# Joint estimation of trend in bulk and extreme daily precipitation in Switzerland

Abubakar Haruna, Juliette Blanchet, Anne-Catherine Favre

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Abubakar Haruna, Juliette Blanchet, Anne-Catherine Favre. Joint estimation of trend in bulk and extreme daily precipitation in Switzerland. Weather and Climate Extremes, 2025, 48, pp.100769. 10.1016/j.wace.2025.100769. hal-05110200

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# Joint estimation of trend in bulk and extreme daily precipitation in Switzerland

Abubakar Haruna ID*, Juliette Blanchet, Anne-Catherine Favre

Univ. Grenoble Alpes, CNRS, INRAE, IRD, Grenoble INP <fnref n="1" />, IGE, 38000 Grenoble, France

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

**Keywords:**

Switzerland

Climate change

EGPD

Precipitation

Nonstationarity

## ABSTRACT

Precipitation is crucial for water supply and energy generation in the Alps. However, heavy precipitation can also lead to natural disasters. It is therefore essential to understand the changes in both mean and extreme precipitation in order to develop effective adaptation and mitigation strategies. This study jointly models the observed long-term trends in both the bulk and extremes of daily precipitation distribution in Switzerland by employing a non-stationary version of the Extended Generalized Pareto distribution (EGPD). The EGPD allows us to model the entire non-zero precipitation range while remaining consistent with extreme value theory in its lower and upper tails.. We incorporated the non-stationarity by allowing the parameters of the distribution to vary with two covariates, time and sea surface temperature, and used a bootstrap approach for uncertainty assessment and to assess the significance of the modeled trends. The results indicate that extreme precipitation has increased in all seasons, while mean precipitation has only significantly increased in winter in northern Switzerland. This increase in winter precipitation is attributed to both a positive trend in the frequency and in the intensity of wet days precipitation.

## 1. Introduction

The impact of climate change on precipitation patterns is a critical issue with far-reaching implications for water resource management, urban planning, and hydropower generation. Indeed, several studies have shown that human activities have affected the global water cycle (e.g. Slater et al., 2021), thereby leading to large-scale changes in mean and extreme precipitation (IPCC, 2023). In Switzerland, where climate change manifests at rates exceeding the global average (MeteoSwiss, 2024), understanding these trends is crucial for informed decision-making and long-term sustainability.

Accurately quantifying changes in mean and extreme precipitation, however, remains a challenging task due to the inherent variability and scarcity of extreme events. Yet, it is crucial, as increased frequency and intensity of extreme precipitation can result in severe flooding, landslides, and erosion, causing substantial damage to infrastructure, properties, and natural landscapes. Meanwhile, changes in mean precipitation can have significant effects on a wide range of environmental and societal systems. For instance, an increase in mean precipitation can lead to enhanced water availability, benefiting agriculture and water supply but potentially increasing the risk of flooding. Conversely, a decrease in mean precipitation can exacerbate drought conditions, stress water resources, and negatively impact agricultural productivity

and ecosystem health. Therefore, understanding and predicting changes in both mean and extreme precipitation is crucial for safety, managing water resources, ensuring food security, and protecting natural ecosystems.

Traditionally, analyses on changes in mean and extreme precipitation have been conducted separately (e.g. Moberg and Jones, 2005; Scherrer et al., 2016a; Ménégoz et al., 2020). For mean precipitation, evidence of trend is investigated using ordinary linear regression (e.g. Schmidli and Frei, 2005; Moberg and Jones, 2005) or non-parametric approaches such as Mann-Kendall test (Mann, 1945; Kendall, 1975). Subsequently, the trend is modeled using a parametric distribution like the gamma distribution which is fitted to the series, with changes assessed based on the parameters of the fitted distribution (e.g. Ben-Gai et al., 1998; Villarini et al., 2010). For extreme precipitation, analyses often focus on heavy precipitation events defined as exceedances of a high threshold (e.g. Frei and Schär, 2001; Schmidli and Frei, 2005; Moberg and Jones, 2005; Olmo et al., 2020) or maxima of a chosen block (e.g. Ménégoz et al., 2020; Bauer and Scherrer, 2024). The trend is then modeled using robust approaches based on theoretically justified Extreme Value Theory (EVT) distributions fitted to the series. EVT approaches typically rely on modeling block maxima series using the generalized extreme value distribution (GEV) or exceedances above

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: abubakar.haruna@univ-grenoble-alpes.fr (A. Haruna).</footnote>

<footnote n="1">¹ Institute of Engineering and Management Univ. Grenoble Alpes.</footnote>
high thresholds using the generalized Pareto distribution (GPD). Non-stationarity is incorporated by allowing the distribution parameters to vary with time or other climate covariates. Examples of applications of the non-stationary GEV can be found for example in Rajczak et al. (2013), Blanchet et al. (2021), Chagnaud et al. (2021), Jayaweera et al. (2024) and those of non-stationary GPD for example in Beguería et al. (2011), Tramblay et al. (2011), Lee et al. (2020).

While EVT provides a robust framework for assessing the tails of the distribution, it has significant limitations. EVT-based methods use only a small fraction of the available data, ignoring the bulk of the precipitation distribution. This limitation hinders a comprehensive understanding of precipitation behavior, which is necessary for practical applications in water resource management. Additionally, the separate modeling of trends in mean and extreme precipitation poses challenges in cases where information on the entire distribution is required, such as in weather generators that need a marginal distribution for the whole precipitation simulation. Incorporating potential non-stationarity in these cases requires jointly modeling the trend in the entire distribution.

Recent studies have attempted to bridge this gap by modeling non-stationarity in the whole precipitation distribution. For example, Benet-et al. (2019) used a time-varying exponential distribution for the non-zero precipitation to study the changes in the mean and heavy precipitation. A major drawback is that since precipitation can be heavy-tailed, the exponential distribution will underestimate the tail of the distribution. More recently, André et al. (2024) used a Weibull distribution to study changes in the whole precipitation distribution between two fixed periods. A potential limitation of the study is that due to the lack of continuity, the overall conclusions may be sensitive to the windows considered, in addition to the stationarity assumption within the periods considered. Krock et al. (2022) used the bulks and tails (BATs) model of Stein (2021) to jointly model the trends in bulk and extremes of daily mean temperature using CO₂ as a covariate. However, even in a stationary context, BATs model has six free parameters, which is already high, and allowing for nonstationarity will inflate the number of parameters of the model.

In this study, we apply the three-parameter model, based on power law, from the families of Extended Generalized Pareto Distribution (EGPD) (Papastathopoulos and Tawn, 2013; Naveau et al., 2016) to jointly model the trends in bulk and extremes of daily precipitation in Switzerland, using either time (year) or sea surface temperature as covariates. Several studies in this region have identified significant positive trends in annual and winter precipitation, along with an increase in the intensity and frequency of heavy precipitation, especially in the northern regions of Switzerland (Widmann and Schär, 1997; Schmidli et al., 2002; Schmidli and Frei, 2005; Begert et al., 2005; Masson and Frei, 2016; Scherrer et al., 2016a,b; Bauer and Scherrer, 2024). Moreover, future projections suggest that these trends will continue, with more intense winter and summer precipitation due to climate change, see the technical report on climate change scenarios for Switzerland (CH2018, 2018) and the work by Vergara-Temprado et al. (2021). Similar patterns were observed/expected across the European Alps, with increased precipitation intensity, particularly in the autumn (Gobiet et al., 2014; Ménégoz et al., 2020; Zeder and Fischer, 2020; Zittis et al., 2021). However, all the aforementioned studies have examined changes in mean and extreme separately, rather than jointly.

The EGD models, proposed by Papastathopoulos and Tawn (2013) and further developed by Naveau et al. (2016), extends the classical GPD by modeling the whole distribution of non-zero precipitation while remaining consistent with EVT in the lower and upper tails. This makes them versatile tools for modeling moderate and extreme precipitation changes. In particular, the three-parameter model has been widely applied in stationary frameworks (e.g. Evin et al., 2018; Le Gall et al., 2022; Gamet and Jalbert, 2022; Milojevic et al., 2023; Rivoire et al., 2022; Beneyto et al., 2024; Haruna, 2024), and has been previously applied in regression settings for space and time modeling

of precipitation (e.g. Haruna et al., 2022; Carrer and Gaetan, 2022; de Carvalho et al., 2022), and wildfire (e.g. Cisneros et al., 2024) using deep graphical regression.

In contrast with de Carvalho et al. (2022) and Carrer and Gaetan (2022) that focused on the distribution of wet days precipitation, we model the entire distribution of all days precipitation (both dry and wet days) using a mixed distribution of a discrete component for the dry days and a continuous component for the wet days. This enables us to model changes not only in the wet day distribution as done in the aforementioned studies but in the whole distribution as well as in the return levels.

The remainder of this article is organized as follows: Section 2 presents the data and the study area. The methodology, model inference, and selection, as well as uncertainty analysis, are presented in Section 3. The results are presented and discussed in Section 4, and finally, the conclusions and some relevant perspectives are given in Section 5.

## 2. Data

We use daily precipitation data from the Swiss Federal Office for Meteorology and Climatology (MeteoSwiss) shown in Fig. 1. The measuring instruments have evolved with time, from completely manual rain gauges (400 cm² ombrometers to 200 cm² Hellmann raingauge) recorded at 06:00 UTC, to fully automatic gauges (tipping-buckets and weighing gauges) with a sampling resolution of 10 min (Fukutome et al., 2018). The data are not homogenized (Scherrer et al., 2016b; Bauer and Scherrer, 2024), but subjected to substantial and very rigorous quality control checks by MeteoSwiss to ensure the reliable quality of the data (MeteoSwiss, 2017; Fukutome et al., 2018). These include checks to ensure that recorded values are within climatologically physical limits, consistent with those from nearby gauges, and satisfy inter-parameter consistency and variability tests. We consider only stations with at least 100 years of records from 1901 to 2022. Such a long record is required for a robust trend analysis with minimal influence of interannual and decadal variability. While measurements in Switzerland are available as far back as 1864, high-quality data is available at a large number of stations only from 1901 (Scherrer et al., 2016a). This leads to 181 stations shown in Fig. 1 with record lengths ranging from 100 to 122 years. For our subsequent analysis, we use a wet days threshold of 1 mm, thereby considering all days with precipitation amounts less than this threshold as dry days. This threshold of 1 mm is commonly used in the literature (e.g. Rivoire et al., 2021) to reduce the uncertainty in the recording procedure. Additionally, we perform a seasonally-based analysis by stratifying the data into four seasons of three months each. Winter includes Dec–Jan–Feb, spring Mar–Apr–May, summer Jun–Jul–Aug, while autumn includes Sep–Oct–Nov. This approach is applied here due to the seasonality of precipitation in the study area as done in previous studies (see e.g. Scherrer et al., 2016b).

The sea surface temperature (SST) we use as a covariate is shown in Fig. 2. We sourced the data from NOAA-NCDC Extended Reconstructed Sea Surface Temperature Dataset (ERSST), Version 5. We use seasonal values of the SST anomaly spatially averaged over the Mediterranean, covering longitude: 6° W to 40° E, and Latitude: 30° to 46° N. Since we focus on long-term trends, we smoothened the SST series before using it as a covariate using the nonparametric locally weighted scatter plot smoothing (lowess). The process was achieved using the lowess function in R.

The study area, Switzerland (Fig. 1), covers an area of 41,285 km² with diverse topography. Approximately 30% of the land is located above 1,500 m. The country's climate is influenced by the Alps, Jura, the Atlantic Ocean, and the Mediterranean Sea, resulting in significant seasonal and regional variations in precipitation intensity and occurrence (see e.g. Sodemann and Zubler, 2009; Giannakaki and Martius, 2015; Scherrer et al., 2016a). Switzerland can be divided into four main
<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Map of Switzerland, the study area. The background color denotes the topography in meters above sea level, and the red-colored circles indicate the location of the 181 daily stations. The main regions are also indicated on the map. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Seasonal sea surface temperature anomaly averaged over the Mediterranean sea. The lines show the smoothed curves obtained using the nonparametric locally weighted scatter plot smoothing (lowess).</figcaption></figure>

climatic zones (see e.g. Frei and Schär, 1998; Molnar and Burlando, 2008). The northeast plateau has little variability, while the northwest, influenced by the Jura mountain, experiences higher accumulations in comparison. The central part which houses the Alps is mainly affected by orography and shadowing of the inner valleys. Ticino (south of the Alps), heavily influenced by the Mediterranean Sea, experiences the highest accumulations and the most intense precipitation. Areas such as

the Alps, the Jura region (northwest), and the Ticino region receive the highest annual precipitation totals, exceeding 2,000 mm. On the other hand, shielded valleys such as the Rhône and Inn experience minimal annual precipitation, between 400 and 800 mm. Summer is the primary precipitation season throughout Switzerland, except for Ticino where the main season is autumn. Conversely, winter records the lowest precipitation across all regions. Ticino experiences the largest intensities
of heavy precipitation, with maximum 1-day totals surpassing 130 mm in autumn. In other regions, heavy precipitation mainly occurs during summer.

# 3. Methodology

## 3.1. Marginal distribution for non-zero precipitation intensities

Let us define $R_t$ as the daily precipitation recorded in a given year $t$ and a given season. Under the stationary assumption, the distribution of $R_t$ does not change with time. In that case, the probability that a given precipitation amount $r$ will not be exceeded is defined mathematically as

$$ \mathbb{P}(R_t \le r) = p_d + (1 - p_d) \times \mathbb{P}(R_t \le r | R_t > 0), \quad (1) $$

where $p_d = \mathbb{P}(R_t = 0)$ is the dry day probability, and $\mathbb{P}(R_t \le r | R_t > 0)$ is the wet days precipitation distribution. In this study, we assume EGPD as the distribution for the wet days precipitation. Eq. (1) can therefore be written as

$$ \mathbb{P}(R_t \le r) = p_d + (1 - p_d) \times G(r), \quad (2) $$

where $G$ denotes the cumulative distribution function (CDF) of the EGPD given by Naveau et al. (2016) as

$$ \mathbb{P}(R_t \le r | R_t > 0) = \left[ H_{\xi} \left( \frac{r}{\sigma} \right) \right]^k. \quad (3) $$

$H_{\xi} \left( \frac{r}{\sigma} \right)$ is the CDF of GPD defined as

$$ H_{\xi} \left( \frac{r}{\sigma} \right) = \begin{cases} 1 - (1 + \xi \frac{r}{\sigma})^{-1/\xi} & \text{if } \xi \neq 0 \\ 1 - \exp(-\frac{r}{\sigma}) & \text{if } \xi = 0 \end{cases}, \quad (4) $$

where $a_+ = \max(a, 0)$, $\sigma > 0$ is the scale parameter, and $\xi \ge 0$ is the shape parameter that controls the upper tail of the distribution. The flexibility parameter, $\kappa > 0$, controls the lower tail. With the addition of only one parameter ($\kappa$) to the CDF of GPD, the distribution can model the full range of non-zero precipitation (see applications in Evin et al., 2018; Le Gall et al., 2022; Haruna et al., 2024).

Given a time series of precipitation, Eq. (2) can be applied by estimating a total of four parameters: $p_d$ for the dry day frequency and $\kappa$, $\sigma$, $\xi$ for the wet days precipitation distribution. Under stationary assumptions, Eq. (2) is assumed to be invariant with time, and $p_d$ is estimated from the proportion of dry days, while the three EGPD parameters are estimated by fitting the model to the non-zero precipitation amounts using the method of maximum likelihood for instance. However, when the stationary assumption is doubted, that is the precipitation distribution has changed over time, the classical approach is to let the parameters of the model depend on time or some time-dependent covariate. In our case, to account for this potential non-stationarity, we propose to model the parameters of the model in Eq. (2) as a function of two covariates; the year $t$ and the sea surface temperature SST. This means that the distribution of $R_t$ changes with time and Eq. (2) is now expressed as

$$ \mathbb{P}(R_t \le r) = p_{d_t} + (1 - p_{d_t}) \times G_t(r), \quad (5) $$

with

$$ G_t(r) = \begin{cases} \left[ 1 - (1 + \xi \frac{r}{\sigma_t})^{-1/\xi} \right]^{\kappa_t} & \text{if } \xi \neq 0 \\ \left[ 1 - \exp(-\frac{r}{\sigma_t}) \right]^{\kappa_t} & \text{if } \xi = 0. \end{cases}, \quad (6) $$

To account for the possible trend, we model the dependence of the parameters with temporal covariate $c_t$ using a linear model

$$ \left\{ \begin{array}{l} \log(\kappa_{c_t}) = \kappa_0 + \kappa_1 c_t, \\ \log(\sigma_{c_t}) = \sigma_0 + \sigma_1 c_t, \\ \logit(p_{d_{c_t}}) = p_{d_0} + p_{d_1} c_t. \end{array} \right. \quad (7) $$

Note that the covariate $c_t$ in Eq. (7) can either be the year $t$ or SST. More so, since there is a high correlation between the two covariates (minimum of 0.81 in autumn, and up to 0.93 in spring), we do not combine them in a single model but rather consider one at a time. More so, we use the same covariate for all the model parameters. We also use a log link for $\sigma$ and $\kappa$ to ensure that they remain positive, and the logit link for $p_d$ defined as $\logit(p_d) = \frac{p_d}{1-p_d}$ to ensure that the probability remains between zero and one. Due to the difficulty in estimating $\xi$ even in a stationary context, we keep it stationary (e.g. Blanchet et al., 2018) which signifies that the non-stationarity in the wet days precipitation distribution is driven by $\sigma$ and/or $\kappa$.

We consider a stationary case and five cases of non-stationarity for each covariate, depending on the stationary assumption in $p_d$, $\kappa$, and $\sigma$. The cases are described below:

1. $M_0$: This is the stationary model where all the model parameters are stationary.

2. $M_{p_d}$: This is the case where only the dry-day frequency is non-stationary, while the wet-day distribution is stationary.

3. $M_\sigma$: This model assumes a trend model in only $\sigma$, whereas both $\kappa$ and $\xi$ are assumed constant. Since the distribution is driven by the trend in only $\sigma$, the trend is uniform in magnitude and direction for the whole wet days precipitation distribution (low, medium, and extreme intensities). See Fig. 3a.

4. $M_{\sigma,\kappa}$. This model assumes a trend model in both $\sigma$ and $\kappa$, while $\xi$ remains stationary. This model includes added flexibility compared to $M_\sigma$, and different trend magnitudes can be observed in the bulk and extreme precipitation. Additionally, depending on the trend sign in the two parameters, opposite trends could be observed in the bulk and extreme precipitation. For instance, a positive trend in high quantiles, and a negative trend in the bulk (medium) intensities. See Fig. 3b where the bulk decreases and the extremes increase.

5. $M_{p_d,\sigma}$: This is the case where both the dry-day frequency and the wet-day distribution are non-stationary. However, the change in the wet day distribution is accounted for by a trend in only $\sigma$.

6. $M_{p_d,\kappa}$: Here also, both the dry-day frequency and the wet-day distribution are non-stationary. However, the change in the wet day distribution is accounted for by both a trend in $\sigma$ and $\kappa$.

Finally, since $\kappa$ controls mainly the lower tail, we do not consider the case with a trend in only this parameter. Accounting for the two different covariates, this gives a total of 10 non-stationary models to be compared.

Another way to express the non-stationarity is in the form of trends in the mean precipitation or a given precipitation quantile, usually extreme quantiles that are associated with a given return period $T$ expressed in years. A $T$-year return level is expected to be exceeded once every $T$-years. Typical return periods in years could be $T \in \{2, 10, 20, 100\}$.

Starting with the mean of all days precipitation $m$, it can be shown that

$$ m_t = \mathbb{E}[R_t] = p_{w_t} \times \mu_t, $$

where $p_{w_t} = (1 - p_{d_t})$ is the wet days frequency and $\mu_t$ is the mean of the wet days precipitation. $\mu_t$ can be computed from the expectation of $G_t$, and since $G_t$ is the EGPD, the expectation is given by Naveau et al. (2016) as

$$ \mu_t = \mathbb{E}[R_t | R_t > 0] = \frac{\sigma_t}{\xi} [\kappa_t B(\kappa_t, 1 - \xi) - 1], \quad (8) $$

where $B(\cdot)$ denotes the beta function.

The non-stationary $T$-year return level for year $t$ denoted as $r_{T,t}$ is given as

$$ r_{T,t} = \frac{\sigma_t}{\xi} \left\{ \left[ 1 - \left( 1 - \frac{1}{T p_{w_t} \delta} \right)^{\frac{1}{\kappa_t}} \right]^{-\frac{1}{\xi}} - 1 \right\}, \quad (9) $$
<figure><img src="image_6.png" /><figcaption>Fig. 3. Illustration of non-stationary EGPD densities with year as covariate, between year $t = 1901$ and $t = 2000$. Green is the beginning of the period ($t = 1901$), red is the middle ($t = 1950$), and blue is the end ($t = 2000$). The circles indicate the mean, while the diamonds indicate the 50-year return level. (a) Example of $M_σ$ where $σ_t$ increases with time, and (b) $M_{σ,x}$ where $σ_t$ increases and $κ_t$ decreases with time. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

where $δ \in \{90, 92, 92, 91\}$ is the number of days in a given season, respectively winter, spring, summer, and autumn.

Finally, we express the percent relative trend per 10 years as

$$ \frac{10}{n} \times \frac{v_n - v_1}{\bar{v}} \times 100, \qquad (10) $$

where $n$ denotes the number of year, $v$ is the statistics of interest, for example, the mean or a given return level, and $\bar{v}$ is the average value for the years $t = 1$ to $t = n$.

## 3.2. Inference

For a given stationary model, the log-likelihood function for the mixed model in Eq. (2) with $s$ the number of dry days, and $w$ the number of wet days (total days = $s+w$), is given by Kedem et al. (1990) as

$$ \ell(p_d, \theta | r) = s \log(p_d) + w \log(1 - p_d) + \sum_{j:r_j>0} \log[g(r_j, \theta)], \qquad (11) $$

with $g$ being the density function of the EGPD (i.e; the derivative of Eq. (3) with respect to $r$), and $\theta = \{\kappa, \sigma, \xi\}$.

For the non-stationary case, let us call $\theta_*$ the vector of the non-stationary parameters of the EGPD, and $p_{d_t}$ from the logit transformation in Eq. (7), then Eq. (11) becomes

$$ \ell(p_{d_t}, \theta_* | r) = \sum_{t=1}^{T} \left\{ s_t \log(p_{d_t}) + w_t \log(1 - p_{d_t}) + \sum_{j:r_{j,t}>0} \log[g(r_{j,t}, \theta_*)] \right\} \qquad (12) $$

As proposed by Naveau et al. (2016) we modify Eq. (12) (the third term in particular) to account for the left-censoring in the wet day distribution.

$$ \ell(p_{d_t}, \theta_* | r) = \sum_{t=1}^{T} \left\{ s_t \log(p_{d_t}) + w_t \log(1 - p_{d_t}) + \sum_{j:r_{j,t}<c} G(c; \theta_*) + \sum_{j:r_{j,t}\ge c} g(r_{j,t}; \theta_*) \right\} \qquad (13) $$

where $G$ and $g$ are respectively the CDF and density function of the EGPD and $c \ge 0$ is the left censoring threshold applied to the non-zero precipitation data (see below).

Therefore the parameters of a given model are estimated by maximizing the log-likelihood in Eq. (13), which accounts for the left-censoring of the EGPD model. This is achieved by using the optim function in R using the Nelder-Mead optimization algorithm. For the rest of the article, we will call $\hat{\psi} = \{\hat{p}_{d_0}, \hat{p}_{d_1}, \hat{\theta}_*\}$ as the fitted parameters for the dry and wet day distribution.

The left censoring in the EGPD model is applied to reduce the influence of the small intensities on the MLE during inference, which could influence the parameter estimation, thereby resulting in a large overestimation of the upper tail shape parameter ($\xi$) (Naveau et al., 2016). This is likely due to the insufficient flexibility of the three-parameter EGPD model to adequately model the left tail of the distribution or the associated uncertainty in the instrumental recording of very small intensities. A usual censoring approach is to apply a uniform censoring threshold (e.g. $c = 2$ mm for all the daily data), but as highlighted by Haruna et al. (2023), this is not usually sufficient and so an appropriate $c$ has to be obtained for each station. We follow their footstep and estimate the value of $c$ for the stationary model that minimizes the squared error between the modeled and empirical quantiles at each station. We checked that this approach resulted in an adequate fit for all the stations. For a given station, we use the same value of $c$ while fitting the stationary and non-stationary models. The left censoring threshold is only used during the inference procedure in Eq. (13), where we use all the non-zero data (inclusive of the data below the threshold) to fit the model, and so the fitted model applies to all the non-zero data. This is different from the classical GPD where the distribution applies only to the data above the threshold.

## 3.3. Model selection

We assess the significance of the trend using the likelihood ratio test, which is based on the deviance statistics defined as

$$ D = 2 \left[ \ell_{M_*}(\hat{\psi}_*) - \ell_{M_0}(\hat{\psi}_0) \right], \qquad (14) $$

where $\hat{\psi}_*$ is the vector of fitted parameters of the non-stationary model $M_*$ (with $M_*$ being any of the five non-stationary models.). A given non-stationary model is said to be significant at the $p\%$ level if $D > D_{crit}$, where $D_{crit}$ is the $p\%-quantile of the $\chi^2$ distribution with $k$ degrees of freedom, and $k$ is the difference in number of parameters between the non-stationary models and the stationary model. In this study, we consider a level of 95%, corresponding to an alpha value of 5%. This means we accept a 5% chance of committing a type-I error, thereby accepting a non-stationary model when in reality a stationary
model is sufficient. At a given station, we retain the model with the lowest p-value among all the models with p-value < 5%. By doing so we can either select the best model for a given covariate (considering five non-stationary models) or the best model irrespective of the covariate (considering ten non-stationary models). If none of the models has p-value < 5%. we retain M₀ and conclude that there is no evidence of trend at the given station.

## 3.4. Uncertainty and trend significance

The likelihood ratio test described in the preceding section tells us whether or not a given non-stationary model of wet days precipitation should be preferred over the stationary model. However, it does not tell us which quantiles have changed. We only know that the distribution as a whole has changed, even though it could be that only the extremes have changed. To address this, we employ a bootstrap approach for uncertainty analysis and to determine the trend significance in any given statistics (mean or quantile). This will allow us to know whether a given return level, e.g. the 1-year or 50-year level has significantly changed. The bootstrap procedure is described below.

Recall that $\hat{\psi}$ is the fitted parameters on the observed series $r^{orig}$. Then we let $r_{i,t}^{orig}$ to be the non-zero precipitation in $r^{orig}$. To obtain a bootstrap series, the following steps are followed:

1. Create a new series $r_{i,t}^{boot}$ with the same dimension as $r^{orig}$, then for each day i in year t,

$$r_{i,t}^{boot} = \begin{cases} 0 & \text{if } u_{i,t} \le \hat{p}_{d_i}, \\ 1 & \text{if } u_{i,t} > \hat{p}_{d_i} \end{cases},$$

where u is a realization of a uniform random variable between 0 and 1, and $\hat{p}_{d_i}$ is the estimated dry day probability for year t. $r_{i,t}^{boot} = 0$ signifies a dry day and $r_{i,t}^{boot} = 1$ signifies a wet day.

2. Apply the CDF of the EGPD in the non-stationary model $M_*$ on $r_{+}^{orig}$ to obtain a stationary sample x, where

$$x_{j,t} = \left[ 1 - \left( 1 + \hat{\xi} \frac{r_{+j,t}^{orig}}{\hat{\sigma}_t} \right)^{-1/\hat{\xi}} \right]^{\hat{\xi}_t},$$

j = 1, 2, ..., $n_{wet,t}$ are the indices of the non-zero precipitation, and $n_{wet,t}$ is the number of non-zero precipitation of year t.

3. Create a bootstrap sample $x_{j,t}^{boot}$ by sampling with replacement b values of x, where b is the number of wet days in $r^{boot}$. Next rescale $x_{j,t}^{boot}$ by applying the quantile function of the EGPD in $M_*$ to create $x_{j,t}^{rescaled}$ according to

$$x_{j,t}^{rescaled} = \frac{\hat{\sigma}_t}{\hat{\xi}} \left\{ \left[ 1 - \left( x_{j,t}^{boot} \frac{1}{\hat{\kappa}_t} \right)^{-\hat{\xi}_t} \right]^{-\hat{\xi}_t} - 1 \right\},$$

where j = 1, 2, ..., $n_{wet,t}$ and $n_{wet,t}$ is the number of non-zero precipitation in year t.

4. Next, fill in the non-zero precipitation on the wet days according to

$$r_{s_d}^{boot} = x_{rescaled,d}^{boot}$$

for d = 1, 2, ..., m where $s_1, s_2, ..., s_m$ are the wet days (times steps where $r_{j,t}^{boot} = 1$).

5. Fit a non-stationary model on $r_{j,t}^{boot}$ and estimate the trends in mean and return levels from Eq. (10).

6. Repeat steps 1 to 5 a total of 500 times to obtain the empirical bootstrap distribution of the trends in mean and return levels. A given trend is significant with a 95% confidence level if the value zero is not within the 2.5% and 97.5% confidence bounds of the distribution.

# 4. Results

## 4.1. Model diagnostics

We begin with a visual inspection of the performance of the models. Since classical quantile-quantile (QQ plots) do not apply in a nonstationarity context due to the distributional assumption of each observation, following Coles et al. (2001), we started by transforming the data of each year using the CDF obtained from the fitted nonstationary model. Assuming that the model is correctly specified, this transformation should yield data that are approximately uniformly distributed on the interval (0, 1). Thereafter, we transform this uniformly distributed data using the inverse CDF of the exponential distribution, to have a better visualization of the upper tail. By doing so, the data is now approximately identically distributed (exponentially), and so QQ plots can be shown. Fig. 4 shows the obtained QQ plots at 12 randomly selected stations and seasons. It can be seen from the plots that the points generally align on the diagonal, which signifies a good level of performance.

To further investigate the quality of the model fits, we compare the quantile estimates from our models to those from other quantile regression methods. Fig. 5 shows the time series of summer non-zero daily precipitation (gray-colored points) at Pfaffikon in the canton of Zurich. The lines show the time evolution of some quantiles (broken lines) as well as the mean (continuous line). The blue lines show the predicted lines using non-parametric quantile regression with total variation roughness penalties (Koenker and Mizera, 2004) obtained using the rqss function in R. The red-colored lines on the other hand are obtained by fitting the non-stationary EGPD model, $M_{\sigma,k}$. In both models, we use SST as the covariate. The figure shows a good agreement between the two approaches. Note also the flexibility of the $M_{\sigma,k}$ to give different evolution of the quantiles, with lower quantiles decreasing and higher quantiles slightly increasing post 1980. Another example is depicted in Fig. 6 which shows the case of a station in Basel for the spring season. In this case, the quantiles predicted with $M_{\sigma,k}$ are compared to those from another nonlinear non-parametric quantile regression method, MCQRNN (Monotone Composite Quantile Regression Neural Network (Cannon, 2018)). It is again obvious that there is a good agreement in the evolution of the quantiles predicted from both approaches, albeit some differences before 1925.

The two given examples illustrate the performance of the non-stationarity EGPD models and the ability to give reasonable quantile estimates that agree well with other quantile regression methods. In addition to yielding consistent and non-crossing quantile estimates, the extrapolation of higher quantiles can be achieved with much lesser uncertainty and through a theoretically justified approach.

## 4.2. Model selection

The result of the model selection based on the likelihood ratio test described in Section 3.3 is presented in Fig. 7. Fig. 7(a) shows the result when time is used as the covariate, while Fig. 7(b) is for SST. In both cases, the circles indicate the locations of the stations. The colored-filled circles are those where the stationary assumption is rejected. The yellow-colored stations are those where $M_{p_d}$ is statistically the best, which signifies that only the dry day frequency has changed, but there is no evidence of any trend in the wet day distribution. On the contrary, brown-colored stations are those where the trend is only in the wet day distribution, while green-colored stations are those where the trend is evident in both the dry day frequency as well as the wet day distribution. In both cases, the light colors involve a trend in only $\sigma$ while dark colors involve a trend in both $\kappa$ and $\sigma$. Finally, locations where the stationary assumption is not rejected are shown by the transparent circles. At these locations, there is no evidence of a trend in either the dry day frequency or the wet day precipitation distribution, and so $M_0$ is statistically better.
<figure><img src="image_7.png" /><figcaption>Fig. 4. Model diagnostics: Sample quantile-quantile plots of the nonstationarity EGPD model in exponential space at some randomly selected stations and seasons.</figcaption></figure>

Looking at the time covariate (Fig. 7(a)), winter and spring are the seasons with the highest number of stations exhibiting significant non-stationarity, with more than 60% of the stations being significant. A spatial pattern is also observed in both seasons, with most of the stations showing significant non-stationarity located in the north and the Wallis. Summer has 40% while autumn has 33% of the stations with significant non-stationarity. In summer these stations are mostly located along the northern-alpine rim and the Bernese Alps. In autumn, however, they are located in the Ticino region and the northeastern alpine slope.

Similar result is obtained when SST is used as a covariate (Fig. 7(b)), although the cases with nonstationarity in autumn are nearly halved and more cases of nonstationarity are observed in spring. Comparing both maps also shows some similar agreement in the spatial pattern of the nonstationarity, although for spring in the northeast, stations now show a trend in both the dry day frequency as well as the wet day distribution (green vs maroon colors).

Finally, to decide on which covariate to adopt, we apply the likelihood ratio test at each station irrespective of the covariate (i.e; for the ten non-stationary models) and retained the model with the least *p*-value as explained in Section 3.3. Fig. 8 shows the result of the final model selection. Triangles denote SST, squares denote time, while circles denote stations where stationarity is not rejected. Season-dependent performance can be observed, with SST being dominant in spring, time in winter, and autumn, and slightly more in summer. The bar graph in Fig. 9 highlights this result, while Table 1 presents the proportion of sites per season, where the best model suggests a given type of non-stationarity. To summarize, the results in this section showed that the linear model in time is sufficient to explain the trend in all the seasons, except spring where the trend is better explained using SST. Therefore, for the remainder of the article, we will retain time as a covariate for all seasons except spring where we will use SST.
<figure><img src="image_8.png" /><figcaption>Fig. 5. Model diagnostics: Evolution of quantiles predicted at a station in Pfäffikon using $M_{\sigma,x}$ and a nonparametric quantile regression (Koenker and Mizera, 2004).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Model diagnostics: Evolution of quantiles predicted at a station in Basel using $M_{\sigma,x}$ and Monotone Composite Quantile Regression Neural Network (Cannon, 2018).</figcaption></figure>

<caption>Table 1</caption>

Proportion of sites per season, where the best model suggests no trend (Column 3), trend in only the dry-day frequency (Column 4), trend in only the wet-day distribution (Column 5), or trend in both components, i.e. both the dry-day frequency and the wet-day distribution (Column 6). The type of covariate considered is given in Column 2.

<table><thead><tr><th>Season</th><th>Covariate</th><th>No trend<br>(M<sub>0</sub>)</th><th>Trend in dry-day frequency only<br>(M<sub>p<sub>0</sub></sub>)</th><th>Trend in wet-day distribution<br>(M<sub>σ</sub> and M<sub>σ,x</sub>)</th><th>Trend in both component<br>(M<sub>p<sub>0</sub>,σ</sub> and M<sub>p<sub>0</sub>,σ,x</sub>)</th></tr></thead><tbody><tr><td rowspan="2">Winter</td><td>Time</td><td>21.0</td><td>13.3</td><td>21.0</td><td>44.7</td></tr><tr><td>SST</td><td>34.3</td><td>16.0</td><td>17.7</td><td>32.0</td></tr><tr><td rowspan="2">Spring</td><td>Time</td><td>17.1</td><td>14.4</td><td>24.8</td><td>43.7</td></tr><tr><td>SST</td><td>4.4</td><td>24.9</td><td>6.1</td><td>64.6</td></tr><tr><td rowspan="2">Summer</td><td>Time</td><td>30.9</td><td>21.5</td><td>16.6</td><td>31.0</td></tr><tr><td>SST</td><td>29.8</td><td>27.6</td><td>13.3</td><td>29.3</td></tr><tr><td rowspan="2">Autumn</td><td>Time</td><td>61.9</td><td>3.9</td><td>25.4</td><td>8.8</td></tr><tr><td>SST</td><td>81.2</td><td>3.31</td><td>14.4</td><td>1.1</td></tr></tbody></table>
<figure><img src="image_10.png" /><figcaption>Fig. 7. Likelihood ratio test. The color-filled circles are those where the stationary assumption is rejected at the 95% level.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 8. Likelihood ratio test. The color-filled shapes are those where the stationary assumption is rejected at the 95% level. Triangles denote stations where SST is the preferred covariate, squares denote time, and circles denote stations where stationarity is not rejected.</figcaption></figure>

## 4.3. Trends analysis

Following the model selection in the previous section, we proceed to the assessment of the observed trends. We specifically look at the trends in (i) the wet days frequency $p_{w,t} = (1 - p_{d,t})$, (ii) mean of the wet days precipitation $\mu_t$, (iii) mean of all day precipitation $m_t$, and (iv) T-year return levels $r_{T,t}$, for $T \in \{10, 100, 200\}$. The trends are always expressed as percent decadal relative trend (see Eq. (10)) and the significance is assessed using the bootstrap procedure described in Section 3.4.

### 4.3.1. Trends in mean precipitation

Fig. 10 shows the relative trends in the mean of wet days precipitation (column a), wet days frequency (column b), and mean of all days precipitation (column c) for each season (rows of the figure). The stations are colored according to the trend direction, with red

being positive and blue being negative. Filled circles indicate significant trends at the 95% level according to the bootstrap procedure in Section 3.4. The upper and lower bounds of the 95% confidence interval are shown in Fig. 11. Red colors indicate positive trends, while blue colors indicate negative trends. At the stations where the trends are significant, the upper and lower bounds remain in the same direction, increasing or decreasing. Conversely, at stations where the trend is not significant, the upper and lower bounds are in opposite direction, indicating that a trend of zero is contained in within the 95% confidence interval.

From Fig. 10, it can be seen that the relative trends show distinct seasonal and regional differences both in magnitude and direction. In particular, the results show that winter has experienced a significant increase in the mean of all days precipitation. It is also seen that this significant increase is mainly in the northern part of the study
<figure><img src="image_12.png" /><figcaption>Fig. 9. Bar plot showing the number of stations where a given covariate (Time or SST) is selected according to the likelihood ratio test.</figcaption></figure>

region, whereas no significant increase or decrease is observed in the south. This increase in the mean of all days precipitation (a mean value of 1.6% relative trend per decade) is seen to be a contribution of increasing wet days precipitation intensity and frequency (Fig. 10a and b). In spring on the other hand (second row from the top), while the mean of wet days precipitation has increased significantly (column a), a significant decrease in the wet days frequency (column b) resulted in weak trends in the mean of all days precipitation (column c), with a mean value of -0.1% relative trend per decade. The observed trends in summer and autumn are much weaker compared to those observed in the other seasons. Similar to spring, both seasons experienced an increase in the mean of wet days precipitation and a decrease in wet days frequency. As a consequence, while summer showed a decrease in the mean of wet days precipitation (-0.1% relative trend per decade), autumn experienced a mean increase of 0.2% relative trend per decade.

The results agree well with the trends' magnitude and significance obtained using the non-parametric Theil-Sen and Mann-Kendall approach (results not shown here). For instance, in the case of the mean of all days precipitation, a mean relative decadal trend of 1.6% (non-stationary EGPD model) versus 1.5% (Theil-Sen) in winter, -0.1% versus -0.1% in spring which shows opposite signs, -0.1% versus -0.2% in summer, and 0.2% versus 0.1% in autumn. Furthermore, previous studies that considered various periods have also reported a significant increase in winter precipitation in the northern and western part of the study area (e.g. Begert et al., 2005; Schmidli and Frei, 2005; Masson and Frei, 2016; Scherrer et al., 2016a). In spring, the increasing trend in the north and some significant decreasing trends in the south agree well with the results in Masson and Frei (2016) considering 1901–2008, and in line with the results of Scherrer et al. (2016a) for 1901–2013. Our findings also show a few stations with some significant decrease in summer precipitation in the south. This is in line with Masson and Frei (2016), but not with the more recent work of Scherrer et al. (2016a). Finally, in autumn, the non-significant decreasing trend in the mean of all days precipitation in the south agrees well with the findings of Masson and Frei (2016) as well as Scherrer et al. (2016a).

### 4.3.2. Trends in extreme precipitation

Fig. 12 shows the trends in seasonal 10, 100, and 200-year return levels. Similar to Fig. 10, the stations are colored according to the trend direction, with red being positive and blue being negative. Filled circles indicate significant trends at the 95% level, while the upper and lower bounds of the 95% confidence interval are shown in Fig. 13. The result shows that in any season the trend is almost the same whatever the level of extremeness. Additionally, the seasonal patterns are much similar in the extremes compared to those observed in the mean (see Fig. 10).

Looking at winter, the mean trend in the 10-year return level is 1.2% per decade. The trend is positive in most of the stations (85.6%),

with more than half being significant. Significant negative trends are observed in only 1.1% of the stations. While all the regions except the inner Alps exhibit positive trends, the significant trends are mainly observed in the Swiss plateau. In spring, the trends are very similar in magnitude and regional pattern to those observed in winter for all the return periods. In summer, however, a different trend pattern is observed. In this season, while the percentage of positive trends in the 10-year level remains high (81.2%), the percentage of the significant positive trends is lower (27.1%, compared to 51.3% in winter.) Significant negative trends account for only 2.2% of the overall total. Regionally, the significant positive trends are mainly located along the northern alpine slope, especially the Bernese Alps. Finally, in autumn, 78.5% of the stations show a positive trend in the 10-year level, out of which 27.6% is significant. Again the significant negative trend accounts for only 0.6% of the overall total. The trends also show a spatial pattern different from the other seasons. The significant positive trends are mainly observed in the northeastern slope, Grisons canton, and the Ticino region. The northwestern region shows very weak trends that are not significant.

To summarize, compared to the observed trends in the mean of all days precipitation, whose direction depends on the season, a more uniform positive trend is observed in the extremes. All the seasons experienced an increase in extreme precipitation. The percentage of stations with positive trends ranges from 78.5% in autumn to 85.6% in winter. Similar results were obtained by Scherrer et al. (2016b) while analyzing trends in annual maxima of daily precipitation using a nonparametric approach. They reported a seasonal positive trend in between 75% to 83% of the stations, for the 1901–2013 period. A more recent study by Bauer and Scherrer (2024) extending the work of Scherrer et al. (2016b) for the 1901–2023 period also reported positive trends at 83% of the stations in the winter-half year and 89% of the stations in the summer-half year. Additionally, in line with our findings, Blanchet et al. (2021) while applying a non-stationary GEV model reported a decreasing 1958–2018 trend in winter 20-year return level estimates in the Swiss valley (Rhine).

## 5. Conclusions

The aim of our study was to jointly model the observed trends in the bulk and in extreme of daily precipitation in Switzerland using a non-stationary version of the EGPD. We incorporated the non-stationarity by allowing the parameters of the distribution to vary with either time (year) or sea surface temperature (SST) averaged over the Mediterranean sea. We tested five non-stationary models allowing us to model trends in the dry day frequency as well as changes in both mean and extremes. We then applied the models to 181 daily stations in Switzerland with at least 100 years of data and selected the best model for each season and station based on the likelihood ratio test. The result of the model selection revealed a coherent seasonal and regional pattern. Models using time as a covariate where found to be more appropriate in all seasons except spring, where SST was found to be better. This suggests that the mechanisms influencing precipitation vary seasonally, necessitating the consideration of appropriate covariates tailored to each season's dominant processes for more accurate modeling.

We assessed the trend significance using a non-parametric bootstrap approach, and the results showed a season-dependent signal in both the mean of all days of precipitation and extremes. A significant increase in winter precipitation was observed in the northern region, with a mean value of 1.6% relative trend per decade over the whole country. More so, the increase is from an increasing trend in both the wet days precipitation and the frequency. In spring on the other hand, while the mean of wet days precipitation has increased significantly, a significant decrease in the wet days frequency resulted in weak trends in the mean of all days precipitation (a mean value of -0.1% relative trend per decade). The observed trends in summer (-0.1% relative trend per decade) and autumn (mean increase of 0.2% relative trend per decade)
<figure><img src="image_13.png" /><figcaption>**Fig. 10.** Relative decadal trend (%) in (a) mean of wet days precipitation $\mu_t$, (b) wet days frequency $p_{w,t}$, and (c) mean of all days precipitation $m_t$. Each row corresponds to a season. In each case, the trend is computed from Eq. (10). Positive trends are shown in red while negative trends are shown in blue. Filled circles indicate trends that are significant at the 95% level according to the bootstrap procedure in Section 3.4. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

are much weaker compared to those observed in the other seasons. In contrast, extreme precipitation was observed to have increased for all seasons with positive trends in over 78.5% of the stations in all seasons. Regional season-specific patterns were also observed, with Ticino canton expressing a significant increase in autumnal extreme precipitation.

The results obtained using the non-stationary approach agree well with previous works that separately modeled trends in mean and extreme precipitation. This highlights the flexibility and potential of the model, and avoids partitioning the data and conducting separate trend analyses for the mean and precipitation extremes. Although time appeared as a relevant covariate in explaining the observed trends in daily precipitation for certain seasons, it is not valid for projection due to the nonlinearity of the underlying processes. Time serves as a proxy rather than a causal driver, and its relationship with precipitation trends may not hold under future conditions, particularly in a nonstationary climate. Daily precipitation is influenced by complex, nonlinear feedback mechanisms such as atmospheric circulation changes, moisture availability, and temperature shifts. The next step is to explore other climate covariates, such as local temperature, blocking patterns, or other indices. An advantage of using these covariates is the availability of future projections from climate models, which will

undoubtedly provide more relevant extrapolations compared to time. More flexible relationships beyond linear trends can be used to capture potential non-linearity. In such cases, generalized additive models can be a valuable tool. Furthermore, more spatially coherent signals could be obtained by spatially pooling information from nearby sites as well as integrating spatial dependence into the analysis through Bayesian hierarchical models or copula-based frameworks, or a regionalization approach (e.g. Tramblay et al., 2013; Blanchet et al., 2021). This regionalization approach has been previously applied in the study area by Haruna et al. (2022) under stationary context. It would thus be interesting to extend the approach in the case of the non-stationary context. Finally, this model can be used as a marginal distribution in weather generators to incorporate potential non-stationarity in mean and extreme precipitation. This can be achieved by allowing the parameters to vary with relevant climate covariates. Model selection can be applied to select the most relevant and informative covariates. By accounting for the potential non-stationarity, more consistent and reliable synthetic weather data can be generated, leading to robust estimates of potential changes in flood extremes using hydrological models.
<figure><img src="image_14.png" /><figcaption>**Fig. 11.** Upper and lower bound of the point wise 95% confidence intervals in the relative decadal trends shown in Fig. 10. Each column corresponds to a season, while for each statistic in a row, the corresponding bounds are shown.</figcaption></figure>

**CRediT authorship contribution statement**

**Abubakar Haruna:** Writing – review & editing, Writing – original draft, Software, Methodology, Formal analysis, Conceptualization.

**Juliette Blanchet:** Writing – review & editing, Validation, Supervision, Methodology, Funding acquisition, Conceptualization.

**Anne-Catherine Favre:** Writing – review & editing, Validation, Supervision, Resources, Methodology, Funding acquisition, Conceptualization.

**Declaration of competing interest**

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

**Acknowledgments**

This research is part of a postdoc research of Abubakar HARUNA, supported by the Swiss Confederation: Bundesamt für Energie (grant

no. SI/502150-01) and the Bundesamt für Umwelt, Switzerland (grant no. SI/502150-01), through the Extreme floods in Switzerland project.

In addition, part of this study has received funding from the Agence Nationale de la Recherche - France 2030 as part of the PEPR “Transformer la modélisation du climat pour les services climatiques” (TRACCS) program under grant number ANR-22-EXTR-0005

**Data availability**

The raingauge data used in this study are freely available from the Swiss Federal Office of Meteorology and Climatology, MeteoSwiss, and can be freely obtained from IDAWEB (2023). Part of the analysis was done with a freely available R package (Haruna, 2023) with GPL-3 license.
<figure><img src="image_15.png" /><figcaption>**Fig. 12.** Relative decadal trend (%) in extremes (a) 10-year, (b) 100-year, and (c) 200-year return levels. Each row corresponds to a season. In each case, the trend is computed from Eq. (10). Positive trends are shown in red while negative trends are shown in blue. Filled circles indicate trends that are significant as the 95% level according to the bootstrap procedure in Section 3.4.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>Fig. 13. Upper and lower bound of the point wise 95% confidence intervals in the relative decadal trends shown in Fig. 12. Each column corresponds to a season, while for each return period in a row, the corresponding lower and upper bounds are shown.</figcaption></figure>

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