Intensity-Duration-Frequency (IDF) rainfall curves in Senegal

Urbanization resulting from sharply increasing demographic pressure and infrastructure development has made the populations of many tropical areas more vulnerable to extreme rainfall hazards. Characterizing extreme rainfall distribution in a coherent way in space and time is thus becoming an overarching need that requires using appropriate models of IDF curves. Using 14 series of 5-min rainfall records collected in Senegal, a comparison of two GEV&scaling models is carried out, resulting in the selection of the more parsimonious one (four parameters) as the recommended model for use. A bootstrap 5 approach is proposed to compute the uncertainty associated with the estimation of these 4 parameters and of the related rainfall return levels for durations ranging from 1h to 24h. This study confirms previous works showing that simple scaling holds for characterizing the temporal scaling of extreme rainfall in tropical regions such as sub-Saharan Africa. It further provides confidence intervals for the parameter estimates, and shows that the uncertainty linked to the estimation of the GEV parameters is 3 to 4 times larger than the uncertainty linked to the inference of the scaling parameter. From this model, maps of IDF 10 parameters over Senegal are produced, providing a spatial vision of their organization over the country, with a north to south gradient for the location and scale parameters of the GEV. An influence of the distance from the ocean was found for the scaling parameter. It is acknowledged in conclusion that climate change renders the inference of IDF curves sensitive to increasing non-stationarity effects, which requires warning end-users that such tools should be used with care and discernment.

Page 12, lines 23-24: it is not clear from where we can deduce that considering higher moments of return periods the sample size explains 80% of the variance of the confidence interval width for μ, 70% for s, 55% for x and 4% only for iT=100. Therefore this part should be better explained.

Response.
Thank for this relevant remark. In fact, there are two sources of confusion here: First there was a typo: "of" rather than "or" in the first part of the mentioned sentence. We have thus corrected the sentence "However this relation weakens when considering higher moments of return periods:" ; it now reads "However, this relation weakens when considering higher moments or higher return periods:".
Secondly, we agree with the reviewer that the second part of the sentence might be too short to clearly explain what we have in mind. It refers to the linear regression between the relative width of the confidence interval of a given parameter/return level and the number of available years.

Thus we have modified the second part of the sentence:
"the sample size explains 80% of the variance of the confidence interval width for μ, 70% for σ, 55% for ξ and 4% only for iT =100" now reads "the coefficients of correlation between the confidence interval width and the sample size (available number of years) are r²=0.80 for μ, r²=0.88 for σ, r²=0.69 for ξ, r²=0.55 for iT =2 and iT =10, and r²=0.004 only for iT =100." We thank the reviewer for its comments which -we hope -helped improved the clarity of the paper.
The fast growing ::::::::::: fast-growing pressure of mankind on planet Earth makes populations increasingly exposed to hydrometeorological hazards such as torrential rains and floods (IPCC, 2012;Mechler and Bouwer, 2015). Hydrologists are thus more compelled than ever to deal with the problem of assessing the probability of extreme rainfall events at different time scales :::::::: timescales : and for various return periods, depending on the area of the target catchment and the issue at stake : , most notably 5 human life protection and infrastructure dimensioning. A classical way of synthesizing the results of such studies is the production of so-called rainfall Intensity-Duration-Frequency (IDF) curves, providing ::::: which ::::::: provide estimates of rainfall return levels over a range of durations. In doing so, scientists are facing :::: face two sets of difficulties, : : one related to data availability : , and the other to the necessity of a proper methodological framework.
The methodological challenge arises from the complex combination of factors causing ::: that ::::: cause : rainfall to be strongly 15 variable at all scales (from the microphysics droplet scales to synoptic scales :::: scale :: to ::::::: synoptic ::::: scale), as a result of the non-linear :::::::: nonlinear interaction of different atmospheric processes (e.g. Schertzer and Lovejoy, 1987). This implies that it is not at all obvious to find a proper theoretical framework to compute IDF curves in a way that ensures coherency between time scales :::::::: timescales. Early works on IDF proposed empirical methods consisting of first adjusting a frequency distribution models ::::: model fitted to rainfall series {R(D)} for each duration D of interest ; and then ::: and :::: then ::::: fitting ::: the : IDF formula {i T (D)} fitted 20 independently to each series of quantiles derived from the first step and corresponding to a given return period T (see e.g. Miller et al., 1973;NERC, 1975). This has the advantage of being easily implementable and is thus commonly used by hydrological engineers and operational climate/hydrological services. However, because of uncertainties in the computation of the quantiles derived for the different durations, the scaling formulation may be physically inconsistent and may lead to gross errors such as parasitic oscillations or intersections between IDF curves computed for two different durations (see Koutsoyiannis et al.,25 1998, for more details). As a remedy to such inconsistencies, Koutsoyiannis et al. (1998) were the first to propose a general IDF formulation that remains consistent with both the foundations of the probabilistic theories and the physical constraints of scaling across durations. Another notable advance was provided by Menabde et al. (1999) who demonstrated that ::: the changes in rainfall distribution with duration formulated by Koutsoyiannis et al. (1998) can be expressed as a simple scaling relationship, opening the path for using the fractal framework in order to describe the time ::::::: temporal scaling between IDF curves established 30 over a range of durations in various regions of the world (see e.g. Yu et al., 2004;Borga et al., 2005;Gerold and Watkins, 2005; Nhat et al.; Bara et al., 2009;Blanchet et al., 2016;Rodríguez-Solà et al., 2016;Yilmaz et al., 2016).
Focusing on Senegal, a region of contrasted coastal to inland semi-arid climate, our paper : 's : ambitions are both to address the uncertainty issue not dealt with in above mentioned :: the ::::::::::::::: above-mentioned papers and to provide IDF curves for a region located at the western edge of the Sahel, looking at :::::::: evaluating : the spatial variability generated by the transition from the coast to inland. In addition to its methodological bearing, the paper aims at making these IDF curves widely accessible to a large range of end-users in the whole country by mapping the values of the scaling parameters and of the rainfall return 5 levels. Furthermore, selecting an IDF model as less sensitive as ::: that :: is ::: the :::: least :::::::: sensitive possible to data sampling effects and computing the associated IDF confidence intervals make easier the update :::::::: facilitates :::::::: updating of the IDF curves when new data are available.
2 Data and Region 2.1 Senegal Climatological Context 10 Senegal is located at the western edge of the African continent between latitudes 12 • N and 17 • N ( Figure 1a). The climate of Senegal is governed by the West African monsoon (Lafore et al., 2011;Janicot et al., 2011;Nicholson, 2013), resulting in a two-season annual cycle: a dry season marked by the predominance of maritime and continental trade winds in winter : , and a rainy season , marked by the progressive invasion of the West African monsoon (Figure 2a to Figure 2c) during the summer.
The length of the rainy season varies along the :: by : latitude and ranges roughly from 5 months (early of June to the end of 15 October) in the South :::: south : to 3 months (mid-July to mid-October) in the Northern ::::::: northern part of Senegal. Rainfall amounts peak in August and September, coinciding with the period when the ITCZ reaches its northernmost position over Senegal.
There is a strong North-South ::::::::: north-south : gradient of the mean annual rainfall ( Figure 2d) ranging from 300 mm in the North :::: north : to more than 1000 mm in the South :::: south : (Diop et al., 2016). This gradient is mainly explained by the number of rainy days (in average between 20 and 80 from North to South :::: north :: to ::::: south) and to a lesser extent by the mean intensity of 20 rainy days (in average between 10 and 15 mm day −1 ), see Figure 2e and 2f.
Senegal regularly undergoes damaging heavy downpours. A recent example is the rainfall event that occurred in Dakar in the morning of August 26, 2012, causing the largest flood over the last twenty years in the city. An amount of 160 mm was 30 recorded at the Dakar-Yoff stationwhich is large , :: a :::: large ::::::: quantity : but not a historical record :: for ::::: daily :::::: rainfall at this station. In fact ::::: Rather, this event was exceptional because of its intensities at short durations (54 mm were recorded in 15 minutes and 144 mm in 50 minutes) exceeding by far :: by ::: far :::::::: exceeding : the previous records in Dakar-Yoff. Such rainfall intensities and their associated disasters justify the importance of better documenting extreme rainfall distributions at short time scales ::::::::: timescales.

Rainfall Data
The archives of climate / ::: and hydrological services of West African countries sometimes contain large amount :::::: amounts : of sub-daily rainfall records. Howeverthese records are , : most of the time :::: these ::::::: records ::: are : stored in paper strip chart formats : , 5 requiring a tedious task of digitization for using : in ::::: order :: to ::: use : them in numerical applications.
The present study has been made possible thanks to an important work of analyzing and digitizing rain-gauge ::: rain :::::: gauge charts carried out for the main synoptic stations of Senegal. This process was undertaken by the laboratory of hydro-morphology of ::::::::::::::: hydro-morphology ::::::::: laboratory ::: of the Geography Department of :: at the University Cheikh Anta Diop of Dakar (UCAD) in collaboration with the National Agency of Civil Aviation and Meteorology (ANACIM) who provided the rainfall paper charts.
Senegalese synoptic stations are equipped with tipping bucket rain-gauges ::: rain :::::: gauges; the receiving ring is 400 cm 2 and a bucket corresponds to 0.5 mm of rain. The roll rotation is daily. The chart analysis has been performed with the software "Pluvio" developed by (Vauchel, 1992) :::::::::::: Vauchel (1992) allowing the computation of 5-minute time-step :::: time :::: step : digitized rainfall series from the paper diagrams. It is a long and laborious task, which has the advantage of allowing a careful "chart by chart" checking of the quality of the records before digitization. For more information on the digitization process, the reader 15 may refer to the publications of Laaroubi (2007) and Bodian et al. (2016).
A total number of 23 tipping bucket rain-gauges was analysed ::: rain :::::: gauges ::::: were ::::::: analyzed, with data going back to 1955 for the oldest and to 2005 for the most recent. As the assessment of extreme rainfall distributions is known for being much ::::: highly sensitive to sampling effect and erroneous data (Blanchet et al., 2009;Panthou et al., 2012Panthou et al., , 2014b, a particular attention was paid to check and select the most appropriate series. 20 The data selection had to conciliate :::::::: reconcile two constraints: (i) keeping the data set as large as possible and (ii) eliminating series that contain too much missing data.
The procedure for classifying one year-station as valid or not is the following: (i) first, the annual number of 5-min data and the annual amount of rain are computed, (ii) the mean inter-annual ::::::::: interannual : values of these two statistics are computed on the whole series, (iii) a year is classified as valid if either the number of 5-min rain data or the amount of rainfall is comprised 25 between 1/2.5 and 2.5 times their mean inter-annual values, ::::::::: interannual :::::: values, :::: and (iv) other years are classified as missing and removed from the whole series. Since missing years influence the mean inter-annual ::::::::: interannual values, step (ii), (iii) and (iv) are repeated until all remaining years are classified as valid (note that, in fact, no station-year had to be excluded after the initial step). All valid years for all series are plotted on Figure 3. In order to keep the IDF fitting robust, only series with at least 10 years of valid data have been used. This led us to retain 14 stations with record length varying from 10 (Fatick 30 station) to 44 years (Ziguinchor station) with a median of 28 years. This dataset has the advantage of fairly covering the whole country ::::::: spatially :::::::::: representing ::: the :::::: entire ::::::: country, but as the length of the series varies, the quality of the IDF estimates might differ from one station to another. This effect will be more precisely analysed ::::::: analyzed ::::: more :::::::: precisely in section 5.2.1.

Empirical IDF formulations
Intensity-Duration-Frequency (IDF ) ::: IDF : curves provide estimates of rainfall intensity for a range of durations {D} and for several frequencies of occurrence (usually expressed as a return period T ). Each curve corresponds to the evolution of a return 5 level (i T ) as a function of rainfall duration D. Historically, several empirical formulations of IDF curves have been proposed.
All can be described by the following general equation (Koutsoyiannis et al., 1998): where w, θ and η are parameters to be calibrated from rainfall observations.

Koutsoyiannis scaling relationship
10 Koutsoyiannis et al. (1998) have demonstrated that the empirical formulations (Equation 1) can be expressed as: The advantage of Equation 2 as compared to Equation 1, is to separate the dependency on T (return period) from the 15 dependency on D (duration): a(T ) only depends on T , and b Koutso (D) only depends on D. A consequence is that for the particular case of D 0 = 1 − θ : : : Then, it becomes a classical frequency analysis of the random variable I(D 0 ) to estimate the return levels i T (D 0 ) -: -i.e. study :::::: Then, Equation 2 can be reformulated as an equality of distribution of random variables I:

Simple scaling relationship
In the particular case of θ = 0, Equation 5 becomes: where b SiSca (D) is a simple scaling formulation of b.

5
The Extreme Value Theory (EVT ) :::: EVT : proposes two methods to extract samples of extreme values from a time series (Coles, 2001): the Block Maxima Analysis (BMA) which consists of defining blocks of equal lengths (often one year in hydrology) and extracting the maximum value within each block; the ::: and : Peak Over Threshold (POT) which consists of extracting all the values exceeding a given threshold.
Compared to BMA, the POT has the advantage of allowing the selection of more than one value per year, thus increasing the 10 sample size used for inferring the model, but : . :::::::: However, the choice of an appropriate threshold is often difficult (Frigessi et al., 2002). Here the BMA approach was preferred as it is more straightforwardly implementable :::::::::::: straightforward :: to ::::::::: implement.
In BMA, when the block is large enough (which is ensured for annual maxima), the Extreme Value Theory :::: EVT : states that the Generalized Extreme Value (GEV) distribution is the appropriate model for block maxima samples (Coles, 2001). The GEV distribution is fully described by three parameters, : : : the location (µ), the scale (σ), and the shape (ξ), which are respectively 15 related to the position, the spread and the asymptotic behaviour ::::::: behavior : of the tail of the distribution: A positive (negative) shape corresponds to a heavy-tailed (bounded : in ::: the ::::: upper ::: tail) distribution. When ξ is equal ::::: tends to 0, the GEV reduces to the Gumbel distribution (light-tailed distribution): with the extreme value distribution :::::::::: distributions (see also Panthou et al., 2014b;Blanchet et al., 2016). In this approach, the I(D) samples are modeled by a GEV model ::::::::: distribution : for which the location and scale parameters are parameterized as a function of D as follows: The return levels are easily obtained at all durations D as: This formulation is equivalent to the following: With :::: with D 0 = 1 − θ.

IDF model inference
Different fitting methods have been tested to adjust the IDF model parameters to rainfall data: one . :::: One : of them (the two-step method) is applicable to both IDF Koutso and IDF SiSca models.
Note that two other methods specifically dedicated to the IDF SiSca model have also been :::: were :::: also : tested: one based on 25 the moment scaling function (as in e.g. Borga et al., 2005;Nhat et al.;Panthou et al., 2014b), and one based on the global maximum likelihood estimation (as in Blanchet et al., 2016). As they did not perform better than the two-step method, they are not presented here.
The fitting of the scaling b(D) is based on the equality of distribution given in Equation 5 for IDF Koutso and Equation 6 for IDF SiSca . If these equations hold, the scaled random variables I(D)/b(D) have for all durations D the same distribution 30 8 as the random variable I(D 0 ) ::: for :: all ::::::::: durations :: D : . This means that the observed scaled samples i(D)/b(D) have similar statistical properties for each duration D. Based on this property, the parameters of b(D) are calibrated in order to minimize a statistical distance between the different scaled samples i(D)/b(D). As suggested by Koutsoyiannis et al. (1998), the difference in medians computed by the Kruskal-Wallis statistic applied on multi-samples (Kruskal and Wallis, 1952)   The flexibility characterizes the capacity of a model to fit the observed data which are used to calibrate its parameters. To 15 that purpose ::::::: evaluate :::::::: flexibility, the IDF models are fitted at each station; : , : then different scores are computed to assess the fitting performances.
The flexibility and the predictive capacity of the IDF models are quantified based on two types of scores: global and quantilequantile. 30 The two global scores ::: used : are the statistics returned by two goodness of fit (GOF) tests: Kolmogorov-Smirnov (KS) and Anderson-Darling (AD). Each test computes a statistic based on the differences between a theoretical Cumulative Distribution Function (CDF) and the empirical CDF. The null hypothesis is that the sample is drawn from the fitted model. The test returns also ::: also :::::: returns : the corresponding p-value (error of first kind :::::::: probability ::: of :::: type :: 1 :::: error). The p-value is used as an acceptation/rejection criterion by fixing a threshold (here 1%, 5%, and 10%). These tests and p-values were computed for each rainfall duration at each station.
The full presentation of theses :::: these : scores can be found in Panthou et al. (2012). A weighted version of these scores is also used in order to assign greater weight to unusual quantiles, as proposed by Begueria and Vicente-Serrano (2006) and also presented in Panthou et al. (2012).

Uncertainty assessment
10 From a methodological point of view, the central contribution of this paper is its attempt at quantifying the uncertainty associated with IDF calculation in a scaling framework. This involves two distinct aspects. One is the uncertainty linked to the estimation of the scaling parameters. The other is the uncertainty linked to the inference of the GEV parameters. This second component is especially important to consider when applying a scaling model to a location where ::: only : daily rainfall series only are available, which is the ultimate purpose of regional IDF models. Indeed : , in some regional studies, the scaling parameters 15 will have to be inferred from a :: the : very few stations where rainfall is recorded at subdaily time-steps :::::::: sub-daily :::: time :::: steps; if they display variations in space, they then ::: then :::: they : will have to be spatially interpolated so as to provide scaling parameter at any location of interest, notably at the location of daily rainfall stations. At these stations, the scaled GEV distribution is thus estimated from the daily observations only, making the inference far less robust than when using a richer scaled sample obtained from observations ranging from one hour -or less-::: (or :::: less) to one day. 20 Therefore, in the following, the uncertainty assessment at a given location will be addressed separately for the two situations: i) first when observations at this location are available over a whole range of time-steps :::: time :::: steps; ii) secondly when only daily observations are available.
4.3.1 Uncertainty linked to the inference of the scaling :::::: Global :::: IDF model at locations with multi time-scale observations :::::::::: uncertainty ::::: when ::::::::::::: multi-timescale :::::::: samples ::: are :::::::: available 25 Confidence intervals for IDF parameters and return levels are estimated using a non-parametric bootstrap (Efron and Tibshirani, 1994). For each station, it consists of fitting IDF curves to bootstrap samples (i(D) boot ) obtained from the original i(D) samples. The entire process consists of three ::: four : steps: 1. The vector of years is resampled with replacement (Monte Carlo resampling) until its length equals the length of the original vector. When only daily observations are available, the GEV parameters are inferred on the corresponding annual block maxima sample of daily data, which contains far less information that the scaled samples used for fitting a scaled GEV when multi timescale 15 :::::::::::: multi-timescale : observations are available. The GEV parameters for the sub-daily time-steps :::: time :::: steps : are then deduced from the daily GEV parameters using scaling parameters that must be inferred from nearby multi timescale :::::::::::: multi-timescale : observations. In some cases this might generate a significant departure :::: GEV :::::: model :::: that ::::: differs ::::::::::: significantly : from the GEV model that would have been fitted directly on the observations at the proper time-step :::: time ::::: steps if they were available. This effect is studied here by assuming that only the daily data were available for fitting the GEV at our 14 stations and :: by : implementing 20 the bootstrap approach in a way that allows separating the uncertainty linked to the GEV parameter inference and the uncertainty linked to the inference of the scaling parameters. This ::::::::: Analyzing ::: the ::::::::: uncertainty : involves two independent bootstrap resampling processes.

Model evaluation and selection
The results of model evaluation ::::: model ::::::::: evaluation :::::: results are presented in Figure 4, Figure 5 and Table 1. In these figures and 5 table, the subscript a (resp. b) relates to the calibration (resp. validation) results. Figure 4 presents the GOF p-value of the KS test obtained for both models ::::::: (IDF SiSca :::: and :::::::: IDF Koutso ) : in calibration and validation mode at each station (the AD test gives similar results, not shown). In Figure 5, all stations are gathered in one single qq-plot from which global scores are computed. All global results (non-weighted and weighted qq-scores) are reported in Table 1. shows that for all stations and durations, : the KS p-values are higher than 10% (i.e. the risk of being wrong by rejecting the null hypothesis "observations are drawn from the models" is greater than 10%). This means that both IDF models fit the observed data with a reasonable level of confidence in calibration and have thus good flexibility skills. The global scores reported in Figure 5a and Table 1 show that in calibration, IDF Koutso slightly outperforms IDF SiSca . This result was expected as 15 IDF Koutso has an additional degree of freedom (θ parameter) compared to IDF SiSca .
The global qq-plots in Figure 5 and the statistics summarized in Table 1 confirm that the two IDF models perform very 20 similarly in validation. IDF SiSca has slightly smaller biases (mean errors) while RMSE and MAE are slightly better for IDF Koutso .

Model selection
In addition to performing closely to each other in both calibration and validation modes, the two models yield very similar parameters and return levels, as may be seen from Figure 6. It is worth noting that the fitted values of the additional parameter θ of the IDF Koutso model range from -0.02 to 0.39, which is relatively close to zero as compared to the [1h-24h] range of durations 25 considered here. This means that the ::::::: IDF Koutso : model is de facto very close to the IDF SiSca model, which is a simplification of the IDF Koutso model assuming θ being equal to zero.
Consequently, while there is no factual reason for considering one of the models to be better than the other, the IDF SiSca model will be retained, according to the following considerations: 1. it is more parsimonious with no clear advantage brought by the fifth parameter of the IDF Koutso model; 2. it is easier to implement, especially in a perspective of regional studies involving the mapping of the scaling parameters; 3. there is a straightforward link between the formulation of the IDF SiSca model and that of the Montana formula (see appendix) commonly used in national or regional agencies; this makes the formulation of the final IDF product easier to grasp by end-users, thus facilitating its adoption and use. The 90% confidence intervals of the IDF curves are displayed as coloured :::::: colored : stripes in Figure 7. As intuitively expected, for a given station, the higher are the return periods considered, the larger are the confidence intervals. Equally conform to :: in 15 :::::::: agreement ::::: with knowledge and practice is the fact that, for a given parameter, the largest uncertainty intervals are usually obtained for the shortest series (Fatick, Podor, and Thies), while the longest series (Dakar-Yoff, Tambacounda, Kaolack, and Ziguinchor) display the narrowest intervals (Table 2). However : , this relation weakens when considering higher moments of :: or :::::
13 5.2.2 Scaling versus GEV related uncertainty when :::: only : daily samples only are available As previously explained, at stations where daily data only :::: only :::: daily :::: data : are available, the sub-daily GEV distributions have to be estimated from this limited set of 24-hour valueswhich . :::: This : significantly increases the uncertainty as may be seen from :::: seen :: in Figure 8. In this figure, the total uncertainty on the 1-hour GEV distribution is separated between :::::: divided ::: into ::: (i) the uncertainty linked to the initial fitting of the 24-hour distribution -GEV(24h) uncertainty -and ::: (ii) the uncertainty generated 5 by using the scaling relationships of equations 10b and 10c in order to downscale to 1-hour distribution GEV(1h) -scaling uncertainty. This decomposition is carried out by following the procedure presented in section 4.3.2. The results are given for the two longest series of our data set (Dakar Yoff, 38 years; Ziguinchor, 44 years), which happen to display two different behaviours ::::::: behaviors. At Dakar Yoff, the GEV(24h) uncertainty becomes clearly :::::::: distinctly larger than the scaling uncertainty from the 10-year return period onwards; at Ziguinchor, this occurs only from the 100-year return period onwards. Associated 10 with this difference is the fact that the downscaled GEV model (dots in Figure 8) diverges from the reference scaled model  Figure 9 synthesizes the results obtained at all stations, basically ::::::::: essentially confirming that the inference of the daily scale GEV(24h) is a far more important source of uncertainty than the inference of the scaling relationship , when it comes to estimate :::::::: estimating : the GEV(1h). Figure 9 displays the minimum, mean, and maximum uncertainty spread obtained on the 14 stations for GEV(24h) on the one hand (red) and the scaling relationship on the other (blue); the 50% shaded interval contains the 7 central values. In order to be able to compute these spreads, the values are expressed as a percentage of the rainfall value 20 given by the GEV(1h) for each station at a given return level. It turns out :::: was ::::: found that the spreads due to the GEV(24h) fit using daily samples are 3 to 4 times higher ::::: larger than those due to the scaling estimate for the 100-year return level and 5 times larger for the 500-year return level.

IDF curves 25
A typical representation of of IDF curves is given in Figure 7. As a result of the IDF model formulations and the fitting on a unique scaled sample (for both IDF Koutso and IDF SiSca ), the return level curves are parallel (they do not cross) and the intensities decrease as the duration increases. The log-log linearity between return levels and durations comes from the simple scaling formulation (the curves would be bended ::: bent : but still parallel , for the IDF Koutso model). Rainfall return levels are of similar order of magnitude for the four stations, even though a North-South ::::::: although :: a ::::::::: north-south : gradient is apparent, : with rainfall 30 intensities gradually increasing from Saint Louis to Dakar and from Dakar to Ziguinchor. At the 2-year return period, rainfall intensities vary from roughly 40 mm h −1 (between 33 and 60 mm h −1 when considering all 14 stations) for the 1h duration to approximately 3 mm h −1 (between 2 and 5 mm h −1 ) for the 24h duration. For any station, the return levels for the 10-year (resp. 100-year) return periods are approximately 1.5 (resp. 2) times higher than the 2-year return levels; these ratios hold at all time scales :::::::: timescales : (from 1h to 24h) as a result of the log-log linearity of the intensity versus the duration. As already discussed in section 5.2, the novelty of these IDF curves is the fact that they are provided with their confidence intervals, allowing the user to get a representation of the uncertainty surrounding the estimated intensity return levels, which is linked to both the sample size and by the quality of the whole GEV&scaling model. 5

IDF mapping for Senegal
Maps of the 4 IDF parameters (GEV + scaling) over the whole :: all :: of Senegal are plotted in Figure 10. They have been produced by kriging the parameters inferred at each of our 14 stations. Two of these parameters (ξ and η) are independent of the duration D, while µ and σ are functions of D; these two parameters are thus mapped for the reference duration of 1h only :::::::::::: (corresponding ::: thus :: to ::: µ 0 ::: and :::: σ 0 ). They both display a clear North-South ::::::::: north-south increasing gradient, a feature already found by Panthou the occurrence of a rainfall intensity less frequent in the North ::::: north than in the South :::: south, simply because there are fewer rainfall events there (as evidenced for the whole region by Le Barbé et al., 2002). 15 As regarding :::::::: Regarding : the two non-duration dependent parameters :: (ξ :::: and :: η), : the shape parameter ξ does not display any clear spatial organization while the scaling parameter η displays a South-West North-East :::::::: southwest :::::::: northeast gradient (with values ranging from -0.8 to -0.9). This suggests that, added :: in ::::::: addition to the latitudinal effect, the distance to :::: from the ocean might also influence the temporal structure of rainfall events. The values of the scaling parameter are very close to those observed by Panthou et al. (2014b) over the AMMA-CATCH Niger network located near Niamey. 20 The general pattern of the maps of 2-, 10-and 100-year ::::: 2-year ::: and ::::::: 10-year return levels given in Figure 11 is almost totally :::::: entirely : driven by the North-South rainfall gradientfor the 2-and 10-year return period :::::::::: north-south :::::: rainfall ::::::: gradient. The pattern of the 100-year return period :::: level : is a bit less regular, with the distance to the ocean seeming to play a role in the western part of the country and a higher patchiness that is certainly largely due to the sampling uncertainty at such a low frequency of occurrence. 25 6 Conclusions and discussion
One important result in this respect is that the uncertainty produced by the inference of the GEV parameters is 3 to 4 times larger than the uncertainty associated with the inference of the scaling relationship. This means that the scaling relationship 15 requires far less data to be inferred correctly than the GEV model. Secondly, maps of the 4 IDF model parameters and associated intensity return levels have been computed, allowing retrieving the :: for ::: the ::::::: retrieval ::: of ::: the general spatial pattern of these parameters over Senegal. The location (µ) and scale (σ) parameters of the GEV distribution, as well as the rainfall intensity levels for the 2-year and 10-year return periods, display a clear increasing gradient from North to South :::: north :: to ::::: south : in line with the climatological gradient of the mean annual rainfall and of the occurrence of wet days. By contrast, for the temporal 20 scaling parameter η, : the increasing gradient is rather oriented from North-East to South-West :::::::: northeast :: to :::::::: southwest, reflecting the influence of both the occurrence of wet days and of the distance to the ocean. The map of ξ is somewhat patchy : , reflecting the fact that this parameter is usually difficult to estimate, but another important result of this study is that its average value is slightly positive, : suggesting that the rainfall distribution is heavy tailed :::::::::: heavy-tailed as often observed in several regions in the world (Koutsoyiannis, 2004b, a) ::::::::::::::::::::: (Koutsoyiannis, 2004a, b). Also worth noting is the fact that the value of η is close to 25 -1 (ranging from roughly -0.9 and -0.8) indicating a steep decrease of intensities as the duration increases. This is a common feature of short and intense rainfalls :::: such as those produced by convective storms. These values are comparable to those found by Mohymont et al. (2004) in the tropical area of Central Africa, and to those obtained in the Sahelian region of Niamey by Panthou et al. (2014b), close to -0.9 in both cases; they . ::::: They : are larger in absolute value than those found in mid-latitude regions, as already underlined by Van-De-Vyver and Demarée (2010). 30 A final consideration relates to the implementation of such IDF models in operational services. While the theoretical framework of coupling the GEV and scaling models might be considered as difficult to handle outside the world of academic research, their implementation to produce ::::::::::: implementing ::::: them ::: for :::::::: producing : IDF curves is relatively easy, especially when using the simplified approach tested here. This approach has the additional advantage of producing relationships between rainfall return levels ::: that ::: are formally equivalent to the so-called Montana relationship (see appendix), widely used in operational services, making easier ::::::::: facilitating the implementation and usage of our IDF model in meteorological/climatological services and hydrological agencies.

Points of discussion and perspectives
In the perspective of extending this work to other tropical regions of the world where subdaily ::::::: sub-daily : rainfall data might be rare, : it remains to explore the effect of using a fixed window to extract the daily rainfall annual maxima, while :::::: whereas : a 5 moving window was used for all durations (including 24h) in this study. As a matter of fact : , daily records of rainfall are carried out at a given hour of the day (usually 6:00 GMT or local time), producing smaller totals than when a mobile window is used to extract the daily rainfall maximum maximorum of a given year(or month). Since the scaling relationships that are used to deduced subdaily ::::: deduce :::::::: sub-daily : statistics from these fixed-window 24-hour maxima are tuned on multi-temporal maxima extracted with :::: using : mobile windows, there is a potential underestimation bias of the subdaily ::::::: sub-daily : statistics inferred at 10 24-hour stations that deserves to be studied ::::: merits ::::: further ::::: study.
Another critical question relates to using statistical inferences presupposing time stationarity in ::: that ::::::::: pressupose :::::::::: stationarity :: in :::: time :: in a context of a changing climate. Warming is already attested in the Sahel and is bound to increaseinvolving possible changing : , :::::::: involving ::::::: possible :::::::: changes :: in : annual rainfall patterns induced by changes in the positioning of the Bermuda-Azores High and of the Saharian ::::::: Saharan : Heat Low. Indeed, rainfall intensification in this region has already been reported 15 by Panthou et al. (2014a) and by Taylor et al. (2017), likely in connection with an average regional warming of about 0.18 K/decade K decade −1 over the past 60 years. While dealing with this question was far beyond the scope of this paper, it is a major challenge for both end-users and researchers. It requires developing non-stationary IDF curves, one possibility ::::::: possible :::::: solution : in this respect being to use both long historical rainfall series and the information that can be extracted from future climate model projections (see e.g. Cheng and AghaKouchak, 2014). 20 At the same time it is important to underline :::::::: emphasize : that stationarity is an elusive concept whose reality is never guaranteed in Nature ::::: nature, even without climate change. The Sahelian rainfall regime, for instance, : is known for its strong decadal variability (Le Barbé et al., 2002) with potentially great impacts on most extreme rainfall events . The use of long rainfall series (multi-decadal) :::::: rainfall ::::: series to fit IDF curves can thus reduce the sampling effects and reduce the IDF uncertainties but they can also introduce some hidden biases linked to this decadal-scale non-stationarity. This happened 25 with the dams built on the Volta river ::::: River in the seventies, and . :::: The ::::: dams :::: were : dimensioned based on the rainfall information of the previous three decades, two of which being abnormally wet ::::: which :::::::: included ::: two :::::::::: abnormally :::: wet ::::::: decades. The reservoirs never filled up in the eighties and nineties. Therefore, while IDF curves are intended to be disseminated to a large community of end-users, they ::::: users must be warned that they are nothing else than a decision making supporting :::: other :::: than :: a ::::::::::::: decision-making ::::::: support tool to be used with care and to be updated regularly. 30 17 Appendix A: Simple scaling IDF to Montana IDF The IDF Montana formulation is as follows: The underscript m is used to differentiate with the ::: the ::::::: Montana :::::::::: formulation ::::: from ::: the scaling expression b in the main paper ( m stands for Montana). In our case, the scaling function is the simple scaling (Equation 12), thus Equation A1 becomes: The two Montana parameters a and b m can be derived by using the equality between the two formulations: Note that when the simple scaling is verified then: (i) D 0 is equal to 1, and depend :::::: depends : only on the unit chosen to    Scaling GEV Statistic over stations Uncertainty Figure 9. Evolution of the spread of the 90% confidence interval of return levels depending on the return period. The red color is for the spread due to the uncertainty of GEV(24h) fitting, while the blue color is for the spread due to the uncertainty of the scaling. All ::::: Results ::: for :: all stations have been gathered ::::::: combined. The mean, min, max, and 50% confidence interval of the spread obtained at the different stations are also shown.