INTRODUCTION
The design of any hydraulic structure requires a precision of the level of
performance desired. This level of performance is often determined by the potential
damage and severity of weather hazards that could cause failure, malfunction
or overflow structure in question. Thus, in the case of stormwater management,
the dimension of various components of the infrastructure system (case of pipes
and canals sanitation) is based on the return period of heavy rainfall events
(Monhymont and Demarée, 2007; Segond
et al., 2007). This information is often expressed as Intensity-Duration-Frequency
(IDF) curves obtained from a statistical study of extreme events. In West Africa,
very little studies (Oyebande, 1982; Puech
and Chabi-Gonni, 1984) were carried out on the intensity-duration-frequency
curves for precipitation. This is mainly due to the fact that in this part of
the globe, the data of rainfall extremes of short duration are generally rare,
expensive and difficult to obtain.
In Côte d’Ivoire, the model which was used for construction of the intensity-duration-frequency curves is that of Montana, fitted with the data previous to the year 1979 over periods of 10 years on average. A technical document has been realized in 1979 and is used today by all the engineers for the design of urban drainage structures.
The probabilistic approach used in this study is based only on the Gumbel distribution.
Recently, the applicability of this distribution has been criticized both on
theoretical and empirical grounds. Thus, new theoretical arguments based on
comparisons of actual and asymptotic extreme value distributions as well as
on the principle of maximum entropy indicate that the Extreme Value Type 2 distribution
should replace the Gumbel distribution (Koutsoyiannis, 2004a).
In addition, several empirical analyses using long rainfall records agree with
the new theoretical findings (Coles et al., 2003;
Koutsoyiannis, 2004b; Goula et
al., 2007; Muller et al., 2009).
Furthermore, the empirical analyses show that the Gumbel distribution may significantly underestimate the largest extreme rainfall amounts. The choice of statistical models (GEV, Lognormal and Gumbel) is significant as it is directly related to the safety of hydraulic structures, establishment of flood zones and estimation of extreme events.
Moreover, manifestations of global warming across the planet are recognized
today (IPCC, 2001). The fact that this warming will affect
the intensity and frequency of rainfall is well accepted by the scientific community
(De Toffo et al., 2009). To better protect people,
it is necessary to update rainfall intensity-duration-frequency relationships
using by water resources engineers to design and manage infrastructure. This
is the aim of the present study carried out on extreme rainfall of short duration
in Côte d'Ivoire.
MATERIALS AND METHODS
Data rainfall used: The national raingauge network was used in this study. It consists of 14 synoptic stations (Fig. 1). The San Pedro station was not chosen because the large number of missing data. The locations of synoptic stations for which data were available are reported in Table 1. For these 13 rainfall stations, annual maximum series of durations ranging from 15 min to over 4 h were analyzed. The data have been provided by the Côte d’Ivoire Meteorological Service (SODEXAM). The record lengths of each rainfall data series varied form a minimum of 14 years to over 43 years.
Annual maximum series used for statistical analysis must comply with the hypothesis of homogeneity, stationarity and randomness. To verify these hypotheses three non parametric tests were used.
The homogeneity test of Wilcoxon (1945) allows to carry
out comparisons between two subsamples and to check if the averages of the two
subsamples are significantly different (Baudez et al.,
1999). The Kendall (1975) trend test is based on the
correlation between the ranks of a time series and their time order (Yue
and Pilon, 2004). It is a rank-based nonparametric method used to detect
the trends in the series. Non-parametric trend detection methods are less sensitive
to outliers (extremes) than are parametric statistics such as Pearson’s
correlation coefficient (Wang et al., 2008).
As the Kendall test requires independent series, serial dependence was also
tested using the non-parametric Wald and Wolfowitz test
(1943).
The null hypotheses were set to no independence, no stationarity and no homogeneity and these were not rejected at the 5% significance level.
Methodology: The objective of the rainfall intensity duration frequency
curves is to estimate the maximum intensity of rainfall for any duration and
return period. The classical approach for building IDF curves has three steps
(Nhat et al., 2006). In first step, a probability
distribution function is fitted to each duration sample. In second step, the
quantiles of several return periods T are calculated using the estimated distribution
function from step one. Lastly, the IDF curves are determined by fitting a parametric
equation for each return period, using regression techniques between the quantiles
estimates and the duration.
|
| Fig. 1: | Geographical
locations of raingauges and the principal climatic areas of the Côte
d’Ivoire |
| Table 1: | Characteristics
of rainfall stations used |
 |
| Table 2: | Probability
distribution functions used |
 |
NameProbability
density functionParameters |
Fitting of distribution functions: The first step is to fit a distribution
function to each group comprised of the data values for a specific duration.
Random hydrological variables that are extremes, such as maximum rainfall and
floods are described by several extremes value distributions (Koutsoyiannis,
2004a; Overeem et al., 2008; Huard
et al., 2009) or often Lognormal distribution (Goula
et al., 2007; Raiford et al., 2007).
For this study, three statistical distributions were retained (Table
2). The empirical probability chosen for this statistical analysis is that
of Gringorten (Koutsoyiannis et al., 1998). The
maximum likelihood method was used to determine distribution parameter of maximum
precipitation.
Adequacy testing: There are different methods to compare and select
the distribution that best fits a given sample. It is possible to visually examine
the quality of the fit between the empirical probability exceedance and a distribution,
both plotted on probability paper. However, this method is based only on the
judgment of the hydrologist and can be sometimes subjective. For this study,
two selection criteria were used, both based on the likelihood function: The
Akaike (1974) and the Bayesian (Schwarz,
1978) information criteria, respectively given in Eq. 1
and 2:
where, AIC is the Akaike information criterion; BIC is the Bayesian information criterion; L is the likehood function; k is the number of parameters; N is the sample size.
Equation 1 and 2 both include k the number
of parameters. Thus, parsimony is taken into account when selecting the best
distribution using these two criteria.
The best fit is the one associated with the smallest BIC and AIC values (Rao
and Hamed, 2001). The BIC criterion tends to penalize three parameter distributions
more severely than the AIC and sometimes the optimal fitted distribution differs
form one criterion to another. In the case of different selections by the AIC
and BIC criterion, the distribution identified by the BIC criterion was selected
to emphasize parsimony.
Empirical IDF formula: In the third step, the empirical formulas are used to construct the rainfall IDF curves. The least-square method is applied to determine the parameter to empirical IDF equation that is to represent the intensity-duration relationships.
For this present study, Montana equation is used to describe the rainfall intensity-duration
relationships. This power-law was used to establish IDF curves for precipitation
in West Africa (Puech and Chabi-Gonni, 1984) and in others
regions of the world (Koutsoyiannis et al., 1998;
De Michele and Salvadori, 2005). Montana equation is
defined as (Ben-Zvi, 2009):
where, I(T) is the rainfall intensity (mm h-1); d is the duration (h); a(T) and b(T) are the constant parameters related to the metrological conditions; T is the return period.
RESULTS AND DISCUSSION
Results
Hypotheses tests: The hypotheses of independence, homogeneity and stationarity
were checked respectively using Wald-Wolfowitz, Wilcoxon
and Kendall tests. All these tests indicate that these annual maximum intensities
series obtained are independent and identically distributed.
Best fitted probability distributions: The Gumbel (EV1) and Lognormal (LN2) distributions are the best models for fitting series. The Gumbel and Lognormal models fit well respectively 48 and 47 annual maximum intensities series. The remaining 9 series were fitted by the Generalized Extreme Value distribution (GEV).
Figure 2a-d show the Gumbel, GEV and Lognormal
distributions fitted by Maximum Likelihood method for annual maximum intensities
series of four rainfall stations.
The Lognormal and Gumbel distributions have nearly the same asymptotic behavior.
Indeed, the Lognormal distribution belongs to the domain of attraction of the
Gumbel distribution. This two distribution functions have an exponential asymptotic
behavior. Figure 3a-h show the locations
of the best fitted distributions. There is geographical area where the distributions
are predominant. The Log-Normal and Gumbel are predominant in the North, East
and West regions. The Generalized Extreme Value distribution is predominant
in central region. There is no distribution that predominates in south Côte
d’Ivoire.
Parameters of distribution: The parameters of distributions estimated
by Maximum Likelihood method form the annual maximum intensities series of Abidjan,
Bouake, Man and Korhogo rainfall stations are shown in Table 3.
|
| Fig. 2: | Log
normal, GEV and Gumbel distributions fitted by Maximum Likelihood method
for annual maximum intensities rainfall series of Abidjan, Man, Bouake
and Korhogo stations (duration 1 h). (a) annual maximum intensities for
abidian station fifted with model: Log-normal 2 (Maximum likelihood),
(b) annual maximum intensities for Man station fifted with model: Log-normal
2 (Maximum likelihood), (c) annual maximum intensities for bouake station
fifted with model: GEV (Maximum likelihood) and (d) annual maximum intensities
for Korhogo station fifted with model: Gumbel (Maximum likelihood) |
|
| Fig. 3: | Locations
of the best fitted distributions of (a) 15, (b) 30, (c) 45, (d) 60, (e)
90, (f) 120, (g) 180 and (h) 240 min (EV1: Gumbel distribution; LN2: Log
Normal two parameters; GEV: Generalized extreme value distribution) |
The location parameter (μ) of Lognormal distribution (LN2) decreases with
rainfall duration while its scale parameter (σ) increases with the rainfall
duration. The value of location parameter varies between 4.79 and 2.73 mm h-1.
Its scale parameter fluctuates on average around 0.30 mm h-1. The
scale parameter (u) of Gumbel distribution (EV1) varies between 7.91 and 43.18
mm h-1. The location parameter of Generalized Extreme Value distribution
(GEV) and its particular case (Gumbel distribution) fluctuates on average around
53 mm h-1. The shape parameter (k) of Generalized Extreme Value distribution
describes the type of asymptotic distribution of annual maximum intensities.
At Abidjan Station (duration of 15 and 30 min), this shape parameter is negative.
These rainfalls of 15 and 30 min duration belong to attraction domain of the
Frechet distribution. At Bouake station, the shape parameter is positive, so
the rainfalls of 15, 45, 60 and 240 min belong to the domain attraction of the
Weibull distribution.
| Table 3: |
The parameters of distributions estimated by Maximum Likelihood
method form the annual maximum intensities series of Abidjan, Bouake, Man
and Korhogo rainfall stations |
 |
Empirical quantiles: The empirical quantiles estimated from parameters
of distributions are shown in Fig. 4a-d.
It is observed that the intensities generated (quantiles) decrease with duration
and increase with the return period. At Bouake, Korhogo and Man stations, the
difference between quantiles of return period 1 year and those associated of
recurrence interval 2 years is great. This seems linked to the difficulty of
statistical distributions to estimate quantiles of little return period.
Rainfall intensity-duration-frequency relationships: The Intensity-Duration-Frequency
(IDF) curves for the precipitation were established using the parameters of
Montana model and the results for Abidjan, Bouake, Korhogo and Man stations
are displayed in Fig. 5a-d on double logarithmic
axes.
|
| Fig. 4: | Maximum
rainfall intensities for different time intervals and return periods obtained
form the cumulative density function: (a) Abidjan, (b) Bouake, (c) Korhogo
and (d) Man |
|
| Fig. 5: | Intensity-duration-frequency
curves for Abidjan (a), Bouake (b), Korhogo (c) and Man (d) rainfall stations
on a log-log scale |
On double logarithmic axes, the intensity-duration-frequency curves are represented
by parallel lines. These curves establish a relationship between the average
rainfall intensity (mm h-1), the duration of this rainfall (minutes)
and rainfall event frequency (years). Generally, the rainfall intensity decreases
with time aggregation and increases with the return period. In addition, there
is a separation of intensity-duration-frequency curves when the return period
decreases.
Comparison of parameters Montana model: The parameters obtained in this
study are widely above those currently used by hydraulic engineers for the design
of urban drainage system (Table 4). For 10 years return period,
the values of parameter a(T) doubled (Abidjan, Adiake, Bondoukou, Dimbokro,
Man, Sassandra stations) or even tripled in some cases (Yamoussoukro station).
In terms of relative bias, the gap between the news parameters and the previous
parameters vary form 44 to 102% (Fig. 6a, b).
| Table 4: | News
and previous parameters Montana model for a return period 10 years |
 |
As we consider the root mean square error, the differences between the parameters
vary form 160 to 370%. For the parameters b(T), the differences between the
parameters (news and previous) fluctuates around 0.20 on average.
|
| Fig. 6: | (a)
Relative Bias (RB) and (b) Root Mean Square Error (RMSE) calculated for
calculated from the previous parameters (DCAD, 1979)
and those of the present study |
|
| Fig. 7: | IDF
curves for return periods 1, 2, 5 and 10 years form the technical document
(DCAD, 1979) and those of the present study for
the Abidjan station on a log-log scale. Return period (a) 1 year, (b)
2 year, (c) 5 year and (d) 10 year |
In terms of relative bias, the differences between the parameters vary form 34 to 44%. For the root mean square error, the gap between the parameters varies form 123 to 149%.
Comparison of intensity-duration-frequency curves: The impact of parameters
evolution on the Intensity-Duration-Frequency (IDF) curves was analyzed. It
finds a significant gap between the IDF curves currently used by hydraulic engineers
and those provided by this present study (Fig. 7a-d
and 8a-d). The annual maximum intensities
of duration lower than 20 min are currently underestimated (Fig.
7, 8). This underestimation of intensities from the IDF
curves decreases with the return period of rainfall. The annual maximum intensities
of duration higher than 20 min are currently overestimated.
|
| Fig. 8: | IDF
curves for return periods 1, 2, 5 and 10 years form the technical document
(DCAD, 1979) and those of the present study for
the Bouake station on a log-log scale. Return period (a) 1 year, (b) 2
year, (c) 5 year and (d) 10 year |
This overestimation is amplified when the return period decreases (Abidjan
and Man stations).
DISCUSSION
Distribution function of annual maximum intensities rainfall: The results
of this present study show that only 9% the annual maximum intensities series
were best fitted by the Generalized Extreme Value distribution (GEV). This distribution
function does not seem be appropriate to describe the maximum intensities of
Côte d’Ivoire. This inappropriateness of the GEV distribution was
observed by Raiford et al. (2007) in South and
North Georgia (United States). However, other recent studies (Ben-Zvi,
2009; Kigumbi and Mailhot, 2009; Veneziano
et al., 2009; Overeem et al., 2008)
showed that GEV law well describes the tail of distribution of annual maximum
intensities rainfall.
In Côte d’Ivoire, the inappropriateness of the GEV distribution for fitting of annual maximum intensities may be due in part to the comparison criteria (BIC and AIC) used for selection of statistical laws. Indeed, those criteria because of principle of parsimony tend to favor the two-parameter laws (Gumbel and Lognormal distribution).
To this is we can add the difficulty of estimation the Generalized Extreme Value distribution (GEV) law shape parameter (k) when the rainfall series are below 30 years.
The statistical analysis shows that most series are well fit by the Gumbel
and the Lognormal distributions. Thus, the majority of annual maximum intensities
have a tail that belong the domain attraction of Gumbel. This asymptotic behavior
(exponential type) was also observed at the statistical distribution of annual
maximum intensities rainfall of short duration of Poland (Twardosz,
2009), North Eastern Algeria (Benabdesselam and Hammar,
2009), Cyprus (Endreny and Pashiardis, 2007) and
Yangambi in Congo Kinshasa (Monhymont and Demarée,
2007).
Parameters of Montana model and intensity-duration-frequency curves for precipitation:
The parameters of Montana provided by previous study were compared to those
obtained in the present study. The analysis shows that these previous parameters
are largely outdated. On the 13 synoptic stations used, the parameters have
almost doubled and even tripled in some cases. These results converge with those
of Kouassi et al. (2007) at the Abidjan area.
According to Soro et al. (2008) several factors
could give plausible explanation the parameters evolution. The sample size is
one factor that could explain the parameters evolution. Indeed, annual maximum
intensities series used in the present study are higher than those used by the
previous study. For example, at Bondoukou station, the previous parameters were
obtained by fitting 5 years of rainfall observations (1974-1978) while this
present study used 30 years of rainfall observations (1970-2000). Similarly,
at Odienne station, the previous parameters (DCAD, 1979)
were calibrated with 10 years of data (1969-1978) while this study used 33 years
(1969-2001). The small sample size used to estimqate the previous parameters
has the effect of introducing considerable error during the estimation of extreme
quantiles (Puech and Chabi-Gonni, 1984).
Moreover, these authors also suggest caution in using the results of technical document for Bondoukou Dimbokro stations. The previous parameters of those stations were obtained respectively with samples of 5 and 9 years of rainfall observations.
The goodness of fit through the choice of statistical model can also explain
the parameters evolution. The parameters currently used by hydraulic engineers
have been fitted with quantiles form only the Gumbel distribution. This arbitrary
choice of the Gumbel distribution may lead to an underestimation of extreme
quantiles. This is confirmed by several studies (Wilks,
1993; Koutsoyiannis, 2004a,b;
Muller et al., 2008) which reject the prevalence
of the Gumbel distribution. These authors affirm that use of Gumbel distribution
without reasoning or possible comparison with other models can lead to bad estimation
of risk.
The parameters evolution of Montana model has had an impact on the shape of
intensity-duration-frequency curves for rainfall. Thus, one noted an underestimation
of annual maximum intensities rainfall for duration lower than 20 min and an
overestimation for those higher 20 min. In Ghana, Endreny
and Imbeah (2009) found that the intensity-duration-frequency curves evolution
of Accra and Ho stations was mainly due to the size of the data used.
CONCLUSION
This present study has enabled the updating of intensity-duration-frequency curves for precipitation of Côte d'Ivoire. The statistical analysis shows that most series are well fit by the Gumbel and the Lognormal distributions. The Generalized Extreme Value distribution is quite unlikely to apply to annual maximum intensities rainfall because the sample size used (often below 30 years). This study shows that the parameters of Montana currently used by hydraulic engineers for the design of urban drainage system are largely exceeded.
Indeed, the values of these parameters have almost doubled and even tripled in some stations. This leads inevitably to an underestimation or in some cases an overestimation of project for flow urban drainage system.
ACKNOWLEDGMENTS
Direction of National Meteorology of Côte d’Ivoire is gratefully acknowledged for providing the data rainfall.
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