Trend and Return Level Analysis of Extreme Rainfalls in Senegal

In recent years, Senegal has been confronted with increasingly frequent and damaging extreme events. In the context of climate change, we conducted this study to characterize the trends of rainfall extremes in Senegal. In this work, we used daily rainfall data from 27 stations in Senegal from the period 1951 to 2005 (55 years). To study their linear trends, non-stationary extreme value models with time as a covariate are fitted to evaluate them. Our results indicate a decreasing trend of extreme rainfalls at most of the stations except for 5 stations. However, the decreasing trends are only significant for two stations (Thiès and Kidira), however, this can only be taken as information that climate change may have already impacted extreme rainfalls. For the 20-year and 30-year return periods, the results show that they have undergone changes, in fact for almost all stations, the trends in return periods are decreasing.


Introduction
Over the last two decades, there has been an upsurge in extreme weather events such as droughts, heat waves, extreme rainfall events, and floods with socioeconomical and environmental impacts not recorded in over 30 years [1]. In 2002, a cold wave accompanied by unusual rains and uninterrupted winds hit Senegal causing the death of 30 persons and about 600,000 head of livestock, not to mention the 2012 flood that caused the death of 26 people; 264,000 people were affected and more than 5000 families had to be relocated. Furthermore, the World Bank in its 2010's report states that developing countries will be hit hard by the effects of climate change, even as they strive to overcome poverty and promote economic growth [2]. However, few studies have been done on the evolution of extreme climate in Senegal, most of them based on indices developed by the Expert Group for Detection and Monitoring of Climate Change (ETCC-DMI), nevertheless, the index method remains a descriptive method and does not allow to assess the statistical properties of extreme events [3]. On the other hand, [4] used the extreme value theory to detect nonstationarity in daily rainfall in the Sahelian region. Indeed, this theory which is the most well-founded to study the evolution of extremes remains unexplored or little used. Thus studying the trends of extremes and calculating return levels with the extreme value theory would be necessary for a good understanding of the evolution of these phenomena in our study area. An important assumption of this theory refers to the stationarity (temporal) on our time series, which implies that the model parameters do not change with time [5], but according to the IPCC report [6] statistically, significant trends were observed in extreme values on hydroclimatological series in different areas of the world which imply that the climate series are known to be non-stationary. This is why different studies have considered time as a covariate. This new approach has been widely used on climate data. [4] introduced a trend in the location parameter in order to detect a break in the series, precipitation [7] and temperatures [8].
As Senegal was hit by an unprecedented drought [9], an open question is whether and to what extent extreme rainfall events have already been affected by this drought. The first answer in this paper will be given by using non-stationary extreme value models to study the evolution of rainfall extremes with available daily data for the period from 1951 to 2005. Our methodology consists of fitting non-stationary extreme value models to the annual maxima of our time series for each station. First, we will use Kendall's test to detect possible trends, then for each station showing a trend, the Akaike Information Criterion (AIC) statistical criterion will be used to select the best model and after analyzing the trends of the extremes. Finally, we will compare the non-stationary return levels (20 years and 30 years) in 2005 with the stationary return level. This paper is organized as follows. Section 2 presents our data and explains la methodology. Section 3 presents the results. Discussion and conclusions are introduced in Section 4 and Section 5, respectively.

Data
The daily rainfall data used in our study comes from the database of the Regional Study Center for Drought Adaptation Improvement (CERAAS). It is collected from various network observations, including the National Agency for Civil Aviation and Meteorology (ANACIM) and the one set up by ISRA/CERAAS as part of the agro-sylvopastoral monitoring. These data concern 27 stations repre-Journal of Water Resource and Protection sentative of different rainfall regimes in Senegal ( Figure 1) and cover the period 1951-2005. Table 1 presents different characteristics of the rainfall stations located in the area study (Senegal).

Methods and Tools
In this framework, the extreme value theory (EVT) will be adopted to characterize the evolution of extreme rainfalls. The Kendall test will be performed to  detect the existence of trend within our time series of rainfall for each station.
For stations without trend, a stationary Generalized Extreme Values (GEV) model will be fitted. The case where the trend is detected in some stations, three (03) non-stationary GEV models will be fitted in order to choose the best model among these three with the AIC statistical criterion. The likelihood ratio test will be also used to find the goodness of fitin on extreme value distributions (Fréchet; Gumbel; Weibull), in order to better estimate the 20-year and 30-year return levels.
Finally, to analyze the trends of extreme rainfalls, relative values will be calculated to quantify decreasing or increasing of such extremes. Therefore, we will study what would be the impact of using a stationary model to the detriment of a non-stationary one to predict the return levels.

Stationarity Test
The Mann-Kendall test was applied to our daily precipitation data [10]. This test is used to examine the existence of a linear trend (upward or downward) in a time series. It compares the null hypothesis H 0 tested with "there is no trend" to the alternative hypothesis H 1 "presence of a trend". If the p-value p < α chosen significance threshold, the H 0 hypothesis is rejected and it is concluded that there is a significant trend, at the chosen threshold (100 × (1 − α)% = 95% confidence).

Extreme Value Theory (EVT)
The approach followed in this paper is based on the extreme value theory. According to the IPCC's report in 2012, most studies rely on climate extreme in-Journal of Water Resource and Protection dices to study moderate extremes [11]. However, in rarer events, it is recommended to use the extreme value theory [12] in order to describe the behavior of such events. The purpose of this theory is to study the asymptotic distribution of the maximum of a sequence of random variables. It states that the maxima of independent and identically distributed data can be modeled by the generalized extreme value distribution [13]. This model merges the three distributions (Gumbel, fréchet, Weibull) into the following single parameterization.
, , where ξ, μ and σ are the shape, location and scale parameters, respectively. The shape (ξ) describes the behavior of the tail, the location (μ) specifies where the distribution is centered, and the scale (σ) its spread or diffusion. If ξ > 0 one has a Fréchet distribution, a light-tailed (Gumbel) distribution if ξ = 0, and a finitetailed (Weibull) distribution if ξ < 0.
Different methods are used to estimate these parameters, the two most commonly used are: maximum likelihood and the L-moments method. In this work, the maximum likelihood method is preferred because it allows us to incorporate covariates in order to detect trends in the extreme rainfalls. In the non-stationary framework, these parameters vary as a function of GEV time (μ(t), σ(t), ξ(t)) to account for variations due to climate change.
The location and scale parameters are assumed to be depending on the time t.
However, the shape parameter (ξ) is difficult to estimate accurately in practice, it will remain constant in this work. Four (04) GEV models are used to fit the annual maximum precipitation for each station.
The basic Stationary GEV (SGEV) model (called Model 1) is fitted with the location (μ), the scale (σ) and the shape (ξ) that are constant over time in contrast to the nonstationary case (NSGEV model) where location μ and scale σ depend on the time t for the three (03) different models (See: Table 2). The so-called

Akaike's Information Criterion (AIC)
The AIC criterion [14] is applied to the estimated models by a maximum likelihood method. It is defined by: where K is the number of parameters in the model and L is the likelihood function. For each station showing a trend, three models (model 2, model 3 and model 4) will be fitted. The choice of the best model is given by the lowest AIC among these three (03) non-stationary GEV models estimated for each station. This goodness of fit model will be taken to study the trends of extremes. The use of the AIC criteria is justified by the fact that when the best model has a linear trend in location and scale parameters, the AIC outperforms all other selection criteria and also it is suitable for small sample sizes [15].

Return Level (RL)
The T-year return level for a given random variable is its quantile that exceeds on average once every T years. An event has a return period of N years if this event is statistically expected every N years. The relationship between the probability of occurrence of an event corresponding to the quantile of level p and its return period T = h × N (h = 365) in the case of annual maximum), is defined by: The estimation of the return level is obtained by inverting the distribution function of the GEV distribution In the non-stationary setting the parameters μ and σ can depend linearly on time t.

Likelihood Ratio Test (Deviance Test)
The likelihood ratio test will be used to see to which distribution (fréchet, Gumbel, Weibull) our maxima belong in order to properly estimate the parameters of the model chosen for each station. The statistic of this test is defined by: where 0 H L and 1 H L are respectively the likelihood functions of the Gumbel distribution and the GEV distribution calculated for the parameter values estimated with the maximum likelihood method. This ratio follows a chi-square distribution with 1 degree of freedom [12] used to accept H 0 (maximum rainfalls are from a Gumbel distribution) or reject H 1 (maximum rainfalls are from a GEV distribution with ξ ≠ 0). Journal of Water Resource and Protection We reject the H 0 hypothesis when the calculated statistic is greater than the quantile of Chi-squared with one degree of freedom: if ξ < 0, we can conclude that we have a Weibull distribution and if ξ > 0, we have the Fréchet distribution. When the statistic is less than the quantile of Chi-squared with one degree of freedom, the hypothesis H 0 is accepted: the observations are then distributed according to a Gumbel distribution. In our study, the p-values are calculated (95% confidence) with the hypothesis H 0 the observations come from a Gumbel distribution.

Trend and Return Level Analysis
First, for trend analysis, it will be based on the sign of 1 µ or 1 σ , if it is positive so the trend is increasing and decreasing, if it is negative and to check the significance of the trends using the likelihood ratio test. For example, if the maxima of rainfalls for a given station follows a non-stationary Gumbel model, to know the significance of its trend, we test from a stationary Gumbel model.
Then, to have a more general vision of the evolution of the extreme rainfalls and to make a comparison of all the stations under study, we will calculate the relative value of the evolution of the parameters μ(t) and σ(t) as follows: A relative change of −40% implies that at the end of the period, the parameter (μ(t) or σ(t)) was 40% lower than at the beginning of the period. This very interesting approach was used by [5] to characterize long-term changes in annual maximum snow depth and snowfall.
Finally, since the return levels are also time dependent, the same approach will be used to analyze the N = 20-year and N = 30-year return levels. A key issue is the importance of using time-dependent return levels, so the N = 20-year and N = 30-year return levels estimated by Stationary GEV and Nonstationary GEV will be compared.  where a trend has been identified. Figure 3 shows the model chosen for each station with the AIC criterion for parameter estimation. The analysis shows us that the annual maxima can be modeled by either Model 2 or Model 3, we can also notice that no station is modeled by Model 4. This shows that statistically testing the time dependence with a single parametric model will provide much more robust results than with two parameters in our study area. The NSGEV model selected for each station will therefore be further investigated throughout this paper.

Likelihood Ratio Test
The GEV model includes the following three distributions: Gumbel, Fréchet, and Weibull. The likelihood ratio test is used to choose which of these distributions is adapted to estimate the parameters (μ, σ, and ξ) for the chosen model for each station. The p-values calculated from this test (95% confidence level) for these 27 stations with the assumption H 0 , the observations are from a Gumbel distribution showed that only two stations (Table 3) will be modeled by a Fréchet distribution (ξ > 0).

Analysis of the Evolution of Rainfall Extremes
The analysis of trends in location or scale parameters shows a diverse picture. Journal of Water Resource and Protection

Choice of a Non-Stationary Model over a Stationary Model, Impact on Return Levels
In this section, we will compare the relative difference between the return levels calculated from two methods: the return levels obtained with NSGEV (trend in extreme rainfalls) versus SGEV (no trend in extreme rainfalls). Journal of Water Resource and Protection

Discussions
Through the analysis of daily rainfall series over the period 1951-2005 at 27 stations throughout Senegal, the objective of this work was mainly to characterize trends in order to determine the evolution of rainfall extremes. The Kendall test was performed on our series to verify stationarity. The results showed that for the 27 stations studied, only 3 stations (Bakel, Kédougou, Kaffrine) are stationary. On the basis of these results 04 GEV models are used, first to detect trends in the location and scale parameters, then the AIC criterion for the choice of the adequate model for each station and finally, studied the evolution of the extreme rainfalls. The results showed that 11 stations can be modeled by Model 2 (trend in location parameter) and 13 by Model 3 (trend in scale parameter). The results of the linear trend and relative value analysis showed a decrease in extreme rainfall at almost all stations. These results are also in agreement with the work of [4], which highlights a downward trend of extreme rainfalls in the Sahelian zone, particularly in Senegal, where we have recorded the most significant decreases. Indeed, during the 1970's, West Africa was hit by a severe drought, which explains the long duration of these dry sequences, leading to a decrease in the intensity and frequency of these extremes at almost all of our stations. To our knowledge, this is the first time that a decreasing trend in the variance (scale parameter) of extreme precipitations has been found in our study area using the extreme value theory. In addition, other works have shown this decrease in extreme rainfalls in West Africa, this is the case of the study of [16] who performed a trend analysis with extreme rainfall indices. The differences obtained by com-paring the 30-year return levels of the NSGEV and SGEV models show that this difference is quite significant for only one Podor station (19%) out of 24 (estimated with the NSGEV model). From these results, it could be said that it is unnecessary to modify the calculation of the return levels (use of a stationary model). The Block Maxima approach, applied to annual precipitation maximums, can lead to a loss of information (of extreme values), which can make it difficult to identify the significance of trends. To overcome this problem, the frequency exceeding a threshold could be used in other studies. This method was used by [17] to study extreme rainfalls in southeastern South America. The results of this article constituting the proof that climate change could already have an impact on these precipitation extremes as other studies have pointed out by adopting this methodology (non-stationary GEV model); [5] to characterize the long-term changes in annual snow depth showed that half of the stations show significant decreasing trends, or the work of [4] who studied Nonstationarity in series of extreme rainfall, these results showed that for all the zones (West Africa) the most probable rupture is a negative rupture between 1966 and 1970. Furthermore, with the increase of extreme events in recent decades, we could expect an increase in the trends of the parameters of location, scale and return levels, so it would be important to deepen this work in order to study the future evolution of the intensity and frequency of these events in a context of climate change.

Conclusions
The characterization of the trends of rainfall extremes with different models of non-stationary extreme values reveals that the linear trends of extremes are decreasing on 19 stations against 05 increasing. However, these trends are only significant for two stations, Thiès and Kidira. The stations of Bakel, Kédougou and Kaffrine do not show any trend and are stationary, which shows that these three stations are not considerably affected by the droughts of the 1970s and 1980s. In fact, this decrease in extreme rainfalls is mainly caused by the two droughts that hit the country. The NSGEV model proves to be useful for calculations of 20-year and 30-year return levels because the most devastating phenomena occur for very large events. Our results show that the trends of the 20-year and 30-year return levels have decreased on the major part of the stations, moreover the comparison.
The 30-year return levels estimated by the NSGEV and SGEV seem to differ little significantly except for one Podor station (−19%), so using one or the other method has little influence on the calculation of return levels. However, with climate change disrupting the climate regime with dramatic consequences on socio-economic development and the environment, the NSGEV model could be the most adequate or appropriate for the estimation of return levels. The great unknown that remains now is the future evolution of the intensity and frequency of these events, in a context of climate change for those extensions of this work could be considered. First, make our models much more complex by incorporating non-linear covariates (quadratic model) with much more advanced non-Journal of Water Resource and Protection stationary return levels [18]. Second, use simulated data from regional climate models with spatial dependence between extremes.