Articles | Volume 29, issue 19
https://doi.org/10.5194/hess-29-4811-2025
© Author(s) 2025. This work is distributed under
the Creative Commons Attribution 4.0 License.Statistical estimation of probable maximum precipitation
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- Final revised paper (published on 30 Sep 2025)
- Preprint (discussion started on 22 Aug 2024)
Interactive discussion
Status: closed
Comment types: AC – author | RC – referee | CC – community | EC – editor | CEC – chief editor
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RC1: 'Comment on egusphere-2024-2594', Anonymous Referee #1, 20 Sep 2024
- AC1: 'Reply on RC1', Jonathan Jalbert, 20 Dec 2024
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RC2: 'Comment on egusphere-2024-2594', Anonymous Referee #2, 18 Nov 2024
- AC2: 'Reply on RC2', Jonathan Jalbert, 20 Dec 2024
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RC3: 'Comment on egusphere-2024-2594', Anonymous Referee #3, 19 Nov 2024
- AC3: 'Reply on RC3', Jonathan Jalbert, 20 Dec 2024
Peer review completion
AR: Author's response | RR: Referee report | ED: Editor decision | EF: Editorial file upload
ED: Reconsider after major revisions (further review by editor and referees) (05 Jan 2025) by Nadav Peleg

AR by Jonathan Jalbert on behalf of the Authors (16 Feb 2025)
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ED: Referee Nomination & Report Request started (20 Feb 2025) by Nadav Peleg
RR by Anonymous Referee #1 (03 Mar 2025)
RR by Anonymous Referee #2 (26 Mar 2025)

ED: Publish subject to revisions (further review by editor and referees) (29 Mar 2025) by Nadav Peleg

AR by Jonathan Jalbert on behalf of the Authors (29 May 2025)
Author's response
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ED: Referee Nomination & Report Request started (03 Jun 2025) by Nadav Peleg
RR by Anonymous Referee #2 (04 Jul 2025)

ED: Publish as is (07 Jul 2025) by Nadav Peleg

AR by Jonathan Jalbert on behalf of the Authors (08 Jul 2025)
I agree with the authors when they state that "translating the definition of PMP into a statistical model is interesting" (line 350). They could also say "estimating PMP is a really hard problem". To the authors' credit, they begin with the commonly accepted definition of PMP as an upper bound, and then construct a statistical model which fits this definition. Fitting a model with a finite upper bound is challenging because precipitation data usually suggests that the distribution is unbounded, and further has a heavy tail. It is probably not surprising that the authors ultimately find their approach to be unsuitable for implementing in practice, and conclude that the best statisical approach is to eschew the upper bound requirement and instead implement extreme value (EV) methods.
Unfortunately, I think the manuscript's structure does not tell its story well. Primarily I view the paper as an interesting way to discuss the challenges of PMP estimation, and talking about their particular model is one part of this larger story. It strikes me that the take away message does not appear in the abstract or in the body until Section 6. I think it would be better to move these messages up front. The story I imagine is something like this:
1. Statistically estimating PMP is hard because its definition assumes a bounded tail, but precipitation data suggests the tail is unbounded. Because statistical estimation is hard, other methods like moisture maximization and Herschfield's scaling get used. Uncertainty and climate change are hard to incorporate into these non-statistical methods and frequently-used moisture maximization approaches involve several subjective judgements.
2. Starting with the ideas which underlie moisture maximization, we develop a sensible statistical model which assumes an upper bound.
3. We perform simulation studies and use the method to fit PMP at two locations in Quebec, but find that estimates for the upper bound have unsuitable uncertainty.
4. We conclude with a discussion and offer our suggestion for best practices.
I think all the pieces of this story are in the paper, but I do not think the current focus of the paper gets the essential message across very well.
I find the notation in the paper to be inconsistent. In equation (1), Y_i denotes precipitation of storm i, but in equation (2) I believe Y_i has been replaced by P_i. Equation (4) supposedly comes from Eq (1), but has quantities EP_i and EP_max, which are presumably PW_i and PW_max in equation (1)? The ratio EP_i/EP_max is known/assumed to be less than 1, correct? If so, please say this explicitly. I believe equation (6) is used as the basis for the statistical model: Y_i = EP_i/EP_max * r_i * PMP. If I am following correctly, Y_i is random and observed. I think EP_i and r_i which underlie Y_i are random, but unobserved. EP_max is a parameter but not known, and PMP is the parameter we wish to estimate. So in the end, the authors propose a model for the observed precipitation Y_i, but use moisture maximization logic to include PMP as a parameter. They choose a beta/Pearson 1 as their distribution to fit. A cynical comment could be "the authors use a data-independent argument to conclude the data arise from a distribution, but which fits the data poorly." I think the story to be told here is that if one begins with a supposition of an upper tail, and one tries to then fit a model based on that assumption, things are really hard.
If I understand correctly, the authors propose a beta/Pearson 1 distribution and fit *all* of the nonzero rainfall data to it. There is talk of thresholding on page 8, but it seems to be more tied to the discrete nature of the measurements rather than to thresholding for focusing on extremes. An EV approach would pick a high threshold or take block maxima and fit an EV model, presumably a reverse-Weibull guaranteeing an upper bound. Would such a method be better suited for estimating an upper bound than fitting a beta to the entire distribution?
The authors show qq plots for the EV models in Figure 6. QQ plots for the beta fit are noticeably absent.
l173. Why is beta known to be greater than 1?
l304: Figure??