the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 19 Mar 2026
Spatio-Temporal Non-Stationarity of Flood Risk in the European Alps over the last 1,450 Years
Abstract. Floods are a major source of losses from natural hazards, yet modelling their occurrence and severity is challenging due to complex spatial dependencies and non-stationary behaviour over time, both of which are increasingly affected by climate change. In this study, we characterise spatial and temporal non-stationarity in flood risk and quantify the dependence of flood occurrences under minimal modelling assumptions. We analyse a unique dataset of flood records from 27 Alpine lakes spanning 1,450 years, applying penalised additive mixed models to capture the empirical spatio-temporal dependence structure of flood events. Our results reveal pronounced regional and temporal variations in flood risk and highlight periods of elevated susceptibility. The model further allows extrapolation of flood occurrence probabilities to unobserved locations across the European Alps, providing a robust tool for hazard assessment under changing climatic conditions.
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Status: open (until 08 May 2026)
- RC1: 'Comment on egusphere-2026-1016', Anonymous Referee #1, 09 Apr 2026 reply
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- 1
The analysis extend the work of Wilhelm et al. (2022) by using GAMM models to investigate the space-time dynamic of flood frequency in the Alps in the last 1450 years. I liked reading the paper although I’ve found the theoretical treatment at a level higher than my knowledge, so consider my comments on that part as the view of a typical reader of NHESS (i.e., not a mathematician).
My main concerns are:
- The manuscript in its current form gives the impression that the available dataset is primarily being used as a vehicle to showcase a statistical method, rather than that the methodological choices are guided by well-defined hydrological questions.
- The data description is insufficiently detailed to support several of the interpretations, particularly those related to the apparent irrelevance of catchment characteristics (see the detailed comments below).
- Several methodological passages are not sufficiently clear, at least to me, particularly with respect to notation, the meaning of individual model components, and the role of the penalty terms (see the detailed comments below).
- Finally, the discussion of the results also needs to be strengthened. Some claims are insufficiently substantiated, potential biases and artefacts in Figure 7 are not fully addressed, and the relative importance of temporal, spatial, and space-time effects is not discussed clearly enough.
Overall, I believe the manuscript has potential, but it would need substantial revision in terms of framing, clarity, data description, and interpretation before it can be considered for publication in NHESS.
Detailed comments:
Title: I wonder whether the wording “Flood Risk” is misleading. Indeed the analysis is on the flood frequency, not touching on the consequences. Probably “Flood Hazard” is a better wording.
Line 8: what does “susceptibility” mean?
Line 9: since climate variables are not used as covariates, how can the model be used for hazard assessment under changing climate conditions?
Lines 21-27: relevant literature is cited here. While Blöschl et al. (2017, 2019) are certainly relevant, I was surprised that Blöschl et al. (2020, https://doi.org/10.1038/s41586-020-2478-3) is not considered, since it could be partially used as a benchmark study (the analysis spans “only” 500 years and is at a larger spatial scale).
Line 83: maybe I would already state here what typical link functions are (e.g., the logarithm). I was puzzled for a while on what could g be while reading the method section.
Line 113: I see the importance of penalizing too flexible models. However I am a bit lost in where equation (6) (or 7) relates to the model (Eq. 2). The same issue (maybe just mine) applies to the penalty term at line 133.
Eq. (9): please define all terms, including sigma and gamma.
Line 189: I would state already what other approach is used instead.
Lines 190-200: the database is described here. I would suggest to give more quantitative details. What ranges do catchment area and elevation span? In the results you will show that these characteristics do not matter, is it because they are all very small catchments at high elevation? I think having an idea on the catchments involved could be useful in the interpretation of the results by the readers.
Eq. (14): where does this variance come from? From the theory in Selch et al. (2018)?
Line 235: what does “originate from” mean?
Eq. (17): please define all terms. I had an hard time to retrieve their meaning.
Lines 237-238: I do not understand the sentence, sorry. Maybe it could be expanded.
Line 262: here the penalty matrix is finally revealed. Maybe a hint on what P could be could have been given before, when P is first introduced.
Line 275: this is true, but what is the size of the catchments in here? And the elevations? More intense convective precipitation is expected to occur at low elevation, for instance.
Line 284: what is the default number of basis functions in mgcv
Lines 293-294: is penalization to 0 the result of the analysis? Can we see it in Table 1? I see edf=0 for ca, caalt and alt, but not for season. And since Table 1 seems to indicate that the 4 terms are not significant (p-value), shouldn’t they be removed from the model, like I would do in a linear regression?
Table 1: the terms in the table should be defined. E.g., what is edf? And Ref.df? What do they mean?
Lines 301-307: are these lake-specific deviations on top of the smooth spatial deviation discussed later? If so, shouldn’t they be discussed later?
Line 310-325: this is an interesting result. It seems coherent with Blöschl et al. (2020, https://doi.org/10.1038/s41586-020-2478-3) for the last part of the series.
Lines 326-329: Figure 5 puzzles me. It seems not to be consistent with Figure 4. Of course Figure 4 contains only the temporal effect. Is Figure 5 also affected by the fact that not all lakes have series going to the early period? In other words, if we count few event in the past because we have data from few lakes, isn’t the figure misleading because of a systematic bias?
Lines 330-346: Figure 4 ranges from 0.7 to 1.4, while Figures 6 and, more so, 7 present even larger ranges. Does it mean that spatial variability is dominant with respect to time? And that space-time clusters are even more important?
Besides, the locations where in Figure 7 the darker colours occur are far from the data (the little black points). Doesn’t this indicate that there may be artifacts? Would these artifacts have an effect on the results in Figure 4 and 6 too?
Also, for the discussion, wouldn’t it be useful to have a figure like Figure 7 where all smooth functions are added (and maybe also the global average term)? This would clarify whether spatial or temporal variability dominates.
Figures 9 and 10: the axes labels are not nicely written.
Line 377: I would use “long-term flood frequency dynamics” instead of “flood dynamics”, which are usually referring to the dynamic of individual events.
Line 379: the irrelevance of catchment area may be due to the fact that the catchments analyzed here are not so different in size. Please provide the numbers. Same comment regarding the remarks at lines 421-422.
Line 406: wouldn’t other climate indices than temperature be useful covariates here?