We model monthly precipitation maxima at 132 stations in Germany for a wide range of durations from 1 min to about 6 d using a duration-dependent generalized extreme value (d-GEV) 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 of the summer months exhibit a more rapid decrease of intensity with duration, as well as higher intensities for short durations than the IDF curves for the remaining months of the year. Thus, when short convective extreme events occur, they are very likely to occur in summer everywhere in Germany. In contrast, extreme events with a duration of several hours up to about 1 d are conditionally 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 long-lasting 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 in 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.

Extreme precipitation events can potentially cause significant damage

The characteristics of extreme precipitation on different timescales can be summarized in terms of intensity–duration–frequency (IDF) curves. These are a standard tool in hydrology for designing hydrological structures and managing water supplies

Extreme value theory offers several approaches to describe the occurrence probability of extreme events

Since extreme events are by definition rare, the estimation of quantiles (return levels) corresponding to small exceedance probabilities (return periods) is always associated with the problem of limited data. When modeling IDF curves, the limitations of the observations are firstly the spatial coverage and secondly the temporal resolution

In order to assess extreme precipitation observations of different aggregation times simultaneously, it is possible to use a duration-dependent extreme value distribution

In this study, we implement monthly covariates analogously for the parameters of the d-GEV distribution. Hence, we model intra-annual variations of extreme precipitation for a wide range of durations from 1 min to approximately 6 d at 132 stations in Germany. This not only allows us to estimate and compare IDF curves of different months, but we also expect to obtain more reliable annual IDF curves due to the more efficient use of the available data. Furthermore, we anticipate that accounting for seasonality and reducing uncertainties in parameter estimation will provide a better understanding of the underlying processes. Hence, we expect to gain new insights into the empirical dependencies of GEV parameters on duration, which are in turn relevant for the modeling of annual maxima. This study addresses the following research questions:

How does the IDF relationship at different stations in Germany evolve throughout the year?

To what extent do the annual IDF curves based on monthly and annual maxima differ?

Does explicit modeling of seasonal variations allow us to draw conclusions aimed at improving the modeling of annual maxima?

We aim to model the intra-annual variations of extreme precipitation on different timescales. For this purpose, we use observations with high temporal resolution from stations in Germany. We use a duration-dependent GEV (d-GEV) distribution with monthly covariates to describe the monthly maxima over a range of durations collectively in one model. Thereby, appropriate models for the intra-annual variations of the d-GEV parameters are selected through stepwise forward regression. This approach allows us to examine how the IDF curves vary throughout the year in different areas of Germany. From this seasonal model, we can derive annual IDF curves as well. We compare these annual IDF curves with those resulting from directly modeling the annual maxima via a verification procedure using the quantile skill index. Finally, we verify whether modeling monthly maxima allows for a more precise estimate of the relationships between GEV parameters and duration. Therefore, we model each duration separately using the GEV distribution with monthly covariates. Details of the data as well as all methods involved are described in the following section.

We use precipitation measurements at 132 stations in Germany that provide a temporal resolution of 1 min. Their locations are presented in Fig.

Map of Germany with positions of all 132 stations considered. Colors represent the length of the available time series with minute resolution. The longest observation period of 51 years exists for station Bever-Talsperre (dark blue).

The observations were accumulated to the following durations:

The challenge in modeling extremes is to estimate probabilities of very rare events or those not even observed yet. Here, we apply the block maxima approach which is commonly used for this purpose. It is based on the Fisher–Tippett–Gnedenko theorem, which essentially states that under certain assumptions the probability distribution of block maxima can be modeled by the generalized extreme value (GEV) distribution

More precisely, let

The GEV distribution is thus likely to be a well suited model for the distribution of annual precipitation intensity maxima of one selected
aggregation duration. In order to model the distribution for different durations simultaneously,

According to the Fisher–Tippett–Gnedenko theorem, the GEV distribution
is an adequate model for block maxima if the block size is sufficiently large. For geophysical applications, such as modeling extreme precipitation, it is common to choose a block size of 1 year, as explicit modeling of seasonality is thereby avoided. However, this results in two major disadvantages: large portions of the data are lost for the analysis if only the annual maxima are used, and the assumption that precipitation events originate from an identical distribution is violated if a distinct intra-annual cycle exists. Therefore, the use of a smaller block size is worth considering. Multiple studies suggest that the GEV distribution is well suited to model monthly block maxima of daily precipitation sums in the midlatitudes

To account for any form of variability in the GEV model (Eq.

The parameters of the GEV distribution can be estimated from a time series of observed block maxima. For this purpose, we apply the widely used maximum likelihood estimator (MLE)

Since the logarithm of the likelihood reaches the maximum at the same value but is easier to calculate, the parameters are estimated by optimizing the log-likelihood numerically:

Nevertheless, Eqs. (

To obtain a parsimonious model, we use a selection procedure consisting of two steps: in the first step, we determine for which of the GEV/d-GEV parameters the modeling of the intra-annual variations is not appropriate and which should therefore remain constant. In the second step, we select which terms of the harmonic series in Eq. (

When modeling intra-annual variations of GEV parameters, the shape parameter

To be consistent, we also use a constant shape parameter in the d-GEV case. The estimation of the duration offset parameter

For the maximum order of the harmonic series in Eq. (

All possible models in the first step of the stepwise regression (GEV case) with

For the station Bever-Talsperre, the resulting estimated d-GEV parameters are presented in Fig.

Estimated d-GEV parameters

When modeling the annual maxima with the d-GEV distribution according to Eq. (

However, we can also derive annual IDF curves, i.e., quantiles of the distribution of annual maxima, from modeling the monthly maxima. Assuming the maxima of all months in a year as independent, the non-exceedance probability

We determine the uncertainties of the estimated IDF curves using the ordinary nonparametric bootstrap percentile method

To visualize the differences between the annual IDF curves resulting from the different models, it can be useful to compare the parameters of the respective distributions of annual maxima. Unfortunately, these are only directly available when modeling the annual maxima. However, we can assume that the distribution of the annual maxima resulting from modeling the monthly maxima is for each duration again a GEV distribution, due to its max-stability property.
Thus, we estimate the GEV parameters of the distribution of annual maxima by firstly using Eq. (

We apply a verification procedure in order to assess the estimated
quantiles, i.e., IDF curves. At a given station, we aim to compare the
annual IDF curves obtained by modeling the monthly maxima with those obtained by modeling the annual maxima. We model the monthly maxima using the
d-GEV distribution with monthly covariates according to Eqs. (

The 0.9 quantiles for station Bever-Talsperre for each month

To provide a detailed analysis we follow

We first show the results for the monthly IDF curves at the station Bever-Talsperre. Based on the probability that the annual

Probability of the annual 0.9 quantile

We obtain quantile estimates, i.e., IDF curves, for each month using the d-GEV distribution (Eq.

Due to the exponential decrease of intensity with duration, a comparison of the

To investigate the intra-annual variations across Germany, we calculate the probability that the annual

Maximum probability of the annual 0.9 quantile

From Fig.

Regarding the month with the highest probability in the case of long durations, we derive a rough division of the stations within Germany into three types. For each type we choose two stations, for which we present the probabilities in detail in Fig.

Station (1) is Bever-Talsperre, which has already been discussed in detail. Similarly to (1), the maximum at the station Saarbrücken-Ensheim (2) also shifts from summer for short durations into late autumn to winter for long durations. The stations differ, however, insofar as the maximum probability for short durations at station (2) in July varies only slightly from the probability in the months of June and September. In addition, the probability for long durations remains rather high in summer, in contrast to station (1). Stations that show a similar shift of the maximum exceedance probability, from summer for short durations to late autumn or winter for long durations, are located exclusively in the western half of Germany and also occur mostly at higher altitudes. The exception are two stations in northern Germany. The location of these stations coincides well with the results of

As examples for stations where the maximum occurs in September or October, for long durations, we present the monthly exceedance probabilities for Cuxhaven (3) and List auf Sylt (4). At both stations the width of the probability peak increases for long durations, while its maximum shifts. This shift is more pronounced at station (3). The probabilities in the interval between December and May are relatively low at both stations for all durations. Some stations of this type are located scattered throughout Germany. However, a clear cluster of these stations exists on the North Sea coast. This group is also characterized by extreme convective precipitation events occurring most likely in August, which could be related to the water temperature in this region reaching its maximum during this month. Accordingly, a possible explanation for the high probability of long-lasting heavy precipitation in the following months might be that extratropical cyclones transport air, which was warmed over the North Sea and thus features a high water content, into this region.

Examples of stations where the probability maximum for all durations occurs in summer are Berlin-Tempelhof (5) and Mühldorf (6). An essential difference to the stations (3) and (4) is that in this case a second maximum occurs in winter for long durations. At station (6) this second maximum is even almost as high as the one in summer for

The monthly exceedance probability is a useful indicator of the months from which the annual maxima of different durations originate. For all stations, the peak of the probability is relatively narrow for short durations, with the maximum in summer. The probability that the annual 0.9 quantile occurs in one of the other seasons is negligible. This fact contradicts the assumption of the block maxima approach that precipitation intensities are identically distributed within the block of 1 year. In other words, a block size of 1 year for short durations results in a much smaller effective block size of about 4 to 6 months. With respect to longer durations, the stations differ greatly, but it can be generally stated that the effective block size increases for long durations. Thus, the annual maxima at a station for different durations originate from effective blocks of different sizes, which might even be in different seasons, depending on the station's location. This effect is further emphasized in Fig. S2. Modeling the monthly maxima, on the other hand, avoids this problem. Therefore, in the following section we compare the annual IDF curves derived from annual maxima with those derived from monthly maxima.

We obtain the annual IDF curves and their confidence intervals from modeling annual and monthly maxima, using the respective methods described in Sect.

Annual IDF curves for three example stations estimated via two models: modeling annual maxima with the d-GEV distribution and modeling monthly maxima using the d-GEV distribution with monthly covariates. The shaded areas represent the respective 95 % confidence intervals. The distributions of the observed annual maxima are shown as box-and-whisker plots, where the whiskers cover the complete data range. In the upper panels the corresponding QSI values are presented, indicating the comparison of the models' performances, where positive values indicate an increase in the skill of the monthly d-GEV model compared to the annual d-GEV model.

For the station Bever-Talsperre, the IDF curves resulting from the two different models are almost identical over a long duration range

At station Saarbrücken-Ensheim, the differences in the IDF curves of the two models are more pronounced throughout the entire duration range: the estimated quantiles of the monthly model are higher for very small and very large durations but lower in the range

This appears even more prominent at the Cuxhaven station. Since the high-resolution time series at this station covers only 19 years, the uncertainties of the annual model are considerably wider than those of the monthly model. The estimates of the monthly model for the 0.9 quantile and the 0.99 quantile are below the respective estimates of the annual model. The estimates for the 0.5 quantile differ only slightly. The quantiles of the monthly model roughly parallel those of the annual model for longer durations. Thus, the monthly model does not deviate essentially from a power law at this station. The QSI does not provide a clear indication regarding which model better represents the data but fluctuates between positive and negative values. This seems to be in agreement with the observations. The spread of the boxes and whiskers first increases and then decreases over duration. As a result, in the duration ranges with narrower box-and-whisker plots, the monthly model better represents the data, especially for higher quantiles, while the annual model is more suitable particularly for

Overall, we find that the differences between the annual and the monthly model are very heterogeneous for individual stations. However, two general statements can be made:

Modeling monthly maxima provides a clear improvement in terms of the quantile estimates' uncertainties, especially for stations with short observational time series.

Although a power-law behavior for long durations is assumed for the monthly IDF relations, the resulting annual IDF curves can deviate from this behavior and are therefore more flexible.

Annual GEV parameters

In terms of model performance, we likewise cannot draw general conclusions. At some stations, such as Cuxhaven, we find that modeling monthly maxima improves the estimates of the annual IDF curves for almost all probabilities and durations. However, at many stations the improvement in the estimation is limited to a selected range of probabilities and durations, and there are also stations at which the estimated quantiles of the monthly model are always worse than those of the annual model. Since the objective of this study is to model the seasonal variations at all stations by applying a uniform framework, the model selection was performed identically for all stations, e.g., choosing

To model the IDF relationship, we have so far assumed that the GEV parameters depend on duration according to Eqs. (

modeling annual maxima using

a separate GEV distribution for each duration (annual GEV)

one d-GEV distribution (annual d-GEV)

modeling monthly maxima using

a separate GEV distribution with monthly covariates for each duration (monthly GEV)

one d-GEV distribution with monthly covariates (monthly d-GEV).

Distribution of shape parameter

For the location parameter

The shape parameter

Since the shape parameter estimates of the four different models vary substantially at the individual stations, we summarize the information for all stations in Fig.

To summarize, we find that modeling the monthly maxima allows new conclusions to be drawn about the behavior of the parameters of the distribution of annual maxima depending on duration. Instead of using the more complex modeling of monthly maxima to estimate annual IDF curves, one might also try to implement the resulting characteristics directly into the model for annual
maxima. We find that for some stations the location

This study focuses on modeling the intra-annual variations of extreme
precipitation on different timescales. For this purpose, we employ a
duration-dependent generalized extreme value (d-GEV) distribution with
monthly covariates. Using this approach allows for the following:

investigation of seasonal variations in the intensity–duration–frequency (IDF) relationship;

the obtaining of more reliable estimates for the annual IDF curves by utilizing information on extreme events more efficiently;

a better understanding of the underlying processes, i.e., the dependence of the parameters on the duration.

Regarding the seasonal variations, we find that everywhere in Germany, the short convective extreme events are most likely to occur in the summer months, whereas there are regional differences for the seasonality of long-lasting stratiform extreme events. Our findings will allow future studies to identify meaningful factors accounting for these regional differences.

Furthermore, our results show that the annual IDF curves based on the monthly maxima constitute a major improvement in terms of uncertainties of the estimates. Using the quantile skill index (QSI), we compare the performance of the models based on the annual and monthly maxima and show that, for some stations, modeling the monthly maxima also leads to a considerable improvement in this regard. A limitation of this study is the strict assumptions that are imposed on the seasonal variations of the distribution parameters. Subsequent studies should therefore investigate the degree to which relaxing these assumptions might further improve the performance of
the model based on monthly data. For example, in the framework of a vector generalized additive model

Finally, we can demonstrate that at some stations the annual IDF curves based on the monthly maxima deviate from the assumption of scale invariance for long durations. We illustrate that this behavior can be captured by a different parameterization of the location and scale parameter. For future research, it might be of interest to compare the monthly model employed in this study with an annual model that uses different parameterization, e.g., the one proposed by

In conclusion, the use of monthly maxima can be beneficial in several respects when estimating IDF curves, even when information on seasonal variations is not required.

The station data are mostly publicly available via the Climate Data Center of the DWD (

The supplement related to this article is available online at:

HWR and JU conceptualized this study. HWR provided supervision. JU developed the software, carried out the statistical analysis, and evaluated the results. FSF supported the code implementation. HWR and FF contributed to the interpretation of the results. JU prepared the original draft. HWR and FSF revised the draft.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors would like to thank the Wupperverband as well as the Climate Data Center of the DWD, for providing and maintaining the precipitation time series. Jana Ulrich kindly appreciates the support and motivation offered by Lisa Berghäuser and Oscar E. Jurado. Finally, we would like to thank Juliette Blanchet and two anonymous referees for their valuable feedback and suggestions to improve the manuscript.

This research has been supported by the Deutsche Forschungsgemeinschaft (grant nos. GRK2043/1, GRK2043/2, and CRC 1114) and the Bundesministerium für Bildung und Forschung (grant no. 01LP1902H). We acknowledge support from the Open Access Publication Initiative of Freie Universität Berlin.

This paper was edited by Nadav Peleg and reviewed by J. Blanchet and two anonymous referees.