
Research Article


Frequency Analysis and New Cartography of Extremes Daily Rainfall Events in Côte d’Ivoire


T.A. Goula Bi,
G.E. Soro,
A. Dao,
F.W. Kouassi
and
B. Srohourou


ABSTRACT

The extreme rainfalls are meteorological events that cause much damage and many casualties. In a disrupted climate context by human activities, it is necessary to consider the probability distribution of rainfall extremes to protect the population. In this context, 43 rainfalls stations over the period 1947 to 1993 were analyzed. Frequency analysis has shown that Gumbel distribution and the Lognormal distribution fit well with a series of annual maximum daily rainfall. It was also shown a link between the probability distribution of annual maximum daily rainfall and climate patterns. Indeed, none of them do not obey a specific rainfall. The present study has also proposed new maps of daily annual maximum rainfall for return periods of 5, 10, 20 and 100 years.





Received:
February 04, 2010; Accepted: April 17, 2010;
Published: June 26, 2010 

INTRODUCTION
The Côte d'Ivoire is located in West Africa and is characterized by humid
equatorial climates and dry tropical climates (Fig. 1). The
West Africa has experienced a decrease in annual rainfall since 1970 (Goula
et al., 2006a). In certain areas of West Africa, the annual rainfalls
have decreased on average from 20 to 40% (Goula et al.,
2009). The flows decreased from 10% in wetlands and subwet and 30% in the
Sahel (Goula et al., 2006b). Despite this climate
context, Côte d'Ivoire as well as many countries in the tropical areas
(Burkina Faso, Mali, Senegal, Niger and Ghana) are facing serious flooding problems.
The flood risk assessment is based on knowledge of extreme precipitation. Other
factors such as land cover and land slope are also determinants. The analysis
of maximum rainfall can be from very different perspectives but complementary.
The statistical approach (frequency analysis) has often been used in major studies
on extreme rainfall, which have been realized on the Cote d'Ivoire (Soro
et al., 2008), the other states of Western and Central Africa (CIEH,
1985; Puech and ChabiGonni, 1984) and Nigeria (Oyebande,
1982). Many studies have been devoted worldwide (Koutsoyiannis
and Baloutsos, 2000; Gellens, 2002; Zalina
et al., 2002; Ramon et al., 2005;
Sisson et al., 2006; Zahar
and Laborde, 2007; Twardosz, 2009; Bodini
and Cossu, 2010).
The present study constitutes the second phase of the study of extreme rainfall
in Côte d'Ivoire. It aims to analyze a series of annual maximum daily
rainfall available in the country beyond the periods used in previous studies.
Indeed, the statistical study of extreme values has always been conducted in
a conventional manner using the Gumbel distribution. Recently, several studies
(Coles and Perrichi, 2003; Koutsoyiannis,
2004; Bacro and Chaouche, 2006; Goula
et al., 2007) have challenged the predominance of the Gumbel distribution
for the quantification of risk associated with extreme rainfall. Some studies
have shown that the probabilities distributions characterizing the floods seem
to have the tails of distributions heavier than the Gumbel distribution (Farquharson
et al., 1992; Turcotte, 1994). Other studies
(Wilks, 1993; Coles and Perrichi.,
2003) extended the skepticism for the Gumbel distribution in the case of
precipitation, showing that it greatly underestimated the extreme values.
This disadvantage is very important for the design of engineering structures. The stakes in the debate between the choice of the Gumbel distribution or any other distributions (GEV, Lognormal, Pearson III, LogPearson III) is important because it is directly related to the reliability of hydraulic structures and roads. Also, is it considered necessary to verify, taking into account these new data, the validity of this distribution of probability (Gumbel) on the entire territory. The aim of this paper is to provide a better estimation of design rainfall order to achieve a new mapping of extreme daily rainfall.
 Fig. 1: 
Principal climatic areas of the Côte d’Ivoire 
MATERIALS AND METHODS Data rainfall used: Daily rainfall data form 34 pluviometric stations located in Côte d'Ivoire have been considered in this study. Their geographical locations are shown in Fig. 2. Daily data form these stations were selected because they are of a high quality and are most complete. The data have been provided by Hydrology Laboratory of Institute of Research for Development. The record lengths of each rainfall data series varied form a minimum of 36 years to over 47 years and cover the period from 1947 to 1993. Methodology of the frequency analysis: Frequency analysis is a statistical approach commonly used in hydrology to relate the magnitude of extreme events to a probability of occurrence. The main objective of frequency analysis is to infer the probability of exceedence of all possible events, in the case the extremes daily rainfall, from observed values (a sample parent population).
Test of independence, stationarity and homogeneity: All series obtained
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 trend test (Kendall, 1975) is based on the
correlation between the ranks of a time series and their time order (Yue
and Pilon, 2004). It is a rankbased nonparametric method used to detect
the trends in the series. Nonparametric 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 nonparametric WaldWolfowitz test (Wald
and WaldWolfowitz, 1943).
Fitting of distribution functions and estimation methods: 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
(Overeem et al., 2008; Huard
et al., 2009) or often Lognormal distribution (Soro
et al., 2010). For this study, five statistical distributions were
retained (Table 1). Several formulas exist to calculate this
probability. In this study, the empirical probability chosen for this statistical
analysis is that of Hazen. In the humid tropical area, it has been used by several
authors (Puech and ChabiGonni, 1984; Goula
et al., 2007). The distribution parameters are determined by the
maximum likelihood method except the log Pearson III, which required the use
of methods of weighted moments.
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.
 Fig. 2: 
Geographical locations of raingauges and the record lengths
of series 
Table 1: 
Probability distribution functions used 

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 equation 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.
RESULTS AND DISCUSSION
Hypothesis testing: Application of the Kendall test showed that there
was no trend in the annual maximum daily rainfall with a risk error of 5% (significance
level of 0.05) except for stations of Adzopé, Agboville and Adiaké
where, the risk is 1% (significance level of 0.01). The Wilcoxon test shows
that the average of the two subsamples is equal at the significance level of
5%. This indicates that the rainfall observations are from the same population.
The Wald test indicates that the rainfall observations are independent with
a significance level of 5%; therefore there is no link between successive observations.
Fitting of probability distributions: Figure 3ad
present the empirical probability of some series of annual maximum rainfall.
From the graph (Fig. 4), it is very hard to select the best
probability distribution. The criteria of comparison was used to solve this
problem by choosing the distribution with the lowest values of BIC and AIC.
In the maximum precipitation serie analyzed, the lowest AIC and BIC was achieved
by the Gumbel distribution (Table 2). Figure
5ad depict the Gumbel, Pearson III, Generalized Extreme
Value and Lognormal distributions fitted by Maximum Likelihood method for annual
maximum daily rainfall of four stations.
Area of validity (distribution of probability): Overall, the probability
distributions of annual maximum daily rainfall do not obey a specific climate
regime (Fig. 6). The Generalized Extreme Value (GEV) distribution
is best suited to Tiassalé, Soubré and Agboville regions. The
Pearson III distribution well fits the extreme NorthEast and in the region
Beoumi.
 Fig. 3: 
Empirical distribution for the annual maximum series of Tabou,
Abidjan, Bouaké and Man 
 Fig. 4: 
Generalized Extreme Value (GEV), Pearson III (PIII), Gumbel
(EV1), Lognormal (LN2) and Log Pearson III (LPIII) distributions fitted
by the method of Maximum Likelihood method for the annual maximum daily
rainfall series of Abidjan 
Table 2: 
The Akaike (AIC) and the Bayesian (BIC) information criteria
calculated for the annual maximum daily rainfall series of Abidjan 

The Lognormal distribution applies well to the SouthEast, West (DananeGuiglo),
in the Center (M'BahiakroDimbokro). This distribution is appropriate to describe
the extreme rainfalls of north regions (Ouangolo, Korhogo and Odienné).
The Gumbel distribution applies in many regions of SouthWest (Tabou, Daloa
Toulépleu). We find this distribution to the East (Abengourou and Bondoukou).
The results at level of these stations converge with other recent studies such
as those by Goula et al. (2007).
Estimation and comparison of quantiles: The quantiles obtained (Table 3) were compared with those of previous study on the Côte d’Ivoire. It was chosen 10 stations which have been simultaneously in previous studies and the current study. At stations Odienné, Gagnoa and Dimbokro (Table 3), we note that the Gumbel distribution was not suitable for the estimation of extreme quantiles. The calculations with the lognormal distribution give very high relative errors (Table 4).
These errors vary between 38 and 5.5% for rainfall return period of 100 years
(P_{100}), between 34 and 4.6% for rainfall return period of 50 years
(P_{50}), between 24 and 2.4% for rainfall return period of 10 years
(P_{10}) and between 18 and 1.3% for rainfall return period of 5 years
(P_{5}). The quantiles were overestimated with the Gumbel distribution
by BrunetMoret (1967). By cons at Station Soubré,
the relative errors vary between 28.9 and 4.32% for P_{100}, P_{50},
P_{10} and P_{5}. The negative sign assigned to different percentages
shows that the Gumbel distribution used for the previous study underestimated
the quantiles from the GEV distribution used in this study. For stations of
Abengourou, Adzopé, Bondoukou, Bouake and Man, the current study confirmed
the Gumbel distribution. However, the errors are not zero. This indicates that
the current design tools are better developed than those used by BrunetMoret
(1967).
Mapping daily record rainfall: The mapping of the different quantiles
for return periods for all stations was conducted. Figure 7ad
show those of the return periods 100, 50, 10 and 5 years. Isohyets vary greatly
in a SouthNorth to latitude 6°N. This area corresponds to the equatorial
system of transition under the influence of the Atlantic Ocean and West African
Monsoon.

Fig. 5: 
Gumbel (EV1), Pearson III (PIII), Log Normal (LN2) and Generalized
Extreme Value (GEV) distributions fitted by Maximum Likelihood method for
annual maximum intensities rainfall series of (a) Abidjan, (b) Beoumi, (c)
Korhogo and (d) Soubre 

Fig. 6: 
Delimitation of validity of probability distribution 
Table 3: 
Parameters of distributions probability and quantile values
(mm) estimated for return periods of 5, 10, 50 and 100 years 

Table 4: 
Comparison of quantiles (mm): Brunet Moret
(1967) and the current study for return periods of 5, 10, 50 and 100
years 

The highest values of centennial quantiles (return period of 100 years) are
estimated on the coastal stations Tabou (304 mm), Sassandra (305 mm), Abidjan
(301 mm) and Adiaké (290 mm). From latitude 6°N to latitude 11°N,
the isohyets vary likeky in a SouthWest and a NorthEast. This monotony is
broken between the latitudes 6.5° and 7.5° N especially in the mountainous
region and around the lake Kossou. The highest values of quantiles in the region
of Man highlight the role of mountains on rainfall extremes. Indeed, the mountains
of western Côte d'Ivoire are advancing a broad eastern mountain range
centered on Guinea and called the backbone of Guinea. The highlights are aligned
NorthWest to SouthEast the Fouta Djallon (1515 m), Mount Loma (2100 m) and
Mount Nimba (1750 m) lie at the intersection of Cote d’Ivoire. This vast
mountain range can contribute to the rainfall amount by opposition to the penetration
of normal monsoon in the continent by promoting and uplift of air masses over
these regions.

Fig. 7: 
Isohyet of annual maximum daily rainfall for return periods
of 100 (a), 50 (b), 10 (c) and 5 years (d) 

Fig. 8: 
Comparison of decennial isohyets (10 years): (a) present study,
(b) previous study (CIEH, 1985) 
 Fig. 9: 
Comparison of decadal isohyets (100 years): (a) present study
and (b) previous study (CIEH, 1985) 
Thus, the foehn effect may partly explain in the influence of this factor (orography)
on rainfall extremes. Other studies have shown the influence of the orography
(Kieffer and Bois, 1997; Zahar and
Laborde, 2007). At the region of Beoumi, daily rainfall return period of
100 years is 212 mm. This value can be explained by the presence of Lake Kossou.
The intense evapotranspiration in this equatorial climate transition may cause
heavy rains in this area. The low quantiles values are observed stations around
Gagnoa, Lakota, Tiassalé, Agboville and Dimbokro (Table
3). In these areas, quantiles vary between 120 and 142 mm for a return period
of 100 years. It is also noted low quantile values at stations of Abengourou
and Agnibilékrou in the East.
Comparison isohyets: The decennial and centennial isohyets of daily
rainfall developed in this present study were compared to those currently used
for the design of hydraulic structures in Côte d'Ivoire entirely (Fig.
8a, b). It appears a significant difference between isohyets
proposed by the CIEH and those of the present study. Indeed, the isohyet proposed
by the CIEH does not cover the entire Ivory Coast. Moreover, these isohyets
are widely spaced compared to those in this study. At the level of centennial
daily rainfall, the isohyet (300 mm) includes only the Tabou area and an infinite
part of the Southeast, while covering the entire coastal strip of the map at
CIEH. The isohyet (200 mm) around the mountainous region of Man appears in the
map proposed by the CIEH. At the level of daily rainfall decennial isohyet (120
mm) could not be traced by the CIEH (9ab).
CONCLUSION The study of extreme rainfall events lets start an update of previous studies with daily series ranging from 1947 to 1993. Frequency analysis has shown that Gumbel distribution and the Lognormal distribution fit well with a series of annual maximum daily rainfall. The GEV fits well with data sets of Agboville and Soubré Tiassalé stations. The Pearson III distribution fits well to the series of extreme rainfall of Beoumi. It was also shown a link between the probability distribution of annual maximum daily rainfall and climate patterns. The present study has also proposed new maps of annual maximum daily rainfall.

REFERENCES 
Akaike, H., 1974. A new look at the statistical model identifical. IEEE Trans. Automatic Control, 19: 716723. Direct Link 
Bacro, J.N. and A. Chaouche, 2006. Uncertainty estimation of extreme rainfall in the Mediterranean region: an illustration with data from Marseille. Hydrol. Sci. J., 51: 389405.
Baudez, J.C., C. Loumagne, C. Michel, B. Palagos, V. Gomendy and F. Bartoli, 1999. Modelisation hydrologique et heterogeneite spatiale des bassinsvers une comparaison de lapproche globale et de lapproche distribuee. Etude et Gestion des Sols, 6: 105184.
Bodini, A. and Q.A. Cossu, 2010. Vulnerability assessment of CentralEast Sardinia (Italy) to extreme rainfall events. Nat. Hazards Earth Syst. Sci., 10: 6172. CrossRef 
BrunetMoret, Y., 1967. Etude Generale Des Averses Exceptionnelles en Afrique Occidentale. Interafricain d’Etudes Hydrauliques, Publications ORSTOM, Ouagadougou, pp: 20.
CIEH, 1985. Etude des pluies journalieres de frequence rare dans les Etats membres du CIEH. Rapport de synthèse. Série hydrologique, Ouagadougou, pp: 20.
Coles, S. and L. Perrichi, 2003. Anticipating catastrophes trough extreme value modeling. J. Roy. Stat. Soc. Ser. Applied Statistics, 52: 405416. Direct Link 
Farquharson, F.A.K., J.V. Meigh and J.V. Sutcliffe, 1992. Regional flood frequency analysis in arid and semiarid areas. J. Hydrol., 138: 487501. CrossRef 
Gellens, D., 2002. Combining regional approach and extension procedure for assessing GEV distribution of extreme precipitation in Belgium. J. Hydrol., 268: 113129. CrossRef 
Goula B.T.A., F.W. Kouassi, V. Fadika, K.E. Kouakou and G.B. Kouadio, 2009. Impacts du changement et de la variabilite climatiques sur les eaux souterraines en zone tropicale humide: Cas de la Côte d`Ivoire. IASH., 334: 190202.
Goula, B.T.A., B. Konan, Y.T. Brou, I. Savane, V. Fadika and B. Srohourou, 2007. Estimation of daily extreme rainfall in a tropical zone: Case study of the Ivory Coast by comparison of Gumbel and Lognormal distributions. Hydrol. Sci. J., 52: 4967.
Goula, B.T.A., I. Savane, B. Konan, V. Fadika and G.B. Kouadio, 2006. Impact of climate variability on water resources of the watersheds N`Zo and N;Zi in Cote d`Ivoire (Africa wet tropical). Vertigo, 7: 112.
Goula, B.T.A., V. J. Kouassi and I. Savane, 2006. Impacts of climate change on water resources in humid tropical zone: Case study of the Bandama Watershed in Côte d`Ivoire. Agronomie Africaine, 18: 111.
Huard, D., A. Mailhot and D. Sophie, 2009. Bayesien estimation of intensitydurationfrequency curves and the return period associated to a given rainfall event. Stoch. Environ. Res. Risk Assess., 23: 17.
Kendall, M.G., 1975. Rank Correlation Methods. 4th Edn., Charles Griffin, London, ISBN: 0195205723.
Kieffer, A. and P. Bois, 1997. Variabilite des caracteristiques statistiques des pluies extremes dans les Alpes françaises. Rev. Sci. Eau, 2: 199216. Direct Link 
Koutsoyiannis, D. and G. Baloutsos, 2000. Analysis of a long record of annual maximum rainfall in Athens, Greece and design rainfall inferences. Nat. Hazards, 22: 2948. Direct Link 
Koutsoyiannis, D., 2004. Statistics of extremes and estimation of extreme rainfall: II. Empirical investigation of long rainfall records. Hydrol. Sci. J., 49: 591610.
Overeem, A., A. Buishand and I. Holleman, 2008. Rainfall depthdurationfrequency curves and their uncertainties. J. Hydrol., 348: 124134.
Oyebande, L., 1982. Deriving rainfall intensitydurationfrequency relationships and estimates for regions with inadequate data. Hydrol. Sci. J., 27: 353367.
Puech, C., and D. ChabiGonni, 1984. Rainfall depthsdurationfrequency curves for precipitation of duration 5 minutes at 24 hours. Hydrological series, (CIEH), Ouagadougou, Burkina Faso, pp: 155.
Ramon, D., C. Bouvier, N. Luke and N. Helen, 2005. Regional approach for estimating distributions of point daily rainfall in the LanguedocRoussillon. Hydrol. Sci. J., 50: 1729. CrossRef  Direct Link 
Rao, A.R. and K.H. Hamed, 2001. Flood Frequency Analysis. CRC Press, New York, United States, pp: 350.
Schwarz, G., 1978. Estimating the dimension of a model. Ann. Stat., 6: 461464. Direct Link 
Sisson, S.A., L.R. Perrichi and S. Coles, 2006. A case for a reassessment of the risk of extreme hydrological hazards in the Caribbean. Stoch. Environ. Res. Risk. Asses., 20: 296306. CrossRef 
Soro, G.E., T.A. Goula Bi, F.W. Kouassi and B. Srohourou, 2010. Update of intensitydurationfrequency curves for precipitation of short durations in tropical area of West Africa (Cote D'ivoire). J. Applied Sci., 10: 704715. CrossRef  Direct Link 
Soro, G.E., T.A. Goula Bi, F.W. Kouassi, K. Koffi and B. Kamagate et al., 2008. Courbes Intensite Duree Frequence des Precipitations en climat tropical humide: Cas de la region d'Abidjan (Cote D'Ivoire). Eur. J. Scientific Res., 21: 394405.
Turcotte, D.L., 1994. Fractal theory and the estimation of extreme floods. J. Res. Nat. Inst. Standards Technol., 99: 377389.
Twardosz, R., 2009. Probabilistic model of maximum precipitation depths for Krakow (Southern Poland, 18862002). Theor. Appl. Climatol., 98: 3745.
Wald, A. and J. Wolfowitz, 1943. An exact test for randomness in the non parametric case based on serial correlation. Ann. Math. Stat., 14: 378388.
Wang, W., X. Chen, P. Shi and P.H.A.J.M. van Gelder, 2008. Detecting changes in extreme precipitation and extreme streamflow in the Dongjiang River Basin in Southern China. Hydrol. Earth Syst. Sci., 12: 207221.
Wilcoxon, F., 1945. Individual comparisons by ranking methods. Biometr. Bull., 1: 8083. CrossRef 
Wilks, D.S., 1993. Comparison of threeparameter probability distributions for representing annual extreme and partial duration precipitation series. Water Resour. Res., 29: 35433549.
Yue, S. and P. Pilon, 2004. A comparison of the test ttest, MannKendall and bootstrap test for trend detection. Hydrol. Sci. J., 49: 2137.
Zahar, Y. and J.P. Laborde, 2007. Modelisation statistique et synthese cartographique des pluies journalieres extremes de Tunisie. Rev. Sci. Eau, 20: 409424.
Zalina, M.D., A.H.M. Kassim, M.N.M. Desa and V.T.V. Nguyen, 2002. Statistical analysis of atsite extreme rainfall processes in peninsular malaysia. Proceedings of the fourth International Conference Cape Town, Mar. 1822, South Africa, pp: 6168.



