Open AccessProceeding Paper

Extreme Rainfall Analysis Including Seasonality in Athens, Greece^†

Konstantinos Vantas

^1,*

and

Athanasios Loukas

Department of Mathematics, Faculty of Science, University of Western Macedonia, 52100 Kastoria, Greece

Department of Rural and Surveying Engineering, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece

Author to whom correspondence should be addressed.

^†

Presented at the 8th International Electronic Conference on Water Sciences, 14–16 October 2024; Available online: https://sciforum.net/event/ECWS-8.

Environ. Earth Sci. Proc. 2025, 32(1), 1; https://doi.org/10.3390/eesp2025032001

Published: 15 January 2025

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Abstract

Extreme rainfall analysis is essential for accurate flood hazard assessment. Traditional approaches, such as the use of annual maxima, may overlook seasonal variations and lead to underestimated precipitation extremes, compromising effective flood risk management strategies. This study applies a point process model to uninterrupted daily rainfall records (1901–2023) from the National Observatory of Athens meteorological station in Thiseion. This method analyzes both the frequency of exceedances above a given threshold and the values of those exceedances, incorporating seasonality into the modeling process. Preliminary analysis using annual maxima revealed no statistically significant trend but indicated clear monthly seasonality in precipitation extremes. By incorporating seasonality, the point process method yielded estimates up to 22% higher than those obtained using traditional annual maxima approaches, such as those employed in Greece’s National Flood Risk Management Plans. These findings highlight the need for a revision of current methodologies, which could significantly impact flood risk assessments and management strategies.

Keywords:

extreme value analysis; extreme rainfall; Greece; Athens; point process method

1. Introduction

Extreme rainfall analysis is crucial for climate modeling, floodplain management and hazard assessment [1,2]. As Earth’s climate changes, there is a significant impact of extreme weather effects around the world [3], highlighting the need for robust methods to assess their impacts. Among these tools, probabilistic methods based on Extreme Value Theory (EVT) play a central role in predicting such events [4]. Currently, two main EVT-based approaches are widely used [5]: (a) the block maxima (BM) method and (b) the point process (PP) method, also known as peaks-over-threshold (POT). The BM method utilizes maximum values from fixed data blocks, usually annual, while the PP method analyzes excesses above a predefined high threshold within continuous time series data [5].

Although convenient and straightforward, the BM method has certain limitations, namely its inability to include information on precipitation’s seasonal cycles and sensitivity to parameter estimation errors. For instance, Katz et al. [4] illustrated how the exclusion of an extreme point can disproportionately affect BM parameter estimates. A comprehensive review of these methods, including parameter estimation and model selection, is provided by Nerantzaki and Papalexiou [6], and a review of recent developments in software implementations is provided by Bezlile et al. [7].

In 2023, new parameters for ombrian curves in Greece were developed as part of the National Flood Risk Management Plans (NFRMP), aligning with the EU Directive 2007/60/EC [8]. These parameters were calculated using the BM method without incorporating continuous precipitation data, limiting their ability to estimate extreme values.

In this study, uninterrupted daily rainfall records ranging over a century (1901–2023) from the National Observatory of Athens meteorological station in Thiseion are analyzed. This research examines the existence of trends in annual maxima and seasonality in daily precipitation values. An appropriate high threshold is selected and two different PP models are fitted using the maximum likelihood method to assess whether incorporating seasonality is statistically significant. Finally, the results are compared with those from the NFRMP. This study aims to address gaps in the NFRMP methodology by improving the understanding of non-stationarity in extreme rainfall analysis. It emphasizes the uncertainties of predictions and models, contributing to a more effective revision of the NFRMP.

2. Materials and Methods

2.1. Data

Continuous daily rainfall records from 1901 to 2023, obtained from the National Observatory of Athens meteorological station at Thiseion (latitude: 37.972° N; longitude: 23.717° E; altitude: 107 m above mean sea level), were used in the analysis. Notably, the station has not been relocated over the years. Daily precipitation values do not show apparent long-term trends over time. Also, no large-scale clustering patterns were found, including neither prolonged periods of intense rainfall nor diminished rainfall that could indicate temporal dependence (Figure 1).

2.2. Non-Parametric Trend Analysis

The presence of a monotonic trend in the annual maxima series of daily precipitation was examined using Kendall’s Tau rank correlation test as implemented in the R language for statistical computing and graphics [9]. Kendall’s Tau measures the strength of the monotonic relationship between two variables, is resistant to the effect of outliers and is a classic statistical method applied to hydrological data [10]. Tau is expressed as

τ = S / D

, where is S is

S = \sum_{i < j} s i g n (x_{j} - x_{i})

(1)

where

(x_{1}, x_{2}, \dots, x_{n})

is the annual maxima for a period of n years, and

D = n \cdot (n - 1) / 2

. In the presence of ties, the formula for

D

is more complicated [11]. The p-value of Tau under the null hypothesis of no association is computed, in the case of no ties, using the algorithm AS71, given by Best and Gipps [12]. When ties are present, a normal approximation, with continuity correction, is used by taking

S

normally distributed with mean zero and variance, as given by Kendall [11].

2.3. Extreme Value Distribution Functions

As mentioned, the two main approaches based on Extreme Value Theory for analyzing extremes are the BM and PP methods. The Generalized Extreme Value (GEV) distribution is used to fit the BM:

G (z) = e x p [- {\{1 + ξ (\frac{z - μ}{σ})\}}_{+}^{- 1 / ξ}]

(2)

where

{x}_{+} = m a x (x, 0)

μ

is the location parameter,

σ > 0

is the scale parameter and

ξ

is the shape parameter [13].

The generalized Pareto (GP) distribution is used to fit excesses over a high threshold:

H (x) = 1 - {\{1 + ξ (\frac{x - u}{σ_{u}})\}}_{+}^{- 1 / ξ}

(3)

where

u

is the high threshold,

x > u

σ_{u}

is the scale parameter (depends on

u

) and

ξ

is the shape parameter.

2.4. Non-Stationarity of Extremes

If the data are believed to be non-stationary, it is possible to integrate this information into the parameters of the distribution functions (Equations (2) and (3)) using a regression-like approach [4,14]. For example, in

μ (t) = μ_{0} + μ_{1} t + μ_{2} t^{2}

(4)

the location parameter

μ

varies with time

t

, where

μ_{0}

μ_{1}

and

μ_{2}

are the regression coefficients.

2.5. Point Process Approach

In the presence of continuous time series data, methods such as the PP approach, which incorporate all available information, are more effective because they (a) utilize both the frequency and magnitude of the threshold exceedances, (b) account for temporal patterns such as seasonality and (c) provide a comprehensive statistical framework for modeling extreme events while being less sensitive to parameter estimation errors [13].

Modeling extreme values using a PP approach was introduced probabilistically by Leadbetter et al. [15,16] and as a statistical method by Smith [17]. A PP model consists of two components: (a) a Poisson process, which models the exceedance of a high threshold, and (b) the generalized Pareto distribution, which describes the excesses above the threshold. In brief, following Coles [13] and Katz [4], if a process is stationary and the data above a threshold do not cluster, the limiting form of the process is non-homogeneous Poisson with intensity measure

Λ

. Given a set

A = (t_{1}, t_{2}) \times (x, \infty)

Λ

is:

Λ (A) = (t_{2} - t_{1}) \cdot {[1 + ξ \frac{x - μ}{σ}]}^{- 1 / ξ}

(5)

where the

μ

σ

and

ξ

parameters have the same meanings as in Equation (2). Maximum likelihood estimation (MLE) can be used to estimate the parameters of a PP by optimizing the log-likelihood function using numerical methods [14]:

l (μ, σ, ξ; x_{1}, \dots, x_{n}) = - k \cdot l n σ - (\frac{1}{ξ} + 1) \sum_{ι = 1}^{n} {[1 + \frac{ξ}{σ} (x_{i} - u)]}_{x_{i} > u} - n_{y} {(1 + \frac{ξ}{σ} (u - μ))}^{- \frac{1}{ξ}}

(6)

where

n_{y}

is the number of years in the data, so that the parameters represent the annual maxima. Similarly, Equation (6) uses excesses, where the parameters are in terms as those in Equation (2). More details about the method can be found in Coles (Chapter 7 in [13]).

2.6. Selection of High Threshold

Before fitting the parameters of a PP, an appropriate threshold must be selected. This choice involves a trade-off between low variance (a lower threshold provides more data) and reduced bias (a higher threshold yields less biased estimates). The ideal threshold should produce parameter estimates that are consistent with those obtained from any higher threshold, within uncertainty bounds (e.g., confidence intervals) [13,14].

2.7. Statistical Test of Nested Models

To determine whether adding covariates to the parameters in the aforementioned regression-like manner improves the model, the likelihood ratio test can be applied [13,18]. Let

x

be the negative log-likelihood of the base model

m_{0}

and

y

be that of the nested model

m

. The likelihood ratio statistic is then given by:

D = - 2 (y - x)

(7)

When testing the null hypothesis

H_{0} : D = 0

, the significance level

a

can be determined using the quantile

c_{a} = (1 - a)

of the

χ^{2}

distribution, with the degrees of freedom equal to the difference in the number of model parameters. The null hypothesis is rejected when

D > c_{a}

3. Results and Discussion

The daily precipitation values per month were analyzed using four key central moments, namely the mean, standard deviation, skewness and kurtosis, as well as additional statistical properties such as the median and coefficient of variation (Table 1). These analyses aimed to identify seasonality in the data. The results show that there is a clear pattern of higher precipitation during the months from October to March. Notably, the highest recorded value in September 1950 was 143 mm. Positive skewness indicates that the data are asymmetric, while positive kurtosis indicates a heavy-tailed distribution.

Annual maxima also did not reveal any apparent long-term trends over time (Figure 2). Kendall’s Tau rank correlation test result indicates that the null hypothesis that annual maxima do not change over time could not be rejected for a significance level α = 5% (

p_{v a l u e} = 0.086

A high threshold of 10 mm was selected for the PP model by examining the stability of the parameter estimates across a range of values. Then, MLE was employed on 1509 rainfall data points that fell over the 10 mm threshold. Two different models were used: (a) a stationary model and (b) one that incorporates the observed seasonality into the location

μ

and shape

σ

parameters of the PP model:

μ (t) = μ_{0} + μ_{1} \cos (\frac{2 π \cdot t}{365.25}) + μ_{2} \sin (\frac{2 π \cdot t}{365.25}) l o g (σ (t)) = σ_{0} + σ_{1} \cos (\frac{2 π \cdot t}{365.25}) + μ_{2} \sin (\frac{2 π \cdot t}{365.25})

(8)

where the day of the year is

t = 1, 2, \dots, 365 .

The logarithm is used in order to keep the scale σ values positive.

A likelihood ratio test for the two nested models confirmed that incorporating seasonality is statistically significant (

p_{v a l u e} < 2 \cdot 10^{- 16}

), as expected.

Figure 3 depicts the effective design value, with 95% confidence intervals, for a 100-year return period (probability p = 0.01) from the non-stationary PP model (b). In this way, a general sense of the annual cycle in extreme precipitation is given for each day of the year. The rainfall records align with the effective 100-year design values, revealing the seasonal pattern, with higher rainfall extremes occurring in winter compared to summer.

Finally, the results from the PP method were compared to those from the NFRMP (Table 2). The stationary PP model’s estimates for various return periods were consistent with the NFRMP values for the upper 95% confidence interval (CI). Notably, the non-stationary PP model, which accounted for seasonality, produced estimates up to 22% higher for the upper 95% CI. It should be noted that the reported CIs were calculated using a normal approximation; however, the use of profile likelihoods would allow skewed intervals and produce even higher values for the upper CI bound [13,20].

4. Conclusions

The results of this study indicate that incorporating seasonality (non-stationarity) into probabilistic extreme analysis of rainfall leads to a significant difference in the rainfall design values. This is because statistical methods, such as the point process approach, can more precisely account for uncertainties involved in modeling and prediction. As a result, these means produce more accurate assessments of future extremes that are less likely to be contradicted by observed hydrological events, like the recent Storm Daniel in Greece. Overall, our findings underscore the need to revise the methodology currently employed in the National Flood Risk Management Plans. Such a revision could have a substantial impact on flood risk assessments and management strategies in Greece.

However, some limitations should be acknowledged. To begin with, this study focuses on a single long-term rainfall record, and extending this approach to regional datasets could reveal spatial dependence in extreme rainfall, which is essential for nation-wide applications. Moreover, the point process models require the careful selection of thresholds and covariates, introducing potential uncertainties in parameter estimation. Finally, the impacts of climate change on extreme rainfall remain a modeling challenge that should be incorporated.

Future research should focus on integrating non-stationary regional models with dynamic climatic projections. Incorporating these methodologies into flood risk management plans will require collaboration among researchers, policymakers and stakeholders to address technical, economical and regulatory challenges.

Author Contributions

Conceptualization, methodology, K.V.; writing—original draft preparation, K.V.; writing—review and editing, A.L. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The original data presented in the study are openly available under license CC-BY 4.0 at https://climpact.gr/ (accessed on 14 January 2025).

Acknowledgments

The data importing, analysis and presentation were carried out using the open source R language for statistical computing and graphics [9] using the packages hydroscoper [21], hyetor [22], eXtremes [14] and ggplot2 [23].

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Daily precipitation values recorded in Athens, Greece.

Figure 2. Annual maximum daily precipitation values recorded in Athens, Greece. Smooth lines are marked in blue, and the gray band marks the standard error variance produced by means of Local Polynomial Regression Fitting [19].

Figure 3. Daily precipitation values recorded in Athens, with the effective 100-year return level of the non-stationary PP model with seasonality in the location and shape parameters.

Table 1. The average statistical properties of the maximum daily observed precipitation values per month (mm). SD, standard deviation; CV, coefficient of variation (the ratio of the standard deviation to the mean).

Precipitation (mm)	Min	Mean	Median	Max	SD	Skew	Kurtosis	CV
January	0.5	19.5	16.4	54.6	13.0	0.86	0.00	0.67
February	0.4	18.5	14.2	98.2	15.3	2.43	8.36	0.83
March	0.8	16.7	12	116	15.0	3.20	15.91	0.90
April	0.1	11.6	8.1	82	10.8	2.81	13.64	0.93
May	0.1	10.4	8.15	49.8	9.2	1.40	2.53	0.88
June	0.1	8.9	4.95	54.8	10.6	1.96	4.38	1.19
July	0.1	9.2	5.1	91	13.9	3.51	15.96	1.51
August	0.1	7.7	3.2	51.5	10.6	2.20	4.80	1.38
September	0.1	14.0	7.1	142.9	19.9	3.46	17.18	1.43
October	0.1	22.1	17.3	83.8	17.8	1.06	0.73	0.80
November	0.1	25.2	20.45	114.9	19.5	1.98	4.80	0.78
December	0.4	24.9	21.9	71.8	14.9	0.86	0.50	0.60

Table 2. A comparison of the values from the NFRMP and the two PP models used in this study. T is the return level.

T (Years)	NFRMP (mm)	Stationary PP		Non-Stationary PP
T (Years)	NFRMP (mm)	95% CI (mm)		95% CI (mm)
5	67	55	64	65	81
10	81	65	80	76	99
50	122	89	120	100	146
100	144	125	142	110	170
500	206	125	201	135	238

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Vantas, K.; Loukas, A. Extreme Rainfall Analysis Including Seasonality in Athens, Greece. Environ. Earth Sci. Proc. 2025, 32, 1. https://doi.org/10.3390/eesp2025032001

AMA Style

Vantas K, Loukas A. Extreme Rainfall Analysis Including Seasonality in Athens, Greece. Environmental and Earth Sciences Proceedings. 2025; 32(1):1. https://doi.org/10.3390/eesp2025032001

Chicago/Turabian Style

Vantas, Konstantinos, and Athanasios Loukas. 2025. "Extreme Rainfall Analysis Including Seasonality in Athens, Greece" Environmental and Earth Sciences Proceedings 32, no. 1: 1. https://doi.org/10.3390/eesp2025032001

APA Style

Vantas, K., & Loukas, A. (2025). Extreme Rainfall Analysis Including Seasonality in Athens, Greece. Environmental and Earth Sciences Proceedings, 32(1), 1. https://doi.org/10.3390/eesp2025032001

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Extreme Rainfall Analysis Including Seasonality in Athens, Greece^†

Abstract

1. Introduction