Climate change has been identified as the cause of increasing risks from meteorological extreme events affecting almost all areas of the economy, nature, and human life, and those will be even more endangered in the future and the references therein. One of the main targets of current and future generations is to avoid further changes and to develop adaptation strategies to reduce risks and burdens.

While climate change can be measured very reliably for the surface temperature, for other variables like extreme precipitation the connection is not yet clear. For regions with good data availability, it has already been shown that frequency and intensity of heavy precipitation have likely increased on the global scale . Furthermore, climate projections show that future extreme precipitation will continue to intensify e.g.,. Since the consequences of heavy precipitation are extensive and can lead to different threats and damages, for example, due to flash floods, river floods, mudslides or soil erosion, an accurate assessment of extreme precipitation changes is crucial for an adequate adaptation. The potential risk due to extreme precipitation is not only dependent on its magnitude, but it also can be related to a change in its seasonal cycle. For example, a shift of strong precipitation from summer to spring leads to an increased flood risk due to a larger likelihood of strong rainfall and snowmelt occurring at the same time . Furthermore, crop losses may rise, since plants are more vulnerable during earlier growing stages .

Analyses of extreme precipitation in Germany for different seasons have already been done(; ). and analyzed quantiles of daily precipitation sums separately for the seasons DJF, MAM, JJA and SON, while used available data more efficiently by modeling monthly maxima of daily precipitation sums for all months simultaneously. This approach has been proven to lead to more robust and reliable results than considering months separately. extended this method by including different durations to efficiently estimate intensity–duration–frequency curves. Furthermore, analyzed the effect of climate change on seasonal extreme precipitation and found a positive connection to the northern-hemispheric temperature rise. In our approach we combine the simultaneous modeling of available data for all months with interannual variations, thus accounting for potential changes in the seasonality due to climate change and natural variability. Here, we point out that when referring to interannual variations, we are not addressing differences between successive years, but rather the trend over the entire observation period, which could be potentially non-linear.

Extreme value statistics (EVS) e.g., are used to quantify the magnitude and occurrence probabilities of these seasonally–interannually varying extremes. EVS have been applied in many different research fields e.g.,. One way to analyze extremes is the block maxima approach, where the observations are divided into blocks with equal lengths. The probability distribution for the maxima of these blocks is represented by the generalized extreme value (GEV) distribution. Instead of considering annual maxima of precipitation, which are frequently used in risk assessment, we take a monthly block size to resolve the seasonal cycle. Contrary to a stationary approach with an individual extreme value model for each calendar month, we take advantage of the smooth variations in the probability distributions of the block maxima across adjacent calendar months. Because of the periodic nature of the seasonal changes, a series of harmonic functions is an appropriate choice for describing the corresponding variations in the GEV parameters. This modeling strategy has already been widely applied e.g.,. It has been shown to provide more accurate monthly and annual return levels (quantiles of the GEV) .

Interannual variations in precipitation have been shown to be associated with its natural variability e.g.,, increased air temperatures , and other effects influencing large-scale atmospheric circulations and precipitation characteristics . Most of these effects are highly non-linear, and their roles are difficult to quantify. Here, we use time as a proxy to combine those different unknown effects. One possibility to model non-linear interannual changes is polynomial regression e.g.,. Orthogonal polynomials are used to reduce multicollinearity and to improve the parameter estimation . Here, we use Legendre polynomials up to an order of 5 to describe the variations across years. On the one hand, this enables the reflection of changes potentially associated with climate change, and, on the other hand, this allows for modeling of natural variability in extreme precipitation. The concept of using higher-order Legendre polynomials has also been applied to assess spatial variations . As the seasonal and interannual covariates are conceptually equal, we combine both approaches. Additionally, interactions between the covariates allow the seasonal cycle to change across years.

The goal of this paper is to assess the performance of the seasonal–interannual modeling with a special attention to a flexible shape parameter ξ. This parameter is difficult to estimate as it interferes with the scale parameter and requires long records for reliable results . Nevertheless, it describes the behavior of the very rare events and consequently plays an important role for assessing extreme precipitation changes. Furthermore, the possible impact of climate change on the seasonal cycle of extreme precipitation is analyzed. We formulate three research questions to be addressed in this study. RQ1

Can a model with interannual variations better represent the observations than a seasonal-only model?

We carry out this investigation for observations from Germany with more than 500 long (≥ 80 years) records, presented in Sect. . The seasonal–interannual modeling is described in Sect. . Model selection and validation tools are covered in Sect. . The gain of modeling seasonal–interannual variations with respect to a just seasonal model (RQ1) and the importance of a flexible shape parameter ξ (RQ2) are assessed in Sects.  and . The impact of climate change on the seasonal cycle of heavy precipitation (RQ3) is tackled in Sect. . Finally, we discuss the results in Sect. .

A dataset of almost 5700 rain gauges measuring daily precipitation amounts is provided by the German Meteorological Service (Deutscher Wetterdienst, DWD) via the continuously updated Open Data Server . Those observation stations are set up according to the WMO guidelines . The daily sums of precipitation are obtained from amounts accumulated between 05:50 and 05:50 UTC of the following day and have been checked for spatial consistency .

For investigating long-term trends a sufficiently long time series is crucial; thus, we only consider the most recent stations with at least 80 years of observations lasting until 31 December 2021. We allow for missing values and larger gaps of several consecutive years, often occurring for the years of the second world war. The 519 stations fulfilling the mentioned criteria are depicted in Fig. ; the color coding shows the station's altitude. The locations are not homogenously distributed in space: some areas are closely covered, while for other areas, for example, in the east of Germany or in the state of Saarland in western Germany, long time records which are still being updated are missing. The common time period for all 519 observation records covers the years from 1941 to 2021. The four stations Krümmel (1 January 1899 until 31 December 2021), Mühlhausen/Oberpfalz-Weihersdorf (1 January 1931 until 31 December 2021), Rain am Lech (1 January 1899 until 31 December 2021) and Wesertal-Lippoldsberg (1 January 1931 until 31 December 2021) are highlighted in Fig.  and will be discussed exemplarily in this study. We have selected these stations as they are characterized by different changes in seasonality (see Sect. ) represented by divergent model setups (see Sect. ). Additionally, their interannual changes are more pronounced than for other stations. We consider monthly maxima of daily precipitation sums while months with less than 27 measured days are discarded from the analysis.

In order to describe the changes in seasonality of extreme precipitation, we build a statistical model. This can be done with concepts of extreme value statistics (EVS), which are widely explored and applied in different scientific fields (e.g., for the financial sector, ; or for geosciences, ). One major strategy in EVS is the block maxima approach leading to an asymptotic model for extreme values: the generalized extreme value (GEV) distribution, briefly described in the following.

We address RQ1: can a model with interannual variations better represent the observations than a seasonal-only model? As mentioned in Sect. , a model with at least one interannual component in any of the GEV parameter was chosen only for 338 of 519 stations (∼ 65 %). To assess the importance of the interannual variations of these 338 stations, we analyze the skill with respect to the seasonal-only model. Table  shows the total QSS for different non-exceedance probabilities (return periods). Skill is positive but small ≲2%, increasing with non-exceedance probability (return period). The latter has to be interpreted with care as there are very few observations in the range of the upper quantiles. Return levels with a return period higher than the time range of the data should be treated cautiously, since the quantile score cannot reasonably evaluate those values . As we consider for each station at least 80 years of observations, this only matters for non-exceedance probabilities (return periods) of 0.99 and 0.995 (100 and 200 years). The small increase in skill due to the inclusion of interannual variation is expected as most of the signal can be described already with the strong seasonal cycle.

We analyze whether interannual variations improve the estimates of return levels for a particular month and stratify the QSS along months, Fig. a. To understand the importance of the monthly subset scores, the reference weighting QSi,ref/QSref (Fig. b) and the contribution to the total skill score (Fig. c) are shown. The frequency weighting Ni/N is (almost) identical for all subsets (as the records generally contain complete years) and is not shown. Averaged over all 338 stations, the monthly QSS (Fig. a) is positive for all months, and non-exceedance probabilities for spring (March, April) and summer (July, August) stand out. For the contribution to the total QSS, the reference weighting (Fig. b) gives more importance to the summer months, leading to the strongest contribution to total QSS in July. The structure of the reference weighting term (Fig. b) indicates that the seasonal-only model does not represent the observations in summer as well as in winter. This probably indicates a stronger need for taking interannually varying return levels in summer into account. Adding the values of Fig. c by row leads to the values depicted in Table . The monthly QSS averaged over 338 stations leads to a consistently positive skill, but the performance varies strongly for the different stations.

We now stratify verification also along stations (K=338) and average over all time steps. The subset QSS for the non-exceedance probabilities of p=0.9,0.98,0.99,0.995 are plotted in Fig. , and the reference weighting is exemplarily illustrated for p=0.99 (since the patterns are similar for all non-exceedance probabilities) in Fig. . The frequency weighting and the contribution to the total QSS are not shown, since the first one does not exhibit any spatial pattern and the last one does not visually distinguish from the figure of the subset QSS. For most of the stations the seasonal–interannual model can represent the observations better than the seasonal-only model.

Only for a few records and higher non-exceedance probabilities/return periods do the variations with the years lead to more uncertain return levels, for example station Wesertal-Lippoldsberg. The monthly contribution to the QSS for this station is depicted in Fig. b. The negative skill mainly arises from overestimated return levels for the summer months, especially for the recent years (visually verified in Sect. ). This merely occurs for higher return periods due to the interannually varying shape parameter ξ (see Fig. ). However, a worse skill for stations with an interannual component in ξ cannot be detected in general (not shown). We discuss the change in the seasonal cycle of Wesertal-Lippoldsberg in more detail in Sect. .

Compared with the location heights of Fig. , the reference weighting for the station-wise analysis in Fig.  shows a clear relationship to station altitude and specifies that the seasonal-only model cannot reflect the data in mountainous regions as well as in lowlands. Analyzing the skill of the seasonal–interannual model with respect to the altitude does not show an improvement, especially for higher located stations (b, blue crosses). This might indicate that both model setups miss important mechanisms for extreme precipitation in mountainous regions (e.g., convection due to lifting or flow direction). Thus, these processes cannot be approximated by solely including temporal covariates but need to be modeled directly and/or via appropriate spatial covariates. This weak point cannot be seen for the example stations, since they are located in the lowlands (Krümmel: 64 m; Wesertal-Lippoldsberg: 176 m) or in the low mountain ranges (Rain am Lech: 409 m; Mühlhausen/Oberpfalz-Weihersdorf: 420 m).

Besides the monthly contribution to the station-wise skill for Wesertal-Lippoldsberg, Fig.  also shows the results for the other three example stations. Rain am Lech serves as an example with a very high skill mainly dominated by a better reflection of the data for the months of May and September. At Mühlhausen/Oberpfalz-Weihersdorf the return levels for spring can be estimated slightly better than for autumn, and Krümmel is dominated by a positive skill for June and July.

In summary, it can be noted that modeling interannual variations is beneficial for estimating return levels for all months, especially for the summer season. However, at a few stations the flexible modeling leads to a partly worse representation, in particular for larger return periods. Both seasonal modeling and seasonal–interannual modeling may have difficulties capturing mechanisms for precipitation formation in alpine regions.

Analyzing the selected models of the 519 considered stations shows that about 34 % (178/519 stations) prefer a model with interannual and/or seasonal variations in ξ. Figure  illustrates the spatial occurrence and the kind of variation (seasonal, interannual or interaction). It is noticeable that the density of stations with a flexible ξ is much higher in the north and east of Germany than in the south. The reason for a location-dependent variable shape parameter is an interesting question for further studies. We assume that different meteorological processes play a major role, e.g., the influence of weather types or the kind of precipitation (stratiform or convective). A dependence of flexibility in ξ on the record length is not obvious (not shown). Most of the stations (106/178, about 60 %) are represented by a model including seasonal variations, whereby many of them (92/106 stations) do not favor an interannually varying shape parameter at all. Only a few stations (17/178, about 10 %) prefer a model with direct interannual changes. Nevertheless, two regions with a slight agglomeration of direct interannual changes can be detected: in the middle of Bavaria (federal state in the southeast) and in the northeast around the Mecklenburger Seenplatte (also known as Mecklenburg Lake Plateau) represented by the example station Krümmel. About 39 % of the stations (69/178) show an interannually varying seasonal cycle (interactions); these stations are almost uniformly distributed across Germany, with a somewhat higher density around the example station Wesertal-Lippoldsberg.

We (a) quantify the gain from a flexible shape parameter with respect to a model with constant ξ and (b) analyze the contributions of the seasonal, interannual and interacting variations. To this end we use four model selection setups with a focus on ξ: setup (1) with constant ξ; setup (2) with seasonal components in ξ; setup (3) with seasonal and interannual components in ξ; and setup (4) with seasonal and interannual components, as well as their interactions in ξ. All other parameters are allowed to have seasonal, interannual and interacting components in all setups. Figure  illustrates the gain in performance for the different steps. The monthly skill for seasonal–interannual variations including interactions against a constant ξ averaged over the 178 stations expressed as the contribution to the total QSS is depicted in the top panel. There is positive skill for all months and return periods (with some exceptions with slightly negative values). The highest contribution to the total skill arises from the summer months, for which the reference weighting is increased (Sect. ).

To analyze the contribution of the seasonal component to the skill, we use setup 2 (seasonal-only in ξ) against a constant ξ (setup 1). Setup 2 results in a variable shape parameter for 106 of 178 stations; for the rest of the records a model with no variations in ξ was preferred. The skill of these 106 stations with respect to setup 1 is depicted in the left panel of Fig. . Seasonal flexibility in ξ improves in particular the return levels for summer; there is a very small gain in winter. For the transitional months March/April and September the increased flexibility led to a slightly worse model. A change in the shape parameter could indicate a change in the dominating precipitation type (convective in summer, stratiform in winter). The flexible modeling does not benefit months characterized by the transition of the precipitation regime, since no dominating precipitation type exists. A variable ξ for the season mainly improves the return levels of the higher return periods, while the skill for 2 and 5 years is slightly decreased.

Setup 3 evaluates the gain by adding an interannual component to ξ with respect to a seasonal-only model (setup 2). Only 17 stations preferred this type of model, but for those we found an improvement of the return level estimates for all months (bottom panel). The interannual variations do not improve the performance for the transitional months of March and September. Additionally, the lower return periods of 2 and 5 years do not benefit from this flexibility.

In setup 4 we allow additionally for interactions and compare the selected models with those chosen in setup 3 (seasonal–interannual model without interactions). The skill averaged over 69 stations with interaction terms for ξ is shown in the right plot of Fig. . The interactions improve the return levels for all months and return periods, especially for the summer months, and are able to faintly compensate for the lack of skill for the lower return periods and the transitional months. The skill shown in Fig.  is averaged over the respective stations; however, the performance for the individual stations can differ. For example, as already mentioned in Sect.  and analyzed in more detail in Sect. , the 100-year and 200-year return levels at the example station Wesertal-Lippoldsberg are overestimated (visually obtained by comparing return levels and observations) for the last decades, resulting in a worse representation of the most recent data. The overestimated return levels can be explained by considering the seasonal cycle of the shape parameter ξ for different years (1931, 1976, 2021) in Fig. . As depicted in Fig.  the shape parameter at this station can be expressed with ξ=ξ0+ξ1,1,cos+ (according to Eq. ), i.e., a linear rise in the amplitude. Thus, for the first observational year 1931 the amplitude is modeled to be zero and increases linearly with time, reaching its maximum for the last year in the record (2021). While for earlier years this linear change represents the extreme precipitation quite well, for the end of the record the values of ξ, especially for summer, become very large, which cannot be supported by the sparse database.

Additionally, Fig.  shows the seasonal cycle in ξ for the example station Krümmel, whose shape parameter is composed of ξ=ξ0+ξ1cos+ξ3P. Seasonality remains unchanged while the direct effect of the third Legendre polynomial leads to a shift of the cycle to smaller/larger values of ξ in 1976/2021 compared to 1931, leading to pronounced variability in the return levels of this station. The seasonal cycles for the example stations will be discussed in more detail in Sect. .

A negative shape parameter is unusual for describing the GEV distribution of extreme precipitation () since the resulting distribution is characterized by an unnatural upper bound. In our analysis the shape parameter is able to change with time such that negative values for ξ for a certain period are considered to be unproblematic.

In general, a varying shape parameter leads to a better representation of the data for all months and return periods, in particular for the very extreme events in summer; the flexibility leads to a worse skill of return level estimates only for very few stations.

In this section we aim to assess the impact of climate change on seasonal extreme precipitation (RQ3). With a simple linear model for each month and station, we quantify the interannual variation of return levels for a given non-exceedance probability. We compare the time period from 1941 to 2021 where all stations have data. Note that estimating linear trends for fixed (and short) periods of time can yield very different results, depending on the considered time period due to decadal variability. Thus, the trend estimates presented here for the given time period serve as a rough indicator for climate change effects; for a more detailed analysis, all the datasets should be taken into account for each station. Appendix  explains the calculation of the linear trend in more detail. Figure  illustrates the proportion of stations with a positive, negative or no trend for (a) the 2-year, (b) 10-year and (c) 100-year return levels. The trends are stated in relative changes from 1941 to 2021. Changes are mainly very weak (< 5 %); only 15 % to 35 % (depending on the month and occurrence probability) show more pronounced trends. In general, an increase in the return levels occurs more often than a decrease for all return periods, especially in June (more than 3 times more often). A decline slightly prevails only for the return levels in April. The patterns of the 5- to 200-year return levels are similar but with smaller trends for the shorter return periods. The characteristics of the 2-year return level differ: an increase is more often visible for the months of March and September to November with a less pronounced signal for the summer months. In contrast to that, the trends of the 100-year return level are stronger in summer.

About 35 % to 50 % of the considered 338 stations (121/338 2-year return level, 164/338 10-year return level, 170/338 100-year return level) show a change larger than 5% for at least 1 month of the year. The trends are regionally very different (maps for 2-year and 100-year return levels are given in Appendix ). Despite the partly very small-scaled characteristics, uniform behavior for several regions can be detected. Two of these regions with more pronounced changes are considered in more detail: one in southern Germany represented by the station Rain am Lech and the other one in the center of Germany exemplified by the station Wesertal-Lippoldsberg.

The 2-, 10-, and 100-year return levels for the station Rain am Lech are depicted in Fig. a. Besides the interannual changes (a.1), the seasonal cycle for the first/last record year (1899/2021) and the first year of the common time period (1941) is shown (a.2). The return levels of this region are characterized by an increase for all months and return periods. For instance, the largest 100-year return level in the year occurring in summer rose from 54.6 mm d-1 in 1899 to 86.0 mm d-1 in 2021, corresponding to an increase of about 58 %. Considering the 100-year return levels of the seasonal-only model demonstrates that a non-interannual approach leads to highly underestimated values, especially for the first record decades. The model selection reveals that not only the location parameter changes linearly with the years but also the scale parameter (Fig. ). The model verification (Fig. ) confirms that the trend in the return levels is necessary for an adequate description of the observations, especially for the transitional months of May, September and October. Thus, an increase in extreme precipitation amounts as expected from the anthropogenic climate change can be seen very clearly for this region.

The second region, which is considered in more detail, is characterized by a decrease in return levels in winter and an increase in summer, leading to a rise of the seasonal cycle's amplitude. Figure b shows the 2-, 10-, and 100-year return levels for the station Wesertal-Lippoldsberg. Since the interannual change for this station is best described by a model with a flexible shape parameter only (Fig. ), the 2-year return levels remain constant with the years. Towards higher return periods, changes are more prominent. They are pronounced for summer and winter, while the transitional months of March/April and September/October remain unaltered. The change in the seasonal cycle could be attributed to a combination of different processes. On the one hand, a higher water content of the air due to a temperature rise leads to an increased potential for extreme precipitation, particularly pronounced in summer. On the other hand, climate change can affect the characteristics of weather types and large-scale atmospheric circulations (e.g., NAO), which could result in a change in extreme precipitation as well in winter. The model verification (Fig. ) confirms that a model with a changing seasonal cycle better represents the data observed in summer for return periods of 10 to 50 years, while the 100- and 200-year return levels are strongly overestimated with respect to the observations, especially for the most recent decades. In contrast, the seasonal–interannual model is more beneficial for estimating winterly return levels with return periods longer than 30 years. These characteristics can be seen as well by comparing the 100-year return levels of the seasonal–interannual model with those of the seasonal-only model.

In addition to a change in the precipitation's magnitude, a phase shift can influence the risk of damage as well. Therefore, we also analyze the linear change in the phase expressed as the day in the year with the highest return level for the time period 1941–2021. Here, a simple linear model is adequate for the cyclic variable since a shift of the day with the highest precipitation from December to January or vise versa does not happen at all. The change of the phase in days for different return periods is illustrated in Fig. a. More than two-thirds of the stations show less pronounced changes (< 5 d). In general, a shift to earlier times in the year appears more frequently for almost all return periods except of the 2-year return level. For the latter, several different regions with strong shifts to later times and only one large contiguous area in the north with a shift to earlier times can be detected (Fig. b). For the 100-year return level (Fig. c), such distinct regions do not appear, but in general a shift to earlier times prevails for the whole country. The latter behavior can be appreciated in station Mühlhausen/Oberpfalz-Weihersdorf, whose 2-, 10- and 100-year return levels are depicted in Fig. c. The shift of the seasonal cycle to earlier times leads to increased return levels for the first half of the year, such that the 100-year return level in spring is about 13 mm d-1 higher in 2021 than it was in 1931. The decrease in autumn appears weaker, with a maximum change in the 100-year return level of about -8 mm d-1. During the 90 years of observations the annual maxima of the 100-year return level has shifted forward by 35 d. Since only a shift and not a rise of the seasonal cycle occurs, the analysis of annual maxima would not show any changes. In general, a shift of the seasonal cycle to earlier times leads to an increased risk potential. The probability of flooding events rises since snow melting and heavy precipitation coincide in spring. Additionally, higher crop losses may occur since plants are more vulnerable to extreme precipitation during early growing stages. Although differences between the 100-year return levels of the seasonal-only and the seasonal–interannual model are not very pronounced, the shift from late summer to early summer, which might be continued in the future, cannot be detected with the non-interannual approach.

The example station Krümmel (Fig. d) shows neither a linear change in the return levels nor a phase shift but points out other interesting features. It serves as a representative of the region Mecklenburger Seenplatte, since several neighboring records show similar characteristics. Here, pronounced climate variability can be detected in the seasonal cycle of extreme precipitation, which might be important for risk assessment and the design of hydraulic structures. Due to that climate variability, it can be shown that the commonly used stationary approach for a fixed historical time period can lead to erroneous return levels. For example, in Germany the stationary return levels based on the observations since 1951 have been used for infrastructure planning (). That means that for the example station Krümmel the heavy precipitation events from the 1930s have been discarded, potentially leading to underestimated return levels and to hydraulic systems with dimensions that are too small for the precipitation of recent years. A non-stationary approach including the whole dataset can improve the accuracy of the return levels. For this example, the seasonal-only approach applied to the whole record might be beneficial in terms of long-term risk assessment and hydraulic design since natural variability does not play a key role for longer planning horizons. However, for short- to mid-term risk assessment, e.g., for agriculture or tourism sector, the natural variability might be of relevance.

We sum up that monotonous trends are spatially different and mainly weak compared to return level uncertainties (not shown). Nevertheless, we detect regions with common and more pronounced changes. In general, the characteristics of the 2-year return levels differ from those of longer return periods.

We analyze seasonal–interannual variations of extreme precipitation at 519 stations (with at least 80 years of observations until 31 December 2021) in Germany using a non-stationary block maxima approach. The three parameters of the generalized extreme value (GEV) distribution are allowed to vary with the months (seasonal variation) and the years (interannual variation), whereby the seasonal variations are captured with a series of harmonic functions and the interannual variations with Legendre polynomials with a maximum power of 5. Interactions between seasonal terms (months) and interannual terms (years) allow the description of an interannually varying seasonal cycle. Since we consider higher polynomial orders than linear trends, the models are able to reflect other than linear trends, e.g., more complex climate variability. A stepwise model selection based on the Bayesian information criterion (BIC) identifies a suitable model for each station separately, which is used to calculate seasonally–interannually changing return levels for different return periods (non-exceedance probabilities). To validate the models, we use a leave-one-year-out cross-validated quantile score to measure the model performance for individual quantiles (return-levels). The quantile skill score (QSS) and its decomposition for stratified verification provide additional information about the skill of the model with respect to only a seasonally varying non-stationary GEV. We addressed three research questions: