06 Jul 2021
06 Jul 2021
Modeling seasonal variations of extreme rainfall on different time scales in Germany
 Institute of Meteorology, Freie Universität Berlin, CarlHeinrichBeckerWeg 612, 12165 Berlin, Germany
 Institute of Meteorology, Freie Universität Berlin, CarlHeinrichBeckerWeg 612, 12165 Berlin, Germany
Abstract. We model monthly precipitation maxima at 132 stations in Germany for a wide range of durations from one minute to about six days using a durationdependent generalized extreme value (dGEV) distribution with monthly varying parameters. This allows for the estimation of both monthly and annual intensity–duration–frequency (IDF) curves: (1) The monthly IDF curves are steeper in summer and exhibit higher intensities for short durations than in the rest of the year. Thus, everywhere in Germany short convective extreme events occur very likely in summer. In contrast, extreme events with a duration of several hours up to about one day are more likely to occur within a longer period or even spread throughout the whole year, depending on the station. There are major differences within Germany with respect to the months in which longlasting stratiform extreme events are more likely to occur. At some stations the IDF curves (for a given quantile) for different months intersect. The meteorological interpretation of this intersection is that the season at which a certain extreme event is most likely to occur shifts from summer towards autumn or winter for longer durations. (2) We compare the annual IDF curves resulting from the monthly model with those estimated conventionally, that is, based on modeling annual maxima. We find that adding information in the form of smooth variations during the year leads to a considerable reduction of uncertainties. We additionally observe that at some stations, the annual IDF curves obtained by modeling monthly maxima deviate from the assumption of scale invariance, resulting in a flattening in the slope of the IDF curves for long durations.
Jana Ulrich et al.
Status: final response (author comments only)

RC1: 'Comment on hess2021336', Anonymous Referee #1, 07 Aug 2021
General comments:
The manuscript provides a novel analysis on seasonal variation of IDF curves for Germany. For estimation of the IDF curves wellchosen state of the art approaches are used. The results are interesting; they reveal the seasonality of extremes for different durations and show smaller uncertainties for IDF estimations when monthly models are used. The manuscript is well written. I have only one general point and a four minor comments for improvement (see specific comments).
The general point concerns the small block size of one month for the extreme value analyses. Even if the GEV is fitting well as shown by the authors the basic assumption of sufficient large n for the validity of Fisher–Tippett–Gnedenko theorem might be questionable also considering the fact that in certain months of some years no extremes might be observed. This problem becomes especially relevant given the results from which months the maxima from different durations originate with respect to the effective block size. Especially the latter is much larger than one month for long durations. I would suggest to discuss this a little further, may be also considering future research regarding specific analyses periods/ seasonal subdivisions depending on the duration.
Specific comments:
 Please indicate the specific GEV parameters on the figures of relate a) – e) to the specific parameters in the caption.
 Lines 233ff: Please add some more explanation how the bootstrap is carried out.
 Equation 16: The equation is not completely clear to me. The variable u is not explicitly defined. However, if I assume u=(o_{n}q_{p}) then it should read e.g. ρ_{p}=pu and not ρ_{p}(u)=pu?
 Lines 425ff: The uncertainties are estimated with different methods, the Fisher information matrix and the bootstrap method. Are these results comparable? Why not using bootstrap for all uncertainty assessments?
 AC1: 'Reply on RC1', Jana Ulrich, 20 Aug 2021

RC2: 'Comment on hess2021336', Anonymous Referee #2, 10 Aug 2021
REVIEW OF THE ARTICLE « Modeling seasonal variations of extreme rainfall on different time
scales in Germany» by Ulrich et al.
SUMMARY: The authors model monthly precipitation maxima in Germany for a wide range of time scales ranging from 1 min to 6 days. They propose a durationdependent GEV (dGEV) with monthly covariates to allow monthly varying parameters. This allows them to show that short duration maxima occur mainly in summer, while longer duration maxima are spread throughout the year. They also show that the annual distribution derived from the monthly distributions are much less uncertain than that derived from annual maxima.
OVERALL COMMENT : The article is well written and very clear. I definitively enjoyed reading it. My main comments are:
 the article is quite long and there are mainly repetitions. In particular the “Discussion” section is very mainly an extended summary of the results. There are actually few sentences of discussion. Would it be possible to drastically shorten this part?
 I’m a bit puzzled by the fact that a constant xi is considered in the dGEV distribution. As Fig. 8 shows, the xi parameter seems to be decreasing with duration (as expected). The authors claim that the shape parameter in difficult to estimate, which is true, but don’t you think that a simple model on xi (e.g. loglinear wrt duration) could be managable?
DETAILED COMMENTS
 l 4: “IDF curves are steeper”: I think this is not understandable in the abstract
 l 5: “short convective events occur very likely in summer” : the “very likely” may be confusing because it is stille rare (probability vs conditional probability)
 l 24: Kuntz et al is in german so I could not check
 l 95: you consider both 1 and 2 minutes. As shown later in your figures, the distributions for 1 and 2 minutes are very similar so I’m not sure that the 2 minute is necessary.
 l 178: Actually to the best of my understanding, Jurado et al 2020 conclude that accounting for dependence gives better results when d<=10h, which is the case for 10/14 (71%) of the considered durations. So I’m not convinced by your justification (but I agree that accounting for dependence increases much model complexity)
 l 205206: I’m not familiar with the crossvalidated likelihood method so I missed this part. E.g. what is the number of folds? Please consider being more specific here.
 l 233238: to be sure: do you use the same sampling years also for deriving the annual GEV from the monthly GEV ? (i.e. do all monthly GEV use the same sampling years?)
 l 297303: If I understood correctly, the sum of the ordinates of the dots of a given duration is equal to 0.1 (due to the 0.9quantile). So dividing the ordinates by 0.1 gives the proportion of exceedances that occur a given month. Wouldn’t it be easier to interpret Fig 4 this way? For example in Bever 45% of the exceedances occur in July.
 Fig 5: I don’t understand the legend for Pr max. For example what do you mean by “<2%”? Isn’t it “12%”? Also I would find it easier to interpret if you divide Pr by 0.1, as said above.
 l 364366: I don’t understand how you deal for cases p>11/T. What is the observed quantile in this case? I guess you consider the maxximul value by I don’t thin that’s correct. So I suggest removing the cases without dots in Fig 6.
 l 426: I agree that Fisher information matrix is correct in this case but for comparison puropose, I suggest using a bootstrap method as for the other cases.
 Fig 8: As said above, the monthly GEV seems loglinearly decreasing with duration
 Section 4: As said above, this is actually almost only an extended summary. Please consider shortening it.
 l 562563: As a first try, I wouldn’t try the multi scaling model but I’d rather consider xi function of d.
TYPOS:
 l 121 eq 4: xi(p) should be xi(d)
 l 151 : remove the brackets
 l 171: the the
 Fig 6 caption: OSI → QSI
 AC2: 'Reply on RC2', Jana Ulrich, 24 Aug 2021

RC3: 'Comment on hess2021336', Anonymous Referee #3, 10 Aug 2021
Review: Modeling seasonal variations of extreme rainfall on different time scales in Germany
General comments
The paper evaluates differences in seasonal extreme rainfall intensity depending on the method used to estimate IDF curves. They primarily evaluate IDF curves created using block maxima at the monthly level, and then compare these curves to those created using annual block maxima, which is the more common approach. The paper is well written and interesting, but rather long. Some suggestions for shortening are listed below.
While the results are informative and supported by extensive analysis, the motivation for this very detailed investigation was not clear from the introduction. IDF curves are typically used for design of longlived infrastructure systems where monthly variations are essentially irrelevant. The authors discuss a few reasons that monthly IDF curves could be valuable, including to support agricultural or water resources stakeholders. However, these stakeholders typically care about monthly average rainfall, and there is no reasoning in the paper that supports why they would be interested in extremes. Could the authors elaborate and/or find a reference that supports this?
It seems instead that the motivation for the paper is to “increase understanding” of monthly extremes and examine “underlying mechanisms.” If so, then what new information do monthly IDF curves bring? Is this simply a convenient way to evaluate monthly extremes and also account for storm duration? Or could monthly IDF curves bring added value to engineering analyses?
The introduction does mention that monthly IDF curves could bring added value compared to annual block maxima by including more data in the analysis. However, there is an existing technique, called “peaks over threshold” (POT), that evaluates all storms in a year over a certain threshold. It is unclear whether the monthly maximum technique brings added value compared to the POT method, but it is clear that the monthly maxima method is not the only way to include more data in the analysis. There are drawbacks to POT, of course, including that the annual return period is no longer directly interpretable since more than one storm per year can be included in the extreme event series. But the POT technique should be mentioned in the introduction as an alternative way to include more data. A comparison of the monthly maximum technique to POT should also be mentioned in the conclusions/future work section.
Based on these points, a distinct motivation for the creating of monthly IDF curves seems to be missing. After reading the results, it seems that monthly IDF curves could bring some added value in terms of uncertainty evaluation and potentially even for parameter fitting. This is of interest to an engineering audience who are developing and using IDF curves. Suggest restructuring the introduction to ensure the typical IDF curve audience understands this before reading the entire, very detailed study.
The motivation or added value for creating monthly IDF curves could also be discussed further (in results or discussion section). Is it worth it to use monthly maximum instead of annual maxima? If so, in which cases: for annual IDF curves in general, or only when we are interested in monthly extremes? Why?
The discussion section, which repeats a lot of the results, could be condensed, or merged with the results section.
Specific comments
Line 25 and line 32 Similar to general comment. Why it is “critical” to provide information about extremes on a monthly basis?
Line 47 – 52 – This is a common problem, not just in Germany. Many places (like the US, NOAA NCDC) have 50 years or more of daily data, with data at subhourly resolutions only available in the past decade or so. I suggest making this statement more generalizable, and say this is also the case for Germany, the focus of this study. Many others will also be interested in using the available data more efficiently through pooling information.
Line 59 – 64 – Yes, block maxima typically are only used for annual maximum because other methods like peaks over threshold (see Coles) are used if you want to capture more data and extremes within a year. Why not use the peaks over threshold method instead of monthly block sizes? If monthly variation is relevant, why are periodic functions needed as covariates (instead of a GEV distribution dependent on duration and month)? It seems that later on you clarify this – parameters can be reduced. Suggest clarifying this in the introduction as well and that you will compare the two techniques later on.
Line 65 – Did Fischer et al compare this method to peaks over threshold? More precise quantile estimates compared to what?
Line 74, research question 3 – this question is unclear and should be briefly introduced in the introduction.
Line 137 – 138 – unclear what is meant by “identically distributed precipitation cannot be motivated if an annual cycle exists…” Please clarify. Is it an interannual cycle or intraannual? Also, do you mean independent identically distributed?
Line 139 – what is meant by “sufficient”? Meaning it can be used? It is a compromise? Also, this sentence is repeated from the introduction. Suggest rewording, shortening, or removing.
Line 215 – choice to keep the shape and theta parameters constant is justified. How so? It seems these parameters are varying in the same fashion as mu, the modified location parameter? A bit more explanation here would be useful.
Line 350 – could you comment on what this implies for IDF created with annual block sizes? Does it matter?
Lines 352 – 354. The authors state that the annual maxima originate from effective blocks of different sizes, seasons, etc. Could you comment more on why this is a problem when annual block sizes are used? Wouldn’t the annual maxima still be captured? Does it matter when it occurred?
Editorial remarks
Line 47 – suggest “weather station” instead of “station”
Lines 103 – 132 – these derivations are rather standard textbook information for extreme values and IDF curves. Suggest shortening this section to provide enough information to introduce equation 8, and putting the rest in Appendix where details can be read if desired. Same comment for MLE, lines 161 to 175.
Line 280 – could you state the 0.9 quantile in terms of probability of occurrence in a given month, or the return period, as in line 295?
Figure 4 – suggest adding the legend again to this figure.
Paragraph lines 322 – 345 could be shortened
Lines 363 – 364 – could you put this information on the figure or in its caption? Would be helpful to interpret the figure.
 AC3: 'Reply on RC3', Jana Ulrich, 26 Aug 2021
Jana Ulrich et al.
Data sets
Monthly precipitation intensity maxima for 14 aggregation times at 132 stations in Germany Jana Ulrich; Felix S. Fauer; Henning W. Rust https://doi.org/10.5281/zenodo.5025657
Jana Ulrich et al.
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