the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 23 Sep 2025
Polynomial depth-duration-frequency curves
Abstract. Depth-duration-frequency (DDF) curves depict how much precipitation occurs on average in a given location during various time intervals once in a given return period. The standard approach to the construction of these curves assumes that the parameters governing the scaling behaviour of rainfall intensity with duration remain constant. We show that in regions where different meteorological processes control short- and long-duration extreme precipitation events, this approach is applicable only in limited time intervals. If the range is as wide as several minutes to several days, three parameters are not sufficient for representing the complexity of the DDF curve shapes. In fact, the curves are wave-shaped because convective and cyclonic precipitation occur for limited lengths of up to several hours and several days, respectively. Thus, we suggest applying polynomial functions of the sixth degree to generate smooth DDF curves that fit design precipitation totals for individual time intervals. Nevertheless, return values need to be fitted against logarithmic time intervals instead of only time. These polynomial DDF curves suitably represent extreme precipitation statistics even in orographically influenced locations where already the precipitation maxima of several hours can be caused by cyclonic precipitation events.
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RC1: 'Comment on egusphere-2025-4268', Hans Van de Vyver, 17 Oct 2025
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AC1: 'Reply on RC1', Miloslav Müller, 31 Dec 2025
Dear reviewer,
Thank you for reviewing our article. Our responses to your individual comments are highlighted in bold.
The manuscript attempts to challenge an established framework for IDF (Intensity-Duration-Frequency) relationships, specifically the work of Koutsoyiannis et al. (1998), which is widely recognized in the field. However, the criticism is based on a fundamental misunderstanding of important statistical concepts. As a result, the arguments put forward are not scientifically convincing.
Apart from the conceptual issues, the manuscript suffers from poor structure and a lack of clarity. Several important methodological steps are omitted or insufficiently explained, further undermining the credibility and readability of the work.
Given these serious shortcomings – both in terms of scientific content and presentation – I do not consider the manuscript suitable for publication in HESS. I therefore recommend rejection.
I shall explain this in more detail below. Since IDF and DDF curves are equivalent, I will use "IDF" in my argument.
The terms IDF and DDF are sometimes used interchangeably. In our study, however, we explicitly distinguish between rainfall intensity (in mm h⁻¹ or l s-1 ha-1) and rainfall total/depth (in mm) as a function of duration. Since our analysis is based on rainfall totals rather than intensities, we consistently use the term DDF curve throughout the manuscript.
1) The general IDF-relationship of Koutsoyiannis et al. (1998) has been confirmed worldwide, certainly for precipitation duration up to 24 hours (and up to 72 hours for Western Europe). On the one hand, these relationships are based on the “simple scale property” of precipitation intensity, and on the other hand, the parameterization results in IDF-curves that do not intersect for different return periods (i.e. physical consistency). Improvements are of course possible, such as using the “multiscaling properties” (Burlando & Rosso, 1996), but these results in rather modest improvements (Van de Vyver, 2018). Extensions to longer rainfall durations have been successfully developed in, for example, Willems (2000) or Fauer et al. (2021).
We acknowledge that the IDF framework of Koutsoyiannis et al. (1998) is widely used and has been successfully validated in many regions. We initially considered applying this approach in our analysis of design precipitation in the Czech Republic as well. However, an analysis of observed precipitation maxima reveals a systematic deviation from a simple power-law scaling with duration. This behaviour is not limited to a few stations, but is consistently observed in the mean maxima derived from 160 Czech stations. Specifically, we identify a pronounced deceleration in the growth of maxima at hourly durations, followed by an acceleration at durations of several tens of hours. While this undulation could be removed by smoothing, we consider it physically meaningful, as it corresponds to a transition from predominantly convective precipitation at short durations to stratiform precipitation at longer durations.
2) Based on the above, the IDF-relationship presented in the manuscript are fundamentally flawed and do not offer any meaningful contribution to the field.
- The polynomial relationships are fitted separately for each return period and do not preclude IDF curves from intersecting. This is precisely the key to the IDF-relationships of Koutsoyiannis et al. (1998).
We agree that the theoretical possibility of polynomial DDF curves crossing each other could be a significant problem. However, since the design precipitation total for a given duration always increases with increasing return period, the crossing could only occur between the durations at which the return levels are given. Since we use a sufficiently dense series of considered durations, we did not encounter this problem.
- I see no connection between the existing IDF-relationships and the polynomial approach. Fundamental log-log scaling properties for precipitation (or at least a good approximation thereof, within a relevant range of rainfall duration) cannot be immediately derived from the new model.
Our results show that log-log scaling is approximately preserved in our model, but the new model also allows us not to ignore the meteorologically-reasonable undulations of the curves.
3) There is no point in introducing a sixth-degree polynomial that approximates the estimated return levels almost perfectly, as shown in Figure 4. After all, the return levels are subject to considerable uncertainty, particularly for longer return periods. The absence of any statistical assessment of overfitting raises serious concerns about the robustness and generalizability of the proposed model. First, one must propose a physically consistent family of IDF-relationships, without overfitting, as already proposed in Koutsoyiannis et al. (1998).
Yes, the sixth-degree polynomial approximates the estimated return levels almost perfectly for stations with long data series when the uncertainty of design totals was reduced by the region-of-influence method. Therefore, we are convinced that it can improve the results for stations with much shorter data series, as presented by the example in Fig. 1 in the supplement (Brno-Žabovřesky station, 36 years).
4) It is not sufficient to test the method at only two locations. Ideally, this should be done on at least one continent. However, global datasets are available for extreme precipitation amounts on a sub-daily basis (GSDR-I).
It is a misunderstanding that the method was tested only at two stations – in fact, it was tested at 160 stations in the Czech Republic; the two stations are only used to present the results (see Chapter 1, first paragraph). For other stations, we refer to https://www.perun-klima.cz/srazky (unfortunately only in Czech), see "Data availability" at the end of the article. If we get the opportunity to edit the article, we will expand it with an overview of the results from all stations. At this moment, we present the average 100year precipitation totals from all Czech stations in Fig. 2 in the supplement to support our results.
Verifying the extent of the area where our results apply could be a next step in the research. Anyway, GSDR-I cannot be used for this purpose due to its hourly resolution only.
5) The DDF-curves presented in Figure 4 for the Czech Republic differ substantially from those of other European countries, and I have no immediate explanation for this. In addition, the estimates shown in Figure 3 for the Desná-Sous station deviate abnormally (for d >10h) and are well outside the confidence interval. The wave-like behavior for station Tábor may be due to sampling uncertainty. Perhaps this is also due to the fact that no log-log plot was used. I would also like to point out that the article does not mention anything about the quality of the data or whether it has been tested for homogeneity and outliers.
The wave-like behaviour of the DDF curves is a key aspect of our study. Based on our analysis, we do not attribute this feature to deficiencies in the data. Instead, we interpret it as a meteorologically meaningful characteristic. Support for this interpretation can be found in a paper by Pöschmann et al. (2021) who present the temporal scaling of extreme rainfall in Germany. Their results indicate that the anomaly we described is not limited to the Czech Republic – see, for example, their figure 9b.
In Figure 3, we have selected three duration ranges for two return periods (10 and 100 years) for which we constructed standard DDF curves. We agree that it will be necessary to include an interval covering longer durations in the selection to show that even in this case the curve does not fit points over the entire duration range.
Regarding data quality, we apologize for not providing details. All maxima above the selected thresholds were manually checked for relevance. The data were also tested for possible inhomogeneity due to the transition from recording ombrographs to automatic rain gauges.
6) As is generally accepted, the different types of precipitation (convective/stratiform) can lead to a flattening of the IDF curves. The authors are therefore not claiming anything new. Furthermore, Figure 5 does not provide a sound scientific explanation for this: a few extreme events have been selected and the precipitation amounts for different rainfall durations are shown (within each chosen single event, some durations correspond to an annual maximum, while others do not). This cannot explain why different behaviors (e.g. a power law) apply to extreme precipitation maxima of short and long duration.
There must be some misunderstanding here, because we are not pointing to the flattening of the DDF curves, but rather to their undulation.
We agree that Fig. 5 does not provide an unquestionable proof of our concept. Its purpose is illustrative aiming to highlight the fundamentally different temporal structure of convective and stratiform precipitation events and serving to provide a physical context for the interpretation of the undulation of the DDF curves. If we get the opportunity to edit the article, we will expand this part of the paper.
7) In addition, the authors Identify the two types of precipitation in a very rudimentary way, and it is much more scientific to identify precipitation types with synoptic conditions (in the form of weather types).
The difference in short-term intensity is one of the reasonable ways to distinguish stratiform and convective precipitation, which we have chosen because of its close connection to the studied issue. However, we have no problem expanding the work to include an analysis of weather types, or applying the method that we used in Müller et al. (2018).
8) Finally, I would like to point out that their estimation method is outdated. More recent developments (i.e. Muller et al. 2007, and later authors) model the GEV-parameters as a function of rainfall duration and thus estimate the IDF-relationships directly. Here, the authors used a two-stage estimation procedure, first estimating the return levels for individual durations and then fitting the IDF model to these. This is known to be a flawed inference, because nothing can be said about the uncertainties (or how they are distributed across the two steps) and because no model choice can be made using statistical tests (see also point 3 regarding overfitting).
Estimating return levels separately for individual durations allows the duration dependence of extremes to be explored directly without imposing a predefined functional form on the GEV parameters and thereby revealing features that could otherwise be partially smoothed out.
Moreover, Figs. 3 and 4 in the supplement show that both location and scale parameters exhibit a dependence on precipitation duration similar to that detected for the individually constructed DDF curves. This indicates that the identified dependence originates from the data and is not created by the subsequent fitting.Additional references
Müller, M., Bližňák, V., Kašpar, M., 2018: Analysis of rainfall time structures on a scale of hours. Atmos. Res., 211, 38–51. https://doi.org/10.1016/j.atmosres.2018.04.015
Pöschmann, J. M., Kim, D., Kronenberg, R., Bernhofer, C., 2021: An analysis of temporal scaling behaviour of extreme rainfall in Germany based on radar precipitation QPE data. Nat. Hazards Earth Syst. Sci., 21, 1195–1207. https://doi.org/10.5194/nhess-21-1195-2021
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AC1: 'Reply on RC1', Miloslav Müller, 31 Dec 2025
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RC2: 'Comment on egusphere-2025-4268', Anonymous Referee #2, 21 Nov 2025
Dear Editor,
The article attempts to address identified drawbacks of the standard three-parameter Intensity-Duration-Frequency (IDF) formulation proposed by Koutsoyiannis et al 1998 by introducing a six-degree polynomial model. While the presented work falls within the scope of the journal, major issues should be addressed before the paper can be considered worthy of publication.
My comments are presented below:
Justification for the Proposed Framework: In the introduction, the authors correctly note the existence of other works in the literature that have provided modifications to the standard approach to address some of its limitations. Examples include the multi-scaling models of Van de Vyver and Demarée (2010) and Faure et al. (2011). Wouldn't it be more natural and methodologically sound to consider and compare their results against these existing, more parsimonious modifications before proposing a significantly more complex approach? The paper should objectively justify the novel, high-degree polynomial framework in comparison to these more parsimonious alternatives.
Comparison Framework and Model Complexity: It is expected that a more flexible model with a large number of parameters (e.g., 42 parameters, assuming 7 return periods multiplied by a 6th-degree polynomial plus intercept) will inherently provide a better fit to the observed data compared to a simple three-parameter model. I strongly suggest that the authors justify the suitability of their highly parameterized model with respect to the standard approach using a robust model selection criterion. This should ideally be either an information criterion (e.g., AIC or BIC), which penalizes complexity, or a robust cross-validation framework, which tests the model's predictive skill on independent data.
Model Formulation and Statistical Inference: The proposed polynomial approach relies on an outdated statistical inference procedure where the parameters are estimated in two separate steps: Estimation of the GEV model parameters for each rainfall duration. Estimation of the polynomial model parameters (return level curve). This two-step approach is known to introduce inconsistencies and underestimate uncertainty. How do the authors propose to address the issue of using this decoupled, sub-optimal estimation approach?
The authors should also justify why a 6the degree polynomial is the coorect model, arther than a lower degree model
Uncertainty Assessment: Given the large number of parameters and the use of a two-step estimation approach, the uncertainty in the proposed formulation will undoubtedly be substantial. The paper should include a dedicated component to thoroughly assess and quantify the uncertainties associated with their model parameters and derived return levels.
Equation 1 and 2: Units and Consistency: I find the units for IN (precipitation intensity) in Equation 1 to be surprising. Is the standard convention not to use mm/hr or mm/min for rainfall intensity? How did the authors arrive at the specified units? The same question applies to the units of R in Equation 2 Are the authors dealing with areal rainfall or point rainfall? In line 57, the authors state that they will use a "standard approach" to refer to the method of Koutsoyiannis. Does the formulation of Equation 1 and the definition of its parameters exactly correspond to the specific formulation proposed by Koutsoyiannis (1998)? This consistency needs to be confirmed.
Citation: https://doi.org/10.5194/egusphere-2025-4268-RC2 -
AC2: 'Reply on RC2', Miloslav Müller, 31 Dec 2025
Dear reviewer,
Thank you for reviewing our article. Our responses to your individual comments are highlighted in bold.
The article attempts to address identified drawbacks of the standard three-parameter Intensity-Duration-Frequency (IDF) formulation proposed by Koutsoyiannis et al 1998 by introducing a six-degree polynomial model. While the presented work falls within the scope of the journal, major issues should be addressed before the paper can be considered worthy of publication.
My comments are presented below:
Justification for the Proposed Framework: In the introduction, the authors correctly note the existence of other works in the literature that have provided modifications to the standard approach to address some of its limitations. Examples include the multi-scaling models of Van de Vyver and Demarée (2010) and Faure et al. (2011). Wouldn't it be more natural and methodologically sound to consider and compare their results against these existing, more parsimonious modifications before proposing a significantly more complex approach? The paper should objectively justify the novel, high-degree polynomial framework in comparison to these more parsimonious alternatives.
We agree that it is necessary to add to the article a comparison of our approach with the multiscaling model, focusing on their ability to represent the observed undulation in shape of the duration dependence.
Comparison Framework and Model Complexity: It is expected that a more flexible model with a large number of parameters (e.g., 42 parameters, assuming 7 return periods multiplied by a 6th-degree polynomial plus intercept) will inherently provide a better fit to the observed data compared to a simple three-parameter model. I strongly suggest that the authors justify the suitability of their highly parameterized model with respect to the standard approach using a robust model selection criterion. This should ideally be either an information criterion (e.g., AIC or BIC), which penalizes complexity, or a robust cross-validation framework, which tests the model's predictive skill on independent data.
We acknowledge the suggestion. Due to the two-step estimation procedure and the fact that the polynomial coefficients are not independent, we will address this comment through a robust block-based cross-validation framework that evaluates predictive performance across durations.
Model Formulation and Statistical Inference: The proposed polynomial approach relies on an outdated statistical inference procedure where the parameters are estimated in two separate steps: Estimation of the GEV model parameters for each rainfall duration. Estimation of the polynomial model parameters (return level curve). This two-step approach is known to introduce inconsistencies and underestimate uncertainty. How do the authors propose to address the issue of using this decoupled, sub-optimal estimation approach?
It should be emphasized that although our estimates of design precipitation are based on a two-stage approach, the associated uncertainty is mitigated by applying the region-of-influence method. We will explicitly quantify this uncertainty in the revised article.
The authors should also justify why a 6the degree polynomial is the correct model, rather than a lower degree model.
We have conducted experiments with lower degree polynomials, which systematically fail to reproduce the observed undulation in the shape of the duration dependence. We agree that the analysis results need to be added to the article. In the form of an electronic supplement, we will also present graphs with lower degree polynomials.
Uncertainty Assessment: Given the large number of parameters and the use of a two-step estimation approach, the uncertainty in the proposed formulation will undoubtedly be substantial. The paper should include a dedicated component to thoroughly assess and quantify the uncertainties associated with their model parameters and derived return levels.
We agree and will add a dedicated uncertainty assessment. Uncertainty will be quantified via a bootstrap scheme that re-runs the full ROI-based GEV estimation and subsequent DDF fitting in each replicate, allowing uncertainty propagation from regional estimation to derived return levels. Given the computational cost of recomputing the ROI regions and regional estimates, this analysis will be conducted for a representative subset of stations.
Equation 1 and 2: Units and Consistency: I find the units for IN (precipitation intensity) in Equation 1 to be surprising. Is the standard convention not to use mm/hr or mm/min for rainfall intensity? How did the authors arrive at the specified units? The same question applies to the units of R in Equation 2 Are the authors dealing with areal rainfall or point rainfall? In line 57, the authors state that they will use a "standard approach" to refer to the method of Koutsoyiannis. Does the formulation of Equation 1 and the definition of its parameters exactly correspond to the specific formulation proposed by Koutsoyiannis (1998)? This consistency needs to be confirmed.
The standard unit of precipitation intensity is mm/h. However, in water management practice in the Czech Republic, the unit l/ha/s is traditionally used, which is why we used it in our work (1 l/ha/s = 0.36 mm/h). We acknowledge that this may be confusing, so we will edit the relevant passage of the article to correspond to the standard units and to ensure consistency with the standard approach.
Citation: https://doi.org/10.5194/egusphere-2025-4268-AC2
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AC2: 'Reply on RC2', Miloslav Müller, 31 Dec 2025
Status: closed
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RC1: 'Comment on egusphere-2025-4268', Hans Van de Vyver, 17 Oct 2025
The manuscript attempts to challenge an established framework for IDF (Intensity-Duration-Frequency) relationships, specifically the work of Koutsoyiannis et al. (1998), which is widely recognized in the field. However, the criticism is based on a fundamental misunderstanding of important statistical concepts. As a result, the arguments put forward are not scientifically convincing.
Apart from the conceptual issues, the manuscript suffers from poor structure and a lack of clarity. Several important methodological steps are omitted or insufficiently explained, further undermining the credibility and readability of the work.
Given these serious shortcomings – both in terms of scientific content and presentation – I do not consider the manuscript suitable for publication in HESS. I therefore recommend rejection.
I shall explain this in more detail below. Since IDF and DDF curves are equivalent, I will use "IDF" in my argument.
1) The general IDF-relationship of Koutsoyiannis et al. (1998) has been confirmed worldwide, certainly for precipitation duration up to 24 hours (and up to 72 hours for Western Europe). On the one hand, these relationships are based on the “simple scale property” of precipitation intensity, and on the other hand, the parameterization results in IDF-curves that do not intersect for different return periods (i.e. physical consistency). Improvements are of course possible, such as using the “multiscaling properties” (Burlando & Rosso, 1996), but these results in rather modest improvements (Van de Vyver, 2018). Extensions to longer rainfall durations have been successfully developed in, for example, Willems (2000) or Fauer et al. (2021).
2) Based on the above, the IDF-relationship presented in the manuscript are fundamentally flawed and do not offer any meaningful contribution to the field.
- The polynomial relationships are fitted separately for each return period and do not preclude IDF curves from intersecting. This is precisely the key to the IDF-relationships of Koutsoyiannis et al. (1998).
- I see no connection between the existing IDF-relationships and the polynomial approach. Fundamental log-log scaling properties for precipitation (or at least a good approximation thereof, within a relevant range of rainfall duration) cannot be immediately derived from the new model.3) There is no point in introducing a sixth-degree polynomial that approximates the estimated return levels almost perfectly, as shown in Figure 4. After all, the return levels are subject to considerable uncertainty, particularly for longer return periods. The absence of any statistical assessment of overfitting raises serious concerns about the robustness and generalizability of the proposed model. First, one must propose a physically consistent family of IDF-relationships, without overfitting, as already proposed in Koutsoyiannis et al. (1998).
4) It is not sufficient to test the method at only two locations. Ideally, this should be done on at least one continent. However, global datasets are available for extreme precipitation amounts on a sub-daily basis (GSDR-I).
5) The DDF-curves presented in Figure 4 for the Czech Republic differ substantially from those of other European countries, and I have no immediate explanation for this. In addition, the estimates shown in Figure 3 for the Desná-Sous station deviate abnormally (for d >10h) and are well outside the confidence interval. The wave-like behavior for station Tábor may be due to sampling uncertainty. Perhaps this is also due to the fact that no log-log plot was used. I would also like to point out that the article does not mention anything about the quality of the data or whether it has been tested for homogeneity and outliers.
6) As is generally accepted, the different types of precipitation (convective/stratiform) can lead to a flattening of the IDF curves. The authors are therefore not claiming anything new. Furthermore, Figure 5 does not provide a sound scientific explanation for this: a few extreme events have been selected and the precipitation amounts for different rainfall durations are shown (within each chosen single event, some durations correspond to an annual maximum, while others do not). This cannot explain why different behaviors (e.g. a power law) apply to extreme precipitation maxima of short and long duration.
7) In addition, the authors Identify the two types of precipitation in a very rudimentary way, and it is much more scientific to identify precipitation types with synoptic conditions (in the form of weather types).8) Finally, I would like to point out that their estimation method is outdated. More recent developments (i.e. Muller et al. 2007, and later authors) model the GEV-parameters as a function of rainfall duration and thus estimate the IDF-relationships directly. Here, the authors used a two-stage estimation procedure, first estimating the return levels for individual durations and then fitting the IDF model to these. This is known to be a flawed inference, because nothing can be said about the uncertainties (or how they are distributed across the two steps) and because no model choice can be made using statistical tests (see also point 3 regarding overfitting).
Additional references
- Burlando, P., Rosso, R. (1996) Scaling and multiscaling models of depth-duration-frequency curves for storm precipitation, J. Hydrol. 187, 45-64.
- Muller, A., Bacro, J. N. & Lang, M. (2007) Bayesian comparison of different rainfall depth-duration-frequency relationships. Stochastic Environ. Res. & Risk Assess. 22, 33–46.
- Van de Vyver H. (2018) A multiscaling-based intensity–duration–frequency model for extreme precipitation. Hydrol. Process.32, 1635–1647.
- Willems, P. (2000) Compound intensity/duration/frequency-relationships of extreme precipitation for two seasons and two storm types. J. Hydrol. 33, 189--205.
Citation: https://doi.org/10.5194/egusphere-2025-4268-RC1 -
AC1: 'Reply on RC1', Miloslav Müller, 31 Dec 2025
Dear reviewer,
Thank you for reviewing our article. Our responses to your individual comments are highlighted in bold.
The manuscript attempts to challenge an established framework for IDF (Intensity-Duration-Frequency) relationships, specifically the work of Koutsoyiannis et al. (1998), which is widely recognized in the field. However, the criticism is based on a fundamental misunderstanding of important statistical concepts. As a result, the arguments put forward are not scientifically convincing.
Apart from the conceptual issues, the manuscript suffers from poor structure and a lack of clarity. Several important methodological steps are omitted or insufficiently explained, further undermining the credibility and readability of the work.
Given these serious shortcomings – both in terms of scientific content and presentation – I do not consider the manuscript suitable for publication in HESS. I therefore recommend rejection.
I shall explain this in more detail below. Since IDF and DDF curves are equivalent, I will use "IDF" in my argument.
The terms IDF and DDF are sometimes used interchangeably. In our study, however, we explicitly distinguish between rainfall intensity (in mm h⁻¹ or l s-1 ha-1) and rainfall total/depth (in mm) as a function of duration. Since our analysis is based on rainfall totals rather than intensities, we consistently use the term DDF curve throughout the manuscript.
1) The general IDF-relationship of Koutsoyiannis et al. (1998) has been confirmed worldwide, certainly for precipitation duration up to 24 hours (and up to 72 hours for Western Europe). On the one hand, these relationships are based on the “simple scale property” of precipitation intensity, and on the other hand, the parameterization results in IDF-curves that do not intersect for different return periods (i.e. physical consistency). Improvements are of course possible, such as using the “multiscaling properties” (Burlando & Rosso, 1996), but these results in rather modest improvements (Van de Vyver, 2018). Extensions to longer rainfall durations have been successfully developed in, for example, Willems (2000) or Fauer et al. (2021).
We acknowledge that the IDF framework of Koutsoyiannis et al. (1998) is widely used and has been successfully validated in many regions. We initially considered applying this approach in our analysis of design precipitation in the Czech Republic as well. However, an analysis of observed precipitation maxima reveals a systematic deviation from a simple power-law scaling with duration. This behaviour is not limited to a few stations, but is consistently observed in the mean maxima derived from 160 Czech stations. Specifically, we identify a pronounced deceleration in the growth of maxima at hourly durations, followed by an acceleration at durations of several tens of hours. While this undulation could be removed by smoothing, we consider it physically meaningful, as it corresponds to a transition from predominantly convective precipitation at short durations to stratiform precipitation at longer durations.
2) Based on the above, the IDF-relationship presented in the manuscript are fundamentally flawed and do not offer any meaningful contribution to the field.
- The polynomial relationships are fitted separately for each return period and do not preclude IDF curves from intersecting. This is precisely the key to the IDF-relationships of Koutsoyiannis et al. (1998).
We agree that the theoretical possibility of polynomial DDF curves crossing each other could be a significant problem. However, since the design precipitation total for a given duration always increases with increasing return period, the crossing could only occur between the durations at which the return levels are given. Since we use a sufficiently dense series of considered durations, we did not encounter this problem.
- I see no connection between the existing IDF-relationships and the polynomial approach. Fundamental log-log scaling properties for precipitation (or at least a good approximation thereof, within a relevant range of rainfall duration) cannot be immediately derived from the new model.
Our results show that log-log scaling is approximately preserved in our model, but the new model also allows us not to ignore the meteorologically-reasonable undulations of the curves.
3) There is no point in introducing a sixth-degree polynomial that approximates the estimated return levels almost perfectly, as shown in Figure 4. After all, the return levels are subject to considerable uncertainty, particularly for longer return periods. The absence of any statistical assessment of overfitting raises serious concerns about the robustness and generalizability of the proposed model. First, one must propose a physically consistent family of IDF-relationships, without overfitting, as already proposed in Koutsoyiannis et al. (1998).
Yes, the sixth-degree polynomial approximates the estimated return levels almost perfectly for stations with long data series when the uncertainty of design totals was reduced by the region-of-influence method. Therefore, we are convinced that it can improve the results for stations with much shorter data series, as presented by the example in Fig. 1 in the supplement (Brno-Žabovřesky station, 36 years).
4) It is not sufficient to test the method at only two locations. Ideally, this should be done on at least one continent. However, global datasets are available for extreme precipitation amounts on a sub-daily basis (GSDR-I).
It is a misunderstanding that the method was tested only at two stations – in fact, it was tested at 160 stations in the Czech Republic; the two stations are only used to present the results (see Chapter 1, first paragraph). For other stations, we refer to https://www.perun-klima.cz/srazky (unfortunately only in Czech), see "Data availability" at the end of the article. If we get the opportunity to edit the article, we will expand it with an overview of the results from all stations. At this moment, we present the average 100year precipitation totals from all Czech stations in Fig. 2 in the supplement to support our results.
Verifying the extent of the area where our results apply could be a next step in the research. Anyway, GSDR-I cannot be used for this purpose due to its hourly resolution only.
5) The DDF-curves presented in Figure 4 for the Czech Republic differ substantially from those of other European countries, and I have no immediate explanation for this. In addition, the estimates shown in Figure 3 for the Desná-Sous station deviate abnormally (for d >10h) and are well outside the confidence interval. The wave-like behavior for station Tábor may be due to sampling uncertainty. Perhaps this is also due to the fact that no log-log plot was used. I would also like to point out that the article does not mention anything about the quality of the data or whether it has been tested for homogeneity and outliers.
The wave-like behaviour of the DDF curves is a key aspect of our study. Based on our analysis, we do not attribute this feature to deficiencies in the data. Instead, we interpret it as a meteorologically meaningful characteristic. Support for this interpretation can be found in a paper by Pöschmann et al. (2021) who present the temporal scaling of extreme rainfall in Germany. Their results indicate that the anomaly we described is not limited to the Czech Republic – see, for example, their figure 9b.
In Figure 3, we have selected three duration ranges for two return periods (10 and 100 years) for which we constructed standard DDF curves. We agree that it will be necessary to include an interval covering longer durations in the selection to show that even in this case the curve does not fit points over the entire duration range.
Regarding data quality, we apologize for not providing details. All maxima above the selected thresholds were manually checked for relevance. The data were also tested for possible inhomogeneity due to the transition from recording ombrographs to automatic rain gauges.
6) As is generally accepted, the different types of precipitation (convective/stratiform) can lead to a flattening of the IDF curves. The authors are therefore not claiming anything new. Furthermore, Figure 5 does not provide a sound scientific explanation for this: a few extreme events have been selected and the precipitation amounts for different rainfall durations are shown (within each chosen single event, some durations correspond to an annual maximum, while others do not). This cannot explain why different behaviors (e.g. a power law) apply to extreme precipitation maxima of short and long duration.
There must be some misunderstanding here, because we are not pointing to the flattening of the DDF curves, but rather to their undulation.
We agree that Fig. 5 does not provide an unquestionable proof of our concept. Its purpose is illustrative aiming to highlight the fundamentally different temporal structure of convective and stratiform precipitation events and serving to provide a physical context for the interpretation of the undulation of the DDF curves. If we get the opportunity to edit the article, we will expand this part of the paper.
7) In addition, the authors Identify the two types of precipitation in a very rudimentary way, and it is much more scientific to identify precipitation types with synoptic conditions (in the form of weather types).
The difference in short-term intensity is one of the reasonable ways to distinguish stratiform and convective precipitation, which we have chosen because of its close connection to the studied issue. However, we have no problem expanding the work to include an analysis of weather types, or applying the method that we used in Müller et al. (2018).
8) Finally, I would like to point out that their estimation method is outdated. More recent developments (i.e. Muller et al. 2007, and later authors) model the GEV-parameters as a function of rainfall duration and thus estimate the IDF-relationships directly. Here, the authors used a two-stage estimation procedure, first estimating the return levels for individual durations and then fitting the IDF model to these. This is known to be a flawed inference, because nothing can be said about the uncertainties (or how they are distributed across the two steps) and because no model choice can be made using statistical tests (see also point 3 regarding overfitting).
Estimating return levels separately for individual durations allows the duration dependence of extremes to be explored directly without imposing a predefined functional form on the GEV parameters and thereby revealing features that could otherwise be partially smoothed out.
Moreover, Figs. 3 and 4 in the supplement show that both location and scale parameters exhibit a dependence on precipitation duration similar to that detected for the individually constructed DDF curves. This indicates that the identified dependence originates from the data and is not created by the subsequent fitting.Additional references
Müller, M., Bližňák, V., Kašpar, M., 2018: Analysis of rainfall time structures on a scale of hours. Atmos. Res., 211, 38–51. https://doi.org/10.1016/j.atmosres.2018.04.015
Pöschmann, J. M., Kim, D., Kronenberg, R., Bernhofer, C., 2021: An analysis of temporal scaling behaviour of extreme rainfall in Germany based on radar precipitation QPE data. Nat. Hazards Earth Syst. Sci., 21, 1195–1207. https://doi.org/10.5194/nhess-21-1195-2021
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AC1: 'Reply on RC1', Miloslav Müller, 31 Dec 2025
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RC2: 'Comment on egusphere-2025-4268', Anonymous Referee #2, 21 Nov 2025
Dear Editor,
The article attempts to address identified drawbacks of the standard three-parameter Intensity-Duration-Frequency (IDF) formulation proposed by Koutsoyiannis et al 1998 by introducing a six-degree polynomial model. While the presented work falls within the scope of the journal, major issues should be addressed before the paper can be considered worthy of publication.
My comments are presented below:
Justification for the Proposed Framework: In the introduction, the authors correctly note the existence of other works in the literature that have provided modifications to the standard approach to address some of its limitations. Examples include the multi-scaling models of Van de Vyver and Demarée (2010) and Faure et al. (2011). Wouldn't it be more natural and methodologically sound to consider and compare their results against these existing, more parsimonious modifications before proposing a significantly more complex approach? The paper should objectively justify the novel, high-degree polynomial framework in comparison to these more parsimonious alternatives.
Comparison Framework and Model Complexity: It is expected that a more flexible model with a large number of parameters (e.g., 42 parameters, assuming 7 return periods multiplied by a 6th-degree polynomial plus intercept) will inherently provide a better fit to the observed data compared to a simple three-parameter model. I strongly suggest that the authors justify the suitability of their highly parameterized model with respect to the standard approach using a robust model selection criterion. This should ideally be either an information criterion (e.g., AIC or BIC), which penalizes complexity, or a robust cross-validation framework, which tests the model's predictive skill on independent data.
Model Formulation and Statistical Inference: The proposed polynomial approach relies on an outdated statistical inference procedure where the parameters are estimated in two separate steps: Estimation of the GEV model parameters for each rainfall duration. Estimation of the polynomial model parameters (return level curve). This two-step approach is known to introduce inconsistencies and underestimate uncertainty. How do the authors propose to address the issue of using this decoupled, sub-optimal estimation approach?
The authors should also justify why a 6the degree polynomial is the coorect model, arther than a lower degree model
Uncertainty Assessment: Given the large number of parameters and the use of a two-step estimation approach, the uncertainty in the proposed formulation will undoubtedly be substantial. The paper should include a dedicated component to thoroughly assess and quantify the uncertainties associated with their model parameters and derived return levels.
Equation 1 and 2: Units and Consistency: I find the units for IN (precipitation intensity) in Equation 1 to be surprising. Is the standard convention not to use mm/hr or mm/min for rainfall intensity? How did the authors arrive at the specified units? The same question applies to the units of R in Equation 2 Are the authors dealing with areal rainfall or point rainfall? In line 57, the authors state that they will use a "standard approach" to refer to the method of Koutsoyiannis. Does the formulation of Equation 1 and the definition of its parameters exactly correspond to the specific formulation proposed by Koutsoyiannis (1998)? This consistency needs to be confirmed.
Citation: https://doi.org/10.5194/egusphere-2025-4268-RC2 -
AC2: 'Reply on RC2', Miloslav Müller, 31 Dec 2025
Dear reviewer,
Thank you for reviewing our article. Our responses to your individual comments are highlighted in bold.
The article attempts to address identified drawbacks of the standard three-parameter Intensity-Duration-Frequency (IDF) formulation proposed by Koutsoyiannis et al 1998 by introducing a six-degree polynomial model. While the presented work falls within the scope of the journal, major issues should be addressed before the paper can be considered worthy of publication.
My comments are presented below:
Justification for the Proposed Framework: In the introduction, the authors correctly note the existence of other works in the literature that have provided modifications to the standard approach to address some of its limitations. Examples include the multi-scaling models of Van de Vyver and Demarée (2010) and Faure et al. (2011). Wouldn't it be more natural and methodologically sound to consider and compare their results against these existing, more parsimonious modifications before proposing a significantly more complex approach? The paper should objectively justify the novel, high-degree polynomial framework in comparison to these more parsimonious alternatives.
We agree that it is necessary to add to the article a comparison of our approach with the multiscaling model, focusing on their ability to represent the observed undulation in shape of the duration dependence.
Comparison Framework and Model Complexity: It is expected that a more flexible model with a large number of parameters (e.g., 42 parameters, assuming 7 return periods multiplied by a 6th-degree polynomial plus intercept) will inherently provide a better fit to the observed data compared to a simple three-parameter model. I strongly suggest that the authors justify the suitability of their highly parameterized model with respect to the standard approach using a robust model selection criterion. This should ideally be either an information criterion (e.g., AIC or BIC), which penalizes complexity, or a robust cross-validation framework, which tests the model's predictive skill on independent data.
We acknowledge the suggestion. Due to the two-step estimation procedure and the fact that the polynomial coefficients are not independent, we will address this comment through a robust block-based cross-validation framework that evaluates predictive performance across durations.
Model Formulation and Statistical Inference: The proposed polynomial approach relies on an outdated statistical inference procedure where the parameters are estimated in two separate steps: Estimation of the GEV model parameters for each rainfall duration. Estimation of the polynomial model parameters (return level curve). This two-step approach is known to introduce inconsistencies and underestimate uncertainty. How do the authors propose to address the issue of using this decoupled, sub-optimal estimation approach?
It should be emphasized that although our estimates of design precipitation are based on a two-stage approach, the associated uncertainty is mitigated by applying the region-of-influence method. We will explicitly quantify this uncertainty in the revised article.
The authors should also justify why a 6the degree polynomial is the correct model, rather than a lower degree model.
We have conducted experiments with lower degree polynomials, which systematically fail to reproduce the observed undulation in the shape of the duration dependence. We agree that the analysis results need to be added to the article. In the form of an electronic supplement, we will also present graphs with lower degree polynomials.
Uncertainty Assessment: Given the large number of parameters and the use of a two-step estimation approach, the uncertainty in the proposed formulation will undoubtedly be substantial. The paper should include a dedicated component to thoroughly assess and quantify the uncertainties associated with their model parameters and derived return levels.
We agree and will add a dedicated uncertainty assessment. Uncertainty will be quantified via a bootstrap scheme that re-runs the full ROI-based GEV estimation and subsequent DDF fitting in each replicate, allowing uncertainty propagation from regional estimation to derived return levels. Given the computational cost of recomputing the ROI regions and regional estimates, this analysis will be conducted for a representative subset of stations.
Equation 1 and 2: Units and Consistency: I find the units for IN (precipitation intensity) in Equation 1 to be surprising. Is the standard convention not to use mm/hr or mm/min for rainfall intensity? How did the authors arrive at the specified units? The same question applies to the units of R in Equation 2 Are the authors dealing with areal rainfall or point rainfall? In line 57, the authors state that they will use a "standard approach" to refer to the method of Koutsoyiannis. Does the formulation of Equation 1 and the definition of its parameters exactly correspond to the specific formulation proposed by Koutsoyiannis (1998)? This consistency needs to be confirmed.
The standard unit of precipitation intensity is mm/h. However, in water management practice in the Czech Republic, the unit l/ha/s is traditionally used, which is why we used it in our work (1 l/ha/s = 0.36 mm/h). We acknowledge that this may be confusing, so we will edit the relevant passage of the article to correspond to the standard units and to ensure consistency with the standard approach.
Citation: https://doi.org/10.5194/egusphere-2025-4268-AC2
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AC2: 'Reply on RC2', Miloslav Müller, 31 Dec 2025
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The manuscript attempts to challenge an established framework for IDF (Intensity-Duration-Frequency) relationships, specifically the work of Koutsoyiannis et al. (1998), which is widely recognized in the field. However, the criticism is based on a fundamental misunderstanding of important statistical concepts. As a result, the arguments put forward are not scientifically convincing.
Apart from the conceptual issues, the manuscript suffers from poor structure and a lack of clarity. Several important methodological steps are omitted or insufficiently explained, further undermining the credibility and readability of the work.
Given these serious shortcomings – both in terms of scientific content and presentation – I do not consider the manuscript suitable for publication in HESS. I therefore recommend rejection.
I shall explain this in more detail below. Since IDF and DDF curves are equivalent, I will use "IDF" in my argument.
1) The general IDF-relationship of Koutsoyiannis et al. (1998) has been confirmed worldwide, certainly for precipitation duration up to 24 hours (and up to 72 hours for Western Europe). On the one hand, these relationships are based on the “simple scale property” of precipitation intensity, and on the other hand, the parameterization results in IDF-curves that do not intersect for different return periods (i.e. physical consistency). Improvements are of course possible, such as using the “multiscaling properties” (Burlando & Rosso, 1996), but these results in rather modest improvements (Van de Vyver, 2018). Extensions to longer rainfall durations have been successfully developed in, for example, Willems (2000) or Fauer et al. (2021).
2) Based on the above, the IDF-relationship presented in the manuscript are fundamentally flawed and do not offer any meaningful contribution to the field.
- The polynomial relationships are fitted separately for each return period and do not preclude IDF curves from intersecting. This is precisely the key to the IDF-relationships of Koutsoyiannis et al. (1998).
- I see no connection between the existing IDF-relationships and the polynomial approach. Fundamental log-log scaling properties for precipitation (or at least a good approximation thereof, within a relevant range of rainfall duration) cannot be immediately derived from the new model.
3) There is no point in introducing a sixth-degree polynomial that approximates the estimated return levels almost perfectly, as shown in Figure 4. After all, the return levels are subject to considerable uncertainty, particularly for longer return periods. The absence of any statistical assessment of overfitting raises serious concerns about the robustness and generalizability of the proposed model. First, one must propose a physically consistent family of IDF-relationships, without overfitting, as already proposed in Koutsoyiannis et al. (1998).
4) It is not sufficient to test the method at only two locations. Ideally, this should be done on at least one continent. However, global datasets are available for extreme precipitation amounts on a sub-daily basis (GSDR-I).
5) The DDF-curves presented in Figure 4 for the Czech Republic differ substantially from those of other European countries, and I have no immediate explanation for this. In addition, the estimates shown in Figure 3 for the Desná-Sous station deviate abnormally (for d >10h) and are well outside the confidence interval. The wave-like behavior for station Tábor may be due to sampling uncertainty. Perhaps this is also due to the fact that no log-log plot was used. I would also like to point out that the article does not mention anything about the quality of the data or whether it has been tested for homogeneity and outliers.
6) As is generally accepted, the different types of precipitation (convective/stratiform) can lead to a flattening of the IDF curves. The authors are therefore not claiming anything new. Furthermore, Figure 5 does not provide a sound scientific explanation for this: a few extreme events have been selected and the precipitation amounts for different rainfall durations are shown (within each chosen single event, some durations correspond to an annual maximum, while others do not). This cannot explain why different behaviors (e.g. a power law) apply to extreme precipitation maxima of short and long duration.
7) In addition, the authors Identify the two types of precipitation in a very rudimentary way, and it is much more scientific to identify precipitation types with synoptic conditions (in the form of weather types).
8) Finally, I would like to point out that their estimation method is outdated. More recent developments (i.e. Muller et al. 2007, and later authors) model the GEV-parameters as a function of rainfall duration and thus estimate the IDF-relationships directly. Here, the authors used a two-stage estimation procedure, first estimating the return levels for individual durations and then fitting the IDF model to these. This is known to be a flawed inference, because nothing can be said about the uncertainties (or how they are distributed across the two steps) and because no model choice can be made using statistical tests (see also point 3 regarding overfitting).
Additional references
- Burlando, P., Rosso, R. (1996) Scaling and multiscaling models of depth-duration-frequency curves for storm precipitation, J. Hydrol. 187, 45-64.
- Muller, A., Bacro, J. N. & Lang, M. (2007) Bayesian comparison of different rainfall depth-duration-frequency relationships. Stochastic Environ. Res. & Risk Assess. 22, 33–46.
- Van de Vyver H. (2018) A multiscaling-based intensity–duration–frequency model for extreme precipitation. Hydrol. Process.32, 1635–1647.
- Willems, P. (2000) Compound intensity/duration/frequency-relationships of extreme precipitation for two seasons and two storm types. J. Hydrol. 33, 189--205.