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
intensity–duration–frequency (IDF) curves. Using a 14 series of 5 min
rainfall records collected in Senegal, a comparison of two generalized
extreme value (GEV) and scaling models is carried out, resulting in the
selection of the more parsimonious one (four parameters), as the recommended model for
use. A bootstrap approach is proposed to compute the uncertainty associated
with the estimation of these four parameters and of the related rainfall
return levels for durations ranging from 1 to 24

The fast-growing pressure of mankind on planet Earth causes populations to be
increasingly exposed to hydrometeorological hazards such as torrential rains
and floods

On the data side, the frequency analysis of extremes requires long and continuous records of rainfall at the same location, something fairly common at a daily time step albeit unavailable in some regions. Moreover, a complicating factor is that, in many cases, it is necessary to consider sub-daily time steps. However, long-term records of sub-daily rainfall are much less numerous or much less reliable and accurate than daily series.

The methodological challenge arises from the complex combination of factors
that cause rainfall to be strongly variable at all scales (from the
microphysics droplet scale to synoptic scale),
as a result of the nonlinear interaction of different atmospheric processes

However, having a consistent scaling framework does not eliminate the
crucial sampling issues associated with the estimation of the parameters of
the IDF model. This involves significant uncertainties in the final
determination of rainfall return levels, a question rarely addressed in the
literature; on that subject, see the pioneering work of

Several recent studies have dealt with the question of IDF calculation for
different West African countries. Some focused on analyzing the behavior of
the extreme rainfall distribution at a given location

More recently,

It is worth noting that both

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 the above-mentioned papers and to provide IDF curves for a region located at the western edge of the Sahel, 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 levels. Furthermore, selecting an IDF model that is the least sensitive possible to data sampling effects and computing the associated IDF confidence intervals facilitates updating of the IDF curves when new data are available.

Senegal is located at the western edge of the African continent between
latitudes 12 and 17

There is a strong north–south gradient of the mean annual rainfall (Fig.

Rainfall regime statistics obtained from daily rain gauges over the
1950–2015 period:

The rains are mainly caused by mesoscale convective systems sweeping the
country from east to west

Senegal regularly undergoes heavy damaging downpours. A recent example is the
rainfall event that occurred in Dakar in the morning of 26 August 2012,
causing the largest flood in the last 20 years in the city. An amount
of 160

The archives of climate and hydrological services of West African countries sometimes contain large amounts of sub-daily rainfall records. However, most of the time these records are stored in paper strip chart formats, requiring the tedious task of digitization in order to use them in numerical applications.

The present study has been made possible thanks to the important work of analyzing and digitizing rain gauge charts carried out for the main synoptic stations of Senegal. This process was undertaken by the hydro-morphology laboratory of the Geography Department at the University Cheikh Anta Diop of Dakar (UCAD) in collaboration with the National Agency of Civil Aviation and Meteorology (ANACIM) that provided the rainfall paper charts.

Senegalese synoptic stations are equipped with tipping bucket rain gauges;
the receiving ring is 400

A total of 23 tipping bucket 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 highly sensitive to
sampling effect and erroneous data

The procedure for classifying 1 station year as valid or not is the
following: (i) the annual number of 5 min data and the annual amount of rain
are computed, (ii) the mean 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 between

Years of network operation. The valid years are represented by black
squares. The total number of valid years is displayed for each station on
the right

IDF curves provide estimates of rainfall intensity for a range of durations

Kolmogorov–Smirnov GOF

Quantile–quantile plots for both IDF models for the different
durations and for all stations (global scores in the legend):

Comparison between IDF

IDF

The advantage of Eq. (

Then, it becomes a classical frequency analysis of the random variable

In the particular case of

In the scaling approach described above, the estimation of rainfall return
levels requires a statistical model of rainfall intensity distribution since
Eqs. (

EVT proposes two methods to extract samples of extreme values from a time
series

Return level plot for the 1

Evolution of the spread of the 90 % confidence interval of
return levels depending on the return period. The red color of the spread is
due to the uncertainty of GEV(24

Maps of IDF

Maps of return levels intensities (from IDF

Compared to BMA, POT has the advantage of allowing the selection of more than
one value per year, thus increasing the sample size used for inferring the
model. However, the choice of an appropriate threshold is often difficult

In BMA, when the block is large enough (which is ensured for annual maxima),
the EVT states that the generalized extreme value (GEV) distribution is the
appropriate model for block maxima samples

A positive (negative) shape corresponds to a heavy-tailed (bounded in the
upper tail) distribution. When

Note that Eqs. (10) to (

In this study, two IDF models are compared: the IDF

Different fitting methods have been tested to adjust the IDF model parameters
to rainfall data. One of them (the two-step method) is applicable to both
IDF

Note that two other methods specifically dedicated to the
IDF

The fitting of the scaling

Once the scale relationship is identified (

With the aim of selecting the best IDF model from the two compared IDF
formulations (IDF

The flexibility characterizes the capacity of a model to fit the observed data that are used to calibrate its parameters. To evaluate flexibility, the IDF models are fitted at each station, then different scores are computed to assess the fitting performances.

The robustness, on the other hand, aims at evaluating whether or not the IDF
model is too flexible due to the model having too many parameters with
respect to the number of observations. It thus depends on the sensitivity of
the IDF model parameters to sampling effects: the less the model parameters
are sensitive to sampling effects, the more robust the model. As the two
models tested here have a different number of parameters (4 for
IDF

The flexibility and the predictive capacity of the IDF models are quantified based on two types of scores: global and quantile–quantile.

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 also
returns the corresponding

GOF tests allow for evaluating the entire distribution but do not guarantee
that all quantiles are correctly estimated. Thus, as a complement,
quantile-based scores are also computed. They characterize the relationship
between theoretical (obtained from the fitted CDF) and empirical (obtained
from the empirical CDF) quantiles. The root-mean-square error (RMSE), the
mean error (ME), and the mean absolute error (MAE) quantile-based scores are
computed. The full presentation of these scores can be found in

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 are available, which is the ultimate purpose of regional IDF models. Indeed, in some regional studies, the scaling parameters will have to be inferred from the very few stations where rainfall is recorded at sub-daily time steps; if they display variations in space, 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 1 h (or less) to 1 day.

Therefore, in the following, the uncertainty assessment at a given location will be addressed separately for the two situations: (i) firstly, when observations at this location are available over a whole range of time steps; (ii) secondly, when only daily observations are available.

Confidence intervals for IDF parameters and return levels are estimated using
a non-parametric bootstrap

The vector of years is resampled with replacement (Monte-Carlo resampling) until its length equals the length of the original vector.

Once a year

The IDF model is fitted on the bootstrap sample

The obtained parameters –

These four steps are repeated 1000 times leading to generate 1000

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 observations are available. The GEV parameters for the sub-daily time steps are then deduced from the daily GEV parameters using scaling parameters that must be inferred from nearby multi-timescale observations. In some cases this might generate a GEV model that differs significantly from the GEV model that would have been fitted directly on the observations at the proper 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 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. Analyzing the uncertainty involves two independent bootstrap resampling processes.

The first bootstrapping method used consists in resampling

In a parallel way, the uncertainty associated with scaling is evaluated by
generating 1000 downscaled samples. The reference GEV(24

The model evaluation results are presented in Figs.

Global quantile–quantile scores results for the different IDF
models:

Figure

Figure

With regards to the validation mode, four stations display

The global q–q plots in Fig.

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 Fig.

Consequently, while there is no factual reason for considering one of the
models to be better than the other, the IDF

The model is more parsimonious, with no clear advantage brought by the fifth parameter of the IDF

The model is easier to implement, especially from the perspective of regional studies involving the mapping of the scaling parameters.

The model a straightforward link between the formulation of the IDF

The bootstrap approach presented in Sect.

The 90 % confidence intervals of the IDF curves are displayed as colored
stripes in Fig.

IDF

When comparing the confidence intervals computed for each parameter of the
scaled GEV, it appears that their width is well correlated between

As previously explained, at stations where only daily data are available, the
sub-daily GEV distributions have to be estimated from this limited set of
24 h values. This significantly increases the uncertainty, as seen in
Fig.

Figure

A typical representation of IDF curves is given in Fig.

Maps of the four IDF parameters (GEV

Regarding the two non-duration dependent parameters (

The general pattern of the maps of 2-year and 10-year return levels given in
Fig.

This study of extreme rainfall over Senegal for durations ranging from 1 to
24

The key advantage of the GEV and scaling approach for computing IDF curves is
twofold: (i) it ensures timescale coherency (for the range of explored
durations) when working at a regional scale, thus allowing for a coherent
spatial interpolation of the IDF model parameters over the region of
interest; and (ii) it offers the possibility of deducing GEV distributions
for shorter durations at locations where only 24

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 difficult to handle outside the world of academic research, 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, facilitating the implementation and usage of our IDF model in meteorological/climatological services and hydrological agencies.

In the perspective of extending this work to other tropical regions of the
world where 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,
whereas a moving window was used for all durations (including 24

Another critical question relates to using statistical inferences that
presuppose stationarity in time in a context of a changing climate. Warming
is already attested in the Sahel and is bound to increase, involving possible
changes in annual rainfall patterns induced by changes in the positioning of
the Bermuda–Azores High and of the Saharan Heat Low. Indeed, rainfall
intensification in this region has already been reported by

At the same time it is important to emphasize that stationarity is an elusive
concept whose reality is never guaranteed in nature, even without climate
change. The Sahelian rainfall regime, for instance, is known for its strong
decadal variability

The data used here belong to the National Agency of Civil Aviation (ANACIM), which is an institution of the state of Senegal. They are not publicly accessible. The procedure to obtain the data can be obtained by contacting ANACIM.

The IDF Montana formulation is as follows:

The underscript

The two Montana parameters

Note that when the simple scaling is verified then: (i)

The authors declare that they have no conflict of interest.

The research leading to these results received funding from the UK's National Environment Research Council (NERC)/Department for International Development (DFID) Future Climate For Africa programme, under the AMMA-2050 project (grant numbers NE/M020428/1, NE/M019969/1, NE/M019950/1, NE/M020126/1 and NE/M019934/1). Edited by: Thomas Glade Reviewed by: Francesco De Paola and one anonymous referee