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:

(1)

(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):

(3)

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.