| 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.
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.