the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 12 May 2025
Transformed-Stationary EVA 2.0: A Generalized Framework for Non-Stationary Joint Extremes Analysis
Abstract. The increasing availability of extensive time series on natural hazards underscores the need for robust non-stationary methods to analyze evolving extremes. Moreover, growing evidence suggests that jointly analyzing phenomena traditionally treated as independent, such as storm surge and river discharge, is crucial for accurate hazard assessment. While univariate non-stationary extreme value analysis (EVA) has seen substantial development in recent decades, a comprehensive methodology for addressing non-stationarity in joint extremes – compound events involving simultaneous extremes in multiple variables – is still lacking. To fill this gap, here we propose a general framework for the non-stationary analysis of joint extremes that combines the Transformed-Stationary Extreme Value Analysis (tsEVA) approach with Copula theory. This methodology implements sampling techniques to extract joint extremes, applies tsEVA to estimate non-stationary marginal distributions using GEV or GPD distributions, and utilizes time-dependent copulas to model evolving inter-variable dependencies. The approach's versatility is demonstrated through case studies analyzing historical time series of significant wave height, river discharge, temperature, and drought, uncovering dynamic dependency patterns over time. To support broader adoption, we provide an open-source MATLAB toolbox that implements the methodology, complete with examples, available on GitHub.
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Status: final response (author comments only)
- RC1: 'Comment on egusphere-2025-843', Anonymous Referee #1, 17 Jul 2025
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RC2: 'Comment on egusphere-2025-843', Sylvie Parey, 10 Oct 2025
General comments
The paper generalizes an approach developed for the estimation of univariate extremes in the non-stationary context to multivariate extremes, through copulas. It is a very interesting approach which deals with a very important and yet not much studied issue, and it is worth being published. Since I have contributed to similar developments in the univariate context, I was excited to read this paper, and I have a couple of comments.
Specific comments
The idea of computing a stationary variable to estimate non-stationary extremes in the univariate context had also been proposed by Parey et al. (2010, 2013, 2019) for temperature and applied to rainfall in Acero et al. 2017. It could be added to the references, and I suggest some additions in the description of the section “Non-stationary marginals”:Lines 154-155: “where 𝑦(𝑡) is the non-stationary series, 𝑥(𝑡) is the assumed stationary series, 𝑇y(𝑡) and 𝐶y(𝑡) are generic terms representing the long-term variation in the mean and amplitude of 𝑦(𝑡), respectively.” Stationarity is not easy to assess globally, however in Parey et al. 2013, a statistical test is proposed for the stationarity of the extremes of x(t) and its application to temperature shows that they can be considered as stationary (when Ty(t) is the trend in mean and Cy(t) the trend in standard-deviation, estimated by LOESS with an optimal smoothing parameter).
In the section “Joint sampling of the extremes”, I agree with the stated limitations of GEV. However, I am not convinced that it can be applied to “relatively slow, seasonal phenomena, such as drought and heat waves” because in those cases, EVT application is not fully justified theoretically. Indeed, it is an asymptotic theory, in that the distribution of the maxima tend to a GEV distribution when the block length tends to infinity. With annual blocks, one assumes that 365 values is a sufficient number for assuming that the distribution of the maximum is a GEV, this can be called a “probabilistic assumption”. This however requires that the phenomenon under study is independent, or slightly depended, so that the block length is not too low (too much lower than 365). The number of heat waves and droughts each year is not very large, so the probabilistic assumption may be too strong. In the case of low flows for example, Parey and Gailhard 2022 use stochastic generation.
Case studies:
Case studies 1 and 2:
1) Are 40 years sufficient to robustly fit the Copulas? How many joint extremes are used on average? Could the uncertainty in the fitting be quantified (uncertainty of the coupling parameter? Is a change from 0.65 to 0.79 significant for example?).
2) The time evolution of the parameter of the copula and of the correlation coefficients is assessed through a Mann-Kendal test. However, they are computed from moving windows of 40 years, with a 1-year sliding, which makes them highly correlated. Doesn’t it undermine the Mann Kendall test?
3) How are the Return Levels computed? In the non-stationary context, there is not any standard definition of a Return Level, and different propositions have been made and compared in the univariate context (Yan et al. 2017). In Parey et al., the Return Level is computed for the stationary variable and back transformed into a Return Level of the variable under study through the changes in mean and standard-deviation; they are therefore representative of a targeted future climate period.
Case study 3: I am not sure that the application of EVT is fully justified in that case, as stated above, especially with monthly SPEI values.References:
Parey S., Dacunha-Castelle D, Hoang TTH.: Different ways to compute temperature return levels in the climate change context; Environmetrics, 2010, DOI 10.1002/env
Parey S., Hoang TTH, Dacunha-Castelle D.: The importance of mean and variance in predicting changes in temperature extremes, Journal of Geophysical Research: Atmospheres, Vol 118, 1-12, 2013, doi:10.1002/jgrd.50629
Acero F.J., Parey S., Hoang T.T.H., Dacunha-Castelle D., Garcia J.A. and Gallego M.C.: Non-stationary future Return Levels for extreme rainfall over Extremadura (SW Iberian Peninsula). Hydrological Sciences Journal, 2017, DOI: 10.1080/02626667.2017.1328559
Parey S., Hoang T.T.H., Dacunha-Castelle D. : Future high temperature extremes and stationarity, Natural Hazards (2019) 98:1115–1134, https://doi.org/10.1007/s11069-018-3499-1
Lei Yan, Lihua Xiong, Shenglian Guo, Chong-Yu Xu, Jun Xia, Tao Du, Comparison of four nonstationary hydrologic design methods for changing environment, Journal of Hydrology, Volume 551, 2017, Pages 132-150, ISSN 0022-1694, https://doi.org/10.1016/j.jhydrol.2017.06.001.
Parey, S.; Gailhard, J. Extreme Low Flow Estimation under Climate Change. Atmosphere 2022, 13, 164. https://doi.org/10.3390/atmos13020164Citation: https://doi.org/10.5194/egusphere-2025-843-RC2
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- 1
The authors present the generalization of a method for statistically analyzing the joint extremes of different environmental variables under non-stationary conditions.
The work is scientifically relevant and addresses a current problem, especially under the conditions of ongoing climate change.
The provision of the code for applying the analysis to other case studies is highly appreciated. I believe the paper could be published after some minor revisions.
In Table 1, I would suggest adding an explanation of the meaning of the third column.
In the description of the first case study, it would be appropriate to add a justification for the choice of a 45-day time window between two extremes. At first glance, this may seem strange, given that it involves the analysis of joint flood probabilities in small basins and wave heights. The choice could make sense if one wishes to consider the joint probability of events with a view to recovery between one event and the next, but I believe this needs to be explained further.
From the text, it appears that in some cases the analyses are based on model results rather than recorded data. This, due to a possible bias between the simulations and the actual data, could impact the quantification of return periods, and I believe it should be explicitly stated. Or, if it has been done, specify whether any form of bias correction was applied to the model data.