigation, Hydrological Sciences Journal, 49 (4), 575–590, doi:10.1623/hysj.49.4.575.54430, 2004a.

Koutsoyiannis, D., Statistics of extremes and estimation of extreme rainfall, 2, Empirical investigation of long rainfall records, Hydrological Sciences Journal, 49 (4), 591–610, doi:10.1623/hysj.49.4.591.54424, 2004b.

Koutsoyiannis, D., Stochastics of Hydroclimatic Extremes - A Cool Look at Risk, Edition 2, ISBN: 978-618-85370-0-2, 346 pages, doi:10.57713/kallipos-1, Kallipos Open Academic Editions, Athens, 2022.

3) Regarding non-stationarity and trend analyses, please discuss that both these are model properties and not timeseries' characteristics, and thus, a timeseries cannot be characterized, for example, as stationary or non-stationary but rather a model can be selected to be stationary and non-stationary and after comparing this model to the timeseries, one may decide which one best fits (i.e., being statistically more significant) the timeseries (e.g., see Koutsoyiannis and Montanari, 2015, for the non-stationarity issue, and Iliopoulou and Koutsoyiannis, 2020, for the trends).

Iliopoulou, T., and D. Koutsoyiannis, Projecting the future of rainfall extremes: better classic than trendy, Journal of Hydrology, 588, doi:10.1016/j.jhydrol.2020.125005, 2020.

Koutsoyiannis, D., and A. Montanari, Negligent killing of scientific concepts: the stationarity case, Hydrological Sciences Journal, 60 (7-8), 1174–1183, doi:10.1080/02626667.2014.959959, 2015.

4) Please conside discussing that the interpolation method to fill in the gaps in the precipitation timeseries will destroy part of the strong dependence of the autocorrelation function of the timeseries, which is known to exhibit a long-range dependence (e.g., see a global-scale analysis on the small-to-large-scale autocorrelation function of precipitation process in Dimitriadis et al., 2021).

Dimitriadis, P., D. Koutsoyiannis, T. Iliopoulou, and P. Papanicolaou, A global-scale investigation of stochastic similarities in marginal distribution and dependence structure of key hydrological-cycle processes, Hydrology, 8 (2), 59, doi:10.3390/hydrology8020059, 2021.

5) Please separate the regular-precipitation and maximum-precipitation analyses, since there seems to be a confusion in the manuscript. I am confused whether the authors fit regular daily precipitation timeseries (through, for example, the Pearson III distribution) or maximum daily precipitation timeseries (through, for exmaple, GEV ditribution) or both.

6) It is mentioned in the manuscript that "...the annual maximum data were found independent."; please discuss that this is probably due to the high bias introduced by the autocorrelation function estimator. For the maximum rainfall data, please see a review in Iliopoulou and Koutsoyiannis (), where it is found that a hidden strong dependence is found in the maximum rainfall records if the correct tools are used.

Iliopoulou, T., and D. Koutsoyiannis, Revealing hidden persistence in maximum rainfall records, Hydrological Sciences Journal, 64 (14), 1673–1689, doi:10.1080/02626667.2019.1657578, 2019.

7) Regarding the impact of the "time-interval" (or else the resolution) of the precipitation timeseries to the serial dependence, please consider discussing that a more appropriate tool to estimate the serial dependence is the climacogram estimator, which is found to have a better performance than the autocorrelation-function estimator (see the first such analysis with these comparisons in Dimitriadis and Koutsoyiannis, 2015).

There are many grammar and syntax mistakes in the manuscript; please consider using a native English speaker to assist you in this (I am often using one myself). Also, there are multiple "NotFound references" in the manuscript.

Author Response

Dear Reviewer#2:

Comment #1: Please consider including for the precipitation probability fitting, the so-called Pareto-Burr-Feller probability distribution (which is a generalized form of the Pareto IV or Burr XII distribution by including a method to fit a complete/continuous precipitation timeseries without separating wet and dry values; see section 4.2 in Dimitriadis and Koutsoyiannis, 2018)

Response to R2C1: Thanks for your suggestion. We have used the PBF distribution for frequency analysis of the complete rainfall time series including both rain and non-rainy days (Dimitriadis and Koutsoyiannis, 2018), which have shown good results (line 129-134, line 240-243).

Comment #2: The presented historical review of precipitation fitting needs major changes, since many mentioned works by the authors include mistakes or older methods of precipitation probability fitting. Please consider looking the review (and mentioned references) presented in a recent extended analysis in the book of Koutsoyiannis (2022). Also, please note that the first global-analyses on the rainfall-extremes probability-fitting was performed by Koutsoyiannis (2004a,b).

Response to R2C2: Thanks for your suggestion. The historical review has been updated according to the provided references between line 69-72 and 79-83 .

One specific note is that the GEV distribution becomes the extreme value type Ⅱ distribution when the parameter >0, which is our case (line 138-139). This view was performed by Koutsoyiannis (2004a).

Koutsoyiannis, D., Statistics of extremes and estimation of extreme rainfall: Ⅰ. Theoretical investigation. Hydrological sciences journal, 2004. Vol.49(No.4): p. 575-590.

Comment #3: Regarding non-stationarity and trend analyses, please discuss that both these are model properties and not timeseries' characteristics, and thus, a timeseries cannot be characterized, for example, as stationary or non-stationary but rather a model can be selected to be stationary and non-stationary and after comparing this model to the timeseries, one may decide which one best fits (i.e., being statistically more significant) the timeseries (e.g., see Koutsoyiannis and Montanari, 2015, for the non-stationarity issue, and Iliopoulou and Koutsoyiannis, 2020, for the trends).

Response to R2C3: Thanks for your suggestion. The authors are holding slightly different opinion, and actually refer to Louise (2021) that the non-stationarity is a characteristic of time series..

Louise J. Slater;Bailey Anderson;Marcus Buechel;Simon Dadson;Shasha Han;Shaun Harrigan;Timo Kelder;Katie Kowal;Thomas Lees;Tom Matthews;Conor Murphy and Robert L. Wilby.Nonstationary weather and water extremes: a review of methods for their detection, attribution, and management[J].Hydrology and Earth System Sciences,2021,Vol.25(7): 3897

Comment #4: Please conside discussing that the interpolation method to fill in the gaps in the precipitation timeseries will destroy part of the strong dependence of the autocorrelation function of the timeseries, which is known to exhibit a long-range dependence (e.g., see a global-scale analysis on the small-to-large-scale autocorrelation function of precipitation process in Dimitriadis et al., 2021).

Response to R2C4: Thanks for your suggestion. We have removed the 10-year data gap from the NOAA dataset. For what is worth, through our analysis, we found that either linear interpolation or zero-padding had little influence on the fitting effect.

Comment #5: Please separate the regular-precipitation and maximum-precipitation analyses, since there seems to be a confusion in the manuscript. I am confused whether the authors fit regular daily precipitation timeseries (through, for example, the Pearson III distribution) or maximum daily precipitation timeseries (through, for exmaple, GEV ditribution) or both

Response to R2C5: Thanks for your suggestion. In this work, the maximum-precipitation analysis was used to verify if the regular precipitation analyses (GEV and Weibull models) were correct. This has been clarified on lines 270-274.

Comment #6: It is mentioned in the manuscript that "...the annual maximum data were found independent."; please discuss that this is probably due to the high bias introduced by the autocorrelation function estimator. For the maximum rainfall data, please see a review in Iliopoulou and Koutsoyiannis (), where it is found that a hidden strong dependence is found in the maximum rainfall records if the correct tools are used

Response to R2C6: Thanks for your suggestion. We have conducted the climacogram analysis based on D. Koutsoyiannis (2010,2019); since our annual maxima followed a normal distribution, the problem of the false alarm of antipersistence for non-Gaussian time series does not exist according to the provided references. This has been discussed on lines 285-288.

Koutsoyiannis D. HESS Opinions" A random walk on water"[J]. Hydrology and Earth System Sciences, 2010, 14(3): 585-601.

Iliopoulou, T., and D. Koutsoyiannis, Revealing hidden persistence in maximum rainfall records, Hydrological Sciences Journal, 64 (14), 1673–1689, doi:10.1080/02626667.2019.1657578, 2019.

Comment #7: Regarding the impact of the "time-interval" (or else the resolution) of the precipitation timeseries to the serial dependence, please consider discussing that a more appropriate tool to estimate the serial dependence is the climacogram estimator, which is found to have a better performance than the autocorrelation-function estimator (see the first such analysis with these comparisons in Dimitriadis and Koutsoyiannis, 2015).

Response to R2C7: Thanks for your suggestion. The results demonstrate that the NSE of the Generalized Extreme Value (GEV) distribution improves with an increase in the time interval. Additionally, the independence of the time series has been verified by incorporating the climacogram method in this paper (line 285-288, line 306-310).

Thank you once again

Best regards,

Wangyuyang Zhai

Author

E-mail: [email protected]

Reviewer 3 Report

The article uses a long time series of precipitation to determine the probability distribution of rain of a certain height. The methodology of the works does not raise any major objections, while referring to the purpose of the municipal drainage and sewage system is problematic in my opinion. This is due to the fact that the sizing of these systems should be based on rainfall intensity and not only on precipitation height - e.g. compare for daily data precipitation with a total of 70mm may occur for an hour or for 12 hours, in the first case the drainage system will be overloaded, in the second case it will not. So, please consider adding this information or comment in the text.

Some of detailed comment below:

line 14 - not only frequency but most important prediction of the rain intensity (see earlier)

line 20- better fit? add some values referred to this statement

line 59 - " few studies" - add references to these

line 105 - how many gaps there was in for the entire population in %?

fig.1 - add wider location map (e.g. country, region?)

line 82, 132 - Ye et al., 2018; He et al., 2023

line 197, 201, etc. - Error! Reference 197 source not found.

line 255 - precipitation depths? rather precipitation height

conclusions - since the phenomenon of a northward shift of precipitation has been observed since 2010, maybe some conclusions should be added from this?

precipitation depth?

Author Response

Dear Reviewer #3:

Comment #1: line 14 - not only frequency but most important prediction of the rain intensity (see earlier)

Response to R3C1: Thanks for your suggestion. The sentence has been modified on line 14.

Comment #2: line 20- better fit? add some values referred to this statement

Response to R3C2: done on lines 22.

Comment #3: line 59 - " few studies" - add references to these

Response to R3C3: done on line 65.

Comment #4: line 105 - how many gaps there was in for the entire population in %?

Response to R3C4: 16%, which is added on line105.

Comment #5: fig.1 - add wider location map (e.g. country, region?)

Response to R3C5: done on line 109

Comment #6: line 82, 132 - Ye et al., 2018; He et al., 2023

Response to R3C6: done on lines 81.

Comment #7: line 197, 201, etc. - Error! Reference 197 source not found.

Response to R3C7: done on lines 221.

Comment #8: line 255 - precipitation depths? rather precipitation height

Response to R3C8: Thanks for your suggestion. It is traditionally named as precipitation depth, as shown in B. Shehu et al., 2023.

Shehu; B. Shehu;W. Willems;H. Stockel;L.-B. Thiele;U. Haberlandt.Regionalisation of rainfall depth–duration–frequency curves with different data types in Germany[J].Hydrology and Earth System Sciences,2023,Vol.27(5): 1109-1132

Comment #9: conclusions - since the phenomenon of a northward shift of precipitation has been observed since 2010, maybe some conclusions should be added from this?

Response to R3C9: Thanks for your suggestion. The phenomenon of a northward shift of precipitation has been observed since 2010. However, it is difficult to determine from our study whether it is related to the northward shift of rain bands due to the lack of enough data.

Thank you once again

Best regards,

Wangyuyang Zhai

Author

E-mail: [email protected]

Round 2

Reviewer 2 Report

The authors have addressed all comments; please see additional suggestions based on the authors' replies:

1) Comment #1: The PBF distribution has been also applied to a global-scale precipitation network (Dimitriadis et al., 2021), and was found to fit precipitation complete probability distribution very well. If I understand correctly, here the authors find that in their case scenario, the Pearson III distribution fits the (regular or maximum?) precipitation timeseries; if yes, then please discuss this in the comparison between Pearson III and PBF.

2) Comment #2: The presented historical review of precipitation fitting needs more changes, since again mentioned works include mistakes (for example, regarding the use of gamma distribution, please see a literature review and the expansion to the use of the generalized gamma distribution for rainfall in Koutsoyiannis, 2005a;b).

Koutsoyiannis, D., 2005a. Uncertainty, entropy, scaling and hydrological stochastics, 1, Marginal distributional properties of hydrological processes and state scaling. Hydrological Sciences Journal, 50(3), 381–404, doi: 10.1623/hysj.50.3.381.65031.
Koutsoyiannis, D., 2005b. Uncertainty, entropy, scaling and hydrological stochastics, 2, Time dependence of hydrological processes and time scaling. Hydrological Sciences Journal, 50 (3), 405–426, doi:10.1623/hysj.50.3.405.65028.

3) Comment #3: Regarding the authors' reply on the stationarity/non-stationarity issue, I understand that most studies in the literature confuse it. I will try to help the authors see my point-of-view, hoping to convince them. I will refer to the work by Dimitriadis and Koutsoyiannis (2018), which you have already mentioned in your work in order to give you an example from there. In section 4.2, there is an application in a daily precipitation timeseries, which both the observed and simulated can be seen in Figure 4d. Now, one may argue that there is a downward trend in the observed timeseries; however, this trend was simulated without using a non-stationary model. In simple words, one may select to use either a stationary or a non-stationary model to capture the observed trend, and therefore, the stationarity/non-stationarity is not a characteristic of a timeseries but rather of a model that is used to capture the timeseries characteristics.

4) Comment #5: It is mentioned in the authors' replies that in this study the maximum-precipitation analysis was performed (and that this is clarified in lines 270-274); however, I still cannot understand the authors' method and novelty/innovation "to verify if the regular precipitation analyses (GEV and Weibull models) were correct". Please consider explaining your method much clearer in the Abstract and Introduction, and please make sure this is reflected in the title: "Developing the Actual Precipitation Probability Distribution based on the Complete Daily Series" (for example, in the current title nothing is mentioned on the use of rainfall extremes). I assume this will be also very confusing for the readers.

5) Comment #7: Please use the reference Dimitriadis and Koutsoyiannis (2015), mentioned in my previous review, to justify the use of the climacogram, since it has been shown that it is more statistically robust than the autocorrelation (the authors can see how smoother the estimation is in their Figure 5).

6) In the Conclusions, it is mentioned that

(1) "The use of complete time series in precipitation frequency analysis gave more realistic estimates of probabilities than the traditional methods relying on the rainy days only.". It is my understanding that the difference in using the complete timeseries (i.e., both wet-dry days) and separated the wet from the 0 rainfall, is a single parameter related to the probability-dry (for example, in Dimitriadis and Koutsoyiannis, 2018, where the mixed PBF is used to fit the complete timeseries of both the wet-dry days, the parameter h is related to the probability-dry). If yes, then the a0 parameter in the Pearson III distribution should be negative, as also h<0 in Dimitriadis and Koutsoyiannis (2018); please discuss that applying the complete rainfall timeseries has been already applied in Dimitriadis and Koutsoyiannis (2018), but with the mixed PBF distributio, where it is also mentioned that this procedure (i.e., not separating wet and dry days) improves fitting performance.

(2) "The return periods of historic rainfall should also be determined from the complete daily series rather than the wet-day only series". It is my understanding that this is equivalent to that the frequency values should be determined from the complete timeseries.

(3) "A clear threshold of 137 days was found in this study to separate the persistent or autocorrelated time series from the antipersistent or independent time series based on the climacogram analysis.". This value was estimated as above 200 days from a global-scale analysis of precipitation timeseries (based on the 1% standardized climacogram; please see Figure 8b in Dimitriadis et al., 2021), which is very close to what the authors found.

(4) "The choice of the starting and ending points had a significant influence on the M-K test and easily led to different mutation points for trend analysis.". Do these trends refer to the regular or maximum rainfall timeseries? Also, is it possible here to give more specific results, as for example, how much does the selection of the starting and ending points affect the trend and its statistical significance? This would be useful to readers dealing with trend-analysis.

(5) "The test of K-permutation sampling revealed that the lack of data would affect the accuracy of the frequency analysis only after the missing data reached 70% of the whole dataset.". I think this conclusion strongly depends on the zero frequency of the used timeseries, and it could easily change if, for example, rainfall timseries from different climate regimes are considered (please, clearly state this issue).

There has been some improvement in the language of the manuscript; however, it is still difficult to comprehend in several points; please consider using a native English speaker to assist you in this (I am often using one myself).

Author Response

Dear Reviewer #2:

Comment #1: The PBF distribution has been also applied to a global-scale precipitation network (Dimitriadis et al., 2021), and was found to fit precipitation complete probability distribution very well. If I understand correctly, here the authors find that in their case scenario, the Pearson III distribution fits the (regular or maximum?) precipitation timeseries; if yes, then please discuss this in the comparison between Pearson III and PBF.

Response to R2C1：Thanks for your suggestion. According to the analysis results, for regular rainfall data, the NSE of the Pearson III was 0.994, and the NSE of the PBF was 0.881. So the Pearson III was better than PBF (Line 230-231).

Comment #2: The presented historical review of precipitation fitting needs more changes, since again mentioned works include mistakes (for example, regarding the use of gamma distribution, please see a literature review and the expansion to the use of the generalized gamma distribution for rainfall in Koutsoyiannis, 2005a;b).

Response to R2C2: Thanks for your suggestion. We have made modifications to the section on the Gamma distribution based on the article by Koutsoyiannis (2005a;b). We have removed the exponential distribution from the texts, as the Gamma distribution is a type of the exponential distribution as the references indicate. However, the references that the reviewers pointed out did not include any information about the generalized gamma distribution (Line 70-73).

Comment #3: Regarding the authors' reply on the stationarity/non-stationarity issue, I understand that most studies in the literature confuse it. I will try to help the authors see my point-of-view, hoping to convince them. I will refer to the work by Dimitriadis and Koutsoyiannis (2018), which you have already mentioned in your work in order to give you an example from there. In section 4.2, there is an application in a daily precipitation timeseries, which both the observed and simulated can be seen in Figure 4d. Now, one may argue that there is a downward trend in the observed timeseries; however, this trend was simulated without using a non-stationary model. In simple words, one may select to use either a stationary or a non-stationary model to capture the observed trend, and therefore, the stationarity/non-stationarity is not a characteristic of a timeseries but rather of a model that is used to capture the timeseries characteristics.

Response to R2C3: Thanks for your suggestion. We have understood stationarity/non-stationarity issue. Since this is not our major concern in this work, we have decided to remove all the contents about the stationarity.

Comment #4: It is mentioned in the authors' replies that in this study the maximum-precipitation analysis was performed (and that this is clarified in lines 270-274); however, I still cannot understand the authors' method and novelty/innovation "to verify if the regular precipitation analyses (GEV and Weibull models) were correct". Please consider explaining your method much clearer in the Abstract and Introduction, and please make sure this is reflected in the title: "Developing the Actual Precipitation Probability Distribution based on the Complete Daily Series" (for example, in the current title nothing is mentioned on the use of rainfall extremes). I assume this will be also very confusing for the readers.

Response to R2C4: Thanks for your suggestion. Our purpose was to find out why the GEV and Weibull distributions performed so badly compared to the PIII and PBF for daily series. As the return periods determined from all four methods were close for the annual maxima, it can be concluded that the models were indeed correct. So If there were the errors should be only caused by the mismatch between the model and the observations on a daily basis (Line 276-282). As this is not a major point of the work, we don’t think it is necessary to be mentioned in the title or abstract.

Comment #5: Please use the reference Dimitriadis and Koutsoyiannis (2015), mentioned in my previous review, to justify the use of the climacogram, since it has been shown that it is more statistically robust than the autocorrelation (the authors can see how smoother the estimation is in their Figure 5).

Response to R2C5: Thanks for your suggestion. We have added the references in the article (Line:164-167). However, the data in this paper could not support this strong argument and it is not the major purpose of this paper to do so either. The readers could refer to the papers to make judgments.

Comment #6: (1) "The use of complete time series in precipitation frequency analysis gave more realistic estimates of probabilities than the traditional methods relying on the rainy days only.". It is my understanding that the difference in using the complete timeseries (i.e., both wet-dry days) and separated the wet from the 0 rainfall, is a single parameter related to the probability-dry (for example, in Dimitriadis and Koutsoyiannis, 2018, where the mixed PBF is used to fit the complete timeseries of both the wet-dry days, the parameter h is related to the probability-dry). If yes, then the a0 parameter in the Pearson III distribution should be negative, as also h<0 in Dimitriadis and Koutsoyiannis (2018); please discuss that applying the complete rainfall timeseries has been already applied in Dimitriadis and Koutsoyiannis (2018), but with the mixed PBF distributio, where it is also mentioned that this procedure (i.e., not separating wet and dry days) improves fitting performance.

Response to R2C6(1): Thanks for your suggestion. From what we have found, the a0 parameter of the Pearson III could be positive considering the dry days. The references have been added (Line 226-228).

Comment #6: (2) "The return periods of historic rainfall should also be determined from the complete daily series rather than the wet-day only series". It is my understanding that this is equivalent to that the frequency values should be determined from the complete timeseries.

Response to R2C6(2): Thanks for your suggestion. We believe that you understand the meaning we want to convey. Conclusion 1 and Conclusion 2 have been merged in our article.

Comment #6: (3) "A clear threshold of 137 days was found in this study to separate the persistent or autocorrelated time series from the antipersistent or independent time series based on the climacogram analysis.". This value was estimated as above 200 days from a global-scale analysis of precipitation timeseries (based on the 1% standardized climacogram; please see Figure 8b in Dimitriadis et al., 2021), which is very close to what the authors found.

Response to R2C6(3): Thanks for your comment. We have carefully reviewed (Dimitriadis et al., 2021) and its Fig. 8b, but could not find a clear threshold on the Fig. 8b (Line 295-296)

Dimitriadis, P., et al., A Global-Scale Investigation of Stochastic Similarities in Marginal Distribution and Dependence Structure of Key Hydrological-Cycle Processes. Hydrology, 2021. Vol.8(No.59): p. 59.

Comment #6 :(4) "The choice of the starting and ending points had a significant influence on the M-K test and easily led to different mutation points for trend analysis.". Do these trends refer to the regular or maximum rainfall timeseries? Also, is it possible here to give more specific results, as for example, how much does the selection of the starting and ending points affect the trend and its statistical significance? This would be useful to readers dealing with trend-analysis.

Response to R2C6(4): Thanks for your comment. The trends refer to the regular time series. Due to the data limit, we were not able to evaluate its significance in this work.

Comment #6 :(5) "The test of K-permutation sampling revealed that the lack of data would affect the accuracy of the frequency analysis only after the missing data reached 70% of the whole dataset.". I think this conclusion strongly depends on the zero frequency of the used timeseries, and it could easily change if, for example, rainfall timseries from different climate regimes are considered (please, clearly state this issue).

Response to R2C6(5): Thanks for your suggestion. We have incorporated the explanations in the article as requested. This pattern holds true for regions with a continental climate as well as drier areas, although the dividing proportion of missing data may slightly vary away from 70%. In the wetter climates with more positive rain data, data missing can be expected to cause more troubles in determining the probability distribution (Line 454-459).

This pattern should hold true for continental climates and dry regions such as the northeastern China with many dry days, though the dividing proportion of missing data may be slightly different from 70%. In the wetter climates with more positive rain data, data missing can be expected to cause more troubles in determining the probability dis-tribution.

Thank you once again

Best regards,

Wangyuyang Zhai

Author

E-mail: [email protected]

Round 3

Reviewer 2 Report

The authors have addressed all comments; please see my response to two of the authors' replies, which require mainly minor adjustments:

Comment #2:

a) Regarding the authors' question, the generalized gamma is linked to the generalized beta prime distribution, which was studied in Koutsoyiannis (2005a,b; see also the internal supplementary report in http://www.itia.ntua.gr/641/, where additional information for some of their characteristics are provided).

b) Moreover, in line 68 some studies are mentioned that do not correspond to historical reviews and the mentioned distributions. For example, for the gamma distribution, one of the first applied it to hydrology is the famous V. Yevjevich (details are mentioned in his book in 1972). However, this famous scientist is not mentioned in the references [19-22] (for example, reference [20] uses J-shaped distribution and just mention application of gamma distribution without proper physical justification; reference [22] has a strange title including "unified theory of stochastics", whereas the theory of stochastics has been mathematically introduced and develloped by the famous A. Kolmogorov in 1931;1933, who is not even mentioned). Please perform a strong check throughout the manuscript to track such isolated references, and replace them with proper old or recent review studies (since the authors are interested in historical review studies for the introduction and not isolated studies of applications). For example, for the application of the gamma distribution in hydrology, please consider using the established review work by Yevjevich (see one of his latest books in 1972) and the recent review by Koutsoyiannis (2005a;b; who also mentions Yevjevich's work and gives some physical justification for the use of other distributions in hydrology, which include gamma); there is also a study by Johnk (1964) who applied the gamma distribution to rainfall but again without performing a review mentioning older studies.

Johnk, M.D., Erzeugung von betaverteilten und gammaverteilten Zufallszahlen,
Metrika, 8: 5-15, 1964.

Kolmogorov, A.N., 1931. Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann., 104, 415-458. (English translation: On analytical methods in probability theory, In: Kolmogorov, A.N., Selected Works of A. N. Kolmogorov - Volume 2, Probability Theory and Mathematical Statistics, ed. by A.N. Shiryayev, Kluwer, Dordrecht, The Netherlands, 62-108, 1992).

Kolmogorov, A.N., 1933. Grundbegriffe der Wahrscheinlichkeitsrechnung, Ergebnisse der Math. (2), Berlin (2nd English Edition: Foundations of the Theory of Probability, 84 pp. Chelsea Publishing Company, New York, 1956).

Yevjevich, V., Probability and Statistics in Hydrology, Water Resources Publications, Fort Collins, Colorado, USA, 1972.

Comment #6(3): Regarding the authors' reply, please see that in Fig.8b, the standardized climacogram has dropped to 1% (in y-y axis) at around 200 days (in x-x axis), which means that the persistence starts to fade out (i.e., below 1%) after this threshold.

Comment #6(4): I understand the authors' difficulty to perform such tests due to data limit. At least, please consider mentioning that limitation and present any rough estimations for the readers.

The English language of the manuscript has been improved but there are still some corrections that need to be made. Such an example (there are more in the text) is:

"The widely used method for precipitation frequency analysis is fitting the observation data to a probability distribution curve intrinsically determined by the corresponding probability density function (PDF) that applies to a specific region." should be rephrased to something like "The widely used method for precipitation frequency analysis is used to fit the observation data to a probability distribution density function (PDF) that is the most appropriate for the selected region.".

Author Response

Dear Reviewer #2:

Comment #1: Regarding the authors' question, the generalized gamma is linked to the generalized beta prime distribution, which was studied in Koutsoyiannis (2005a,b; see also the internal supplementary report in http://www.itia.ntua.gr/641/, where additional information for some of their characteristics are provided).

Response to R2C1：Thanks for your suggestion. We now have a clear understanding of the significance of the generalized gamma distribution， and understand that the generalized gamma is linked to the generalized beta prime distribution

Comment #2: Moreover, in line 68 some studies are mentioned that do not correspond to historical reviews and the mentioned distributions. For example, for the gamma distribution, one of the first applied it to hydrology is the famous V. Yevjevich (details are mentioned in his book in 1972). However, this famous scientist is not mentioned in the references [19-22] (for example, reference [20] uses J-shaped distribution and just mention application of gamma distribution without proper physical justification; reference [22] has a strange title including "unified theory of stochastics", whereas the theory of stochastics has been mathematically introduced and develloped by the famous A. Kolmogorov in 1931;1933, who is not even mentioned). Please perform a strong check throughout the manuscript to track such isolated references, and replace them with proper old or recent review studies (since the authors are interested in historical review studies for the introduction and not isolated studies of applications). For example, for the application of the gamma distribution in hydrology, please consider using the established review work by Yevjevich (see one of his latest books in 1972) and the recent review by Koutsoyiannis (2005a;b; who also mentions Yevjevich's work and gives some physical justification for the use of other distributions in hydrology, which include gamma); there is also a study by Johnk (1964) who applied the gamma distribution to rainfall but again without performing a review mentioning older studies.

Response to R2C2: Thanks for your suggestion. The reference [20] has been removed. And reference to variable V. Yevjevich(1972), Kolmogorov(1931,1933) and Koutsoyiannis(2005a,2005b) has been included.. Furthermore, an understanding of the developmental history of the gamma distribution has been gained, as well as the connection between the generalized gamma distribution and the beta prime distribution.(Line 70-72)

Yevjevich, V., Probability and Statistics in Hydrology, Water Resources Publications, Fort Collins, Colorado, USA, 1972.

Comment #3: Regarding the authors' reply, please see that in Fig.8b, the standardized climacogram has dropped to 1% (in y-y axis) at around 200 days (in x-x axis), which means that the persistence starts to fade out (i.e., below 1%) after this threshold.

Response to R2C3: Thanks for your suggestion. We have understood the meaning of Fig.8b.

Comment #4 I understand the authors' difficulty to perform such tests due to data limit. At least, please consider mentioning that limitation and present any rough estimations for the readers.

Response to R2C4: Thanks for your suggestion. We have been modified this part in the article. We think that our finding could be influenced by the fraction of the dry days within the dataset (72.6% in this study) (Line 457-458).

Thank you once again

Best regards,

Wangyuyang Zhai

Author

E-mail: [email protected]

Round 4

Reviewer 2 Report

Thank you for your replies and for addressing the comments. Please see my last 2 comments based on the authors' reply on the previous review-revision round:

Comment #2: I think the authors have misunderstood my suggestion. To assist the authors, please see my suggestion for their paragraph on the historical review (consider replacing the references in lines 68-72) with this one below, which includes only review papers and not isolated studies (also, the Pearson III distribution is the generalized gamma one, which is already mentioned in the paragraph):

"Historically, precipitation frequency analyses mainly used the incomplete time series of only wet days with non-zero rain data to fit the probability distributions, such as Gamma [19], Log-Pearson III [20], or the power-type distributions [21, 22]. Recently, the generalized gamma distribution (or else Pearson III) has been used to analyze rainfall data, providing insights into the link of the beta prime distribution [21, 22]."

[19] Yevjevich, V., Probability and Statistics in Hydrology, Water Resources Publications, Fort Collins, Colorado, USA, 1972.

[20] El Adlouni, S., Bobée, B., & Ouarda, T. B. M. J. On the Tails of extreme event distributions in Hydrology. Journal of Hydrology, 355, 16–33. https://doi.org/10.1016/j.jhydrol.2008.02.011, 2008.

[21] Koutsoyiannis, D., Uncertainty, entropy, scaling and hydrological stochastics. 1. Marginal distributional properties of hydrological processes and state scaling / Incertitude, entropie, effet d'échelle et propriétés stochastiques hydrologiques. 1. Propriétés distributionnel. Hydrological Sciences Journal, 2005. 50(3).

[22] Koutsoyiannis, D., Uncertainty, entropy, scaling and hydrological stochastics. 2. Time dependence of hydrological processes and time scaling/Incertitude, entropie, effet d'échelle et propriétés stochastiques hydrologiques. 2. Dépendance temporelle des processus hydrologiques et échelle temporelle. Hydrological Sciences Journal, 2005. 50(3).

Comment #3: Please mention the work by Dimitriadis et al. (2021) that is mentioned in previous reviews, which draws a similar conclusion to the authors that persistence starts to fade out after the threshold of around 200 days (the authors suggest 137 days based on their analysis).

Author Response

Dear Reviewer #2:

Comment #1: I think the authors have misunderstood my suggestion. To assist the authors, please see my suggestion for their paragraph on the historical review (consider replacing the references in lines 68-72) with this one below, which includes only review papers and not isolated studies (also, the Pearson III distribution is the generalized gamma one, which is already mentioned in the paragraph):

Response to R2C1：Thanks for your suggestion. We have made the necessary modifications in the article based on your suggestions (Line 68-72).

Comment #2: Please mention the work by Dimitriadis et al. (2021) that is mentioned in previous reviews, which draws a similar conclusion to the authors that persistence starts to fade out after the threshold of around 200 days (the authors suggest 137 days based on their analysis).

Response to R2C2: Thanks for your suggestion.

First of all, this is the journal’s policy: “** There is no need to forcefully cite the references recommended by the reviewers.”

Second, we actually do not agree with the authors’ opinion in the last round of review: “Comment #3: Regarding the authors' reply, please see that in Fig.8b, the standardized climacogram has dropped to 1% (in y-y axis) at around 200 days (in x-x axis), which means that the persistence starts to fade out (i.e., below 1%) after this threshold.” This is based on the following four considerations.

The Fig.8b in the recommended paper included three lines q5, q95, and mean for climacogram, which divided the dataset into three groups. Only one of the three lines (q5(Cl)) seems to have a shape similar to our finding, but our analysis was based on the whole dataset without data subsetting. Further, the mean line in the Fig.8b of the recommended paper was very different from our finding.

Our finding of the threshold was made based on the inflection point of the fitting line, while the reviewer’s statement “which draws a similar conclusion to the authors that persistence starts to fade out after the threshold of around 200 days”, which was not even mentioned at all from the recommended paper, could not be reflected from the Fig.8b. The inflection point of q5 (Cl) line occurred at 1000 days (0.1%), which was far from our finding.

In the previous round of review, the reviewer also mentioned in their Fig.8b that “the standardized climacogram has dropped to 1% (in y-y axis) at around 200 days (in x-x axis)”. Their y-axis was calculated differently from our y-axis (please see the Eq.1 in their paper and Eq.12 in our manuscript). Further, based on our finding, we did not observe a connection between 1% climacogram and the threshold as the reviewer suggested in the recommendation.

Lastly, there was only one short paragraph in the recommended paper briefly discussing Fig.8b, which did not mention the statement about the threshold of 200 days or 1% y-axis value, as the reviewer pointed out in the recommendations. Here is the excerpt:

“In Figures 2–8, which show the climacogram and climacospectrum for each process, we observed similarities in the shape of the dependence structure spanning from strong correlations at the small scales to a power-type behavior at large scales, with a convex shape at the intermediate scales.”

Further, the statement of “200 threshold and 1% climacogram” was also absent from their abstract and conclusion as follows:

“Abstract: To seek stochastic analogies in key processes related to the hydrological cycle, an extended collection of several billions of data values from hundred thousands of worldwide stations is used in this work. The examined processes are the near-surface hourly temperature, dew point, relative humidity, sea level pressure, and atmospheric wind speed, as well as the hourly/daily streamflow and precipitation. Through the use of robust stochastic metrics such as the K-moments and a second order climacogram (i.e., variance of the averaged process vs. scale), it is found that several stochastic similarities exist in both the marginal structure, in terms of the first four moments, and in the second order dependence structure. Stochastic similarities are also detected among the examined processes, forming a specific hierarchy among their marginal and dependence structures, similar to the one in the hydrological cycle. Finally, similarities are also traced to the isotropic and nearly Gaussian turbulence, as analyzed through extensive lab recordings of grid turbulence and of turbulent buoyant jet along the axis, which resembles the turbulent shear and buoyant regime that dominates and drives the hydrological-cycle processes in the boundary layer. The results are found to be consistent with other studies in literature such as solar radiation, ocean waves, and evaporation, and they can be also justified by the principle of maximum entropy. Therefore, they allow for the development of a universal stochastic view of the hydrological-cycle under the Hurst–Kolmogorov dynamics, with marginal structures extending from nearly Gaussian to Pareto-type tail behavior, and with dependence structures exhibiting roughness (fractal) behavior at small scales, long-term persistence at large scales, and a transient behavior at intermediate scales.”

“5. Conclusions

The major innovation of this study is the uniting view of the key hydrological-cycle processes through the analysis of several billions of observations from hundred thousands of stations by robust statistical metrics of (a) the K-moments, for the estimation of the marginal structure of the first four moments, and of (b) the climacogram, for the estimation of the second-order dependence structure. The key examined hydrological-cycle processes are the near-surface temperature, dew point, humidity, sea level pressure, atmospheric wind speed, streamflow, and precipitation, as well as other processes from previous studies, such as shear and buoyant turbulent processes analyzed through small-scale laboratory experiments, and solar radiation, and ocean waves. The main traced stochastic similarities are as follows:

(1) A hierarchy related to the hydrological cycle was identified with the dew point, temperature, relative humidity, solar radiation, and sea level pressure all exhibiting a lower skewness over kurtosis absolute ratio than the turbulent processes, wind speed, and ocean waves, and with a stronger long-term persistence (LTP) behavior in the dependence structure (H > 0.75), followed by streamflow and precipitation, both of which exhibit a smaller skewness–kurtosis absolute ratio and a weaker LTP behavior (H 0.75).

(2) All the examined processes can be adequately simulated by the truncated mixed- PBF distribution, adjusting for probability dry and lower (or upper) truncation, in terms of the first four moments, and ranging from (truncated) nearly Gaussian to Pareto-type tails.

(3) As the sample size increases, different records of the same process from several locations converge to a smaller area of the nondimensionalized statistics (skewness–kurtosis), indicating a common marginal behavior.

(4) All the examined hydrological-cycle processes exhibit a similar dependence structure that extends from the fractal behavior with roughness (M < 0.5) located at the small intermittent scales to the LTP behavior at large scales (H > 0.5), while both indicate large uncertainty and high climatic variability.

(5) Finally, since the above empirical findings are consistent with previous studies and can be justified by the principle of maximum entropy, they allow for a uniting stochastic view of the hydrological-cycle processes under the Hurst–Kolmogorov (HK) dynamics in terms of both the marginal and dependence structures.”

So we are frustrated to add this citation here that could not support the reviewer’s recommendation and had little relevance with our finding.

Thank you once again

Best regards,

Wangyuyang Zhai

Author

E-mail: [email protected]

Round 5

Reviewer 2 Report

Dear Authors,

It is my understanding that there has been a misconception (or frustration) here regarding my review. First of all, please accept my apologies if you think I forced you in any way to cite any references recommended in my reviews. It is my point-of-view that during a review-process the authors can either follow the recommendations or justify their reply if they disagree (regardless of the references the reviewers suggest); this procedure offers a way for the authors-reviewers-editors to communicate with each, know each others' methods related to the manuscript, and exchange knowledge and opinions, all with the aim to improve the manuscript so it can be published in the journal. All my comments in the submitted reviews were based on this concept. And I think it worked, in the sense that the authors have applied new tools (like the climacogram that is found to be more smooth than the autocorrelation function), but also for me (like the impact of the missing data on the distribution).

Also, the reviewers can sometimes suggest some references (even self-references) if they are strongly related to the authors' methods or findings, if they can be found helpful to the authors, or in order to notify the authors of more robust references existed in the literature as compared to any isolated or poor used references.

Therefore, please let me explain in more detail why I think this comment and reference is relevant to the authors' finding (related to the fade out of the dependence structure after the threshold of 137 days). The suggested paper (Dimitriadis et al., 2021) is one of my scientific-team's latest huge works (which lasted several years) that includes the largest (so-far) stochastic analysis of hydrological-cycle ground-stations (in total thousand of stations were processed with several billions of records). One of the main targets of this work was to show whether there were any stochastic similarities among timeseries of the same process (e.g., precipitation) from completely different locations. One of the results was that there were (!); for example, we have depicted in Figure 8 the average dependence structures (please note that the depicted dependence structure has been estimated as the global-average one from thousands of precipitation stations).

Therefore, the relation to the authors' 2nd conclusion (i.e., "A clear threshold of 137 days was found in this study to separate the persistent or autocorrelated time series from the antipersistent or independent time series based on the climacogram analysis.") are the following (in my and my team's humble opinion):

(a) It was very interesting for us to see that independent works have drawn similar conclusions regarding the shape of the climacogram for the daily precipitation. Please note that it is impossible for the "average-global" climacogram to be similar to the climacogram from 1 station (there is a very small probability that a single random value drawn from a distribution is the same as the distribution's mean value). So, I found it interesting that a similar climacogram shape was drawn from 2 independent studies, which actually is in favor of our paper's conclusion (i.e., regarding stochastic similarities in the dependence structures). However, if the authors do not find this interesting and worthy of reporting, then please ignore this review-comment. Also, in your Fig. 5, please include units in the axes (i.e., 'mm' in the y-y axis and 'days' in the x-x axis).

(b) The authors are correct concerning that there was no mention of the 1% threshold in our paper; however, the global-average climacogram is estimated that includes many threshold values depending on the error. For example, for the 1% error, the threshold value is estimated from the Fig. 8b to be around 200 days (from the mean), with lower-limit 100 (q5) and upper-limit 1000 (q95) days. Again, we find it exciting that the conclusion from an independent study (i.e., 137 days) was within the limits of our study (this again is in favor of the existence of stochastic similarities in the precipitation process). However, if the authors do not find this exciting, then please ignore again this review-comment.

(c) Since the climacogram shown in Fig. 5 is drawn from a small sample of timeseries, there is a high possibility that this break at 100 days is caused by the estimation's statistical bias (this issue is discussed in our paper), and so, the Hurst parameter should not be estimated from this part but from the climacogram power-law before the break. In our paper, this does not apply since the climacogram is estimated from thousands of samples, and thus, the estimation bias is rather small, and this is why the global-average climacogram is almost a straight power-type line. If one tries to estimate the Hurst parameter from the upper part of the Fig. 5 climacogram, then a larger Hurst is estimated (close to 0.7; in our paper, we suggest a global-average H=0.6 with a lower limit close to 0.5 and an upper limit reaching 0.7).

Finally, since human relationships are far more important and worthy than papers and scientific results, please note that was never my intention to cause any inconvenience; we spent several years analyzing all these data and I am sometimes carried away from scientific curiosity and excitement.

Wish you all the best with your paper, and please feel free to contact me (or my co-authors) if you wish to further discuss any scientific topic to exchange opinions.

Best Regards,

Panayiotis Dimitriadis ([email protected])

Author Response

Dear Reviewer #2:

We actually learnt a lot for your guidance and recommendations through this review process. We truly appreciate your continuous help and the time spent in revising our manuscript. We are willing to learn new things and add new references into the manuscript based on the facts and our judgments. That’s why we keep citing more papers as you recommended during the reviews. However, only with regards to the last point of your recommendations, we, with all due respect, do not find a similarity between the Fig. 8b in your previous paper and our Fig. 5, which has been explicitly explained in our replies in the last round. Here are our specific responses to each of your points of this round:

Comment #1: It was very interesting for us to see that independent works have drawn similar conclusions regarding the shape of the climacogram for the daily precipitation. Please note that it is impossible for the "average-global" climacogram to be similar to the climacogram from 1 station (there is a very small probability that a single random value drawn from a distribution is the same as the distribution's mean value). So, I found it interesting that a similar climacogram shape was drawn from 2 independent studies, which actually is in favor of our paper's conclusion (i.e., regarding stochastic similarities in the dependence structures). However, if the authors do not find this interesting and worthy of reporting, then please ignore this review-comment. Also, in your Fig. 5, please include units in the axes (i.e., 'mm' in the y-y axis and 'days' in the x-x axis).

Response to R2C1：Thanks for your suggestion. Although it might be interesting to compare two figures, we find that the shapes between two papers are different. The mean line and q95 lines in the figure 8b in your paper included a curve without an inflection point, while ours did have an inflection point. The q5 line in your figure 8b did have an inflection point at 1000 days, which were 10 times as our finding. However, we have made the necessary modifications in the article based on your suggestions (Figure 5). Thanks for pointing this out.

Comment #2: The authors are correct concerning that there was no mention of the 1% threshold in our paper; however, the global-average climacogram is estimated that includes many threshold values depending on the error. For example, for the 1% error, the threshold value is estimated from the Fig. 8b to be around 200 days (from the mean), with lower-limit 100 (q5) and upper-limit 1000 (q95) days. Again, we find it exciting that the conclusion from an independent study (i.e., 137 days) was within the limits of our study (this again is in favor of the existence of stochastic similarities in the precipitation process). However, if the authors do not find this exciting, then please ignore again this review-comment.

Response to R2C2: Thanks for your suggestion. We have comprehended the essence of your suggestion, and based on our calculations using the actual data, we have arrived at a precise threshold that is tailored to our specific dataset.

Moreover, as we explained in the reply of the last round, we did not use 1% error to determine the threshold, while we used the inflection point instead. So we are actually not comparing apple to apple.

Comment #3: Since the climacogram shown in Fig. 5 is drawn from a small sample of timeseries, there is a high possibility that this break at 100 days is caused by the estimation's statistical bias (this issue is discussed in our paper), and so, the Hurst parameter should not be estimated from this part but from the climacogram power-law before the break. In our paper, this does not apply since the climacogram is estimated from thousands of samples, and thus, the estimation bias is rather small, and this is why the global-average climacogram is almost a straight power-type line. If one tries to estimate the Hurst parameter from the upper part of the Fig. 5 climacogram, then a larger Hurst is estimated (close to 0.7; in our paper, we suggest a global-average H=0.6 with a lower limit close to 0.5 and an upper limit reaching 0.7).

Response to R2C3: Thanks for your suggestion. Based on the methodology presented in the Koutsoyiannis, D(2010), we have come to the conclusion that the data exhibits long-term independence. This finding aligns with the results obtained from another method discussed in our own article. The agreement between these two approaches further strengthens our belief that the dataset indeed demonstrates long-term independence.

Further, it is hard to say which result is better when a global study is compared to a local study. It is true to say that a local-scale study could not represent the global pattern, while the results of a global study also may not accurately reflect the real condition of a local station.

Koutsoyiannis, D., HESS Opinions" A random walk on water". Hydrology and Earth System Sciences, 2010. 14(3): p. 585-601.

Once again, we sincerely appreciate the time and effort you have dedicated to the review process. We do admire your tremendous achievements and contributions on this field, and look forward to learning more from you in the future.

Thank you once again

Best regards,

Wangyuyang Zhai

Author

E-mail: [email protected]

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