Selection of a Probability Model Adapted to the Current Climate for Annual Maximum Daily Rainfall in the Benin Mono-Couffo Basin (West Africa)

Abstract

The control of rainfall extremes is essential in the design of hydro-agricultural works, as their performance depends on it. This study aims to determine the best-fit probability model suited to current climatic conditions in the Mono-Couffo basin in Benin. To this end, daily rainfall data from six rainfall stations from 1981 to 2021 were used. The application of the Decision Support System (DSS) with graphical and numerical performance criteria (such as RMSE, SD, and CC represented by the Taylor diagram; AIC and BIC) made it possible to identify the best distribution class and then to select the most suitable distribution for this basin. The results indicate that class C distributions, characterized by regular variations, are the most appropriate for the modeling maximum annual daily precipitation at all stations (78% of cases). Of these, the Inverse Gamma distribution proved to be the most suitable, although its estimation errors ranged from 16.47 mm/d at Aplahoué to 39.80 mm/d at Grand-Popo. The second most appropriate distribution is the Log-Pearson Type III. The use of the Inverse Gamma distribution is, therefore, recommended for hydro-agricultural development studies in the Mono-Couffo basin.

1. Introduction

Controlling extreme precipitation is one of the major challenges of our century [1]. For these authors, these extremes cause damage yearly throughout the world, in general, and in West Africa, in particular. The damage created by these extremes on hydraulic structures is important, requiring their study, especially for agricultural water management, as they are designed to respond to the probability of occurrence of these extremes during their cycle, [2]. According to Agué and Afouda [3] in Benin, the study carried out by the CIEH (1985), often used to design hydraulic structures, concerns periods that are now largely outdated. Indeed, the objective of the methods developed by ORSTOM and CIEH was to control hydrological regimes to contribute to the development of agricultural and hydroelectric potential in the concerned countries [4]. Annual maximum daily rainfall is, therefore, an essential parameter for estimating the design flood using these approaches. However, West Africa has been experiencing climate change since the 1970s and is one of the region’s most vulnerable to climate variability [5]. The Mono-Couffo basin has not remained on the sidelines of these climate changes, as shown by the work of Amoussou et al. [6], Amoussou [7], Amoussou et al. [8] and Amoussou et al. [9]. In this context, the current use of tools developed in the 1960s–1970s for the design of hydraulic structures is undoubtedly impacting the durability of these structures [10]. As a result, these methods for estimating project floods need to be updated. The first step in this updating process is to select an appropriate statistical model that could effectively describe the distribution of the annual maxima series (AMS) of daily precipitation. This task is not easy, however, and remains one of the major challenges of engineering practice due to the significant spatial and temporal variability of precipitation maxima [11]. This estimate is usually based on a precipitation frequency analysis (PFA) [12] using the AMS of precipitation at different scales as required (hourly, daily, etc.). Several studies around the world have shown that in hydrology, the most suitable laws are GEV in Ontario, Canada [11], Log-Pearson Type III in the United States [13], Gumbel in Côte d’Ivoire [14], and Gumbel and Lognormal in Benin [3]. Table 1 summarizes some recent studies on the selection of the most suitable distributions for the analysis of precipitation frequency worldwide. An analysis of these manuscripts shows that the authors were not particularly interested in the tails of distributions but in frequency analysis and the estimation of distribution parameters, comparing their performance based on several criteria and statistical tests. Furthermore, there is no consensus on which distribution to use, let alone on the approach for choosing distribution parameters, as reported by Kouassi et al. [14]. This situation calls for the use of tools to test various distributions, starting by analyzing their distribution tails to select the best-fitting class or family of distributions for use in choosing the most appropriate model. There are several approaches to studying tail behavior, including the Mean Excess Function (MEF) and log–log graphs [15,16], as well as scalar indices [17], the Gini index [18,19] and the obesity index [20]. The work of El Adlouni et al. [15] and Ehsanzadeh et al. [21] combined some of these approaches, such as FEM, log–log, Hill ratio and Jackson statistics, under a Decision Support System (DSS) to characterize distribution tails. This tool allowed these authors to select an appropriate distribution class that adequately represented the annual instantaneous peak flows of the Potomac River but also for the Reference Hydrometric Basin Network (RHBN), precipitation and UNESCO discharge, respectively. Nassa et al. [22] also used DSS to select an appropriate distribution class for maximum annual daily rainfall at 26 stations in the Ivory Coast.

To the authors’ knowledge, no previous publication has dealt with the application of the Decision Support System (DSS) to select the most appropriate class of distributions before proceeding with frequency analysis in Benin and, more specifically, in the Beninese Mono-Couffo basin.

This study, therefore, aims to select the “optimal” model that could provide adequate estimates of extreme precipitation in six (06) rainfall stations located in the Benin Mono-Couffo basin from the DSS. This choice has not only enabled us to design more efficiently but also reduce the extent of damage to hydraulic structures caused by extreme events.

The results of such a study could provide valuable information on the suitability of models in the basin, or even decide on the most appropriate distribution to use according to area and period, following the example of other countries such as the United States and Australia, which have adopted the Log-Pearson Type III distribution for estimating flood frequencies, and the Logarithmic (LN) distribution in China presented in the work of Bobée [23] and Robson and Reed [24].

This article describes the theoretical framework and methodology, the results related to the selection of the class of distributions that adequately represents the samples studied and the most suitable distribution model, and implications for future research.

Table 1. Summary of some recent studies of rainfall frequency analysis around the world.

Study	Regions	No. of Stations	Period of Record	Distribution	Estimation Method	Goodness of Fit	Conclusion
[25]	Montreal (Canada)	9	1963–1974	MEXP, EXP, WEB, GAM	ML	${\chi }^2$, AIC	The mixed exponential was found to be the best model.
[26]	Ilorin, Nigeria	1	1955–1995	LPIII, EV1, LN, EXP, N	-	-	The LPIII distribution best suited the maximum daily rainfall data while the normal distribution best described the maximum monthly rainfall for Ilorin.
[27]	Southern Quebec (Canada)	20	-	BEK, BEP, GEV, GNO, GPA, GUM, PIII, LPIII, WAK	ML, LM	Q-Q plots, CC, RMSE, RRMSE, MAE, Bootstrap	The GEV was recommended as the most suitable distribution.
[28]	Malaysia	70	23 to 28 years	GAM, GNO, GPA, GEV, LPIII, PIII, GUM, WAK	LM	CC, MAE, RMSE, RRMSE	The GEV distribution is the most appropriate distribution for describing the annual maximum rainfall series in Malaysia.
[29]	Nigeria	20	54 years (1952–2005)	GUM, LGUM, N, LN, PIII, LPIII	-	${\chi }^2$, F, CC, $R^2$	The LP3 distribution performed the best, followed by PIII and LGUM distributions.
[30]	Chott-Chergui basin (Algeria)	27	1970–2004	GEV, GUM, LPIII, LN	ML	${\chi }^2$	GEV law showed a good fit to the series of maximum daily rainfall of the “Chott Chergui” basin.
[31]	India	4	Fatehabad and Hansi (1954–2011) Hissar (1969–2011) and Tohana (1951–2011)	GUM, EV2	LM	${\chi }^2$, K-S, D-Index	The study shows the GUM distribution is better suited for rainfall estimation for the stations under study.
[32]	Brazil	342	2001–2010 (10 years)	GEV, Kappa, GUM, LN	LM	K-S, ${\chi }^2$, Filliben test $A^2$	The Kappa distribution presented the best performance, followed by the GEV.
[33]	Poland	1	1960–2009 (50 years)	GED2, GED3, GAM, GUM, WEB, LN	ML	BIC, RRMSE	GED3, gamma and Weibull distributions were the best for describing.
[3]	Benin	26	1921–2001	GEV, GUM, LN, PIII, LPIII	ML	AIC, BIC	The results showed a predominance of Gumbel and Lognormal laws.
[34]	Qatar	29	(36 years)	N, LN2, LN3, GAM2, GAM3, GGAM, GUM, LLO, GEV, PIII, LPIII, GPA, Beta, WEB	-	K-S, ${\chi }^2$, $A^2$	GEV distribution is found to be the best-fit distribution.
[11]	Ontario (Canada)	21	(63 years)	BEK, BEP, GEV, GLO, GNO, GPA, GUM, LPIII, PIII, WAK	ML, NCMs, LM	Q-Q plots, RMSE, CC, RRMSE, MAE, BIC, AIC, bootstrap	The GEV distribution is the best model for describing the distribution of daily and sub-daily annual maximum rainfalls.
[14]	Ivory Coast	1	1961- 2014	GEV, GUM, LN	ML	AIC, BIC, ${\chi }^2$	The law that best adjusted the annual maximum daily rainfall of the Port-Bouët station (Abidjan) was the law of Gumbel.
[12]	Egypt	31	-	N, LN, PIII, LPIII, GUM, GEV, GAM, EXP, LO, GLO, GPA, WEB	MOM, ML, LM	Q-Q plot, RMSE, RRMSE, CC, AIC, BIC, BIASr	LN, LPIII and EXP are the top three distributions for the frequency analysis of daily annual extreme rainfalls in Egypt.
[35]	lower Ouémé valley (Benin)	9	39 years (1981 to 2019)	GEV, GUM, LN, WEB, GAM	-	AIC, BIC	The results obtained show that the Gumbel, Lognormal, and GEV distributions are the most suitable for the data series in the study area.
[36]	Niger	1	1960–2020	N, GUM, LN	-	K-S, T-S	It was found that the Gumbel model is the most suitable for modeling extreme rainfall and for calculating return periods.
[37]	Romania	14	-	14 probability distributions from Beta families	LM, MOM	RME, RAE	The best-fit distributions are the Kumaraswamy, the Generalized Beta Exponential and the Generalized Beta distributions, which presented stability related to both the length of the data and the presence of outliers.
[38]	Poland	51	1971–2014	PIII, WEB, LN, GEV, GUM	ML	RMSE, R2, PWRMSE	The GEV distribution as recommended for calculating the maximum daily precipitation with the specific probability of exceedance.

Source: Gado [12], updated. MEXP, Mixed Exponential; EXP, Exponential; WEB, reverse Weibull; GAM, Gamma; BEK, Beta-K; BEP, Beta-P; GEV, generalized extreme value; GNO, generalized normal; GPA, Generalized Pareto; GUM, Gumbel; PIII, Pearson Type III; LPIII, Log-Pearson Type III; WAK, Wakeby; EV2, Frechet; LGUM, Log-Gumbel; N, Normal; LN, Log-Normal; GED2, 2 Parameter Generalized Exponential; GED3, 3 Parameter Generalized Exponential; GGAM, Generalized Gamma; LLO, Log-Logistic; GLO, Generalized Logistic; ML, maximum likelihood; LM, L-moment; NCMs, Non-Central Moments; MOM, Method of Ordinary Moments; ${\chi }^2$, Chisquare; AIC, Akike Information Criteria; CC, Correlation Coefficient; RMSE, root mean square error; RRMSE, Relative Root Mean Square Error; RME, relative mean error; PWRMSE, peak-weighted root mean square error; RAE, relative absolute error; MAE, Maximum Absolute Error; F, Fisher’s test; $R^2$, coefficient of determination; K-S, Kolmogorov–Smirnov; $A^2$, Anderson-Darling; BIC, Bayesian Information Criteria; T-S, Test-Student.

Table 1. Summary of some recent studies of rainfall frequency analysis around the world.

Study	Regions	No. of Stations	Period of Record	Distribution	Estimation Method	Goodness of Fit	Conclusion
[25]	Montreal (Canada)	9	1963–1974	MEXP, EXP, WEB, GAM	ML	${\chi }^2$, AIC	The mixed exponential was found to be the best model.
[26]	Ilorin, Nigeria	1	1955–1995	LPIII, EV1, LN, EXP, N	-	-	The LPIII distribution best suited the maximum daily rainfall data while the normal distribution best described the maximum monthly rainfall for Ilorin.
[27]	Southern Quebec (Canada)	20	-	BEK, BEP, GEV, GNO, GPA, GUM, PIII, LPIII, WAK	ML, LM	Q-Q plots, CC, RMSE, RRMSE, MAE, Bootstrap	The GEV was recommended as the most suitable distribution.
[28]	Malaysia	70	23 to 28 years	GAM, GNO, GPA, GEV, LPIII, PIII, GUM, WAK	LM	CC, MAE, RMSE, RRMSE	The GEV distribution is the most appropriate distribution for describing the annual maximum rainfall series in Malaysia.
[29]	Nigeria	20	54 years (1952–2005)	GUM, LGUM, N, LN, PIII, LPIII	-	${\chi }^2$, F, CC, $R^2$	The LP3 distribution performed the best, followed by PIII and LGUM distributions.
[30]	Chott-Chergui basin (Algeria)	27	1970–2004	GEV, GUM, LPIII, LN	ML	${\chi }^2$	GEV law showed a good fit to the series of maximum daily rainfall of the “Chott Chergui” basin.
[31]	India	4	Fatehabad and Hansi (1954–2011) Hissar (1969–2011) and Tohana (1951–2011)	GUM, EV2	LM	${\chi }^2$, K-S, D-Index	The study shows the GUM distribution is better suited for rainfall estimation for the stations under study.
[32]	Brazil	342	2001–2010 (10 years)	GEV, Kappa, GUM, LN	LM	K-S, ${\chi }^2$, Filliben test $A^2$	The Kappa distribution presented the best performance, followed by the GEV.
[33]	Poland	1	1960–2009 (50 years)	GED2, GED3, GAM, GUM, WEB, LN	ML	BIC, RRMSE	GED3, gamma and Weibull distributions were the best for describing.
[3]	Benin	26	1921–2001	GEV, GUM, LN, PIII, LPIII	ML	AIC, BIC	The results showed a predominance of Gumbel and Lognormal laws.
[34]	Qatar	29	(36 years)	N, LN2, LN3, GAM2, GAM3, GGAM, GUM, LLO, GEV, PIII, LPIII, GPA, Beta, WEB	-	K-S, ${\chi }^2$, $A^2$	GEV distribution is found to be the best-fit distribution.
[11]	Ontario (Canada)	21	(63 years)	BEK, BEP, GEV, GLO, GNO, GPA, GUM, LPIII, PIII, WAK	ML, NCMs, LM	Q-Q plots, RMSE, CC, RRMSE, MAE, BIC, AIC, bootstrap	The GEV distribution is the best model for describing the distribution of daily and sub-daily annual maximum rainfalls.
[14]	Ivory Coast	1	1961- 2014	GEV, GUM, LN	ML	AIC, BIC, ${\chi }^2$	The law that best adjusted the annual maximum daily rainfall of the Port-Bouët station (Abidjan) was the law of Gumbel.
[12]	Egypt	31	-	N, LN, PIII, LPIII, GUM, GEV, GAM, EXP, LO, GLO, GPA, WEB	MOM, ML, LM	Q-Q plot, RMSE, RRMSE, CC, AIC, BIC, BIASr	LN, LPIII and EXP are the top three distributions for the frequency analysis of daily annual extreme rainfalls in Egypt.
[35]	lower Ouémé valley (Benin)	9	39 years (1981 to 2019)	GEV, GUM, LN, WEB, GAM	-	AIC, BIC	The results obtained show that the Gumbel, Lognormal, and GEV distributions are the most suitable for the data series in the study area.
[36]	Niger	1	1960–2020	N, GUM, LN	-	K-S, T-S	It was found that the Gumbel model is the most suitable for modeling extreme rainfall and for calculating return periods.
[37]	Romania	14	-	14 probability distributions from Beta families	LM, MOM	RME, RAE	The best-fit distributions are the Kumaraswamy, the Generalized Beta Exponential and the Generalized Beta distributions, which presented stability related to both the length of the data and the presence of outliers.
[38]	Poland	51	1971–2014	PIII, WEB, LN, GEV, GUM	ML	RMSE, R2, PWRMSE	The GEV distribution as recommended for calculating the maximum daily precipitation with the specific probability of exceedance.

Source: Gado [12], updated. MEXP, Mixed Exponential; EXP, Exponential; WEB, reverse Weibull; GAM, Gamma; BEK, Beta-K; BEP, Beta-P; GEV, generalized extreme value; GNO, generalized normal; GPA, Generalized Pareto; GUM, Gumbel; PIII, Pearson Type III; LPIII, Log-Pearson Type III; WAK, Wakeby; EV2, Frechet; LGUM, Log-Gumbel; N, Normal; LN, Log-Normal; GED2, 2 Parameter Generalized Exponential; GED3, 3 Parameter Generalized Exponential; GGAM, Generalized Gamma; LLO, Log-Logistic; GLO, Generalized Logistic; ML, maximum likelihood; LM, L-moment; NCMs, Non-Central Moments; MOM, Method of Ordinary Moments; ${\chi }^2$, Chisquare; AIC, Akike Information Criteria; CC, Correlation Coefficient; RMSE, root mean square error; RRMSE, Relative Root Mean Square Error; RME, relative mean error; PWRMSE, peak-weighted root mean square error; RAE, relative absolute error; MAE, Maximum Absolute Error; F, Fisher’s test; $R^2$, coefficient of determination; K-S, Kolmogorov–Smirnov; $A^2$, Anderson-Darling; BIC, Bayesian Information Criteria; T-S, Test-Student.

2. Materials and Methods

2.1. Study Area

Benin is a West African country located on the Gulf of Guinea. It comprises four major hydrographic units: the Niger basin, the Volta basin, and the coastal basin, which includes the Mono-Couffo unit on the west side, and the Ouémé-Yéwa unit on the east side. The Mono-Couffo group is a transboundary basin shared by Benin and Togo. The present study focuses on the Beninese part of this basin, as shown in Figure 1. It extends between latitudes 06°16′ and 09°20′ north and between longitudes 0°42′ and 2°25′ east [7] and covers an area of 5.554 km², i.e., 20% of the total surface area. The Mono-Couffo complex is made up of the Mono and Couffo sub-basins, hence the term “complex”, and is crossed by two main rivers flowing from northeast to southwest and northeast to northwest and southwest. Enjoying a sub-equatorial climate with two dry seasons (November–March and August) and two rainy seasons (April–July and September–October), the basin benefits from an average annual rainfall over the period 1981 to 2021 of 1026 mm and an average temperature ranging from 25.93 °C to 29.23 °C. Agriculture remains the main economic activity in the basin [39]. This activity, which employs almost 80% of the population, remains essentially rain-fed and therefore dependent on climatic variability, which is increasingly leading to migration to flood plains, valleys, lowlands and rivers to benefit more from the hydrological potential. Good water management in these lowland agro-ecosystems requires the characterization of rainfall extremes, which is essential for setting up the essential hydro-agricultural structures on which agricultural production activities depend.

2.2. Data

This study required the use of climatic data. These mainly concern the annual maximum series (AMS) of daily rainfall within the period 1981 to 2021, i.e., forty-one (41) consecutive years. These data come from six (06) rain gauge stations (Figure 1) and were collected from the Agence Nationale de la Météorologie du Bénin (Météo Bénin).

The Hyfran-Plus software (Hydrological Frequency Analysis) version 2.2. developed by El Adlouni et al. [15] was used to carry out a complete frequency analysis of the various data. It incorporates a Decision Support System (DSS) developed to select the most appropriate class of distributions for estimating quantiles (maximum daily rainfall for a given return period).

2.3. Methods

Identifying an appropriate distribution that can correctly describe the distribution of the annual maximum series (AMS) of daily rainfall is not easy and remains one of the major challenges in engineering practice due to significant spatial and temporal variability of rainfall maxima [11]. Thus, this section presents a procedure for selecting the most appropriate probability model(s) to describe the behavior of AMS in three (03) steps: verification of basic hypothesis tests, application of the Decision Support System (DSS) and an assessment of the validity of the models preselected by the DSS.

2.3.1. Hypothesis Testing for Frequency Analysis

According to Kouton [40], it is essential to check the independence, stationarity and homogeneity of the data before making a statistical adjustment. To this end, El Adlouni and Bobée [41] suggest the use of the following tests:

• Test of independence (Wald–Wolfowitz): This test assesses whether two samples come from the same distribution by verifying the independence of the observations;
• Stationarity test (modified Kendall): This test examines whether data are stationary, i.e., whether their statistical properties (such as mean and variance) do not change over time;
• Annual homogeneity test (Wilcoxon): This test compares the distributions of two samples to determine whether there are significant differences between them.

These tests work on the principle of a null hypothesis (H0) and an alternative hypothesis (H1). H0 represents an assertion to be tested, while H1 is the opposite conclusion. The test aims to validate or reject H0, which is achieved by analyzing the p-value obtained. If the p-value is less than or equal to a significance level α (usually set at 5%), H0 is rejected, indicating significant evidence against the null hypothesis. On the other hand, if the p-value is greater than α, H0 is considered valid. The results of these three tests provide the U, K and W statistics, respectively, as well as the p-value (P), which are analyzed at the α significance level of 5%.

2.3.2. Selecting Distribution Classes

In general, determining the best-fit law has always been tricky and the choice of model can be crucial for estimating precipitation for different extreme value return periods [22]. Thus, it is important to select the class of distributions that adequately represents the studied sample before proceeding with fitting [15].

In fact, these authors distinguish three (03) main classes or families of distributions C, D and E according to whether they are heavy or light tailed, grouping together the ten (10) distributions commonly used in hydrology for the study of extreme values as follows:

• Class C (regularly varying distributions): Fréchet (EV2), Halphen Inverse type B (HIB), Log-Pearson type III (LPIII) and Inverse Gamma (IG);
• Class D (sub-exponential distributions): Halphen type A (HA), Halphen type B (HB), Gumbel (EV1), Pearson type III (PIII) and Gamma (G);
• class E (exponential law: EXP).

The choice of these distribution classes is made through the Decision Support System (DSS) presented in Figure 2, which enables the application of several statistical tests:

• The Jarque–Bera test: considered for testing Lognormality with a priori selection based on the coefficients of variation and skewness (Cv, Cs) diagram;
• The Log–Log plot: used to discriminate between class C, on the one hand, and classes E and D, on the other hand;
• The Mean Excess Function (MEF) to discriminate between classes D and E;
• Two statistics: Hill’s ratio and modified Jackson statistic, for confirmatory analysis of the conclusions suggested by the previous two methods.

More theoretical details of this classification and the criteria are available in the studies of El Adlouni and Bobée [42] and El Adlouni et al. [15].

2.3.3. Distribution Parameter Estimation Methods

The AMS adjustments were carried out using the distributions contained in the chosen class(es). These distributions operate based on several parameters, which are estimated using several approaches: method of moments (MOM), the maximum likelihood method (ML), the L-moments (LM) and Sundry Averages Method (SAM).

The MOM is the simplest but can produce less accurate estimates of the parameters, when compared to ML or LM methods, with both considered more efficient in providing estimates of lower variance [43]. Consequently, these methods are commonly used to estimate the parameters of different distributions in numerous studies, including [27,32,33,44]. Furthermore, reference [12] has shown that the performances of both L-moments and maximum likelihood methods are almost equal and much better than that of the method of moments.

In this work, the method chosen for parameter estimation is the maximum likelihood method for all distributions except the LPIII distribution, for which the SAM method proposed by [45] has been used.

Table 2. The probability density function f (x) and the associated parameters for each selected distribution.

Distributions	$\mathbf{Probability} \mathbf{Density} \mathbf{Function} \mathit{f}\left(\mathit{x}\right)$	Parameters
GEV	$f\left(x\right)={{\displaystyle \frac{1}{\alpha }}\left[1-{\displaystyle \frac{k}{\alpha }}(x-u)\right]}^{{\displaystyle \frac{1}{k}}-1}.exp\left\{-{\left[1-{\displaystyle \frac{k}{\alpha }}(x-u)\right]}^{{\displaystyle \frac{1}{k}}}\right\}$	$u$$. \alpha$$. k$	(1)
IG	$f\left(x\right)={\displaystyle \frac{{\alpha }^{\lambda }}{\mathsf{\Gamma }\left(\mathsf{\lambda }\right)}}{\left({\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$x$}\right.}\right)}^{\mathsf{\lambda }+1}{e}^{\left({\displaystyle \frac{-\alpha }{x}}\right)}$	$\alpha . \lambda .$	(2)
LPIII	$f\left(x\right)={\displaystyle \frac{{\alpha }^{\lambda }}{\mathsf{\Gamma }\left(\mathsf{\lambda }\right)}}{(lnx-m)}^{\lambda -1}{e}^{-a(lnx-m)}$	$\alpha$$. \mathsf{\lambda }. m$	(3)
HIB	$f\left(x\right)={\displaystyle \frac{2}{m^{-2\vartheta }{ef}_{\vartheta }\left(\alpha \right)}}{x}^{-2\vartheta -1}exp[-(\begin{array}{cc}{\displaystyle \frac{m}{x}})2+\alpha & ({\displaystyle \frac{m}{x}}\end{array})$$]; x>0$	$\alpha$$. m$$. \vartheta$	(4)

shows the probability density function f and associated parameters for each selected distribution

The IG and HIB distributions do not have a position parameter. Thus, the parameters $\alpha , \lambda$ of IG represent the scale and shape parameters, respectively. As for the HIB distribution, $m$ is a scale parameter, $\alpha$ and $\vartheta$ are the shape parameters with $m>0$, $\alpha ϵ\mathbb{R}$ et $\vartheta >0$. The different fitting was made at a confidence level of 95%.

The parameters $u$, $\alpha$, $k$ refer, respectively, to the position, scale, and shape parameters of the different distributions. The position parameter $u$ characterizes the order of magnitude of the extreme rainfall series. The shape parameter $k$ indicates the behavior of the extremes or the shape of the distribution. Depending on the value of this shape parameter, three (03) types of GEV distribution can be distinguished:

$k$ = 0: Gumbel distribution (EV1);

$k$ < 0: Fréchet distribution (EV2);

$k$ > 0: Inverted Weibull distribution (WEB or EV3).

2.3.4. Criteria for Determining the Most Appropriate Probability Model

Determining the most suitable model involves selecting the “optimal” distribution from among those resulting from the application of DSS. This selection is based on a comparison of the fits of different distributions using a non-exceedance probability graph.

This graph shows the non-exceedance probability on the x-axis, also known as the non-exceedance frequency, and the values of the variable of interest (in this case, daily precipitation AMS) on the y-axis. However, for Gado et al. [12] and Kouassi et al. [22], these plots are useful for visual assessment; however, it is subjective and cannot exactly describe the statistical significance of the fit, especially in the case of comparison among many statistical models. To this end, five (05) criteria were used: standard deviation (SD), Pearson’s correlation coefficient (CC) (Equation (5)), root mean square error (RMSE) presented by Equation (6) and the AIC and BIC criteria. For a better appreciation of SD, CC and RMSE, the Taylor diagram [46] was used.

Taylor Diagram

The Taylor diagram is a polar diagram in which the correlation is the angular coordinate, and the standard deviation of the database is the radius [47]. This type of diagram summarizes several key statistical data for evaluating complex models, including standard deviations, correlations, and root-mean-square errors (RMSE) of simulations versus observations [48].

The “optimal” model is the one with:

• A high correlation, close to the horizontal axis;
• A standard deviation close to the circle corresponding to the standard deviation of the observations, indicating that the variability of the model is similar to that of the observed data;
• A low RMSE, close to the observation reference point “circle on the x-axis”, meaning that the model’s error is minimized and its performance is better.

Thus, one of the greatest advantages of Taylor diagrams is that they graphically summarize the closeness of the observed data to the estimates of each database, with the similarity between the models quantified in terms of correlation and the amplitude of their variations, represented by the standard deviations [49], satisfying the relationship of Equation (7).

$$r={\displaystyle \frac{\sum _{i=1}^N(S_i-\stackrel{̿}{s})\times (x_i-\stackrel{̿}{x})}{\sqrt{{\sum }_{i=1}^N{(S_i-\stackrel{̿}{s})}^2}\times \sqrt{{\sum }_{i=1}^N{(x_i-\stackrel{̿}{x})}^2}}}$$

(5)

$$RMSE=\sqrt{{\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{(S_i-\stackrel{̿}{s})}^2}$$

(6)

$$E^{\prime 2}={\sigma }_s^2+{\sigma }_x^2-2{\sigma }_s{\sigma }_{x}r$$

(7)

where $N$ is the sample size, $x_i$ is the maximum daily precipitation value observed in year $i$, $S_i$ is that of the model under consideration. $\stackrel{̿}{x }; \stackrel{̿}{s}$ et ${\sigma }_x ; {\sigma }_{s }$ are means and standard deviations, respectively.

AIC and BIC criteria

Proposed by Schwarz (1978), the Bayesian Information Criterion (BIC) is used in the Bayesian context of probabilistic model selection [3]. The objective here is to find the statistical model that maximizes the posteriori distribution of models, i.e., the most likely model given the data [40]. The expression of this Bayesian criterion is:

$$BIC=-2\times \mathit{log}\left(L\right)+2\times K\times \mathit{log}\left(N\right)$$

(8)

Akaike’s Information criterion is based on a pseudo-distance between an unknown distribution and an arbitrary distribution. Akaike’s comparison criterion (AIC) selects the model that achieves the best bias-variance compromise for the number of data N available. Such a compromise respects the parsimony principle of theoretical frequency distribution laws [22]. The expression of the AIC criterion is defined as follows:

$$AIC=-2\times \mathit{log}\left(L\right)+2K$$

(9)

With:

$L$: likelihood;

$K$: number of parameters;

$N$: sample size.

The best statistical model is the one with the lowest AIC and BIC values [31,33].

3. Results

3.1. Analysis of Hypothesis Tests Applied to Annual Maximum Daily Rainfall

The results of the hypothesis tests prior to the frequency analysis carried out on the annual maximum series of daily rainfall are presented in

Table 3. Results of frequency analysis hypothesis tests applied to annual maximum series of daily rainfall over the 1981–2021 period.

Rain Gauge Station	Independence Test		Stationarity Test		Homogeneity Test
	U	P (α = 5%)	K	P (α = 5%)	W	P (α = 5%)
Dogbo	1.46	0.14	1.22	0.22	1.29	0.19
Grand-Popo	2.28	0.02	1.2	0.23	0.38	0.70
Aplahoué	0.59	0.55	0.10	0.92	1.13	0.26
Bopa	0.65	0.51	1.33	0.18	1.32	0.19
Toffo	1.23	0.22	1.68	0.09	0.98	0.33
Abomey	0.11	0.91	1.62	0.11	1.03	0.30

.

An analysis of the table shows that all stations test the independence hypothesis at the 5% risk. The modified Kendall test showed that there was no trend in the observations at the 5% threshold for all stations. Application of the Wilcoxon–Mann–Whitney test indicates that the means of the two sub-samples are equal at the 5% threshold at all stations. It, therefore, follows that the annual maximum rainfall series (AMS) for Dogbo, Grand-Popo, Aplahoué, Bopa, Toffo and Abomey stations are independent, homogeneous, and stationary.

As the hypothesis testing approach is an essential step in performing frequency analysis, the verification performed on the data set shows that it meets these criteria, enabling the analysis to continue.

3.2. Selecting the Most Appropriate Distribution Class

The identification of the most appropriate distribution class to represent the AMS of the six (06) rainfall stations was carried out by applying the DSS.

Table 4. DSS results for the 1981–2021 period applied to the AMS of six (06) stations.

	Distributions	Stations
		Abomey	Toffo	Dogbo	Grand-Popo	Bopa	Aplahoué
	Log-Normal (LN)	X	X	X	X	X	X
Class C	Fréchet (EV2)	V	V	V	V	V	V
	Halphen Type B Inverse (HIB)	V	V	V	V	V	V
	Log-Pearson Type III (LPIII)	V	V	V	V	V	V
	Inverse Gamma (IG)	V	V	V	V	V	V
Class E	Exponential (EXP)	V	V	X	X	X	V
Class D	Halphen Type A (HA)	X	X	X	X	X	X
	Gamma	X	X	X	X	X	X
	Pearson Type III (PIII)	X	X	X	X	X	X
	Halphen Type B (HB)	X	X	X	X	X	X
	Gumbel (GMB)	X	X	X	X	X	X
Tests	Log-Normal test	NA	NA	NA	NA	NA	NA
	JB Test	NA	NA	NA	NA	NA	NA
	Log-log plot	C	C	C	C	C	NA
	Mean Excess Function (MEF)	E	E	NA	NA	NA	E
	Hill ratio plot	C	C	C	C	C	C
	Jackson Statistic	D-E	D-E	C	C	C	D-E
Results	Class	C-E	C-E	C	C	C	C-E
Percentage (%)	C	67	67	100	100	100	33
	D	0	0	0	0	0	0
	E	33	33	0	0	0	67

X: Reject; V: Accepted; NA: No Applicable.

shows the results of the DSS application by station.

A analysis of the table shows that log normality is not verified at all six (06) stations. The log–log plot test indicates that the AMS belongs to class C (distribution with regular variations) except at the Aplahoué station, where the concavity is more remarkable (Figure 3), leading to the application of the Mean Excess Function (MEF) method. The latter shows that the curve (Figure 4) of the MEF is linear (at the level of the highest observations), and the slope observed is practically zero (0.009), implying that the underlying distribution is exponential (class E).

The confirmation of this result through analysis of the Hill ratio curve shows that the latter converges to a constant non-zero value, suggesting class C laws, and that the Jackson statistic does not tend towards 2 (class D or E), as shown in Figure 5. However, the FEM had already discriminated class D from E. These analyses lead us to conclude that the distributions of Class E (67%) and Class C (33%) per AMS of daily precipitation are well matched.

At the Dogbo, Grand-Popo and Bopa stations, the log–log plot test indicates that the curves are linear (Figure 6), particularly for high values of log(u) and that the distributions, therefore, belong to class C (100%) of regular variation distributions. The Hill ratio and the Jackson statistic confirm these results, as the Hill graph and the Jackson curve clearly and consistently converge towards 2 (Figure 7 and Figure 8). However, at Toffo and Abomey, although the log–log plot indicates class C, the Hill ratio and Jackson statistic indicate class C and D or E, respectively, necessitating the application of the Mean Excess Function (MEF) method to discriminate between class D or E. The MEF suggests class E for these stations. The AMS of daily rainfall at Toffo and Abomey are well matched to the distributions of classes C (67%) and E (33%).

Of the three (03) classes (C, D and E) studied for the 1981–2021 period on the six (06) stations, classes C and E emerged with an average percentage of apparition of 78% and 22%, respectively. Thus, class C, distributions with regular variation (such as EV2, HIB, LPIII and IG), is the one that offers a good fit over the 1981–2021 period in the basin. These probability distributions were used to fit the series of annual daily precipitation maxima for the six (06) stations, and the results are presented in the following section.

3.3. Selection of Best-Fit Distribution

The choice of the best law or “optimal” model is based on an examination of the fits and numerical analyses. The parameters of the different models, estimated by the maximum likelihood (ML) method and applied to the AMS of daily precipitation of the six (06) stations, are presented in

Table 5. Parameter analysis of various models applied to AMS of daily precipitation.

Stations	Model	Parameters
		Location	Scale	Shape
Dogbo	GEV	68.13	18.26	0.03
	LPIII	−1.68	240.16	854.89
	IG		938.30	12.97
Grand-Popo	GEV	79.70	30.79	−0.08
	LPIII	−3.73	177.68	1011.60
	IG		515.85	6.09
Aplahoué	GEV	80.16	20.15	0.13
	LPIII	2.72	−65.16	51.04
	IG		1297.30	15.48
Bopa	GEV	70.75	19.13	−0.13
	LPIII	1.46	24.76	11.08
	IG		862.87	11.24
	HIB		132.38	0.29 and 2.96
Toffo	GEV	64.52	18.13	−0.172
	LPIII	1.49	18.25	6.83
	IG		714.09	10.13
	HIB		127.05	1.43 and 2.25
Abomey	GEV	72.94	17.93	−0.08
	LPIII	1.45	32.04	14.87
	IG		1085.46	13.79
	HIB		148.83	0.20 and 3.64

.

An analysis of the table reveals variability in the various estimated parameters. Values for the position parameter range from −3.729 (Grand-Popo) to 80.156 (Aplahoué). The scale parameter also shows a wide range of variation, from −65.16 (Aplahoué) to 1297.30 (Aplahoué). A closer look at the shape parameter of the GEV distribution reveals that it is below zero (0) for stations in Grand-Popo (−0.08), Bopa (−0.13), Toffo (−0.17) and Abomey (−0.08). On the other hand, the values for Dogbo and Aplahoué are positive, at 0.03 and 0.13, respectively. These results reveal a Fréchet-type distribution (positive values) for Dogbo and Aplahoué, while the other stations show an inverted Weibull-type distribution (negative values). Although the HIB distribution belongs to class C, it could not be fitted to the Dogbo, Grand-Popo and Aplahoué stations. Consequently, the parameter values of this distribution are missing from the table. This is because the coefficients of variation and skewness (Cv, Cs) of these stations are not within the zone of influence of this distribution. Appendix A presents graphs of the coefficients of variation and asymmetry (Cv, Cs) of the stations. The results of AMS fitting for the 1981–2021 period at each station with the four (04) class C distributions are shown in Figure 9.

An examination of the graphs reveals that the fit of the distributions varies considerably from station to station. In Abomey, instability in the central part of the distributions leads to both under- and over-estimates for all distributions. In general, Abomey, Toffo and Bopa show similar behavior in terms of fitting probability distributions. The IG distribution (in green) is often overestimated by LPIII (in blue), which in turn is overestimated by EV2 (in red), then by HIB (in orange) as the case may be. Overall, these three distributions offer a good fit for the AMSs, but IG tends to deviate from observations at the extremes. However, at the Aplahoué, Dogbo and Grand-Popo stations. IG seems to fit the most extreme values better than the other distributions. As a result, all the distributions fit more accurately at low precipitation (left-hand tail), as well as in the central part. On the other hand, they have difficulty modeling the most extreme values, i.e., the right-hand tail of the distributions.

The choice of the most suitable distribution model is, therefore, complex and cannot be reliably made based on these graphical analyses alone. This underlines the need for a more objective assessment, using numerical comparison criteria. To this end, Figure 10 shows a Taylor diagram reflecting the performance of the four models on the annual maxima series of daily rainfall recorded for each station.

An analysis of the graphs shows that at the Abomey station, the Fréchet model (EV2) correlates well (0.61) with AMSs, but its standard deviation (19.3 mm/d) underestimates their variability (26.11 mm/d), meaning that this model does not fit the data optimally. The same applies to the HIB and IG models, except that the IG model (24.53 mm/d) underestimates the variability of AMSs less than the others and shows a very good correlation (0.72). The LPIII model, on the other hand, shows a good correlation (0.62) and captures the variability (26.63 mm/d) of AMSs well. In theory, the LPIII model should be the most appropriate for this station, but its correlation is weaker than that of the IG model. Consequently, the IG and LPIII models are, respectively, the most suitable for the Abomey station.

At the Toffo station, all models show a high correlation (0.68–0.75) with the AMSs, suggesting that they can capture the general rainfall trend. In terms of variability, the IG model captures the variability of the data more accurately (standard deviation of 31.81 mm/d close to that of the observed data 31.41 mm/d) than the other models. In descending order, the IG and EV2 models are the most suitable for this station. In contrast to the other stations, at Bopa, although the HIB, LPIII and EV2 models show a good correlation (0.62, 0.61 and 0.59, respectively), they underestimate AMS (standard deviations of 26.25. 22.82 and 27.24 mm/d, respectively, vs. 29.92 mm/d); however, the EV2 model underestimates it less. As for the IG model, it offers a good correlation (0.67) and captures AMS variability well (standard deviation of 30.28 mm/d).

At the Aplahoué, Grand-Popo and Dogbo stations, the IG model offers the best fit, although it slightly underestimates AMSs at Aplahoué. This model is followed by LPIII in Aplahoué and Dogbo, and by EV2 in Grand-Popo.

It is clear from the analysis of the diagrams that the IG model offers superior performance on all stations compared with the other models. However, despite being the best-performing model, it still has errors ranging from 16.47 mm/d (Aplahoué) to 39.80 mm/d (Grand-Popo), as shown by the blue contours representing RMSE values.

The IG model is followed by the LPIII and EV2 distributions. These conclusions are confirmed by the AIC and BIC criteria applied to all stations, as shown in

Table 6. BIC and AIC results for the 1981–2021 period.

Stations	GEV		Log-Pearson Type III		Inverse Gamma (IG)		Halphen Type B Inverse (HIB)
	BIC	AIC	BIC	AIC	BIC	AIC	BIC	AIC
Dogbo	377.75	372.61	377.64	372.5	374.03	370.60	-	-
Grand-Popo	425.74	420.60	425.69	420.55	422.68	419.25	-	-
Aplahoué	379.99	374.85	379.93	374.79	378.12	374.69	-	-
Bopa	388.54	383.40	388.67	383.53	385.45	382.03	388.58	383.44
Toffo	386.07	380.92	386.07	380.93	383.23	379.80	386.01	380.87
Abomey	381.32	376.18	381.32	376.18	378.10	374.68	381.35	376.20

.

An analysis of the table shows that the values of the criteria are close for the LPIII and GEV (Fréchet) distributions on several stations, regardless of the criterion used. As the best model is the one that simultaneously presents the lowest values for both criteria, only the GI distribution respects this condition on all stations, confirming previous analyses. Thus, the Inverse Gamma (IG) distribution is the one that best represents the behavior of the annual maximum series (AMS) of daily rainfall in the Benin Mono Couffo watershed. It is, therefore, considered the “best-fit” or “optimal” model, followed by the LPIII and EV2 distributions. The identification of the Inverse Gamma distribution as the best fit also suggests that the annual maximum series of daily rainfall has a low return frequency. The quantiles estimated from this distribution are presented in Appendix B.

4. Discussion

The annual maximum series of daily rainfall over the period 1981 to 2021 are all stationary, homogeneous, and independent, which ensures that the statistical models applied are appropriate and that the results obtained are representative of actual conditions. Application of the DSS tool shows that class C distributions (78%) are best suited to extreme precipitation on all rain gauge stations. These results are in line with those of Nassa et al. [22], who showed that class C appears at a rate of 57.47% and 46.84%, respectively, in the 30-year and 40-year series in Côte d’Ivoire. It should, therefore, be noted that this tool offers greater potential for selecting suitable distribution classes. However, the work of Gupta and Chavan [50] points out that this tool assumes that a probability distribution belongs to either a light-tailed or a heavy-tailed distribution class, an assumption that is not always verified. Indeed, recent studies, notably those by Wietzke et al. [51] and Gupta and Chavan [52], have shown that the same distribution can be characterized as light-tailed or heavy-tailed depending on its shape parameter or tail index. For example, the Log-Normal (LN) distribution lies at the boundary between classes C and D [15,21]. Despite this, it has been integrated into class D in the Decision Support System (DSS), for two main reasons: (i) it produces more conservative results (notably overestimation) in quantile estimation studies, and (ii) there is a lack of appropriate criteria to clearly differentiate classes C, D and LN [52]. This classification could nevertheless lead to errors in the estimates. These observations underline the need to further improve the performance of the DSS tool. The integration of new discrimination criteria, or a more flexible approach to managing border distributions, could enable the development of a more comprehensive and powerful tool for climate risk management and the design of hydraulic or hydro-agricultural structures.

Indeed, class C distributions are regular variation distributions, a subclass of heavy-tailed distributions. The tails of class C distributions decrease asymptotically according to the power law [41,42]. The implication of this decay is that extreme events are not only rare but can occur with significant intensity. In this way, characterizing the behavior of the tail helps to understand the probability of occurrence of extremes [50].

Furthermore, in selecting the most suitable distribution model, the LPIII, IG, WEB, EV2 and HIB distributions show a certain variability from one station to another, as Yuan et al. [2] point out in their analyses. In addition, Yuan et al. [2], Amoussou et al. [8] and El Adlouni et al. [53] point out the poor ability of models to represent the most extreme precipitation, which is the most important region for engineering design and planning applications [12]. Since we are talking about the safety of structures, combined analyses of several parameters were used to maximize correlation while reducing errors and standard deviation variability in the Taylor diagram representation of AMSs. This highlighted the Inverse Gamma distribution as the best-suited model in the basin, offering superior performance over all stations compared with other models. This distribution offers correlations between 0.61 (Dogbo) and 0.75 (Toffo). However, despite being the best-performing model, its errors range from 16.47 mm/d (Aplahoué) to 39.80 mm/d (Grand-Popo).

Indeed, Agué and Afouda [3], over the period 1921 to 2001, showed that the best laws in ascending order throughout Benin (using 26 stations) are GUM, LN, GEV, PIII and LPIII. However, in the Mono-Couffo basin, only the Gumbel, LN and PIII laws emerged as the most suitable for adjusting maximum daily rainfall, respectively. These differences in laws could be explained by two factors. Initially, several authors, such as Gupta and Chavan [50]. Moccia et al. [54] and Allé [55], highlighted a decline in precipitation from the 1970s onwards, which became more marked in the early 1980s, with a slight upturn towards the end of the 1990s. However, the work of Seydou [56] and Nakou et al. [57], for the periods 1981–2017 and 1970–2020, respectively, has shown an upward trend marked by high rainfall variability in the form of alternating deficit and surplus years, predisposing the basin to hydro-climatic risks. This indicates the installation of a new mode of variability after the 1980s. The second factor is the criterion for choosing the laws used for the frequency analysis, which is based on the predominant laws derived from previous studies (also used by many, including Habibi et al. [30] in Algeria; Kouassi et al. [58] in Côte d’Ivoire; Avossè et al. [35] in Benin and Amadou Abdou et al. [36] in Niger). However, the DSS has been developed to enable the most suitable class of distributions to be chosen as a prelude to the various adjustments [59] as carried out in the present first. It is also important to note that the laws obtained by Agué and Afouda [3] are different from those obtained. There is, therefore, a change in the adjustment laws from the period 1921–2001 to the period 1981–2021.

5. Conclusions

This study examined the frequency analysis of annual maximum daily rainfall in Benin’s Mono-Couffo watershed, using data from six (06) rainfall stations for the period 1981 to 2021. The application of the Decision Support System (DSS) as well as graphical and numerical performance criteria enabled us to successively determine the best distribution class and select the most suitable distribution in the Benin Mono-Couffo basin.

The results show that daily precipitation AMSs preferentially follow the Inverse Gamma distribution, with estimation errors ranging from 16.47 mm/d at Aplahoué to 39.80 mm/d at Grand-Popo. The second most suitable distribution is Log-Pearson Type III. Changes in the distribution laws were observed between the periods 1921 and 2001 and 1981 to 2021, pointing to instability in the climate. It is imperative to no longer rely on the quantiles of the ORSTOM and CIEH approach, as these are now obsolete.

For efficient management of water resources in the basin, the use of Inverse Gamma distribution is recommended when designing hydro-agricultural schemes.

Finally, a nationwide study is needed to clearly establish optimal distributions for each region, as the USA and Australia have done. Future research should also focus on the evolution of quantiles so that they can be better integrated into the sizing of hydraulic structures and improve decision-making.