CHARACTERIZATION AND FREQUENCY ANALYSIS OF ONE DAY ANNUAL MAXIMUM AND TWO TO FIVE CONSECUTIVE DAYS MAXIMUM RAINFALL OF ACCRA, GHANA

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1 VOL., NO. 5, OCOBER 007 ISSN ARPN Joural o Egeerg ad Appled Sceces Asa Research Publshg Network (ARPN). All rghts reserved. CHARACERIZAION AND FREQUENCY ANALYSIS OF ONE DAY ANNUAL MAXIMUM AND WO O FIVE CONSECUIVE DAYS MAXIMUM RAINFALL OF ACCRA, GHANA Xelde Seth Kwaku 1 ad Ophor Duke 1 1 Departmet o Earth ad Evrometal Studes, Motclar State Uverst, Upper Motclar, New Jerse, USA E-mal: eldes1@mal.motclar.edu ABSRAC Aual oe da mamum raall ad two to ve cosecutve da s mamum raall correspodg to a retur perod o to 100 ears has bee coducted or Accra, Ghaa. hree commol used probablt dstrbutos; ormal, logormal ad gamma dstrbuto have bee tested to determe the best t probablt dstrbuto that descrbes the aual oe da mamum ad two to ve cosecutve das mamum raall seres b comparg wth the Ch-square value. he results revealed that the log-ormal dstrbuto was the best t probablt dstrbuto or oe da aual mamum as well as two to ve cosecutve das mamum raall or the rego. Based o the best t probablt dstrbuto a mamum o mm 1 da, mm das, mm 3 das, mm 4 das ad mm 5 das s epected to occur at Accra ever two ears. Smlarl a mamum raall o mm, 40.49, 7.77 mm, 9.07 mm ad mm s epected to occur 1 da,, 3, 4 ad 5 das respectvel ever 100 ears. he results rom the stud could be used as a rough gude b egeers ad hdrologsts durg the desg ad costructo o draage sstems the Accra metropols as poor draage has bee deted as oe o the major actors causg loodg Accra. Kewords: retur perod, requec, probablt dstrbuto, gamma, cosecutve das mamum raall. INRODUCION he procedure or estmatg the requec o occurrece (retur perod) o a hdrologcal evet such as lood s kow as (lood) requec aalss. hough the ature o most hdrologcal evets (such as raall) s erratc ad vares wth tme ad space, t s commol possble to predct retur perods usg varous probablt dstrbutos (Upadhaa ad Sgh, 1998). Flood requec aalss was developed as a statstcal tool to help egeers, hdrologsts, ad watershed maagers to deal wth ths ucertat. Flood requec s utlzed to determe how ote a storm o a gve magtude would occur. It s a mportat tool or the buldg ad desg o the saest possble structures (e.g. dams, brdges, culverts, draage sstems etc.) because the desg o such structures demads kowledge o the lkel loods whch the structure would have to wthstad durg ts estmated ecoomc useul le (Bruce ad Clark, 1966). I partcular, aalss o aual oe da mamum raall ad cosecutve mamum das raall o deret retur perods ( tpcall to 100 ears) s a basc tool or sae ad ecoomc plag ad desg o small dams, brdges, culverts, rrgato ad draage work as well as or determg draage coecets (Bhakar et al., 006). he devastatg loodg Accra recet tmes amog other thgs has bee attrbuted to poor draage (wumas ad Asoma-Boateg, 00). Eve though some studes have bee carred out to map out areas wth Accra that are susceptble to loodg (e.g. Narko, 00; wumas ad Asoma-Boateg, 00) othg s kow o the requec o occurrece o such evets. I ths preset stud, aual 1 da ad to 5 cosecutve das mamum raall data o Accra (Accra arport stato) have bee aaltcall tted wth three theoretcal probablt dstrbuto uctos, vz., ormal, logormal, ad gamma dstrbuto. Frequec aalss (correspodg to to 100 ears) o aual 1 da ad to 5 cosecutve das mamum raall has also bee carred out usg the best tted probablt dstrbuto. MAERIALS AND MEHODS he dal raall data recorded at the Accra arport stato (05 36 N lattude, W logtude ad 66.7 m above sea level) or a perod o 30 ears ( clusve) were used or ths aalss. Aual to 5 das cosecutve das raall were computed usg the method descrbed b Bhakar et al. (006), b summg up raall o correspodg prevous das. Mamum amout o aual 1 da to to 5 das cosecutve das raall or each ear was used or the aalss. Statstcal parameters o aual 1 da as well as cosecutve das mamum raall have bee computed ad are show able- 1.Oe da to ve das mamum raall data were tted wth three ma probablt dstrbutos (able-). estg the goodess o t o probablt dstrbuto For the purpose o predcto, t s usuall requred to uderstad the shape o the uderlg dstrbuto o the populato. o determe the uderlg dstrbuto, t s a commo practce to t the observed dstrbuto to a theoretcal dstrbuto. hs s doe b comparg the observed requeces the data to the epected requeces o the theoretcal dstrbuto sce certa tpes o varables ollow specc dstrbuto (lahu, 006). Oe o the most commol used tests or testg the goodess o t o emprcal data to spec theoretcal 7

2 VOL., NO. 5, OCOBER 007 ISSN ARPN Joural o Egeerg ad Appled Sceces Asa Research Publshg Network (ARPN). All rghts reserved. requec dstrbuto s the ch-square test (Haa, 1977). I applg the Ch-square goodess-o-t test, the data are grouped to sutable requec classes. he test compares the actual umber o observatos ad the epected umber o observatos (epected values are calculated based o the dstrbuto uder cosderato) that all the class tervals. he epected umbers are calculated b multplg the epected relatve requec b the total umber o observato. he sample value o the relatve requec o terval s computed accordace wth equato 1 (Bhakar et al., 006) as: ) (.(1) s = he epected relatve requec a class terval ca also be appromated usg equato = p ) () ( he Ch-square test statstc s computed rom the relatoshp c m = ( O E ) / E..(3) = 1 O s the observed ad E s the epected (based o the probablt dstrbuto beg tested) umber o observato the th class terval. he observed umber o observato the th terval s computed rom equato 1 as: ( ) =. Smlarl, s the correspodg epected umbers o occurreces terval. he dstrbuto o c s the ch-square dstrbuto wth ν = k l 1 degrees o reedom. I coductg the goodess o t test usg the ch-square test, a codece level, ote epressed as 1, s chose ( where s reerred to as the sgcace level ). pcall, 95% s chose as the codece lmt. he test poses the ull hpothess (H o ) that the proposed probablt dstrbuto s rom the speced dstrbuto. H o s rejected c > 1, ν,. he value o s determed rom 1 ν publshed tables wth v degrees o reedom at the 5% level o sgcace. I ths stud three commol used probablt dstrbutos were tted wth 1 da ad to 5 das cosecutve das mamum raall. he three dstrbutos are brel dscussed below. Normal dstrbuto he ormal dstrbuto, a two parameter dstrbuto, has bee deted as the most mportat dstrbuto o cotuous varables appled to smmetrcall dstrbuted data (lahu, 006). he probablt dest ucto s gve b 1 1 µ ep or σ π σ.(4) ( ; µ,σ ) = Log ormal dstrbuto Most hdrologcal cotuous radom varables are oted to be asmmetrcall dstrbuted hece t s computatoall advatageous to trasorm the dstrbuto to a logormal dstrbuto (lahu, 006). he trasormato s usuall acheved b takg the logs o the varables questo. A radom varable s sad to ollow a logormal dstrbuto the logarthm (usuall atural logarthm) o s ormall dstrbuted. he probablt dest ucto o such a varable = l : 1 1 µ = ep or 0..(5) σ π σ Gamma dstrbuto I hdrolog the gamma dstrbuto has a advatage o havg ol postve values (lahu, 006). he probablt dest ucto o a gamma dstrbuted radom varable s gve b 1 / β ( ) e β Γ 1 = or 0... (6) he gamma ucto s deed such that the total area uder the dest ucto s ut as: Γ = 0 1 e d...(7) Frequec aalss ad requec actors Values o 1 da,, 3, 4 ad 5 cosecutve das mamum raall ca be estmated statstcall through the use o the Chow (1951) geeral requec ormula. he ormula epresses the requec o occurrece o a evet terms o a requec actor, K, whch depeds upo the dstrbuto o partcular evet vestgated. Chow (1951) has show that ma requec aalses ca be reduced to the orm 1 (8) X = X ( ) + C V K For the ormal dstrbuto the requec actor ca be epressed b equato 10 (whch s the same as the stadard ormal varable z). K = µ σ (9) he value o z could be oud rom stadard ormal dest ucto tables or could be calculated rom equato 10 as below: 8

3 VOL., NO. 5, OCOBER 007 ISSN ARPN Joural o Egeerg ad Appled Sceces Asa Research Publshg Network (ARPN). All rghts reserved. z = w w w w w...(10) where 1 w = I (11) p 1 p s substtuted or p equato 11 whe p > 0.5.he value o z ths case s gve a egatve sg (Bhakar et al., 006). he equato or the parameters terms o the sample momets or the ormal dstrbuto s gve b µ =, σ = S (1 ) For the logormal dstrbuto t s assumed that Y=I X s ormall dstrbuted. he magtude o a evet havg a retur perod, X s obtaed rom the relato X = ep( Y ) where Y = Y 1+ C K )... (13) ad K ( vy µ = σ K...(14) he value o ca be computed usg equato 10 or oud rom the stadard ormal dstrbuto table. he equato or the parameters terms o the sample momet or the logormal dstrbuto s gve b µ =, σ = S.(15) I the case o the gamma dstrbuto requec aalss ca be doe usg the method o momets as descrbed b Haa (1977). he equato or the parameters terms o the sample momet s gve b =, β =.(16) s s RESULS AND DISCUSSION he data as preseted able- revealed that the computed Ch-square values or the three probablt dstrbutos were less tha the crtcal Ch-square value at the 95% codece level or 1 da as well as cosecutve das mamum raall seres (ecept the cosecutve das mamum raall or the ormal dstrbuto). Results o the Ch-square values or the deret dstrbuto as preseted able- dcated that the logormal dstrbuto gave the mmum value or all dal raall seres aalzed. he logormal dstrbuto ucto s deemed the best t ucto or 1da ad to 5 cosecutve das mamum raall the stud rego. able-3 gves the aual 1 da ad cosecutve das mamum raall or deret retur perods as determed b the selected best t dstrbuto. he result show that a mamum o mm 1 da, mm das, mm 3 das, mm 4 das ad mm 5 das s epected to occur at Accra (Arport ad surroudg areas) ever two ears. Smlarl a mamum raall o mm, 40.49mm, 7.77 mm, 9.07 mm ad mm epected to occur 1 da,, 3, 4 ad 5 das respectvel ever 100 ears. It was recommeded b Bhakar et al. (006) that to 100 ears s a sucet retur perod or sol ad water coservato measures, costructo o dams, rrgato ad draage works. he to 100 ears retur perod obtaed ths stud could be used as a rough gude durg the costructo o such smlar structures. I partcular, these values could be ver beecal durg the costructo o draage sstems the Accra metropols as poor draage has bee deted as oe o the major actors causg loodg the area. able-1. Summar statstcs o aual 1 da as well as cosecutve das mamum raall. Statstcal parameters 1 da da 3 da 4 das 5 das Mmum (mm) Mamum (mm) Mea (mm) Stadard Devato (mm) Coecet o Varato Coecet o skewess Kurtoses

4 VOL., NO. 5, OCOBER 007 ISSN ARPN Joural o Egeerg ad Appled Sceces Asa Research Publshg Network (ARPN). All rghts reserved. Cosecutve das able-. Ch-square value or the three deret dstrbutos. Normal Log ormal Gamma Degrees o reedom Oe da wo das hree das Four das Fve das Crtcal Ch square value able-3. 1 da as well as cosecutve das mamum raall or varous retur perods. Retur Perod 1 da das 3 das 4 das 5 das ACKNOWLEDGEMEN hs stud was uded b the College o Scece ad Mathematcs at the Motclar State Uverst, Upper Motclar New Jerse, USA. he authors are grateul to the Ghaa Meteorologcal Servces Departmet or provdg the dal raall data. REFERENCES Bhakar, S.R., Basal, A.N. Chhajed, N., ad Puroht, R.C Frequec aalss o cosecutve das mamum raall at Baswara, Rajastha, Ida. ARPN Joural o Egeerg ad Appled Sceces. 1(3): Bruce, J.P., ad Clark, P.H Itroducto to hdrometeorolog. Pergamo Press, Oord. Chow, V A geeralzed ormula or hdrologc requec aalss. rasactos Amerca Geophscal Uo. 3(): Chow, V.. (ed) Hadbook o appled hdrolog. McGraw-hll, New York. Narko, K.B. 00. Applcato o a ratoal model GIS or lood rsk Assessmet Accra, Ghaa. Joural o Spatal Hdrolog. (1): lahu K he characterzato o raall the ard ad sem-ard regos o Ethopa. Water S.A. 3(3): wumas, Y.A.; Asoma-Boateg, R. 00. Mappg seasoal hazards or lood maagemet Accra, Ghaa usg GIS. Geoscece ad Remote Sesg Smposum. IGARSS apos; 0. IEEE Iteratoal. Vol. 5, pp Upadhaa, A., ad Sgh, S.R Estmato o cosecutve das Mamum Raall b Varous Methods ad ther Comparso. Ida Joural o S. Cos. 6():

5 VOL., NO. 5, OCOBER 007 ISSN ARPN Joural o Egeerg ad Appled Sceces Asa Research Publshg Network (ARPN). All rghts reserved. : umber o observato the th terval Lst o smbols : umber o observato : epected relatve theoretcal requec o the th class p ( ) : md pot o the th class terval. : probablt dest ucto o a radom varable ( the th class terval). m : umber o class tervals z : stadard ormal varable σ : stadard devato o the sample µ : mea o l σ : stadard devato o l l : umber o parameters used ttg the proposed dstrbuto : scale parameter o the gamma dstrbuto β : the shape parameter o the gamma dstrbuto Γ : the gamma ucto. X : Y : C : C V v Y X : K : mea value o X mea o Y coecet o varato or orgal data, X : coecet o varato or l X evet havg a retur perod requec magtude o a actor. 31

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