Statistical estimation of probable maximum precipitation

Martin, Anne; Fournier, Elyse; Jalbert, Jonathan

doi:https://doi.org/10.5194/egusphere-2024-2594

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https://doi.org/10.5194/egusphere-2024-2594

Preprints

22 Aug 2024

| 22 Aug 2024

Statistical estimation of probable maximum precipitation

Anne Martin, Elyse Fournier, and Jonathan Jalbert

Abstract. Civil engineers design infrastructures exposed to hydrometeorological hazards, such as hydroelectric dams, using the estimation of probable maximum precipitation (PMP). The World Meteorological Organization (WMO) defines PMP as the maximum amount of water that can physically accumulate over a given time period and region, depending on the season and without considering long-term climate trends. Current methods for calculating PMP have many flaws: some variables used are not directly observable and require a series of approximations to be used; uncertainty is not always taken into account and can sometimes be complex to determine; climate change, which exacerbates extreme precipitation events, is difficult to incorporate into the calculations and subjective choices increases estimation variability. The goal of this work is to propose a statistical and objective method for estimating PMP that meets the WMO definition and allows for uncertainty estimation and climate change incorporation. This novel approach leverages the Pearson Type I distribution, a generalization of the Beta distribution over an arbitrary interval. The proposed method is applied to estimate the PMP at two meteorological stations in Québec, Canada.

Received: 16 Aug 2024 – Discussion started: 22 Aug 2024

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Journal article(s) based on this preprint

30 Sep 2025

Statistical estimation of probable maximum precipitation

Anne Martin, Élyse Fournier, and Jonathan Jalbert

Hydrol. Earth Syst. Sci., 29, 4811–4824, https://doi.org/10.5194/hess-29-4811-2025,https://doi.org/10.5194/hess-29-4811-2025, 2025

Short summary

Anne Martin, Elyse Fournier, and Jonathan Jalbert

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2024-2594', Anonymous Referee #1, 20 Sep 2024

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??

Citation: https://doi.org/10.5194/egusphere-2024-2594-RC1
- AC1: 'Reply on RC1', Jonathan Jalbert, 20 Dec 2024
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-2594/egusphere-2024-2594-AC1-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2024-2594-AC1
RC2:
'Comment on egusphere-2024-2594', Anonymous Referee #2, 18 Nov 2024
The paper under consideration presents a statistical approach using the Pearson Type I distribution to estimate the upper bound of historical rainfall, incorporating uncertainty bounds. The authors aim to quantify uncertainty and address subjectivity inherent in the various stages of the World Meteorological Organization (WMO)-recommended Probable Maximum Precipitation (PMP) estimation methods. However, the WMO-recommended moisture maximization method focuses on maximizing highly efficient storms based on the physical mechanisms of those storms. From a statistical perspective, precipitation’s tendency to exhibit a heavy-tailed distribution poses challenges in defining an upper bound, and this study similarly encounters this issue. Here are my major concerns:
Previous studies demonstrated that precipitation naturally often exhibits a heavy-tailed distribution (shape parameter greater than 0) that brings rare storms over global scale, the proposed method struggled to estimate upper bound of those places by majority of parameter estimation methods. In this study, the method of moments partially quantifies the range with limited data, but the range is unrealistically large, for example, the range estimated for St-Hubert station varies between 165 to 9006.

A simulation study is conducted with distribution assumed convex density and found more than 40,000 sample size (In arid/semi-arid region that equivalent to more than 1000 years wet days) is required to stabilize the estimate. Given this, it is surprising that the authors did not attempt to expand the sample size for the two stations by incorporating numerical model ensemble precipitation products. Doing so could have supported their findings.

This study compares their estimated upper limit with moisture maximization based PMP value. The PMP values using moisture maximization were found to be 282 mm for Montréal and 436 mm for St-Hubert, whereas the observed 24-hour maximum precipitation for these stations was 81.9 mm and 106.5 mm, respectively. Thus, the maximization ratio will be 3.44 and 4.09. The reason behind the exceptionally high maximization ratio may be due to the selection of storms and/or estimation of storm associated precipitable water. This study includes low magnitude storms (0.9 quantile might give more than 500 samples but previous studies mostly consider the highest 50 or less storms) and did not separate those storms that could lead to higher maximization ratio. Previous studies mostly limit the maximization ratio 2.0 (that only for orographic storms). Imposing a similar limit could provide some physically possible value around 200mm that aligns with 10,000-year return level (POT based) and PMP value would not much different within 26 km distance. Since the moisture maximization method provides an unrealistically high PMP value, comparing with this value to validate the method is questionable. It is recommended to use multiple study sites and consider those sites where maximization ratio lies below 2.0 and compare within those sites.

National Academies of Sciences, Engineering, and Medicine (2024) recommends for risk-informed extreme value analysis methods that account for low exceedance probabilities and provide robust uncertainty and nonstationary quantification. It remains unclear how the proposed method offers advantages over or resolves issues better than these recommended approaches.

The choice of Pearson Type I distribution over other distributions is missing.

Minor comment:
Line 304: The placeholder “Figure??” needs Figure number. Additionally, there is inconsistent notation in Equation 4 compared to Equation 1. The term "EP" and “EP_max” should be clearly defined to maintain consistency and avoid confusion.
Citation: https://doi.org/10.5194/egusphere-2024-2594-RC2
- AC2: 'Reply on RC2', Jonathan Jalbert, 20 Dec 2024
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-2594/egusphere-2024-2594-AC2-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2024-2594-AC2
RC3:
'Comment on egusphere-2024-2594', Anonymous Referee #3, 19 Nov 2024

Interesting paper. However, I feel it may become more of use with additional investigations of refinements needed to the method that has been proposed. See my comments below.
l125: Is EPmax defined for a calendar day for nearby regions or defined as the maximum at that point location? Equation 1 referred to PWmax which is the atmospheric moisture. EPmax is I suspect the same but should be consistent or else defined. If non-seasonal, please refer to WMO guidelines for seasonal variations in PMP.
l140 - interesting. However, the sampled ri are non-iid, which complicates their use in defining the Beta distribution I think, Plus, there is an assumption that the sampled ri has an upper limit of 1. Given this limiting value will dictate/influence the PMP estimate, the uncertainty associated with this assumption is important to characterise.
Also, how can stationarity be assumed given there is a clear temporal trend in precipitable water time series. Would violation of the stationarity assumption distort the beta distribution parameters?
l375 - The authors recommendation makes sense. In addition to the issues they have mentioned, I also feel that the lack of independence (unless they parameters are being fitted using iid data above a threshold) and the presence of a trend are limiting factors. I wonder if using a nonstationary model and regional data can help overcome these limitations.

Citation: https://doi.org/10.5194/egusphere-2024-2594-RC3
- AC3: 'Reply on RC3', Jonathan Jalbert, 20 Dec 2024
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-2594/egusphere-2024-2594-AC3-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2024-2594-AC3

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2024-2594', Anonymous Referee #1, 20 Sep 2024

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??

Citation: https://doi.org/10.5194/egusphere-2024-2594-RC1
- AC1: 'Reply on RC1', Jonathan Jalbert, 20 Dec 2024
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-2594/egusphere-2024-2594-AC1-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2024-2594-AC1
RC2:
'Comment on egusphere-2024-2594', Anonymous Referee #2, 18 Nov 2024
The paper under consideration presents a statistical approach using the Pearson Type I distribution to estimate the upper bound of historical rainfall, incorporating uncertainty bounds. The authors aim to quantify uncertainty and address subjectivity inherent in the various stages of the World Meteorological Organization (WMO)-recommended Probable Maximum Precipitation (PMP) estimation methods. However, the WMO-recommended moisture maximization method focuses on maximizing highly efficient storms based on the physical mechanisms of those storms. From a statistical perspective, precipitation’s tendency to exhibit a heavy-tailed distribution poses challenges in defining an upper bound, and this study similarly encounters this issue. Here are my major concerns:
Previous studies demonstrated that precipitation naturally often exhibits a heavy-tailed distribution (shape parameter greater than 0) that brings rare storms over global scale, the proposed method struggled to estimate upper bound of those places by majority of parameter estimation methods. In this study, the method of moments partially quantifies the range with limited data, but the range is unrealistically large, for example, the range estimated for St-Hubert station varies between 165 to 9006.

A simulation study is conducted with distribution assumed convex density and found more than 40,000 sample size (In arid/semi-arid region that equivalent to more than 1000 years wet days) is required to stabilize the estimate. Given this, it is surprising that the authors did not attempt to expand the sample size for the two stations by incorporating numerical model ensemble precipitation products. Doing so could have supported their findings.

This study compares their estimated upper limit with moisture maximization based PMP value. The PMP values using moisture maximization were found to be 282 mm for Montréal and 436 mm for St-Hubert, whereas the observed 24-hour maximum precipitation for these stations was 81.9 mm and 106.5 mm, respectively. Thus, the maximization ratio will be 3.44 and 4.09. The reason behind the exceptionally high maximization ratio may be due to the selection of storms and/or estimation of storm associated precipitable water. This study includes low magnitude storms (0.9 quantile might give more than 500 samples but previous studies mostly consider the highest 50 or less storms) and did not separate those storms that could lead to higher maximization ratio. Previous studies mostly limit the maximization ratio 2.0 (that only for orographic storms). Imposing a similar limit could provide some physically possible value around 200mm that aligns with 10,000-year return level (POT based) and PMP value would not much different within 26 km distance. Since the moisture maximization method provides an unrealistically high PMP value, comparing with this value to validate the method is questionable. It is recommended to use multiple study sites and consider those sites where maximization ratio lies below 2.0 and compare within those sites.

National Academies of Sciences, Engineering, and Medicine (2024) recommends for risk-informed extreme value analysis methods that account for low exceedance probabilities and provide robust uncertainty and nonstationary quantification. It remains unclear how the proposed method offers advantages over or resolves issues better than these recommended approaches.

The choice of Pearson Type I distribution over other distributions is missing.

Minor comment:
Line 304: The placeholder “Figure??” needs Figure number. Additionally, there is inconsistent notation in Equation 4 compared to Equation 1. The term "EP" and “EP_max” should be clearly defined to maintain consistency and avoid confusion.
Citation: https://doi.org/10.5194/egusphere-2024-2594-RC2
- AC2: 'Reply on RC2', Jonathan Jalbert, 20 Dec 2024
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-2594/egusphere-2024-2594-AC2-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2024-2594-AC2
RC3:
'Comment on egusphere-2024-2594', Anonymous Referee #3, 19 Nov 2024

Interesting paper. However, I feel it may become more of use with additional investigations of refinements needed to the method that has been proposed. See my comments below.
l125: Is EPmax defined for a calendar day for nearby regions or defined as the maximum at that point location? Equation 1 referred to PWmax which is the atmospheric moisture. EPmax is I suspect the same but should be consistent or else defined. If non-seasonal, please refer to WMO guidelines for seasonal variations in PMP.
l140 - interesting. However, the sampled ri are non-iid, which complicates their use in defining the Beta distribution I think, Plus, there is an assumption that the sampled ri has an upper limit of 1. Given this limiting value will dictate/influence the PMP estimate, the uncertainty associated with this assumption is important to characterise.
Also, how can stationarity be assumed given there is a clear temporal trend in precipitable water time series. Would violation of the stationarity assumption distort the beta distribution parameters?
l375 - The authors recommendation makes sense. In addition to the issues they have mentioned, I also feel that the lack of independence (unless they parameters are being fitted using iid data above a threshold) and the presence of a trend are limiting factors. I wonder if using a nonstationary model and regional data can help overcome these limitations.

Citation: https://doi.org/10.5194/egusphere-2024-2594-RC3
- AC3: 'Reply on RC3', Jonathan Jalbert, 20 Dec 2024
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-2594/egusphere-2024-2594-AC3-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2024-2594-AC3

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) Author's response Author's tracked changes Manuscript

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

Dear Jonathan Jalbert,

Thank you for submitting the revised version of your manuscript. It has been re-evaluated by two of the original reviewers as well as myself. The manuscript is substantially improved in comparison with the original version. However, several aspects still require further attention before the manuscript can be considered suitable for publication. Please refer to the detailed comments provided by the reviewer (in the reviewer’s report), as well as my own comments (listed below). I look forward to receiving the revised version of the manuscript.

Sincerely,
Nadav Peleg

Specific comments
• L3: I recommend moving the WMO definition from the abstract to the introduction.
• L16: The subheading "Context" is not necessary; consider removing it.
• L17–25: The introduction currently focuses too heavily on dam safety, whereas PMP has broader applications in hydrological risk assessment. As HESS is not specifically a journal for dam safety research, I suggest revising this section to present PMP in a broader hydrological context, with dam safety as one example among others.
• L35–37: The relevance of PMF is unclear. If it is not directly used in the model development or not specifically mentioned later in the manuscript, I recommend removing it. On a different, more general comment - the introduction could be revised to be less Canada-specific and more broadly applicable to international readers.
• L58–59: Eq. 2 may not be necessary. You could start discussing the ratio on line 60, referencing Eq. 1, which should suffice for clarity.
• Section 1.5: This section might be better integrated into the introduction. If the main point is to emphasize that climate models can be used to estimate changes in PMP (you are not using climate models later in the paper, just discussing their use), I suggest shortening the section and incorporating it earlier in the text.
• L108: Please clarify the acronym "CC" on first use.
• L124–128: Consider removing this text; it does not appear essential.
• L128–129: The sentence regarding data availability would be more appropriately placed in the data availability section at the end of the manuscript. Can be removed here.
• Structural suggestion: You might consider presenting the proposed PMP estimation model (Section 3) before introducing the case study data (Section 2). Since the model is general, introducing it first may improve the manuscript's logical flow. However, this is a suggestion; you may ignore it and choose to retain the current structure.
• Figures 1 and 2: These figures are not “a must” and could be moved to the supplementary material as Figures S1 and S2 (supplementary material in a separate file, not as an appendix).
• Table 2: This table could be merged with Table 1 to avoid redundancy in presenting station-specific information.
• L159–160: Please provide additional information for the EVT analysis: What threshold was selected? Why not use annual maxima? Which distribution was fitted (e.g., GEV)? Consider adding a brief sensitivity analysis on the threshold and distribution choice.
• Section 3.2.1: The method of moments is well established (see your references to Johnson et al.). Consider moving large parts of this section (or entirely) to the supplementary material unless some parts of it are critical to your main argument.
• Section 5: Subsections are not strictly necessary here; the results can be presented as a single, continuous narrative.
• L304–307 and L319–321: The parameter estimates would be more clearly presented in a table.
• Figure 7: Please clarify why quantiles close to 100 are missing in panel (a), and why quantiles exceed 100 in panel (b).
• L314–315: It is somewhat limiting to present the case study using only one of the three proposed estimation methods you described. You demonstrate them using the synthetic data, but it is better also to demonstrate using real case studies. I suggest including an additional case study, possibly using data from another region (you should not limit yourself to Canada), where all estimation methods are applied and compared. This would provide a more robust demonstration of the model's applicability.
• Section 6.2: The EVT results for the observed data are currently presented late in the manuscript. I recommend moving this material to Section 2.2, as it fits naturally with the presentation of other PMP estimates (see my above comments about the EVT).
• L376–377 and for the second station later: Please summarize the GPD parameter estimates for each station in a table for clarity.
• L416–417: The recommendations of the “National Academies of Sciences, Engineering, and Medicine” are specific to the Canadian or U.S. context. I suggest rephrasing such statements to make them more general and globally relevant.

Hide

AR by Jonathan Jalbert on behalf of the Authors (29 May 2025) Author's response Author's tracked changes Manuscript

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)

Journal article(s) based on this preprint

30 Sep 2025

Statistical estimation of probable maximum precipitation

Anne Martin, Élyse Fournier, and Jonathan Jalbert

Hydrol. Earth Syst. Sci., 29, 4811–4824, https://doi.org/10.5194/hess-29-4811-2025,https://doi.org/10.5194/hess-29-4811-2025, 2025

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Anne Martin, Elyse Fournier, and Jonathan Jalbert

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This paper introduces a statistical method to estimate probable maximum precipitation (PMP), addressing flaws in current approaches. The method accounts for uncertainty and incorporates climate change impacts, using the Pearson Type I distribution. Tested at two meteorological stations in Québec, it offers an objective solution for more reliable PMP estimates, crucial for infrastructure like dams exposed to extreme weather.

Statistical estimation of probable maximum precipitation

Journal article(s) based on this preprint

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