The main data source for this study was the National Meteorological Agency of Mali (MALI-METEO), which provided monthly rainfall data from Bamako-Senou rain gauge station for a period 1991 - 2020. As extreme rainfalls are generally of daily rainfall series, African Rainfall Climatology version-2 (ARC2) data was used to model extreme rainfalls in this study. Monthly rainfall data from MALI-METEO was used to validate the gridded ARC2 data, the validation of ARC2 with gauge data is given by ([41]). The location of Bamako-Senou’s rain gauge station is provided in (Table 1).

Table 1. Bamako-Senou’s rain gauge characteristics.

Missing values presented in daily rainfalls data is of 68 out of 11 323 days. A missing rate of 5% or less in the datasets is inconsequential ([43]). The summary statistics of ARC2 data is presented in Table 2below.

Table 2. Summary statistics of ARC2 data from 1991 to 2021.

As the capital and administrative center of the Republic of Mali located at 8˚0'10" W and 12˚38'21" N on both sides on the bank of Niger river (Figure 1), Bamako has a 245 km² land area a population of about 1.3 million, and a population density of 5300 people per square kilometer ([24]). Bamako has one rainy season which usually lasts about six (6) months, beginning mid-May to October, peaking usually in August. However, the dry season occurs between February and the first two (2) weeks of May. The Intertropical Convergence Zone (ITCZ) largely controls Bamako’s rainfall characteristics by oscillations across the North and South of the African continent, which usually brings rainfall to South Mali between June and October each year ([50]).

The average monthly rainfall in Bamako ranges from 54.1 mm in May to 290.2 mm in August, with a mean annual rainfall of 991.3 mm ([24]). The warmest and coolest months are therefore April and May respectively, and December to January representing the months without rain ([24]). Daily maximum and minimum temperature of 40˚C and 17˚C respectively are recorded in April and December, with an average annual temperature of 29˚C usually recorded in March. Floods are common in Bamako in the rainy season, as the city can usually receive more than 600 mm of rainfall. Bamako is located in a valley covered with sandstone deposits and has two (2) types of soil: the soil formed by rock formation, and soil caused by lateralization and alluvial formation occupying the primary and secondary riverbeds and tributaries ([24]). Bamako’s vegetation is largely characterized by savannah forest and rivers ([24]).

Figure 1. Location of the Bamako area with climate station.

2.3.1. Quantile Method for Threshold Selection

The quantile of a distribution is a quantity Q P such that a proportion p of the population must be less than or equal to Q P . For instance, the quantile Q 99 is the value such that 99% of the population is less than or equal to Q 99 .

Consider a population of discrete values or a continuous population density, the k − t h q -quantile for a variable X can be written as:

The ‘‘ p − q u a n t i l e ’’ is based on a real number p with 0 < p < 1 then p replaces k q in the above formulas.

2.3.2. Generalized Pareto (GP) Distribution Method

The peaks over threshold (POT) was used to model the tail of the distribution, the approach consisted in the analysis of the data that are considered as extreme observations, the data that surpass a threshold level u . Let us consider X 1 , … , X n a sequence of independent and identically distributed random variables, having distribution function F, the focus was on conditional probability such as:

With x F is the endpoint of F, the following result gives an approximation to this probability for high values of the threshold u . The distribution function of ( X − u ) , c o n d i t i o n e d t o X > u , is approximately by the generalized Pareto (GP) distribution given as follows:

With

where y ≥ 0 f o r ξ ≥ 0 a n d 0 ≤ y ≤ σ ξ f o r ξ < 0 , ξ a n d σ are the shape and scale of the GP distribution.

Or

Denoting by δ u = P ( X > u ) and from the conditional distribution the expression of

The levels x m that is exceeded on average once every m observations is the solution of

2.3.3. Fitting Probability Distributions to Annual Maximum (AM) Series

For fitting probability distribution to AM series, the Normal, Lognormal (LN), Gamma, Pearson 3 and Gumbel distributions were fitted to AM series variables (Table 3).

Table 3. Summary table of distributions used to fit AM series in Bamako.

where μ x , σ x 2 and γ x are the mean, standard deviation and skewness of x , respectively. Y = l n x ; parameters α , β a n d c represent the shape, scale and location respectively of the P3 distribution, and Γ is the gamma function.

One of the purposes of this article was to explore which distribution functions give the best fit with the AM series sample. The Chi-square test and the Kolmogorov-Smirnov test were used to check the adequacy of the tested distribution functions at 0.05 significance level. The root mean square error (RMSE), and the Akaike information criterion (AIC) were used to define the best fit distribution. Table 4 below gives the test statistics for model selection.

Table 4. Test statistics for model selection.

where F is the actual cumulative relative frequency and F ^ is expected cumulative relative frequency. The estimated value for the test statistic should then be compared with the critical value selected based on the significance level and the number of data points. f i is the actual frequency and f ^ i is the expected frequency, i is the number of classes and k is the total number of categories. This estimation is close enough for testing the goodness-of-fit whenever the expected frequency in any category is equal or greater than 5. χ 2 is chi-square distributed with k − 1 degree of freedom. This form of AIC presented in Table 4 was used by [22].

The threshold estimation was done by the method of quantile, the mean excess plot using the peak over threshold (POT) method was used to set the optimal threshold. The percentage of missing values out of the number of rainy days was 0.6%. A missing rate of 5% or less in the datasets is inconsequential ([43]). The mean excess plot should be linear, and the parameter estimates should be stable above the threshold at which the GP model becomes valid. Confidence intervals were added to take account the effects of estimation uncertainty in the evaluation of optimal threshold. FromFigure 2(a), a threshold above 42.9 mm is not appropriate because the confidence bounds increase significantly. This is visible in Figure 2(b) that there is not enough data above the threshold 42.9 mm. Meanwhile by taking a threshold below 33.8 mm, the variances in Figure 2(b) are too small, which means that the number of observations is too large, and the asymptotic approximation will be violated in GP distribution. Therefore, it can be approximated that the threshold should be around 33.8 - 42.9 mm. In this case, the Poisson distribution may help define the optimal threshold. As the variance of the data is greater than the mean of the observed values, the Poisson distribution (in which the variance is equal to the mean) will not be a good candidate leading to low p-values. In addition, as the mean residual plot must be approximately linear with the threshold, the linearity still exists between 33.8 - 42.9 mm. Therefore, the threshold which corresponds to 99^th percentile of the data was chosen (40.8 mm) as an optimal threshold. The number of exceedances for the threshold 40.8 mm was 112 exceedances corresponding to on average 3.6 peaks per year. When the quantile Q 95 was chosen, the number of exceedances above the threshold was 536, corresponding to on average 17.3 peaks per year. Moreover, when the quantile Q 90 was chosen the number of peaks per year on average was 36.3 peaks per year. However, various studies suggested the threshold value should be high so that the model assumptions are not violated ([26]). [21] and [7] showed some strategies to select the best threshold. The best threshold should be at least greater than the mean but cannot be too high ([18]). Figure 2(c) and Figure 2(d) show the number of peaks-over-threshold 40.8 mm and the generalized Pareto (GP) distribution diagram of the threshold 40.8 mm.

Figure 2. (a) The mean excess plot; (b) the shape parameter of the GP distribution; (c) the peaks over the threshold u = 40.8 m m for the period 1991 - 202 and (d) the GP distribution diagram.

[14] on urban Flash flood and extreme values analysis in Bamako revealed that the threshold at which flood may occur in Bamako was about 47 mm. Moreover, [33] showed that extreme thresholds in Sahel regions were in most cases between 20 to 50 mm. Another study on the debilitating floods in Sahel countries, principally the flash flood which triggered in most Sahel countries, predicted that most floods which occurred in Sahel countries were thresholds of 99^th percentile ([42]). Table 5 below gives parameters estimation of the GP distribution and the goodness-of-fit AIC value.

Table 5. GP distribution parameters estimation using the method of Maximum Likelihood and the AIC value.

It is well documented that the specific choice of the threshold value would affect the magnitude of exceedances beyond the selected threshold and the arrival time of the extreme events ([51]). The threshold u = 40.8 m m should be large enough to ensure that the observations are independent without loss of appropriate information. Furthermore, choosing a low threshold value could introduce serial dependence of both occurrence times and magnitudes, thereby violating the assumption of independence ([51]).

Several studies have been conducted over the year for the best model selection to describe extreme rainfalls in various areas. To the best of our knowledge, we have not been across any specific study online to describe the best model selection for extreme rainfalls in Bamako area. Many probability distribution models have been suggested for representing the distribution of annual hydrologic extremes at a single site ([10]; [25]; [47]; [19]; [35]; [38]; [39]). Distributions such as exponential, Gamma, Weibull, log-normal, logistic, Gumbel, GP and GEV distributions have been commonly employed in hydrological studies as referenced by ([46]; [20]; [1]; [31]; [3]; [11]). For instance, in an extreme rainfalls study in Bangladesh ([30]) found that the Generalized Extreme Value (GEV), Generalized Normal (GNO) and P3 distributions are the best models that can be used for describing accurately the distributions of annual maximum rainfalls in Bangladesh. The national guidelines of different countries suggest the use of different distributions. For instance, Log-Pearson 3 has been recommended in the US in Bulletin 17B ([15]). The generalized extreme value (GEV) distribution is recommended in various countries in Europe, including Australia, Germany, Italy, and Spain ([38]). The GEV distribution and LP3 are also recommended in Australia ([6]). However, many other distributions have also been employed popularly in various regions including the generalized Pareto (GP) distribution in Belgium, Gumbel distribution in Finland and Spain and the generalized logistic (GLO) distribution in the UK ([38]). In Canada, the choice of a specific distribution is not mandatory, however, LP3, Log-normal three parameters (LN3), GEV, Gumbel have been used popularly ([9]; [4]; [17]). Recent study for instance in Kenya found that when applying the Normal, Lognormal, Weibull, GEV, Exponential, Gamma, Logistic, Gumbel, Uniform and the GP distributions to modelling maximum temperature and total rainfall, the result showed that the GEV, Gamma, and Log-Normal distributions were well-suited for both maximum temperature and total rainfall datasets ([32]). In this study, five (5) commonly used probability distribution functions namely, the Normal, Log-normal (LN), Gamma, Gumbel type 1 and Pearson 3 (P3) distribution were used to fit the AM series. Figure 3below shows the empirical and density plots of various distributions using the MLE and MOM as parameters estimations.

Figure 3. Empirical and density plots of distributions using the MLE as parameters estimation in the left and the MOM in the right.

The K-S, Chi-square test, AIC, and the RMSE tests were applied to the AM series data for the period 1991 - 2021. The results of the tests for the case when the parameters were estimated with the MLE and the MOM are shown in Table 6 below. All the distributions except the Normal distribution using both the MLE and the MOM for parameters estimation were passed for the Chi-square test at 0.05 significance level. When parameters were estimated with the MLE, P3 gave better result than other distributions followed the Gumbel distribution and the LN distribution as third best fitted distribution, the gamma and the normal distributions gave weak results compared to other distributions. When the MOM was used for parameters estimation, the LN distribution comes first as the best fitted distribution followed by the P3 distribution as second and the Gumbel distribution as third, the Gamma and Normal distribution gave weak results compared to other distributions. All the distributions passed the K-S test at 0.05 significance level, the critical value for the K-S test when the number of categorical intervals is 9 was 0.43001. The P3 distribution using the MLE and the LN distribution using the MOM can be used for AM series frequency analysis in the study area.

Table 6. Tests results for the AM method for the period 1991 - 2021 and parameters estimated with the MLE and the MOM, the significance level for the test was chosen as 5%.

In comparing the peaks-over-threshold method and the annual maximum method for flood frequency analysis, [8] found that when the distributions parameters were estimated with the MOM and L-moment, the LP3 distribution gave the best results in most of the applied tests. However, when the distributions parameters were estimated using the MLE, the best results were obtained using the P3 distribution, while the lognormal (LN) and LP3 distributions were second and third respectively.

The comparison between the POT and the AM series was based on ARC2 daily rainfall values for the period 1991 - 2021. The POT sample was 3.6 times higher than the AM sample. The truncation level selection is one of the disadvantages of the POT method. [48] found that a POT sample with an average of two peaks per year gave better results than the AM method. [12] reported that an average of at least 1.65 peaks per year should be selected in the POT method to gain advantage over the AM method. Moreover, [28] found that more than 0.91 peaks per year should be selected for the POT model to have advantage over the AM method when the parameters are estimated with the MOM. FromFigure 4, for smaller return periods the POT method and the AM method gave approximately similar results when parameters are estimated with the MLE and the MOM. However, for higher return periods the POT gave higher quantile rainfalls than the AM series method and described better than the AM series. The P3 distribution when parameters are estimated with the MLE and the MOM gave best and higher quantiles than other distributions in AM method. The POT method using the GP distribution also gave best result for quantiles estimation than the P3 distribution in AM series. In conclusion, therefore, the POT method gave best result than the AM series for estimating quantiles. For flood frequency analyses (FFA) in the region, the POT method could be used to inform the return periods of upcoming flash floods in Bamako. The design of most urban water infrastructures mostly depends on return periods of extreme events; this could also be an alternative using the POT method to design those infrastructures in Bamako.

Figure 4. Results of the POT and the AM series quantiles.