the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 06 Mar 2026
Simulated temporal scaling dependencies in sub-daily precipitation
Abstract. Rainfall intensity-duration-frequency (IDF) curves are an essential tool in water management, for instance in urban stormwater handling. They are commonly derived by fitting generalised extreme value distributions to observed annual maximum rainfall values, a process that requires long-term, high temporal resolution (sub-daily) observations of precipitation to ensure robust estimates. Alternatively, IDF curves can be approximated with simplified parametric mathematical expressions fitted to empirical data, providing a possibility to avoid the challenging data requirements. In this case, the parametric expression can be based on two key parameters that specify the shape of the curves: the wet-spell mean precipitation μ and the wet-spell frequency fw. For these two parameters, robust estimates are easier to obtain.
The resulting parametric IDF curves exhibit a fractal dimension and the present study takes a step towards better understanding the conditions influencing this fractal dimension and its spatial and temporal variability. To this end, we explore the dependencies across different timescales, using hourly precipitation data from convection-permitting (3 km) regional climate model simulations carried out with the HCLIM model over northern Europe. The analysis is applied to HCLIM simulations driven by boundary conditions from the ERA-Interim reanalysis, as well as from the EC-Earth and GFDL-CM3 global climate models for current and future climates following the RCP8.5 scenario.
We find that the relationship between wet-spell mean precipitation for different durations, and hence the sub-daily fractal dimension, is influenced by geographical conditions, as is also the wet-spell frequency. The results are consistent across different boundary conditions representing current climate conditions (reanalysis and global climate models), and showed little sensitivity to the driving model, indicating that different meteorological phenomena prevail in different regions and that these are well represented in the models. Future climate projections show changes in the fractal dimension and wet-spell frequency ratios with a general north-south gradient. Overall, the models indicate a shift towards fewer, but more intense wet-hours per wet day.
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Status: final response (author comments only)
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RC1: 'Comment on egusphere-2026-207', Anonymous Referee #1, 30 May 2026
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AC1: 'Reply on RC1', Andreas Dobler, 14 Aug 2026
We thank the first reviewer for the comments which we find useful and constructive. Please find below our point by point answers to the comments. The responses are in bold and the reviewer comments are repeated in italics.
General comments:
The study is motivated by the need for new approaches to generate IDF statistics, not requiring very long time series. But no “IDF results” are presented, making the study feel unfinished in my opinion. I would recommend presenting and evaluating the results in terms of IDF statistics and not only the IDF-related parameters. The data sets have been analyzed by Dyrrdal et al. (2023) and I suggest comparing with these results, to assess the performance of the IDF concept presented.
The study is not merely motivated by the need for new approaches to generate IDF statistics, but also by a wish to improve our understanding using simple (but sound) mathematical formulations that approximate the IDF values for (m)any return periods and durations in a consistent way. While the two parameters β and γ do not directly provide IDF values, maps of β and γ (changes) provide insights into whether short, i.e. sub-daily, or longer time-scales are dominating local heavy precipitation (changes) in terms of intensity and frequency. Thus, they represent information that are the basis for intensity, duration or frequency (change) maps in a condensed manner, providing an overview of geographical differences in temporal scaling relations with only two parameters.
However, we thank the reviewer for the suggestion and realise that providing the actual IDF values will improve the value of this study and help us evaluate our approach. We have therefore calculated several IDF and climate-factor maps (IDF changes) for the model region, using the best-fit parameters given in Benestad et al. (2021), and compared them to literature (Dyrrdal et al., 2015, 2023; Olsson et al., 2022). Maps that correspond to Dyrrdal et al. (2023) will be included in an additional section in the revised manuscript, together with an assessment of the performance of our proposed method. The new results show that generally our proposed method provides results that agree well with the earlier studies.An alternative could be to study the parameters in their own right, and downplay the relationships to IDF statistics. This could perhaps work out as a Technical note.
As suggested (see above), we will include IDFs.Overall I feel the manuscript has a lack of clear focus, but it is sometimes hard to follow the line of argument. For example, both the Introduction and the Discussion jumps between different aspects without obvious connections or a smooth, logical flow. I recommend the authors to spend more time improving the clarity and flow of the text.
Thanks for this feedback. We will spend some time improving the flow in the Introduction and the Discussion part and add more text to bridge between the single aspects.Specific comments:
Introduction: It has quite a lot of detailed methods and derivations that I think should be in a Method section.
We will move the more technical parts from the Introduction to the Method section.32: There are by now many non-stationary models, this limitation is not so crucial I think.
Although we agree that many non-stationary models exist, this limitation is still present in many practical applications due to simplicity and legal frameworks, e.g. in municipal or national design value engineering. We will rephrase this section to make our point more clear.38-42: It is unclear to me exactly how this reference relates to the present study.
We will review the reference and reformulate to make its relevance more clear or remove it entirely.62-64: Why is Caldas-Alvarez et al relevant here if scaling was not assessed? What did they find?
We realised that scaling was addressed in the preprint version of Caldas-Alvarez et al. only where they looked at the scale-dependency of thermodynamical processes influencing extreme precipitation (the original title was “Scale-dependency of extreme precipitation processes in regional climate simulations of the greater Alpine region”, https://wcd.copernicus.org/preprints/wcd-2022-11/wcd-2022-11.pdf). In the final revised version, the scale-dependency parts have been removed. We will change the reference accordingly.
The preprint paper is relevant because it deals with scaling issues and presents new insights on the scale-dependency of precipitation. However, it did not consider its fractal nature as in our case (the dimension D as in xD is non-integer, but a rational number). Nevertheless, it is in the scientific spirit to acknowledge colleagues' past efforts that provide a context for our study.85-86: What is the range of gamma?
Gamma is strictly larger than 0 because sub-daily precipitation frequency can not be larger than the daily one, i.e. fw(L) <= fw(24h). We will also include an upper limit for gamma in the revised study: Since fw(L)*L/24 >= fw(24h), the upper limit of gamma is given by gamma <= fw(24h)*(24-L)/(24*ln(24/L)) and thus, for L<=12, gamma < fw(24h)*0.75 < 1.
(The values we found for gamma are usually < 0.1.)98-104: The last paragraph in the Introduction is important for outlining the objectives of the study, but I feel this paragraph is rather fuzzy. I suggest formulating a number of clear research questions here and then relating back to them in Discussion and Conclusions.
We agree that this is an important paragraph to highlight the objective of our study. We will reformulate it to make it more clear. We will also consider adding research questions.Method: I suggest renaming to Data and methods.
Agreed, we will rename the section to Data and Methods.112-116: I suggest moving these sections to after the subsequent paragraph.
We will restructure the first two paragraphs of the Method section on the HCLIM simulations to have a more logical order.130: 20 years of data is not that short, I think many (most?) IDF calculations made used shorter time series.
Yes, many IDF estimations certainly use shorter time-series. However, in terms of estimating robust climate statistics, longer periods than 20-years would be preferable. Since we are including IDF estimates in the revised study, we will rephrase this paragraph.135-141: In my view this part should be in the previous section.
Thank you for this suggestion. We will move it to the Methods and Data section.Fig. 1: It is really hard to distinguish between the different line styles and symbols, esp. in the blue swarm.
We will redraw Fig. 1 to make it more clear. Following the suggestion of Reviewer 2 we will split Fig. 1 into two panels.Fig. 1: The analysis of data from Oslo is not described in the text. How was this curve produced?
We see now that the information is well hidden in the R markdown script provided in the Code and data availability section. We will include more information on how Fig. 1 is produced in the text directly.165: Consider including Figs. S1 and S2 in the manuscript.
We have considered adding Figs. S1 and S2 to the manuscript. However, we don’t think they add additional information beyond that the GCM driven HCLIM runs are able to reproduce the patterns already shown in Fig. 2 and Fig.3 for the ERA-Interim driven HCLIM runs. As the evaluation of the HLCIM runs is not in the focus of our study, we prefer to leave the figures in the SI and simply state that the patterns are well reproduced. As we will add IDF maps, the number of figures in the main text will already increase.225-227: The final sentence is rather inconclusive, I suggest outlining some more firm implications.
We will reformulate the sentence, making the implications more clear.Fig. 7: Considering showing these locations on a map.
The locations are shown in Fig. 4. We will add this information to the caption of Fig. 7.References:
Dyrrdal A. V., Lenkoski A., Thorarinsdottir T. L., and Stordal F. (2015), Bayesian hierarchical modeling of extreme hourly precipitation in Norway, Environmetrics, 26, pages 89–106. doi: 10.1002/env.2301Dyrrdal, A.V., Médus, E., Dobler, A., Hodnebrog, Ø., Arnbjerg-Nielsen, K., Olsson, J., Thomassen, E.D., Lind, P., Gaile, D., and P. Post (2023) Changes in design precipitation over the Nordic-Baltic region as given by convection-permitting climate simulations, Weather Clim. Extremes, 42, 100604, doi: 10.1016/j.wace.2023.100604
Olsson, J., Dyrrdal, A.V., Médus, E., Södling, J., Aņiskeviča, S., Arnbjerg-Nielsen, K., Førland, E., Mačiulytė, V., Mäkelä, A., Post, P. and Thorndahl, S.L., 2022. Sub-daily rainfall extremes in the Nordic–Baltic region. Hydrology Research, 53(6), pp.807-824. doi: 10.2166/nh.2022.119
Citation: https://doi.org/10.5194/egusphere-2026-207-AC1
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AC1: 'Reply on RC1', Andreas Dobler, 14 Aug 2026
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RC2: 'Comment on egusphere-2026-207', Anonymous Referee #2, 17 Jul 2026
General comments
In general, while I find the paper is very detailed to some rainfall parameters, it is not clear what the broader impacts are. It looks like something can be said about projected rainfall changes, but they are not said directly; what people care are actual rainfall changes but not changes to parameters that are used in IDF curves.
Some part of the text can probably be shuffled a bit. I also have some reservations how the figures are presented.
The original review document as I have typed using a word processor is attached.
Major comments
- Introduction discussion on ζ or β vs γ:
- The description of the above is sometimes hard to follow, so I just wish some clarifications.
- In a nutshell: ζ or β describe how intensity of rainfall changes by accumulation period, while γ describe how frequency of rainfall changes by accumulation period. Would the above be the most direct description of what they mean? This ties it nicely with IDF and should be the way these parameters introduced first in a single sentence, probably right after Equations 2 and 3. The more detailed description and discussion can come after. I appreciate detail discussion, but there should be a balance between detail and directness.
- This is in light of the text between lines 101-104: for example, it is entirely possible that γ can increase (more rainfall events of short duration), but β-hat to decrease (the now more frequent short-duration showers are relatively weaker than longer-duration events after rescaled in IDF). This type of intuition ties in nicely with the IDF again.
- Another suggestion from me is to have this detailed description about frequency and intensity parameters moved to the Method (section 2) instead in the Introduction (section 1). If anything, Method section is relatively short, considering how detailed Section 1 is after line 55; consider moving post line-55 to Method section.
- Minor: The description about γ should occur sooner than it currently is (like right after Equation 2). Right now, γ is not described until multiple paragraphs after Equation 2. Symbols and variables should be explained immediately after the equation is introduced.
- Discussion check for Figure 3B and lines 155-163: Did not you say higher γ indicates higher concentration of rainfall frequency in a few hours during a wet day (lines 85-86)? Why do you now say large γ off the Norwegian coast is associated with “decrease towards shorter duration frequencies is largest” (line 159-160)? What does that statement even mean? Please check.
- General comments about quality and presentation of figures:
- Make sure the captions are consistent with the main text and other figures. The goal is to make all figures directly readable by the reader without referring to the text first. I have commented on this in some minor comments.
- For change plots like Figures 4, 5, 6, S6, they will be easier to read if white (or near-white) colour is used label 0 or small changes (a divergent colour bar that is white-centred on 0). The delta-γ plots are particularly an issue because either side of the 0 is yellow.
- Given the larger changes with some of the parameters in the interior of Scandinavia as well as the Baltic Sea coast, it may be useful to add a grid point in Sweden for Figure 7.
- Conclusions: I guess what the conclusions (as well as the abstract) are missing one key information: given all these rainfall parameters (μ, β-hat, γ, etc.) are changing, are there something general we can say about the model projections? What are the general impacts and stakeholder interpretations of these changing parameters? I know the authors are avoiding the estimation of return levels or IDF directly; however, let’s take Figure 4 as an example, what would I expect when both b-hat and g increase for most of model domain? This is sort of related item 3 above: the chosen Norwegian cities are not particularly “well-located” for β-hat and γ changes but looks like there are interesting results over Sweden that worth talking about. The goal is to translate these diagnosed rainfall parameter changes to something that have broader impacts.
Minor comments:
Figure 1: Perhaps break this plot into two panels. One showing sensitivity of μ vs f_w and other one showing f_w vs ζ. Right now, there are a bit too many lines and marker variation within a single panel.
Figures 2, S1, S2 caption: Add “μ“ and “f_w” next to “wet-spell intensities” and “wet-spell frequencies” to make them consistent with plot labels, line 143, and caption of Figure 3.
Figure S3 caption: R-square for what? The caption should include what are the dependent and independent for the linear regression (i.e. log u_L and β-hat log(L/24)).
Line 181 and Figures S4-S6: Figures S4 and S5 show the actual β-hat and γ of the near as well as the far future simulation. It is only for S6 that you show their change (i.e. to be compared with Figure 4). I would not group S4 and S5 together with S6 in the main text discussion (line 181). Do you even need Figures S4 and S5 since you are primarily interested in the delta for near- and far-future simulation? If you do think it is necessary to keep Figure S4 and S5, make the γ contour levels for S5 to be the same as Figures 3, S2, and S4.
Lines 188-189: “Note that especially hourly frequencies decrease over large parts of the domain” – rephrase this sentence.
Figure 6 caption: “… GFDL-CM3 driven (bottom)…”
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AC2: 'Reply on RC2', Andreas Dobler, 14 Aug 2026
We thank the reviewer for the comments which we find useful and constructive. Please find below our point by point answers to the comments. The responses are in bold and the reviewer comments are repeated in italics.
General comments
In general, while I find the paper is very detailed to some rainfall parameters, it is not clear what the broader impacts are. It looks like something can be said about projected rainfall changes, but they are not said directly; what people care are actual rainfall changes but not changes to parameters that are used in IDF curves.
We understand that the presentation of the two parameters β and γ and their changes seems somewhat theoretical. Thus, following the suggestions by both reviewers, we will include actual IDF estimates in an additional section to provide a more direct picture of the impacts and changes in rainfall return values and how they relate to the two parameters β and γ.Some part of the text can probably be shuffled a bit. I also have some reservations how the figures are presented.
We will reorganise and revise the text and figures as suggested by both reviewers.The original review document as I have typed using a word processor is attached.
Major comments
Introduction discussion on ζ or β vs γ:
The description of the above is sometimes hard to follow, so I just wish some clarifications.
In a nutshell: ζ or β describe how intensity of rainfall changes by accumulation period, while γ describe how frequency of rainfall changes by accumulation period. Would the above be the most direct description of what they mean? This ties it nicely with IDF and should be the way these parameters introduced first in a single sentence, probably right after Equations 2 and 3. The more detailed description and discussion can come after. I appreciate detail discussion, but there should be a balance between detail and directness.
We thank the reviewer for asking these questions and providing his thoughts. This is indeed a good and concise description of the two parameters and we will add it.This is in light of the text between lines 101-104: for example, it is entirely possible that γ can increase (more rainfall events of short duration), but β-hat to decrease (the now more frequent short-duration showers are relatively weaker than longer-duration events after rescaled in IDF). This type of intuition ties in nicely with the IDF again.
Yes, it is entirely possible that γ increases but β-hat decreases. An increase in γ means a concentration of precipitation in time, i.e. it rains for fewer hours on a rainy day. Then, the frequency of short events increases less than the frequency of longer events, i.e. there are actually less (!) rainfall events of short duration (relative to long ones). This can for instance be seen at the Bergen, Trondheim and Tromsø grid-point (Fig. 7, right), especially for the GFDL-CM3 far future, and the corresponding delta-γ maps (Fig. 4). At the same time, β-hat can both increase (Tromsø) or decrease (Bergen), resulting in either a more pronounced intensity increase at short durations (Tromsø, Fig.7 left), or a (slightly) larger relative increase at long durations (Bergen, Fig. 7 left).
These changes directly affect the IDF values. E.g. with the increase of γ, the (relative) sub-daily frequency decreases. Then, the sub-daily IDF values increase less (compared to daily) because these events occur less frequently and the same IDF values occur only at higher return periods (tau). The IDF values that will be added to the revised study show how the γ and β-hat changes affect the IDFs.
Another suggestion from me is to have this detailed description about frequency and intensity parameters moved to the Method (section 2) instead in the Introduction (section 1). If anything, Method section is relatively short, considering how detailed Section 1 is after line 55; consider moving post line-55 to Method section.
This was also suggested by the other reviewer. We will move some parts from the Introduction to the Method section.Minor: The description about γ should occur sooner than it currently is (like right after Equation 2). Right now, γ is not described until multiple paragraphs after Equation 2. Symbols and variables should be explained immediately after the equation is introduced.
We agree. We will move the description of γ further up in the revised study.Discussion check for Figure 3B and lines 155-163: Did not you say higher γ indicates higher concentration of rainfall frequency in a few hours during a wet day (lines 85-86)? Why do you now say large γ off the Norwegian coast is associated with “decrease towards shorter duration frequencies is largest” (line 159-160)? What does that statement even mean? Please check.
Thanks for pointing this out! The sentence was a bit confusing. We will rephrase it to “but the wet-spell frequency decreases rapidly with shorter durations”. Please note that a “higher concentration of rainfall frequency in a few hours during a wet day” actually means a smaller sub-daily rainfall frequency which corresponds to larger values of gamma. This is, it may rain often (as at the Norwegian west-coast), but on a rainy day it does not rain all the time. Small gamma values on the other hand mean more persistent rain on rainy days.
General comments about quality and presentation of figures:
Make sure the captions are consistent with the main text and other figures. The goal is to make all figures directly readable by the reader without referring to the text first. I have commented on this in some minor comments.
Thanks for noting this, we will incorporate this into the corresponding captions.For change plots like Figures 4, 5, 6, S6, they will be easier to read if white (or near-white) colour is used label 0 or small changes (a divergent colour bar that is white-centred on 0). The delta-γ plots are particularly an issue because either side of the 0 is yellow.
Our intention was to clearly differentiate between positive and negative changes. However, we agree that with this small differences around zero may be overemphasized. Thus, we will add white to the divergent colour bars for values around 0. We will also make the first positive value in the delta-γ and delta-fw plots more different from yellow.
Given the larger changes with some of the parameters in the interior of Scandinavia as well as the Baltic Sea coast, it may be useful to add a grid point in Sweden for Figure 7.
The four locations in Norway were mostly selected due to their general difference in their precipitation climate and the differing changes in β-hat and γ. In our opinion, the results provide nice examples of the complex interplay of the changes in the precipitation characteristics. To also include pronounced changes, we will add locations (Kiruna and Mora in Sweden) in the proximity of the largest changes in the revised study.Conclusions: I guess what the conclusions (as well as the abstract) are missing one key information: given all these rainfall parameters (μ, β-hat, γ, etc.) are changing, are there something general we can say about the model projections? What are the general impacts and stakeholder interpretations of these changing parameters? I know the authors are avoiding the estimation of return levels or IDF directly; however, let’s take Figure 4 as an example, what would I expect when both b-hat and g increase for most of model domain? This is sort of related item 3 above: the chosen Norwegian cities are not particularly “well-located” for β-hat and γ changes but looks like there are interesting results over Sweden that worth talking about. The goal is to translate these diagnosed rainfall parameter changes to something that have broader impacts.
Our intention of analysing changes in β-hat and γ rather than IDF changes, was to not only show how the IDFs change for different durations (this has been shown several times already), but also provide insights on why. For instance, an increase in β-hat means that the mean hourly precipitation increases more than the mean daily precipitation, which already has a direct impact beyond IDF changes. Similarly, changes in γ reflect changes in the temporal concentration of rainfall that also, but not only, affect the IDFs. However, as suggested by both reviewers, we have now calculated IDF estimations and will add IDF maps to the study. With this, we can also formulate general impacts related to IDFs. These will be added to the manuscript, making the study easier to interpret directly by e.g. stakeholders, and putting them in context of the β-hat and γ changes.Minor comments:
Figure 1: Perhaps break this plot into two panels. One showing sensitivity of μ vs f_w and other one showing f_w vs ζ. Right now, there are a bit too many lines and marker variation within a single panel.
We thank the reviewer for this suggestion and will split Fig. 1 into two panels.Figures 2, S1, S2 caption: Add “μ“ and “f_w” next to “wet-spell intensities” and “wet-spell frequencies” to make them consistent with plot labels, line 143, and caption of Figure 3.
We will add “μ“ and “f_w” to the captions.Figure S3 caption: R-square for what? The caption should include what are the dependent and independent for the linear regression (i.e. log u_L and β-hat log(L/24)).
We will include the dependent and independent variable of the linear regression in the caption.Line 181 and Figures S4-S6: Figures S4 and S5 show the actual β-hat and γ of the near as well as the far future simulation. It is only for S6 that you show their change (i.e. to be compared with Figure 4). I would not group S4 and S5 together with S6 in the main text discussion (line 181). Do you even need Figures S4 and S5 since you are primarily interested in the delta for near- and far-future simulation? If you do think it is necessary to keep Figure S4 and S5, make the γ contour levels for S5 to be the same as Figures 3, S2, and S4.
Thanks for pointing this out. We think the future maps of β and ζ provide interesting (supplementary) information and will keep the figures S4 and S5. However, we will add a sentence in the main text referring to figures S4 and S5 independent of S6. Note that the γ contour levels for S5 are already the same as Figures 3, S2, and S4, but S5 includes an additional level at the top to cover values > 0.1 which are not present in the other simulations.Lines 188-189: “Note that especially hourly frequencies decrease over large parts of the domain” – rephrase this sentence.
We will rephrase this sentence.Figure 6 caption: “… GFDL-CM3 driven (bottom)…”
We will add “bottom” to the caption.Citation: https://doi.org/10.5194/egusphere-2026-207-AC2
- Introduction discussion on ζ or β vs γ:
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The authors analyze wet-spell mean precipitation and frequency in high-resolution convection-permitting Regional Climate Model simulations (for present and future climate), motivated by relations to IDF statistics. They find that IDF magnitude is more related to wet-day mean precipittaion than frequency. They also find geographical differences as well as future changes, generally towards fewer but more intense wet hours. The methodology is largely based on earlier work but applied in anew context. The analyses appear accurately performed and the presentation is acceptable. I have however a number of comments, described in the following.
General comments:
Specific comments:
Reference:
Dyrrdal, A.V., Médus, E., Dobler, A., Hodnebrog, Ø., Arnbjerg-Nielsen, K., Olsson, J., Thomassen, E.D., Lind, P., Gaile, D., and P. Post (2023) Changes in design precipitation over the Nordic-Baltic region as given by convection-permitting climate simulations, Weather Clim. Extremes, 42, 100604, doi: 10.1016/j.wace.2023.100604.