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Estimating IDF Curves Consistently over Durations with Spatial Covariates
Article
Peer-Review Record

Evaluating the Performance of a Max-Stable Process for Estimating Intensity-Duration-Frequency Curves

Water 2020, 12(12), 3314; https://doi.org/10.3390/w12123314
Reviewer 1: Anonymous
Reviewer 2: Anonymous
Water 2020, 12(12), 3314; https://doi.org/10.3390/w12123314
Received: 27 September 2020 / Revised: 21 November 2020 / Accepted: 23 November 2020 / Published: 25 November 2020

Round 1

Reviewer 1 Report

Evaluating the Performance of a Max-Stable Process for Estimating Intensity-Duration-Frequency Curves

The present paper deals with the assessment of a novel procedure to estimate IDF curves for an ensemble of rain gauges, building on the integration of different processes. According to the Authors, the need to provide a novel method comes from the inadequacy of existing procedures to account for the asymptotical dependence among different durations and at the same time ensuring consistency between quantiles for increasing return periods and durations. In my opinion the paper is clear in the premises and in the assumptions. The methodological part is rigorously explained, and results are effectively shown. I think the paper needs no particular modifications to be published in this journal. I only have a couple of minor suggestions:

  • Since the methodological part is very theoretical and quite hard to follow, I recommend adding a Conclusions section, which is missing at present, to summarize the main findings and to remark the practical repercussions of the research outcomes, for example concerning dealing with multiple rain gauges at the same time.
  • Page 6, Eq. 12: define “u”.
  • Page 14, line 386: I think there are some missing lines. In the same page there are some issues with references.
  • To complete the literature, review the Authors could also mention the storm index method, which is another procedure to estimate IDF curves in a consistent way. See for example: Padulano, R., Reder, A., Rianna, G. (2019) “An ensemble approach for the analysis of extreme rainfall under climate change in Naples (Italy)” Hydrological Processes 33.14 (2019): 2020-2036.

Author Response

Dear Reviewer,

We appreciate your constructive comments, which we have now addressed in the manuscript. We have highlighted all changes in the manuscript.

  • 1. Since the methodological part is very theoretical and quite hard to follow, I recommend adding a Conclusions section, which is missing at present, to summarize the main findings and to remark the practical repercussions of the research outcomes, for example concerning dealing with multiple rain gauges at the same time.

We have implemented this accordingly, thank you for this advice.

  •  2. Page 6, Eq. 12: define “u”.

We have expanded the explanation of the cost function in the text to clarify what is "u" within the definition of the quantile score.

  •  3. Page 14, line 386: I think there are some missing lines. In the same page there are some issues with references.

We regret making such a mistake on the proof reading. We have added the missing part of the sentence. We have also double checked all the references to add any possible missing information.

  • 4. To complete the literature, review the Authors could also mention the storm index method, which is another procedure to estimate IDF curves in a consistent way. See for example: Padulano, R., Reder, A., Rianna, G. (2019) “An ensemble approach for the analysis of extreme rainfall under climate change in Naples (Italy)” Hydrological Processes 33.14 (2019): 2020-2036.

Thank you for the additional insight regarding additional methods for IDF curve estimation. We added a couple of lines in the introduction regarding the storm index method, as well as in the discussion when talking about further studies regarding a nonstationary setting.

We hope you find these revisions sufficient, and we wish to thank you again for your comments.

Reviewer 2 Report

This study investigates performance of Max-Stable processes in estimating Intensity-duration-frequency curves over 6 German stations. Mathematical descriptions of the presented work are sound and the work has value for urban infrastructure design and designing storm water management system. I have moderate suggestion for potential publication:

  1. In lines 24-36, authors have pointed various literature for IDF generation. They have also mentioned the use of spatial co-variates, however, they have not pointed the use of either time or the time series of large-scale oscillation pattern such as ENSO as the covariates. Applying Bayesian inference Ganguli & Coulibaly (2019) employed nonstationary IDFs considering time as a covariate at six stations at Southern Ontario and projected IDF trends at projected time period.
  2. Line 49: pls. include a citation depicting copula-based IDF statistics (Singh and Zhang, 2007).
  3. Line 116-117: This part is elusive & require clear explanation; in the sentence before it is written not a RV; contradictorily, it is written as ordered RV.
  4. Lines 236-240: If time series used is 40 years—it would be wise to extrapolate (which is typically performed in frequency analysis) up to 100 years only. With limited data 200 – 500 years extrapolation would tend to give erroneous values even if emphasis is placed on simulating tail probability.
  5. Line 260: QQ plot could be shown in supplement.
  6. Line 277: Pls. cite relevant literature.
  7. Page 8: Figure 1: It is difficult to differentiate station-by-station, why not to show 6 stations with different bar colors to distinguish each other.
  8. Legends should be placed in the figure in Figure 7. In figure 7 caption, the las line: I presume different distances are computed using MS-GEV vs rd-GEV approaches, then this line has no meaning since distance is calculated b/n MS-GEV vs rd-GEV.
  9. Line 342: I would vote only report RP values for 5, 10, 25, 50 & 100 which has implications for infrastructural designs. Very large RP amounts to be extrapolation of limited number of data say 30-40 years.
  10. Line 344 and Section therein: Right now only skill metrics used here is QSI, can you also show other measures such as uncertainty of estimated quantile of MS_GEV vs rd_GEV approaches? Pls. refer Table 2&3 of Ganguli & Coulibaly (2017). The ratio of upper bound vs LB will provide the estimates of UQ.
  11. Line 404 and statements after that: Throughout the paper discuss the performance b/n MS-GEV vs rd-GEV. Then how come suddenly d-GEV crops up pls. clarify.

References:

  1. Ganguli, P. Coulibaly, P. (2019). Assessment of future changes in intensity-duration-frequency curves for Southern Ontario using North American (NA)-CORDEX models with nonstationary methods. Journal of Hydrology: Regional Studies, 22, 100587.
  2. Ganguli, P. Coulibaly, P. (2017). Does nonstationarity in rainfall require nonstationary intensity–duration–frequency curves? Hydrol. Earth Syst. Sci, 21, 6461-6483.
  3. Singh, VP., Zhang, L. IDF curves using the Frank Archimedean copula. J. Hydrologic Engineering. Vol. 12:6.

Author Response

Dear Reviewer,

We appreciate your constructive comments, which we have now addressed in the manuscript. We have highlighted all changes in the manuscript.

  •  1. In lines 24-36, authors have pointed various literature for IDF generation. They have also mentioned the use of spatial co-variates, however, they have not pointed the use of either time or the time series of large-scale oscillation pattern such as ENSO as the covariates. Applying Bayesian inference Ganguli & Coulibaly (2019) employed nonstationary IDFs considering time as a covariate at six stations at Southern Ontario and projected IDF trends at projected time period.

We thank you for the additional insight regarding additional methods for IDF curve estimation. We added a couple of lines in the introduction regarding the use of a nonstationary approach, as well as in the discussion when talking about further studies.

  • 2. Line 49: pls. include a citation depicting copula-based IDF statistics (Singh and Zhang, 2007)

We have added a couple of lines regarding the use of copulas for a multivariate approach to IDF modelling in the introduction.

  • 3. Line 116-117: This part is elusive & require clear explanation; in the sentence before it is written not a RV; contradictorily, it is written as ordered RV.

We regret not having clearly expresed our explanation in the matter. What we meant to say is that in the chosen IDF relationship for I(d), only I(d) was a random variable, but not d. Furthermore the process in which I(d) is generated makes it an ordered random variable. We have extensively rewritten this paragraph to better clarify this.

  • 4. Lines 236-240: If time series used is 40 years—it would be wise to extrapolate (which is typically performed in frequency analysis) up to 100 years only. With limited data 200 – 500 years extrapolation would tend to give erroneous values even if emphasis is placed on simulating tail probability AND
  • 5. Line 342: I would vote only report RP values for 5, 10, 25, 50 & 100 which has implications for infrastructural designs. Very large RP amounts to be extrapolation of limited number of data say 30-40 years.

Initially, we had chosen such a long return period to compare with previous studies that also used the 200 and 500 year RP. However, after receiving your recommendation and further discussion between us, we have removed all mentions both in the text and in figure (2,3,4,7,8) regarding the 200 and 500 year return period. We agree that this results are very hard to interpret from a 40 year data series. We thank you for your insight on this issue.

  • 6. Line 260: QQ plot could be shown in supplement.

We have added a small sample of the QQ plots from the observations as an appendix. However, as we have one QQ plot for each of the 120 durations x 6 stations, we could not display all of them. We choose to show for 4 different durations (1,3,48,72) h for three different stations.

  •  7. Line 277: Pls. cite relevant literature.

We have included two different previous studies that use up to 120 hours for the accumulation duration.

  •  8. Page 8: Figure 1: It is difficult to differentiate station-by-station, why not to show 6 stations with different bar colors to distinguish each other.

Our original intention was to pool the information from all stations in one single plot as to show the distribution of rainfall for the whole Wupper catchment. We rewrote the caption to make this more clear. We regret that we could not show the stations separately, as this would require a lot of space to be properly appreciated.

  •  9. Legends should be placed in the figure in Figure 7. In figure 7 caption, the las line: I presume different distances are computed using MS-GEV vs rd-GEV approaches, then this line has no meaning since distance is calculated b/n MS-GEV vs rd-GEV.

We have added a legend in the bottom of Figure 7. We also rewrote the caption to clarify what we meant with the different distance measure. In this case it was the choice of using either the euclidean or log-distance for the variogram of the MS-GEV approach.

  • 10. Line 344 and Section therein: Right now only skill metrics used here is QSI, can you also show other measures such as uncertainty of estimated quantile of MS_GEV vs rd_GEV approaches? Pls. refer Table 2&3 of Ganguli & Coulibaly (2017). The ratio of upper bound vs LB will provide the estimates of UQ.

Thank you for this point! We consider the quantile skill score estimated in a cross-validation setting as a well defined and sufficient basis for a comparison of the point estimates resulting from the two approaches. It is, however, very valuable to have additional information about the uncertainty of the two approaches, in the way you suggest. We consider the issue of estimating uncertainty for the MS-GEV approach is so complex that including it in our current study is beyond the scope of our current objective. It is likely that a Bayesian approach for the estimation of the max-stable parameters would yield the desired uncertainty. However, the use of the full likelihood for the Brown-Resnick process is a relatively new topic in the field, and it is still not completely clear how well the use of composite likelihoods (such as the pairwise likelihood used by us in this study) works with Bayesian inference.

Furthermore, the construction of the rd-GEV approach does not permit to easily obtain asymptotic confidence intervals. These uncertainty measures need to be constructed (a bootstrap approach might work here) and their coverage need to be validated before a comparison is possible. We consider this worth another study and prefer to defer the investigation of their idea to a future study.

  •  11. Line 404 and statements after that: Throughout the paper discuss the performance b/n MS-GEV vs rd-GEV. Then how come suddenly d-GEV crops up pls. clarify.

We have replaced the d-GEV for the rd-GEV in the discussion section. Originally we meant to talk about the d-GEV as the other referenced studies allow the theta parameter to vary, but for consistency with our study, we have changed it so we always talk about applications where the theta parameter is set to be zero, that is, the rd-GEV. There is an exception to this, but this is clarified in the text.

We hope that you find these revisions sufficient, and we thank you again for your insightful comments.

Round 2

Reviewer 2 Report

The authors have addressed my queries satisfactorily. I have only a minor suggestion regarding the presentation of figure 1. Again, it is not clear year-by-year temporal evolution of rainfall extreme pattern over six stations. Does each box in the boxplot show annual maxima rainfall at individual station locations? If this is true, can you please show them using 6 different colors/using 'factor separator' option to separate each station group? 

Author Response

Dear reviewer,

We thank you again for your constructive comments. We have done some changes regarding your question on figure 1 of the manuscript:

  • Does each box in the boxplot show annual maxima rainfall at individual station locations? If this is true, can you please show them using 6 different colors/using 'factor separator' option to separate each station group?

In the original Figure 1, the boxplots did not show individual station locations, rather, they showed the distribution of annual maxima when pooling all stations together. We thought this could give the reader an overall impression of rainfall maxima in the Wupper catchment. However, we agree that it is interesting to show how each station varies in time, for which we have added an extra panel with the information from each individual station. We believe this will provide the reader with additional information about the temporal evolution of each station.

We have also added information to the caption to make this information more clear.

For comparison reasons, the original Figure 1 and the new version have been added as an attachment.

Best regards,

The authors.

 

Author Response File: Author Response.pdf

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