Examining the Probabilistic Characteristics of Maximum Rainfall in Türkiye

Abstract

Hydrologists need to predict extreme hydrological and meteorological events for design purposes, whose magnitude and probability are estimated using a probability distribution function (PDF). The choice of an appropriate PDF is crucial in describing the behavior of the phenomenon and the predictions can differ significantly depending on the PDF. So, the success of the probability distribution function in representing the data of extreme value series of natural events such as hydrology and climatology is of great importance. Depending on whether the series consists of maximum or minimum values, the theoretical probability density function must be appropriately fit to the right or left tail of the extreme data, which contains the most critical information. This study includes a combined evaluation of the performance of four different tests for selecting the appropriate probability distribution of maximum rainfall in Türkiye: Kolmogorov–Smirnov (KS) test, Anderson–Darling (AD) test, Probability Plot Correlation Coefficient (PPCC) test, and L-Moments Z^DIST test. Within the scope of the study, maximum rainfall series of seven rainfall durations from 15 to 1440 min, at rain gauge stations in 81 provinces of Türkiye, were examined. Goodness of fit was performed based on ranking using a combination of four different numerical tests (KS, AD, PPCC, Z^DIST). The probabilistic character of maximum rainfall was evaluated using a large dataset consisting of 567 time series with record lengths ranging from 45 to 80 years. The goodness of fit of distributions was examined from three different perspectives. The first is an examination considering rainfall durations, the second is a province-based examination, and the third is a general country-based assessment. In all three different perspectives, the Wakeby distribution was determined as the best fit candidate to represent the maximum rainfall in Türkiye.

1. Introduction

Changes in the conditions in the atmosphere, specifically increases in greenhouse gases, have undesirably affected the weather and climate of our planet in recent decades. According to the Sixth Assessment (AR6) Report of the Intergovernmental Panel on Climate Change [1], “human-caused greenhouse gas emissions have led to an increased frequency and/or intensity of some weather, and climate extremes since pre-industrial times.” Frequency analysis is the most powerful scientific method for estimating the intensity and frequency of hydrological and climatic extremes. Maximum rainfall estimates obtained for certain frequencies or return periods by frequency analysis are significant project criteria in the planning of rainwater network projects, flood protection structures, drainage channels, and some other infrastructure projects such as culverts and bridges, and are also of great importance in erosion analysis.

The first and most important step of frequency analysis is to correctly determine the probability distribution function (PDF) representing the behavior of the relevant hydrological and/or meteorological phenomenon. Thus, a solid basis for future predictions of the studied phenomenon is established. Two important steps should be followed at this stage. First, potential probability distributions that are expected to successfully represent the phenomenon under study should be carefully listed, considering previous experience. The second step is to choose the best fit distribution among the PDFs considered. Rainwater network projects, flood protection structures, drainage channels, and infrastructure projects such as culverts and bridges are designed according to extreme rainfall events whose magnitude and probability are estimated using a PDF. The results may vary considerably from one distribution to another, impacting the design and size of the structure. Overestimation of the design event increases construction and maintenance costs, while underestimation may result in loss of properties and human lives [2,3]. For these reasons, the selection of an appropriate PDF is a matter of great importance.

Many probability distributions have been proposed for representing the distribution of hydrologic and climatic extremes [4,5,6,7,8,9]; however, there is still no general agreement as to which distribution(s) should be used. The most frequently used distributions in hydrological and meteorological frequency analysis are Normal, (N), 2-parameter Lognormal (LN2), 3-parameter Lognormal (LN3), 2-parameter Gamma (GAM), Pearson Type III (PE3), Log Pearson Type III (LP3), Generalized Logistic (GLO), Generalized Pareto (GPA), Gumbel (GUM), generalized extreme value (GEV), and 5-parameter Wakeby (WAK) distributions [7].

The choice of probability distribution function for the frequency analysis of extremes depends on several dynamics, including previous experiences, knowledge of the modeler, national practice, objective of the study, data accessibility, and governmental necessities [9,10]. The national guidelines of different countries recommend the use of different distributions. For instance, Log-Pearson 3 has been recommended in the US in Bulletin 17B [11]. The generalized extreme value (GEV) distribution and LP3 are recommended in Australia [12]. GEV distribution is also a recommended choice in many other countries in Europe, including Austria, Germany, Italy, and Spain [9]. However, many other distributions have also been used popularly, including the Gumbel (GUM) distribution in Finland and Spain, and the Generalized Logistic (GLO) distribution in the UK [9]. Moreover, in Slovenia, the Agency of Environment recommends the use of five distribution models (i.e., N, LN, PE3, LP3, and GUM); Slovakia often uses the GAM, LN3, LP3, and GEV distributions. In Canada, the use of a specific distribution is not compulsory; however, LP3, LN3, GEV, and GUM have been used popularly [13,14,15,16]. Environment Canada currently uses GUM to construct at-site IDF curves for all stations in Canada [17]. This distribution is also recommended for the development of rainfall IDF relations by the Canadian Standard Association [18].

The common method for selecting a proper probability model is mainly based on the best fit of the model to the observed data and the best fit selection approach depends strongly on the characteristics of the existing rainfall record at a given site [19,20,21]. Various methods have been used in the literature to determine the most appropriate probability distribution model for maximum rainfall. Statistical tests (i.e., Kolmogorov–Smirnov, Anderson–Darling, Chi-square, root mean square error (RMSE), Probability Plot Correlation Coefficient tests (PPCC)), L-Moment’s Z^DIST test, criteria (i.e., AIC, BIC, and ADC), and graphical tests (i.e., product moment ratio and L-Moment ratio diagrams, Q-Q plot) are the most commonly used techniques for PDF selection.

Studies on the determination of maximum rainfall of a specific return period constitute a significant study area of hydrology and meteorology. For this purpose, the frequency analysis and therefore the best fit distribution analysis of maximum rainfall have been the subject of several studies in Türkiye and at the global scale. Some of the recent studies focused on frequency analysis and estimation of maximum rainfall [15,16,22,23,24,25,26,27,28,29] while others focused more on the selection of the PDF that best fits the maximum rainfall [3,15,16,21,27,30,31,32,33,34,35,36].

Case studies dealing with the determination of the statistical and probabilistic characterization of extreme rainfall over Türkiye are numerous, but they usually focus on certain parts of the country or consider only the maximum rainfall for a certain duration (e.g., 1 h or 24 h) [28,37,38]. To our knowledge, this is the first contribution that investigates the probabilistic characteristics of annual maximum rainfall both across the country and for various rainfall durations.

In this study, the 3-parameter versions of the 2- and 3-parameter distributions were preferred (LN3 instead of LN2, PE3 instead of Gamma, and GEV instead of GUM). In addition, NOR, GLO, GPA, LP3, and WAK probability distribution functions were taken into account. Goodness of fit was evaluated with a ranking-based methodology using a combination of four different numerical tests (KS, AD, PPCC, Z^DIST). The procedure was performed with the maximum rainfall data obtained from meteorological stations in 81 provinces of Türkiye for seven different durations (15 min, 30 min, 1 h, 3 h, 6 h, 12 h, and 24 h). The data, study area, and methodology are presented in Section 2, followed by results in Section 3, discussions in Section 4, and conclusions in Section 5.

2. Materials and Methods

2.1. Study Area

Türkiye is situated at the crossroads of Europe and Asia, spanning the Anatolian Peninsula and the Thrace region in the northern hemisphere. Geographically, it extends between 36° and 42° north latitudes and 26° to 45° east longitudes. The country is bordered by three major seas: the Mediterranean to the south, the Aegean to the west, and the Black Sea to the north. This unique geographical setting contributes to the country’s remarkable climatic diversity.

The total area of Türkiye is approximately 780,000 km², and the national average annual rainfall is around 643 mm. However, the climate across Türkiye is far from uniform. Due to its varied topography and location, the country encompasses a broad range of climate types. In general, four main climate zones can be identified: the Black Sea climate in the north, with rainfall throughout the year; the Mediterranean climate in the west and south, characterized by hot, dry summers and mild, wet winters; a semi-arid steppe climate in Central and Southeastern Anatolia; and a harsh continental climate in the Eastern Anatolia region, with cold, snowy winters and relatively dry, long summers.

Reflecting this climatic complexity, Türkiye is divided into seven geographical regions, primarily based on climatic characteristics: Marmara, Aegean, Mediterranean, Southeastern Anatolia, Eastern Anatolia, Black Sea, and Central (or Inland) Anatolia. Each region exhibits distinct climatic features.

The Marmara region, located in the northwest, has a climate similar to the Balkans—humid and mild in summer, with cold winters that bring higher-than-average precipitation, often as snow. The Aegean region features a classic Mediterranean climate along the coast, while inland areas become cooler in winter, especially at higher elevations where snowfall is more common.

In the Mediterranean region, the Taurus Mountains stretch parallel to the coastline, typically 40–50 km inland. These mountains significantly influence local climate patterns, especially by enhancing orographic rainfall on their windward slopes. The region experiences hot, humid summers and mild, rainy winters.

Southeastern Anatolia is characterized by a semi-arid climate, with hot, dry summers and mild winters. Precipitation here is generally below the national average. Eastern Anatolia, on the other hand, is mountainous and known for its severe winters with abundant snowfall and rainfall, followed by dry, prolonged summers.

In the north, the Black Sea region features another set of coastal mountain ranges running parallel to the sea. These mountains create a barrier effect, resulting in year-round orographic rainfall on their windward sides. This region records the highest annual rainfall in the country, nearly double the national average.

Finally, the Central Anatolia region, enclosed by high mountain ranges to both the north (Black Sea) and south (Mediterranean), exhibits a continental climate with hot, dry summers and cold winters. Due to limited moisture intrusion, annual rainfall here remains below the national means.

2.2. Meteorological Data Used in Study

Meteorology services of countries use different standard duration rainfalls obtained from continuous rainfall records. The General Directorate of Meteorology (known as MGM) in Türkiye produces a record of annual maximum rainfalls using the overlapping moving window for 14 standard durations (5, 10, 15, 30 min, 1, 2, 3, 4, 5, 6, 8, 12, 18, and 24 h). In this study the AMR series of 7 standard durations (15, 30 min, 1, 3, 6, 12, and 24 h) are obtained from MGM, which have been measured since 1938 [38]. Data from the meteorological stations of 81 cities were used in the study, and the quality of the data, the sufficient length of rainfall records (at least 35 years), and the spatial distribution of the stations to represent different climatic conditions across the country were taken into consideration. A total of 567 extreme datasets (81 stations × 7 durations) were examined. The list of stations and their geographical location and measurement period are given in Appendix A (

Table A1. Meteorological stations in 81 provinces of Türkiye.

Station	Latitude (°N)	Longitude (°E)	Altitude (m)	Annual Rainfall (mm)	Data Recording Period
Adana	37.00	35.34	23	668	1944–2020
Adıyaman	37.76	38.28	672	715	1963–2020
Afyonkarahisar	38.74	30.56	1034	444	1957–2020
Ağrı	39.73	43.05	1646	526	1967–2020
Aksaray	38.37	33.99	970	360	1965–2020
Amasya	40.67	35.84	409	463	1965–2020
Ankara	39.97	32.864	891	392	1940–2020
Antalya	36.55	31.98	6	1040	1964–2020
Ardahan	41.11	42.71	1827	558	1967–2020
Artvin	41.18	41.82	613	695	1965–2020
Aydın	37.84	27.84	56	658	1959–2020
Balıkesir	39.63	27.92	102	604	1957–2020
Bartın	41.63	32.36	33	1063	1966–2020
Batman	37.86	41.16	610	489	1969–2020
Bayburt	40.26	40.22	1584	451	1966–2020
Bilecik	40.14	29.98	539	461	1960–2020
Bingöl	38.89	40.50	1139	945	1966–2020
Bitlis	38.48	42.16	1785	1072	1966–2020
Bolu	40.73	31.60	743	555	1949–2020
Burdur	37.72	30.29	957	428	1964–2020
Bursa	40.23	29.01	100	708	1951–2020
Çanakkale	40.14	26.40	6	623	1958–2020
Çankırı	40.61	33.61	755	415	1959–2020
Çorum	40.55	34.94	776	431	1958–2020
Denizli	37.76	29.09	425	568	1959–2020
Diyarbakır	37.90	40.20	674	491	1940–2020
Düzce	40.84	31.15	146	838	1965–2020
Edirne	41.68	26.55	51	599	1949–2020
Elazığ	38.64	39.26	989	421	1957–2020
Erzincan	39.75	39.49	1216	376	1957–2020
Erzurum	39.95	41.19	1758	431	1956–2020
Eskişehir	39.77	30.55	801	356	1940–2020
Gaziantep	37.06	37.35	854	564	1957–2020
Giresun	40.92	38.39	38	1292	1966–2020
Gümüşhane	40.46	39.47	1216	463	1966–2020
Hakkari	37.58	43.74	1720	793	1956–2020
Hatay	36.20	36.15	104	1153	1957–2020
Iğdır	39.92	44.05	856	259	1966–2020
Isparta	37.79	30.57	997	566	1957–2020
İstanbul	40.91	29.16	18	661	1974–2020
İzmir	38.40	27.08	29	711	1938–2020
Kahramanmaraş	37.58	36.92	572	722	1966–2020
Karabük	41.20	32.63	278	549	1966–2020
Karaman	37.19	33.22	1018	338	1965–2020
Kars	40.60	43.11	1777	508	1965–2020
Kastamonu	41.37	33.78	800	485	1948–2020
Kayseri	38.69	35.50	1094	390	1950–2020
Kırıkkale	39.84	33.52	751	383	1967–2020
Kırklareli	41.74	27.22	232	582	1966–2020
Kırşehir	39.16	34.16	1007	382	1942–2020
Kilis	36.71	37.11	640	499	1966–2020
Kocaeli	40.77	29.92	74	814	1945–2020
Konya	37.98	32.57	1031	328	1950–2020
Kütahya	39.42	29.99	969	328	1941–2020
Malatya	38.34	38.22	950	384	1958–2020
Manisa	38.62	27.41	71	742	1958–2020
Mersin	36.78	34.60	7	610	1958–2020
Mardin	37.31	40.73	1040	673	1966–2020
Muğla	37.21	28.37	646	862	1944–2020
Muş	38.75	41.50	1322	759	1966–2020
Nevşehir	38.62	34.70	1260	422	1965–2020
Niğde	37.96	34.68	1211	343	1959–2020
Ordu	40.98	37.89	5	1050	1965–2020
Osmaniye	37.10	36.25	94	817	1974–2020
Rize	41.04	40.50	3	2301	1940–2020
Sakarya	40.77	30.39	30	844	1962–2020
Samsun	41.34	36.26	4	722	1957–2020
Siirt	37.93	41.94	895	715	1959–2020
Sinop	42.03	35.16	32	692	1965–2020
Sivas	39.74	37.00	1294	430	1958–2020
Tekirdağ	40.96	27.50	4	578	1963–2020
Tokat	40.33	36.56	611	435	1966–2020
Trabzon	40.99	39.77	33	829	1966–2020
Tunceli	39.16	39.54	914	871	1966–2020
Şanlıurfa	37.16	38.79	550	459	1959–2020
Şırnak	37.52	42.45	1375	720	1959–2020
Uşak	38.67	29.40	919	558	1941–2020
Van	38.47	43.35	1675	395	1956–2020
Yalova	40.66	29.28	4	755	1962–2020
Yozgat	39.82	34.82	1301	572	1960–2020
Zonguldak	41.45	31.78	135	1226	1945–2020

). The distribution of meteorological stations across the country is given in the map in Figure 1.

2.3. Probability Distributions Considered

Frequency analysis is the most important tool used by hydrologists and meteorologists when rainfall projections are made in relation to a specific return period or frequency in the design of water structures and especially rainwater drainage systems; in the risk analysis of natural events such as floods and landslides; and in the analysis of droughts and climate change, which have become increasingly important in recent decades. In the frequency analysis of extreme rainfall events, the probability distributions that best represent the characteristics of maximum rainfall data of various standard durations (5, 10, …, 30 min, 1 h, 2 h, …, 24 h) can be determined using theoretical probability distribution functions such as generalized extreme value (GEV) [39,40,41], Pearson type III (PE3) [21,42], Lognormal (LN) [27,37,43], Log-Pearson type III (LP3) [25,36,44], Generalized Logistic (GLO) [38,45], Wakeby (WAK) [46,47,48,49], Normal (NOR) [38,45], and Generalized Pareto distribution (GPA) [50,51].

In the study, the 3-parameter versions of the 2- and 3-parameter distributions were preferred (LN3 instead of LN2, PE3 instead of Gamma, and GEV instead of GUM). Thus, the PE3, GLO, GEV, LN3, LP3, NOR, GPA, and WAK probability distribution functions were considered in accordance with the studies of maximum rainfall in the literature.

Table 1. The probability density functions and the parameters of the distributions.

Distribution	Probability Density Function	Parameters
GEV	$f\left(x\right)={\alpha }^{-1}\mathit{e}\mathit{x}\mathit{p}\left[-\left(1-\kappa \right)y-\mathit{e}\mathit{x}\mathit{p}\left(-y\right)\right]$ $wherey=-{\kappa }^{-1}\mathit{l}\mathit{o}\mathit{g}\left[1-\kappa {\displaystyle \frac{x-\xi }{\alpha }}\right],\kappa \ne 0$ $wherey={\displaystyle \frac{x-\xi }{\alpha }},\kappa =0$	$\xi :location$ $\alpha :scale$ $\kappa :shape$
GLO	$f\left(x\right)={\alpha }^{-1}{\displaystyle \frac{\mathit{e}\mathit{x}\mathit{p}\left(-\left(1-\kappa \right)y\right)}{{\left(1+\mathit{e}\mathit{x}\mathit{p}\left(-y\right)\right)}^2}}$ $wherey=-{\kappa }^{-1}\mathit{l}\mathit{o}\mathit{g}\left[1-\kappa {\displaystyle \frac{x-\xi }{\alpha }}\right],\kappa \ne 0$ $wherey={\displaystyle \frac{x-\xi }{\alpha }},\kappa =0$	$\xi :location$ $\alpha :scale$ $\kappa :shape$
P3	$f\left(x\right)={\displaystyle \frac{{\left(x-\xi \right)}^{\alpha -1}\mathit{e}\mathit{x}\mathit{p}\left(-\frac{x-\xi }{\beta }\right)}{{\beta }^{\alpha }\Gamma \left(\alpha \right)}}$ $wherex\ge \xi$	$\xi :location$ $\alpha :scale$ $\kappa :shape$
LP3	$f\left(y\right)={\displaystyle \frac{{\left(y-\xi \right)}^{\alpha -1}\mathit{e}\mathit{x}\mathit{p}\left(-\frac{y-\xi }{\beta }\right)}{{\beta }^{\alpha }\Gamma \left(\alpha \right)}}$ $wherey\ge \xi andy=lnx$	$\xi :location$ $\alpha :scale$ $\kappa :shape$
LN3	$f\left(x\right)={\displaystyle \frac{1}{\left(x-\gamma \right)\sigma \sqrt{2\pi }}}\mathit{e}\mathit{x}\mathit{p}\left[-{\displaystyle \frac{{\left(\mathit{ln}\left(x-\gamma \right)-\mu \right)}^2}{2{\sigma }^2}}\right]$ $wherey<x<\infty$	$\gamma :location$ $\alpha :scale$ $\kappa :shape$
GPA	$f\left(x\right)={\alpha }^{-1}\mathit{e}\mathit{x}\mathit{p}\left(-\left(1-\kappa \right)y\right)$ $wherey=-{\kappa }^{-1}\mathit{l}\mathit{o}\mathit{g}\left[1-\kappa {\displaystyle \frac{x-\xi }{\alpha }}\right],\kappa \ne 0$ $wherey={\displaystyle \frac{x-\xi }{\alpha }},\kappa =0$	$\xi :location$ $\alpha :scale$ $\kappa :shape$
NOR	$2{\pi }^{-1/2}{\alpha }^{-1}exp\left(\partial y-y^2/2\right)$ $wherey=-{\kappa }^{-1}\mathit{l}\mathit{o}\mathit{g}\left[1-\kappa {\displaystyle \frac{x-\xi }{\alpha }}\right],\kappa \ne 0$ $wherey={\displaystyle \frac{x-\xi }{\alpha }},\kappa =0$	$\xi :location$ $\alpha :scale$ $\kappa :shape$
WAK	$f\left(x\right)={\displaystyle \frac{{\left(1-F\left(x\right)\right)}^{\left(\delta +1\right)}}{\alpha (1-{F\left(x\right)}^{(\beta +\delta )})+\gamma }}$	$\xi :location$ $\alpha ,\gamma :scale$ $\beta ,\delta :shape$

shows the theoretical probability density function $f\left(x\right)$ of the random variable (x, here corresponding to maximum rainfall) and the parameters of the 8 probability distributions considered in the study [7]

2.4. Methods for Selecting the Best Fit Distributions

There are many numerical tests in the literature that are based on different principles for testing the suitability of probability distributions. The comprehensive literature review we provided in the Section 4 discusses numerous goodness of fit tests. The most commonly used of these are the Kolmogorov–Smirnov, Anderson–Darling, and X2 tests. Furthermore, the Z^DIST statistic, proposed in the L-Moment’s frequency analysis procedure, is also frequently used. There are other methods that compare data obtained from empirical probabilities with data from theoretical probabilities of distributions (Q-Q plots, RMSE, RRMSE, MAE, CC, PPCC, BIAS, etc.).

The Chi-square test was not considered due to its drawbacks, such as being sensitive to the number of classes and boundaries, being overly sensitive in large samples, and losing the detailed structure of the original distribution during classification. The methods (Q-Q plots, RMSE, RRMSE, MAE, CC, PPCC, BIAS, etc.) that compare data derived from empirical probabilities with data derived from theoretical probabilities of relevant distributions yield results representing the same idea. So, PPCC was selected to represent this category.

Thus, AD, KS, PPCC, and the L-Moment goodness of fit statistic (Z^DIST) were included in the analyses.

2.4.1. Kolmogorov–Smirnov (K-S) Test

The Kolmogorov–Smirnov (K-S) test is a widely used non-parametric method for assessing the goodness of fit between observed hydrological data and a theoretical probability distribution. The K-S test compares the empirical cumulative distribution function $F^{*}\left(x_i\right)$ of the sample with the theoretical cumulative distribution function $F\left(x_i\right)$, calculating the maximum absolute difference between them. The test statistic is defined as follows:

$${\Delta }_{max}=max\left|F^{*}\left(x_i\right)-F\left(x_i\right)\right|$$

(1)

The computed ${\Delta }_{max}$ value is compared against critical values ${\Delta }_{\alpha }$ of the probability distributions based on sample size and a chosen significance level. If ${\Delta }_{max}$ > ${\Delta }_{\alpha }$, the null hypothesis is rejected, indicating that the observed data do not follow the specified theoretical distribution [52].

2.4.2. Anderson–Darling (A-D) Test

The Anderson–Darling (A-D) test is a powerful non-parametric goodness of fit method used to evaluate the fitness of a theoretical distribution to the observed data. It is similar to the K-S test but is particularly more sensitive to deviations in the tails of the distribution. The test statistic is defined as follows:

$$A^2=-n-{\displaystyle \frac{1}{n}}\sum _{i=1}^n\left[\left(2i-1\right)\left(lnF\left(x_i\right)+ln(1-F(x_{n+1-i})\right))\right]$$

(2)

where n is the sample size, F(x) is the theoretical cumulative distribution function, and $x_i$ are the ordered data values. The calculated $A^2$ value is compared to critical values of the considered theoretical distribution. If the test statistic exceeds the critical value, the null hypothesis is rejected, indicating that the data do not follow the specified distribution [53].

2.4.3. Probability Plot Correlation Coefficient (PPCC) Test

The Probability Plot Correlation Coefficient (PPCC) test, introduced by [54], is widely recognized as a simple yet statistically powerful method for evaluating the goodness of fit between observed data and theoretical distribution.

In the PPCC method, the sample is ordered, and empirical probabilities are matched with the theoretical quantiles of the assumed distribution. The correlation coefficient (r) between the ordered data and the theoretical quantiles is calculated as follows:

$$r={\displaystyle \frac{\sum (x_i-\overline{x}\left)\right(y_i-\overline{y})}{\sqrt{\sum ({x_i-\overline{x})}^2\sum {(y_i-\overline{y})}^2}}}$$

(3)

where $x_i$ are the ordered observations, $y_i$ are the theoretical quantiles, and $\overline{x}$ and $\overline{y}$ are the means of observed data and theoretical quantiles, respectively.

A PPCC value close to 1 indicates a good fit between the observed data and the assumed distribution. Multiple distributions can be evaluated by calculating their respective PPCC values; the highest value typically suggests the best fitting distribution [54].

2.4.4. Goodness of Fit Measure (Z^DIST) of L-Moment Method

Determining the ideal statistical distribution for a dataset is a foundational task, essential for creating reliable predictive models. This technique is grounded in L-Moments, which synthesize information from data order in a way that is less vulnerable to extreme values and more stable with limited samples. The $Z^{DIST}$ measure capitalizes on these properties to deliver a clear, quantitative verdict on which distribution fits best [6,55,56].

The procedure evaluates how closely a dataset’s L-moment ratios align with the expected values from a theoretical probability distribution. The formula that drives this comparison is as follows:

$$Z^{DIST}=(t_4-{\tau }_4)/{\sigma }_4$$

(4)

In this expression, $t_4$ represents the regional average L-kurtosis of the observed data, ${\tau }_4$ is the theoretical L-kurtosis of the proposed distribution, and ${\sigma }_4$ stands for the standard deviation of the sample L-kurtosis, often derived through simulation. The distribution that yields a $Z^{DIST}$ value closest to zero is judged to be the most appropriate match. A lower absolute $Z^{DIST}$ value indicates a closer fit. By computing $Zdist$ for a suite of potential distributions, the model with the smallest absolute $Z^{DIST}$ can be objectively identified as the best fitting distribution for the given dataset, streamlining the model selection process.

2.4.5. Conjunctive Evaluation of Selecting Criteria

In this section, the results of three goodness of fit tests are evaluated together to determine the most appropriate probability distribution for the data. In the K-S, A-D, and Z^DIST tests, the probability distribution that yields the smallest test statistic is ranked as the most appropriate (1st); in the PPCC test, the distribution that yields the largest test statistic is ranked as the most appropriate distribution. In this way, the fit performances of the eight probability distributions are ranked from best to worst (1 to 8). To evaluate the tests together, an average rank number is calculated for each distribution, the smallest of which indicates the most appropriate distribution. The eight distributions in

Table 2. Goodness of fit ranking results of 24 h maximum rainfall for Adiyaman station.

	Rank
Distribution	KS	AD	PPCC	ZDIST	Average
WAK	1	1	1	2	1.25
GEV	1	1	3	6	2.75
LP3	7	7	2	3	4.75
PE3	2	3	4	7	4.00
GLO	3	2	6	1	3.00
LN3	4	4	7	4	4.75
NOR	6	5	5	5	5.25
GPA	5	6	8	8	6.75

are ranked by the conjunctive evaluation of four goodness of fit tests. The results for 24 h maximum rainfall measured at Adiyaman meteorological station are given as an example, and the minimum average ranking grade of the most appropriate probability distribution is shown in bold.

2.5. Stationary Analysis

One of the significant mistakes in frequency analysis of hydro-meteorological extremes is the assumption that the phenomenon under study is constant over time. However, it is an undeniable fact that hydro-meteorological variables are not always stationary under the influence of climate change.

To demonstrate the stationarity of maximum rainfall, a Mann–Kendall trend analysis was conducted for seven rainfall durations across 81 provinces. Mann–Kendall is a non-parametric test for determining whether monotonic trends exist in time series. The null hypothesis (H0) assumes there is no trend and that the data are independent and identically distributed. The alternative hypothesis (H₁) proposes the presence of a trend—either increasing or decreasing. The Mann–Kendall statistic (S) is computed to reveal the trend direction.

$$\begin{gathered} S=\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}sgn(x_j-x_i) \\ sgn\left(x_j-x_i\right)=\left\{\begin{array}{c}+1,if\left(x_j-x_i\right)>0\\ 0,if\left(x_j-x_i\right)=0\\ -1,if\left(x_i-x_j\right)>0\end{array}\right. \end{gathered}$$

(5)

In the equations, n represents the total number of data, and $x_i$ and $x_j$ are the observed data at times i and j, respectively (j > i). The sign (sgn) function is computed with Equation (2). A positive value of S suggests an increasing (positive) trend, while a negative value of S indicates a decreasing (negative) trend.

When ties occur among data values and the dataset contains more than 10 elements, the variance is estimated using the following formula, assuming a normal distribution:

$$Var\left(S\right)={\displaystyle \frac{n\left(n-1\right)\left(2n+5\right)-\sum _{i=1}^P{t}_i(t_i-1\left)\right({2t}_i+5)}{18}}$$

(6)

where P denotes the number of tied groups and $t_i$ is the number of data points in the i-th tied group. The standardized $Z$ value is then calculated as

$$Z=\left\{\begin{array}{c}{\displaystyle \frac{S-1}{\sqrt{\mathrm{Var}\left(S\right)}}};IfS>0\\ 0;IfS=0\\ {\displaystyle \frac{S-1}{\sqrt{\mathrm{Var}\left(S\right)}}};IfS<0\end{array}\right.$$

(7)

3. Results

3.1. Best Fit Probability Distributions for Each Rainfall Durations

The average ranks of three tests are calculated for seven standard rainfall durations of 81 stations, as shown in

Table 3. The mean values of average ranks of 81 stations for seven rainfall durations.

Distribution	15 min	30 min	1 h	3 h	6 h	12 h	24 h
WAK	2.2	2.2	2.6	2.4	2.2	2.4	2.3
GEV	2.9	3.0	3.0	2.9	2.9	2.8	3.0
LP3	3.0	3.2	3.0	3.4	3.4	3.3	3.7
PE3	4.7	4.7	4.5	4.4	4.8	4.7	4.7
GLO	4.7	4.6	4.9	4.7	4.6	4.6	4.1
LN3	5.3	5.2	4.9	4.9	5.0	5.0	4.9
NOR	6.5	6.6	6.6	6.6	6.6	6.6	6.7
GPA	6.7	6.6	6.5	6.7	6.6	6.7	6.7

. The mean values of average ranks are computed for 81 stations and given for seven durations in Table 3 and Figure 2. Figure 2 shows the best probability distribution for each rainfall duration.

According to Figure 2, WAK distribution is the best fit for each duration. GEV, LP3, PE3, GLO, LN3, NOR, and GPA take the second, third, fourth, fifth, sixth, seventh, and eighth places, respectively.

3.2. Best Fit Probability Distributions by Provinces

Considering the 81 provinces of Türkiye, the best fit probability distributions of each province are determined for seven rainfall durations according to the ranking. The best fit distributions by provinces across the country are determined for seven rainfall durations and presented in Appendix B (

Table A2. Goodness of fit of maximum rainfall for seven rainfall durations and overall for 81 provinces.

Province	Rainfall Duration
	15 min	30 min	1 h	3 h	6 h	12 h	24 h	Overall
Adana	GEV	GEV	WAK	WAK	WAK	GEV	WAK	WAK
Adıyaman	WAK	WAK	GLO	GLO	GLO	WAK	WAK	WAK
Afyonkarahisar	WAK	GEV	WAK	WAK	GEV	GEV	WAK	WAK
Ağrı	GEV	WAK	WAK	WAK	WAK	WAK	WAK	WAK
Aksaray	WAK	WAK	GEV	WAK	WAK	GEV	WAK	WAK
Amasya	GLO	WAK	WAK	GLO	GEV	GEV	WAK	GLO
Ankara	WAK	LP3	GEV	WAK	WAK	WAK	WAK	WAK
Antalya	WAK	WAK	GEV	GEV	WAK	WAK	WAK	WAK
Ardahan	GEV	WAK	WAK	WAK	WAK	WAK	LP3	WAK
Artvin	WAK	WAK	GEV	GEV	WAK	WAK	GEV	WAK
Aydın	WAK	WAK	WAK	LP3	LP3	GEV	WAK	WAK
Balıkesir	GEV	WAK	WAK	WAK	WAK	WAK	WAK	WAK
Bartın	GEV	GEV	LP3	WAK	GEV	GEV	WAK	GEV
Batman	WAK	WAK	GEV	GEV	WAK	WAK	GEV	WAK
Bayburt	GEV	WAK	WAK	WAK	GEV	WAK	GEV	WAK
Bilecik	WAK	WAK	LP3	WAK	WAK	GEV	GEV	WAK
Bingöl	WAK	WAK	WAK	WAK	WAK	WAK	GLO	WAK
Bitlis	WAK	GEV	LP3	GLO	GEV	WAK	WAK	WAK
Bolu	GEV	LP3	WAK	WAK	WAK	WAK	WAK	WAK
Burdur	WAK	WAK	WAK	WAK	WAK	WAK	WAK	WAK
Bursa	WAK	WAK	GEV	WAK	GEV	GEV	GLO	WAK
Çanakkale	WAK	GEV	WAK	GEV	GEV	WAK	WAK	GEV
Çankırı	LP3	GEV	WAK	GLO	WAK	WAK	WAK	WAK
Çorum	WAK	WAK	WAK	WAK	GEV	WAK	GEV	WAK
Denizli	WAK	WAK	LP3	WAK	WAK	GEV	WAK	WAK
Diyarbakır	WAK	WAK	WAK	WAK	WAK	WAK	GEV	WAK
Düzce	GEV	WAK	WAK	WAK	GEV	WAK	WAK	WAK
Edirne	WAK	WAK	GEV	WAK	WAK	WAK	GEV	WAK
Elazığ	WAK	WAK	WAK	WAK	WAK	WAK	GEV	WAK
Erzincan	WAK	WAK	WAK	WAK	GEV	WAK	GLO	WAK
Erzurum	LP3	WAK	GEV	GEV	GLO	GLO	GEV	GEV
Eskişehir	LP3	GEV	GEV	WAK	WAK	WAK	GEV	WAK
Gaziantep	LP3	WAK	WAK	WAK	LP3	GEV	WAK	WAK
Giresun	WAK	LP3	WAK	PE3	WAK	WAK	LP3	WAK
Gümüşhane	WAK	WAK	WAK	WAK	WAK	GEV	GEV	WAK
Hakkari	WAK	WAK	WAK	WAK	WAK	WAK	WAK	WAK
Iğdır	WAK	GEV	GEV	WAK	WAK	GEV	WAK	WAK
Isparta	WAK	WAK	WAK	WAK	WAK	GLO	WAK	WAK
İstanbul	WAK	GEV	GEV	GEV	WAK	WAK	LP3	WAK
İzmir	GLO	WAK	GEV	WAK	WAK	WAK	GEV	WAK
Kahramanmaraş	WAK	WAK	GLO	GEV	WAK	WAK	WAK	WAK
Karabük	WAK	WAK	WAK	LP3	WAK	WAK	LN3	WAK
Karaman	LP3	LP3	WAK	GEV	WAK	WAK	WAK	WAK
Kars	GEV	WAK	GEV	WAK	GEV	GLO	GEV	WAK
Kastamonu	WAK	WAK	WAK	LP3	WAK	WAK	LP3	WAK
Kayseri	WAK	LP3	WAK	WAK	WAK	WAK	WAK	WAK
Kırıkkale	WAK	WAK	WAK	LP3	GLO	GLO	WAK	WAK
Kırklareli	WAK	GLO	WAK	WAK	WAK	WAK	GEV	WAK
Kırşehir	GEV	WAK	GEV	WAK	WAK	WAK	WAK	WAK
Kilis	WAK	WAK	WAK	GEV	WAK	WAK	WAK	WAK
Kocaeli	WAK	WAK	LP3	WAK	WAK	WAK	WAK	WAK
Konya	GEV	WAK	WAK	GEV	WAK	GEV	WAK	WAK
Kütahya	GEV	WAK	LP3	WAK	WAK	GEV	GEV	GEV
Malatya	GEV	WAK	GLO	GEV	WAK	WAK	GEV	WAK
Manisa	WAK	GEV	GEV	WAK	WAK	GEV	WAK	WAK
Mardin	LP3	WAK	WAK	LP3	GEV	WAK	WAK	WAK
Mersin	WAK	GEV	GEV	GEV	WAK	WAK	PE3	WAK
Muğla	WAK	WAK	GEV	GLO	GLO	GEV	WAK	WAK
Muş	WAK	GEV	LP3	WAK	GLO	GEV	WAK	WAK
Nevşehir	WAK	WAK	WAK	WAK	WAK	GEV	WAK	WAK
Niğde	WAK	WAK	WAK	WAK	WAK	GLO	WAK	WAK
Ordu	WAK	WAK	WAK	WAK	WAK	GEV	WAK	WAK
Osmaniye	GEV	WAK	WAK	WAK	WAK	GEV	WAK	WAK
Rize	GEV	WAK	WAK	LP3	LP3	WAK	WAK	WAK
Sakarya	GEV	WAK	WAK	WAK	GEV	GEV	WAK	WAK
Samsun	GEV	WAK	WAK	WAK	LP3	WAK	WAK	WAK
Siirt	WAK	GEV	PE3	WAK	WAK	WAK	GLO	WAK
Sinop	WAK	GEV	WAK	WAK	WAK	LP3	WAK	WAK
Sivas	WAK	WAK	WAK	GEV	WAK	GEV	WAK	WAK
Şanlıurfa	WAK	GLO	LP3	PE3	GEV	GLO	WAK	WAK
Şırnak	WAK	GEV	PE3	WAK	WAK	WAK	GLO	WAK
Tekirdağ	WAK	WAK	GEV	WAK	WAK	WAK	WAK	WAK
Tokat	WAK	WAK	WAK	GEV	GEV	GEV	WAK	WAK
Trabzon	WAK	LP3	WAK	PE3	WAK	WAK	LN3	WAK
Tunceli	LP3	WAK	WAK	WAK	WAK	WAK	GLO	WAK
Uşak	WAK	GEV	WAK	GEV	GEV	GEV	WAK	GEV
Van	WAK	WAK	WAK	WAK	WAK	WAK	WAK	WAK
Yalova	WAK	WAK	WAK	WAK	WAK	WAK	LN3	WAK
Yozgat	LN3	LN3	LN3	LN3	GEV	LN3	LN3	LN3
Zonguldak	PE3	PE3	PE3	GEV	WAK	LN3	LN3	LN3

). The ratio of each distribution is calculated and given in

Table 4. Best fit distribution rates of different rainfall durations in Türkiye.

Distribution	15 min	30 min	1 h	3 h	6 h	12 h	24 h	Overall
WAK	65%	67%	59%	62%	68%	59%	65%	90%
GEV	21%	21%	22%	20%	21%	30%	21%	6%
LP3	9%	7%	10%	7%	5%	1%	9%	0%
PE3	1%	1%	4%	4%	0%	0%	1%	0%
GLO	2%	2%	4%	6%	6%	7%	2%	1%
LN3	1%	1%	1%	1%	0%	2%	1%	2%
NOR	0%	0%	0%	0%	0%	0%	0%	0%
GPA	0%	0%	0%	0%	0%	0%	0%	0%

. For example, the WAK distribution for maximum rainfall of 15 min duration was the best fit for 53 of the 81 provinces (65%) across the country. In addition, the probability distribution maps for seven rainfall durations and the overall (average) of 81 provinces are illustrated in Figure 3.

Considering rainfall durations, the best fit distribution in Türkiye is the WAK distribution (approximately in the 60–65% range). This is followed by the GEV distribution (approximately 20%). These two distributions are followed by the LP3 distribution and the GLO distribution, with the ranges between 5% and 10%. Only in a few provinces maximum rainfall is represented by the PE3 and LN3 distributions. NOR and GPA failed to represent maximum rainfall in any province or rainfall duration. On the other hand, in the overall map obtained with minimum rank averages, the WAK distribution dominated the other distributions at a rate of 90%.

3.3. Best Fit Probability Distribution in Türkiye

In this country-based step, the averages of the ranks of the seven standard durations and 81 provinces were calculated. In this way, eight series, each consisting of 81 data points, were obtained for eight probability distributions. Figure 4 shows the overall ranking results as a box plot diagram for eight PDFs.

According to Figure 4, both the box plot and median values show that WAK is the best fit probability distribution in terms of overall performance.

On the other hand, WAK (at the significance level of α = 0.05) was accepted in four numerical tests of a total of 567 data series consisting of seven different rainfall durations of 81 provincial stations.

3.4. Best Fit Probability Distributions According to Goodness of Fit Tests

Finally, to evaluate the effect of the goodness of fit test in determining the best fit probability distribution, the overall ranks given to the distributions by four different tests and their average were obtained and the results are given in

Table 5. Average ranks of KS, AD, PPCC, and Z^DIST tests the probability distributions.

Distribution	KS	AD	PPCC	ZDIST	Average
WAK	2.39	2.82	1.72	1.22	2.04
GEV	3.27	3.08	3.03	2.48	2.96
LP3	3.98	3.55	3.46	2.97	3.49
PE3	4.33	3.79	3.54	5.03	4.17
GLO	4.38	3.98	3.95	5.43	4.44
LN3	4.62	4.51	5.80	5.27	5.05
NOR	6.57	6.58	6.70	6.56	6.60
GPA	6.46	7.69	7.79	7.04	7.24

and Figure 5.

According to Table 5 and Figure 5, the KS, AD, PPCC, and Z^DIST tests showed similar results with minor differences (seventh and eighth places in the ranking of the KS test; fifth and sixth places in the ranking of the Z^DIST test). The WAK distribution is the best fit according to four tests, and the last column, which averages the minimum ranks of all tests, significantly ranks the WAK distribution as first. GEV, LP3, PE3, GLO, LN3, NOR, and GPA take the second, third, fourth, fifth, sixth, seventh, and eighth places in average, respectively.

3.5. Stationary Analysis Results

Due to the influence of global warming, it is important to examine the changes in hydro-meteorological variables over time. In this study, the maximum rainfall series of seven rainfall durations from meteorological stations located in 81 provinces were examined for stationarity. The results are presented in Appendix C (

Table A3. Stationarity analysis of maximum rainfall series.

	15 min	30 min	1 h	3 h	6 h	12 h	24 h
Adana	stationary	stationary	stationary	stationary	stationary	nonstationary	stationary
Adıyaman	stationary	stationary	stationary	nonstationary	stationary	stationary	stationary
Afyonkarahisar	nonstationary	nonstationary	nonstationary	nonstationary	stationary	stationary	stationary
Ağrı	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Aksaray	stationary	nonstationary	nonstationary	nonstationary	stationary	stationary	nonstationary
Amasya	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Ankara	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Antalya	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Ardahan	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Artvin	nonstationary	nonstationary	nonstationary	nonstationary	stationary	stationary	stationary
Aydın	nonstationary	nonstationary	nonstationary	nonstationary	nonstationary	nonstationary	nonstationary
Balıkesir	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Bartın	stationary	stationary	stationary	nonstationary	nonstationary	nonstationary	nonstationary
Batman	stationary	stationary	stationary	stationary	nonstationary	nonstationary	stationary
Bayburt	nonstationary	nonstationary	nonstationary	stationary	stationary	stationary	stationary
Bilecik	stationary	stationary	stationary	nonstationary	nonstationary	stationary	stationary
Bingöl	nonstationary	stationary	stationary	nonstationary	nonstationary	stationary	stationary
Bitlis	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Bolu	stationary	stationary	nonstationary	nonstationary	stationary	stationary	stationary
Burdur	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Bursa	stationary	nonstationary	nonstationary	nonstationary	nonstationary	stationary	stationary
Çanakkale	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Çankırı	stationary	stationary	stationary	stationary	stationary	nonstationary	stationary
Çorum	stationary	stationary	stationary	nonstationary	nonstationary	stationary	stationary
Denizli	stationary	stationary	stationary	stationary	nonstationary	nonstationary	stationary
Diyarbakır	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Düzce	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Edirne	stationary	stationary	nonstationary	nonstationary	nonstationary	nonstationary	nonstationary
Elazığ	stationary	stationary	stationary	stationary	nonstationary	nonstationary	stationary
Erzincan	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Erzurum	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Eskişehir	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Gaziantep	stationary	stationary	stationary	stationary	stationary	stationary	nonstationary
Giresun	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Gümüşhane	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Hakkari	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Hatay	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Iğdır	nonstationary	stationary	stationary	stationary	stationary	stationary	stationary
Isparta	stationary	stationary	stationary	stationary	stationary	stationary	stationary
İstanbul	stationary	stationary	stationary	stationary	stationary	stationary	stationary
İzmir	nonstationary	nonstationary	nonstationary	nonstationary	nonstationary	nonstationary	nonstationary
Kahramanmaraş	stationary	stationary	stationary	stationary	nonstationary	stationary	stationary
Karabük	nonstationary	nonstationary	nonstationary	stationary	nonstationary	nonstationary	nonstationary
Karaman	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Kars	stationary	stationary	stationary	stationary	stationary	stationary	nonstationary
Kastamonu	nonstationary	nonstationary	nonstationary	stationary	nonstationary	nonstationary	nonstationary
Kayseri	nonstationary	nonstationary	nonstationary	nonstationary	nonstationary	nonstationary	stationary
Kırıkkale	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Kırklareli	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Kırşehir	stationary	nonstationary	nonstationary	stationary	nonstationary	nonstationary	nonstationary
Kilis	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Kocaeli	nonstationary	nonstationary	nonstationary	nonstationary	nonstationary	stationary	stationary
Konya	stationary	stationary	stationary	stationary	stationary	stationary	nonstationary
Kütahya	stationary	nonstationary	nonstationary	nonstationary	nonstationary	nonstationary	nonstationary
Malatya	nonstationary	stationary	stationary	stationary	stationary	stationary	stationary
Manisa	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Mardin	stationary	stationary	stationary	stationary	stationary	stationary	nonstationary
Mersin	stationary	stationary	stationary	nonstationary	nonstationary	nonstationary	nonstationary
Muğla	nonstationary	nonstationary	nonstationary	nonstationary	stationary	stationary	stationary
Muş	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Nevşehir	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Niğde	stationary	nonstationary	nonstationary	stationary	stationary	stationary	stationary
Ordu	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Osmaniye	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Rize	nonstationary	nonstationary	nonstationary	nonstationary	nonstationary	nonstationary	stationary
Sakarya	nonstationary	nonstationary	nonstationary	nonstationary	nonstationary	nonstationary	stationary
Samsun	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Siirt	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Sinop	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Sivas	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Şanlıurfa	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Şırnak	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Tekirdağ	stationary	stationary	stationary	nonstationary	stationary	stationary	stationary
Tokat	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Trabzon	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Tunceli	nonstationary	stationary	stationary	nonstationary	nonstationary	stationary	stationary
Uşak	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Van	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Yalova	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Yozgat	stationary	stationary	stationary	stationary	stationary	stationary	stationary
Zonguldak	stationary	stationary	stationary	stationary	stationary	stationary	stationary

), and it was concluded that stationarity conditions were not met in 21 of the 81 provinces (province names in bold). It is recommended that non-stationary frequency analysis be conducted in future frequency analysis studies for Türkiye’s maximum rainfall.

4. Discussion

To understand the characteristics of hydro-meteorological or climatic events, it is of great importance to determine the probability distribution of the considered phenomenon properly. The subject of determining the appropriate probability distribution has been the common objective of many scientific studies in different parts of the world to rationally estimate the frequency and intensity of the event.

Table 6. Summary of recent studies on determination of probability distribution of maximum rainfall.

Study	Rainfall	Site/Region—Country	Goodness of Fit Test	Best Fit PDF
[28]	sub-daily and daily max.	Black Sea region—Türkiye	AD	GEV
[32]	5-day max.	Zhujiang River Basin—China	KS, AD, χ2	WAK
[25]	1 h max.	Japan	χ2	LP3
[41]	daily max.	Lazzio, Sicily—Italy	RMSE, KS	GEV
[46]	5 min, 1 h max.	Southern Quebec—Canada	Q-Q plot, RMSE, RRMSE, MAE, CC	WAK, GEV, NOR
[57]	daily max.	Northern—Algeria	KS, Q-Q plot	GUM, GEV
[44]	sub-daily, daily, 2-,3-day max.	New South Wales—Australia	KS, AD, χ2	LP3, GEV
[58]	daily max.	Upper Vistula Basin, Poland	RMSE, R2,PWRMSE	GEV
[38]	sub-daily and daily max.	Inland (Central) Anatolia—Türkiye	ZDIST	GLO, NOR, GEV
[45]	sub-daily and daily max.	Umbria Region—Italy	ZDIST	GLO, NOR, GEV
[37]	sub-daily and daily max.	Aegean region—Türkiye	KS, AD, χ2	GAM, LN2, GEV
[42]	monthly and annual daily max.	Ngong River Basin—Kenya	KS, AD, Cramér–von Mises	PE3, GEV
[39]	sub-daily and daily max.	Arizona—USA	Cramér–von Mises, Lilliefors, AD	GEV
[21]	sub-daily and daily max.	Ontario region—Canada	RMSE, RRMSE, CC MAE, AIC, BIC	PE3, GEV, GNO
[59]	sub-daily and daily max	Ontario region —Canada	Q-Q plot, RMSE, RRMSE, MAE, CC	GEV
[36]	daily max.	Jhelum River basin—India	KS, AD, χ2, RMSE, Q-Q plot	LP3, GEV
[60]	daily max.	SE and NE USA	AD	WAK
[61]	daily max.	Amman Zara Basin—Jordan	KS	GLO, GEV, NOR
[27]	daily max.	Egypt	RMSE, RRMSE, CC, BIASr, AIC, BIC	LP3, LN2, EXP
[43]	sub-daily and daily max.	Sicily region—Italy	ZDIST	LN3, GEV

EXP, Exponential; GEV, generalized extreme value; NOR, normal; GPA, Generalized Pareto; GUM, Gumbel; PE3, Pearson Type III; LP3, Log-Pearson Type III; WAK, Wakeby; KAP, Kappa; LN2, 2 Parameter Log-Normal; LN3, 3 Parameter Log-Normal; GAM, 2 Parameter Gamma; GLO, Generalized Logistic; χ2, Chi-square; AIC, Akike Information Criteria; CC, Correlation Coefficient; RMSE, Root Mean Square Error; RRMSE, Relative Root Mean Square Error; MAE, Maximum Absolute Error; K-S, Kolmogorov–Smirnov; AD, Anderson–Darling; BIC, Bayesian Information Criteria; Relative BIAS (BIASr); Q-Q Plot; Z^DIST.

provides a summary of recent studies examining the probability distributions of maximum rainfall around the world.

In this study, the best fit probability distribution for Turkey’s maximum rainfall data was evaluated from three different perspectives (duration-based, province-based, country-based). In Section 3, the results indicate the best fit probability distribution for maximum rainfall in Türkiye as WAK, on a rainfall duration basis (Table 3 and Table 4 and Figure 2 and Figure 3), on a province basis (Table 4; Figure 3), and on a country basis (Figure 4).

Ref. [60] showed WAK distribution as the best fit for southeastern and northeastern USA; Ref. [47] presented WAK as the best fit for Zhujiang River Basin, China; and Ref. [46] determined WAK as one of the best representative distributions for Southern Quebec, Canada.

The GEV distribution, which stands out as the second-best distribution in this study, was found to be the best fit by [28,39,41,58] for maximum rainfall data.

Refs. [25,27,44] showed the LP3 distribution as the best fit for maximum rainfall. LP3 distribution is widely used in Japan, Australia, the USA, and Canada and ranked third in our study.

When we look at the studies conducted for Türkiye, the results of our study differ from previous studies. In the study by [38] for Central Anatolia, the Z^DIST method showed GLO, GEV, and GNO; in the study by [37] for the Aegean region, the KD, AD, and X² methods showed GAM, GEV, and LN2; and in the study by [28] for the Black Sea region, the AD method showed the GEV distribution as the best fit. A detailed examination of these studies reveals that the WAK distribution is not considered. Furthermore, the State Meteorological Service of Türkiye does not use the WAK distribution as a candidate for maximum rainfall.

The previous studies in Table 6 are examined in detail, and it is noted that scientists generally tend to consider the GEV distribution as a candidate distribution when examining maximum rainfall. However, Ref. [6] stated that the Wakeby distribution is capable of representing a wide range of distributional shapes due to its high number of parameters. This flexibility increases its success in representing the tails of probability density functions with great precision, especially in extreme value analyses. These tails contain the most important information about extreme value series. This may be due to the mathematical structure of the 5-parameter WAK distribution being more complex than the GEV.

While some studies in the literature use a single test to determine best fit probability distributions, the use of two or more tests is recommended to reduce the bias of the methods.

5. Conclusions

Reasonable estimation of maximum rainfall is crucial for taking precautions against extreme natural events such as floods and erosion and is also important for the design of water structures and, in particular, rainwater drainage projects. Perhaps the most crucial step in estimating maximum rainfall with a specific frequency or intensity is accurately defining the probabilistic characteristics of the maximum rainfall event. Estimates obtained through frequency analysis can vary significantly depending on the PDF. Therefore, rationally determining the probability distribution function is crucial for representing extreme value series data in natural phenomena such as hydrology and climatology. Depending on whether the series consists of maximum or minimum values, the theoretical probability density function should be able to adequately represent the right or left tail of the extreme data containing the most critical information.

In this study, maximum rainfall data of seven standard durations of 81 meteorological stations located in the provinces of Türkiye were examined, with the aim to determine the goodness of fit of eight different probability distributions with the combination of four numerical tests (KS, AD, PPCC, Z^DIST).

Three different perspectives were considered when examining the fitness of the probability distributions. The first was an examination of the maximum rainfall series of seven different rainfall durations across the country; the second was a province-based check of suitability for all 81 provinces; and the last was a general (country-based) examination in which the seven durations and their averages were considered.

From a rainfall duration perspective, maximum rainfall series of 5 min, 10 min, …, 12 h, and 24 h were analyzed separately. The rank number indicating the fit of each standard duration series to the distributions was obtained from the rank averages of all stations for the same duration. The results (Table 3 and Figure 2) show that the WAK distribution is the most suitable distribution (with minimum rank number) for each rainfall duration. GEV, LP3, PE3, GLO, LN3, NOR, and GPA are ranked second, third, fourth, fifth, sixth, seventh, and eighth, respectively. In addition to numerical calculations, graphical analyses were performed to examine suitable distributions for different rainfall durations.

From a province-based perspective, considering all 81 provinces in Türkiye, the best fit probability distribution for each province was ranked by seven rainfall durations and the average rank (overall) of all rainfall durations. The results (Table 4 and Figure 3) show that, in the province-based ranking, the best fit distribution in Türkiye is WAK (approximately in the 60–65% range). This is followed by the GEV distribution (approximately 20%). These two distributions are followed by the LP3 distribution and the GLO distribution, with the ranges between 5% and 10%. Only in a few provinces is maximum rainfall represented by the PE3 and LN3 distributions. NOR and GPA failed to represent maximum rainfall in any province or rainfall duration. On the other hand, in the overall map obtained with minimum rank averages, the WAK distribution dominated the other distributions at a rate of 90%.

From the country-based perspective, when the ranking averages of stations in 81 provinces for seven standard periods are considered, the WAK distribution, which shows the lowest rank, stands out as superior to the other distributions (Figure 4). Both the box plot and median values indicate that the WAK distribution is followed by the GEV, LP3, PE3, GLO, LN3, NOR, and GPA distributions, respectively.

For the three different perspectives examined, the KS, AD, PPCC, and Z^DIST tests showed parallel results with minor differences.

While the WAK distribution appeared to be the most suitable for both the standard duration-based and country-based perspectives, it was not always the most suitable distribution in the province-based assessment. This raises the question of whether WAK is acceptable for all stations in all provinces and all precipitation durations. To clarify this, we would like to note that WAK is significantly accepted for all 567 rainfall series at a significance level of α = 0.05 according to all four numerical tests.

This study used data based on long records, considering the maximum rainfall data of seven different rainfall durations from 15 min to 24 h, and did this by using data from 81 stations located in all provinces across the country. The results were evaluated from four different perspectives for different time periods, different provinces, and the country as a whole, and it was concluded that the WAK distribution can be accepted as the parent probability distribution for Turkey’s maximum rainfall.

It is hoped that the study’s findings will contribute to the literature, assist water resources studies, and support planners, hydrologists, and meteorologists working on maximum rainfall.