Annual maxima of daily precipitation sums can be typically described well with a stationary generalized extreme value (GEV) distribution. In many regions of the world, such a description does also work well for monthly maxima for a given month of the year. However, the description of seasonal and interannual variations requires the use of non-stationary models. Therefore, in this paper we propose a non-stationary modeling strategy applied to long time series from rain gauges in Germany. Seasonal variations in the GEV parameters are modeled with a series of harmonic functions and interannual variations with higher-order orthogonal polynomials. By including interactions between the terms, we allow for the seasonal cycle to change with time. Frequently, the shape parameter

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

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

Analyses of extreme precipitation in Germany for different seasons have already been done(

Extreme value statistics (EVS)

Interannual variations in precipitation have been shown to be associated with its natural variability

The goal of this paper is to assess the performance of the seasonal–interannual modeling with a special attention to a flexible shape parameter

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

How important is a flexible shape parameter to reflect recorded variations?

How does climate change affect the seasonal cycle of extreme precipitation in Germany?

We carry out this investigation for observations from Germany with more than 500 long (

A dataset of almost 5700 rain gauges measuring daily precipitation amounts

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 519 long stations covering at least the years from 1941 to 2021. Station altitude [m] is encode with colors. Additionally, the locations of stations Krümmel (orange rectangle), Mühlhausen/Oberpfalz-Weihersdorf (green rhombus), Rain am Lech (violet triangle pointing up) and Wesertal-Lippoldsberg (blue triangle pointing down) are depicted.

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,

For a sequence of independent and identically distributed (iid) random variables

The choice of the appropriate block size is dependent on the nature of the considered random variable

In the frame idea of vector generalized linear models

To account for the periodic nature of the seasonal cycle, the dependence of GEV parameters on the months can be described with a series of harmonic oscillations with amplitude

To describe the oscillation Eq. (

The harmonic series for location,

To capture interannual variations, polynomials typically provide a good approximation. With orthogonal polynomials such as Legendre polynomials, we avoid dependence between terms which have proven useful for modeling spatial variations

The Legendre polynomials for the orders 1 to 5.

We focus on the interannual changes in the seasonal cycle, which can be incorporated into the statistical model using interactions between the seasonal and interannual terms in the predictor of the vector generalized linear model. It can be thought of as the amplitude and phase of the seasonal cycle changing in time.

Using Eq. (

Standard interaction terms (left-hand side) between the first-order sine and the Legendre polynomials of order 1 (top row, orange), order 2 (middle row, green) and order 3 (bottom row, red). A negative and a positive scaling of the Legendre polynomials lead to the desired interaction terms (right-hand side).

To avoid the bipartite behavior we use two transformations of the Legendre polynomials:

The transformed Legendre polynomials are illustrated in Fig.

Positively (solid) and negatively (dashed) transformed Legendre polynomials of order 1 to 5.

Thus, the interactions with the harmonic functions for the location parameter

These terms show the desired behavior as depicted exemplarily in Fig.

Combining the seasonal and interannual variations with these interactions leads to a flexible model for the location parameter:

Using a VGLM, we allow the scale

The

After introducing the model setup in the previous section, the maximum orders for harmonic functions and Legendre polynomials have to be selected. Here, we set maximum orders

Now, the procedure is repeated for interaction terms starting with the final model from part one. To answer RQ1 (Sect.

Stepwise model selection for example stations. BIC against stepwise-selected covariates for location (yellow), scale (pink) and shape (blue) for each iteration. All covariates listed are included in the final model.

Figure

The model selection procedure was applied for all 519 stations individually. About 65 % of the stations (

Properties of the interannual variability components of those 338 models including at least one.

It can be seen that the selected interannual covariates are partly very variable in space. This can be explained by (1) a large spatial variability in extreme precipitation due to partly small-scaled events and (2) the model selection procedure, which chooses one suitable model, even if other models are comparably appropriate. However, common characteristics can be detected. The GEV's location and scale parameter are mainly affected, and interannual changes in the seasonal cycle (interactions) dominate. Nevertheless, changes in the shape parameter and changes without affecting the seasonal behavior occur, often for several stations of the same region, indicating common local characteristics. The stations with direct effects are mainly characterized by a linear interannual change in the location parameter. For the interactions, the preferred Legendre polynomial is not so obvious.

To answer RQ1 and RQ2 the performance of a model with respect to a reference has to be analyzed. We use the quantile skill score (QSS)

The quantile skill score (QSS) is defined as

For stratifying verification along months or stations we use the decomposition of the QSS

We address RQ1: can a model with interannual variations better represent the observations than a seasonal-only model? As mentioned in Sect.

Total QSS for different non-exceedance probabilities

We analyze whether interannual variations improve the estimates of return levels for a particular month and stratify the QSS along months, Fig.

Subset QSS

We now stratify verification also along stations (

Subset QSS for 338 different stations for the non-exceedance probability/return period of

Map of reference weighting for non-exceedance probability 0.99/return period 100 years

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.

Monthly contribution (

Compared with the location heights of Fig.

Besides the monthly contribution to the station-wise skill for Wesertal-Lippoldsberg, Fig.

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 % (

Spatial distribution of stations with flexible shape parameter

We (a) quantify the gain from a flexible shape parameter with respect to a model with constant

Scheme for analyzing the importance of and performance gain by a flexible shape parameter as contribution to the total QSS. Illustrations (axes, colors, signs) equal to Fig.

To analyze the contribution of the seasonal component to the skill, we use setup 2 (seasonal-only in

Setup 3 evaluates the gain by adding an interannual component to

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

Seasonal cycle of the shape parameter

Additionally, Fig.

A negative shape parameter is unusual for describing the GEV distribution of extreme precipitation (

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

Proportion of stations with a positive (light/dark orange), negative (light/dark blue) or neutral (white) relative change from 1941 to 2021 for the

About 35 % to 50 % of the considered 338 stations (

Observations and return levels for the stations Rain am Lech (1 January 1899 until 31 December 2021)

The 2-, 10-, and 100-year return levels for the station Rain am Lech are depicted in Fig.

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

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.

Phase shift in days from 1941 to 2021 for the non-exceedance probability/return period of

The example station Krümmel (Fig.

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:

For

As the shape parameter

To quantify the consequences of climate change for the seasonal cycle, we obtain linear trends of the interannually varying return levels and the phase of the seasonal cycle (day in year with the highest return level) for the common analysis period from 1941 to 2021. A unambiguous signal in these trends which could be related to climate change cannot be found since only about one-fifth to one-third of the 519 considered stations (2-year return level: 23 %; 10-year return level: 32 %; 100-year return level: 33 %) show a stronger linear change (

Since extreme precipitation is highly variable in time and space and long datasets are rare, coherent outcomes of different research studies are crucial for a suitable risk assessment and risk adaptation.

The pronounced climate variability in extreme precipitation which can be detected at the example station Krümmel partly fits the results of

An understanding of physical mechanisms leading to the observed results was not in the focus of this study but needs to follow. We imagine a combination of increased convection due to higher surface temperatures and moisture

Seasonal and interannual variation in extreme precipitation can be described with a combination of harmonic functions and orthogonal polynomials like the Legendre polynomials. For this investigation but also for previous studies, the latter has proven to be helpful to approximate highly non-linear variations. However, their nature of having the highest/lowest values at the borders of the time period potentially leads to very high or low return levels for the beginning and the end of the time series. This could mislead the analysis of trends. A possible strategy to prevent the boundary problem is to select a slightly larger scaling area than the period observed for obtaining the Legendre polynomials.

A possible application of the presented seasonal–interannual approach in the field of risk adaptation could be realized by calculating design-life levels. This concept has been introduced by

Extreme precipitation is influenced by many different effects (e.g., location, air temperature, large-scale atmospheric circulation, lifting effects), and most of them are highly non-linear and difficult to quantify in terms of their role. In this study, we utilize the time as a covariate since it can be seen as a proxy combining those different unknown effects. Based on our results, the consequences of climate change could be assessed in more detail by using surface temperature, greenhouse gas emissions or indices of large-scale atmospheric circulation patterns as terms in the predictor. This offers also an opportunity to evaluate the climate variability of extreme precipitation and the processes associated with it.

As discussed above, the interannual variability of one example station visually matches the results of

Additionally, trends might differ for different durations of the precipitation events, changes for, e.g., hourly or sub-hourly extreme precipitation are worthwhile to consider apart from daily precipitation sums. Typically, observation records of higher resolved extreme precipitation are shorter, and hence analysis of interannual variability is more uncertain. One possibility to improve accuracy is to use a smooth relationship between different durations directly in the formulation of the GEV

Furthermore, a different approach for modeling the interannual variations could be considered to overcome the boundary problem of the Legendre polynomials; it might be worthwhile to consider different orthogonal polynomials, e.g., the first kind of the Chebyshev polynomials, or to use a vector generalized additive model (VGAM,

In our investigation we consider return level estimates. However, analyzing their uncertainties is crucial. For further investigations, confidence intervals, e.g., calculated with the delta method

We introduce a seasonal–interannual modeling approach to assess variations of extreme precipitation, leading to more accurate return levels. The interactive consideration enables a modeling of a changing seasonal cycle in the form of a changing amplitude and/or phase. The approach is able to reflect long-term changes and climate variability. In addition, we show that a flexible shape parameter of the GEV is beneficial. Finally, we use the approach to detect regions in Germany for which extreme precipitation is likely to be affected by climate change. In general, changes are weak; however, an increase is prevalent compared to a decrease. The lower extreme precipitation rises generally in spring and autumn, and its seasonal cycle is shifted to later times in the year; heavy precipitation increases mainly in summer and occurs earlier in the year.

The linear trend in return levels and phase of seasonal cycle is calculated for each station, month and occurrence probability separately using a simple linear model. The relative change from the first to the last year included in the linear model is obtained with

The 100-year return level in millimeters per day (mm d

Figures

Relative change from 1941 to 2021 for the 2-year return level (non-exceedance probability of

Relative change from 1941 to 2021 for the 100-year return level (non-exceedance probability of

According to

The design-life level

Estimated parameter for location

The analysis was carried out using R, an environment for statistical computing and graphics

Daily precipitation sums in Germany are provided by the National Climate Data Center of the German Weather Service (DWD) and are publicly accessible under

MP, HWR and UU designed the study concepts and methodology. MP conducted the analysis, generated the results and wrote the first draft. All authors contributed to writing the manuscript and approved the final version.

At least one of the (co-)authors is a member of the editorial board of

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

We thank Oscar Jurado de Larios and Felix Fauer for proofreading and Theano Iliopoulou and the anonymous referee for their careful reading of our manuscript and their constructive comments.

This research has been supported by the Deutsche Forschungsgemeinschaft (grant no. GRK 2043/1) within the research training group NatRiskChange at Potsdam University.The article processing charges for this open-access publication were covered by the Freie Universität Berlin.

This paper was edited by Piero Lionello and reviewed by Theano Iliopoulou and one anonymous referee.