Bell Formula

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FTP oumes hydrologiques - Orstom -Septembre 1992 121

BELLSORMULAA REAPPRAISAL

R . .MANLEY

Lun des buts principaux dea rgionalisation des donnes mtorologiques ethydrologiques est de permettre lextension, soitansIespacesoitdans le temps,des donnes ponctuelles limites.ne application spcifique de cette techniquea t propose par Bell (Bell F.C., Generalised rainfallduration-frequencyrelationships, Proc ASCE,5,HY1,3 11-327,1969).IIafait lhypothse quetousles orages de courte dure taient dusux cellules convectives dont learactretait similaire partout dans le monde. Utilisant une base de donnes tires deplusieurs pays,l adevelopp une formule gnrale des rapports fiquencelduredorages. Cet rtic le tudie lapplication de cette formule en utilisant des donnpluviomtriques de ix pays et de trois continents. Il dmontre que, duneaongnrale, la formule est de bonne prcision mais que des modifications quieservent des donnes locales peuvent apporter des amliorations.

Consultant inEngineering Hydrology 59,Panton Street CambridgeCB2 1HLUnited Kingdom


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It is a fiquent cornplaint of hydrologiststh& they do not have enough data,or%hathe siatathey do have is not for the site they are interested in.To ive somehelp in these situations various techniqueshave been developped toenabledataesuntries have standard formula for ohis which have the test of t h e .A very commonproblern is definhg the quantity of 1 o be expectedforagiven fiq uen cy of occurrence. early atkmptto produceageneral formulain the United Kingdom was that due to BILW (1936) whkh was ofthe form :

fiom the orne or similar 91 for the purpose in hancl.

YB =J,25 f (Je +6 , 1 )3*55 (1)where :II is the nurnber of events in PO years,R s the ainfall depeh in inches,t is the duration in hours.This equationwas valid for periods fiom 5 minutes o 2hours."IXswas baseclondata fiom only 12 stations with 16 yems record. Using more extensive ataasimgl i f iedversionoftheformula~spr~uc~byHoLL~(1967)whichwas

p3 = f j p 4 (2)which was valid for t up t o 2s hours.

Bell's methodwas developgedafteranmalysis ofrainfall data iomthe UnitedStates, the USSR, Australia and South E c a . JHis method is based on theassumption hat the most intense short urationstonnsare caused by wnvectivestorm " 41s and that such stormshave similarcharacteristicswherevertheyoccurin theworld.For this reason his method is only valid fo r stormsofup to 2 hours


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joumdeshydrologiques- Orstom -Septembre 1992 123duration. In its basic form there are two equations, one of which defines thechanges dueo different storm durations andhe otherthe changes dueo differentreturn periods. Thefirst of these is :R[t,T]=(O. 54 t.25 - O.50) R[6O,T] (3)where :R is the total rainfall inmillimetres,t is storm duration n minutes,

T is return period in years.The second is : R[t,T] =(O. 21 In(T) +O.52) R[t,1O] (4)Thesetwo quations can be combinedo give a generalised formula, whichs :R[t, T] =(O. 54 r25- 0.5) 0.21 In(T) +0.52) R[60,1O] (5)For an evaluation of theormula the followingdata sets were usedMedan, Sumatra (Indonesia)

from Wild andHall (1982) (quoted inSHAW 1988)),Kumasi, Ghana and Oxford, United Kingdomboth quotedinWILSON1983),Maputo, MozambiqueDHV(1981),Yundum Airport, Banjul, Gambiafrom TOWNSEND1977),Niamey, Nigerfrom an unpublished report.

The data are not presentedn a consistent format. n the first four cases theyconsist of values of intensity for different return periods and differenturations.In some caseshe shortest duration is0 minutes andn others it isonly 6minutes.There are also some variationsn the case ofhe return periods used.or Banjul,the maximum storms of different durations over20 year period re recorded.For Niamey he data is presented as he number of times whendifferent valueswere exceeded over23 year period.n his case the return periodsfthe 12mostsevere events were assigned using the Weibull formula whichs :P =n / ( l V+ I ) (6)where :P s the probability of excedence,n is the rank of the event,N is the total number of years.Figures 1 to 6 show the esults using the formula (equation ) and the valuesas presented for the six stations.


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150

IPOO

CunI:.dt

55

O

EI' 50

55

O


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1/2LI'joumbeshydrologiques- Orstom -Septembre 1992 125

150 -EEI2 100(1E

-Cu.*a 50 -

O 50 100 150 200 250 30050minutesFigure 3Rainfall / duration /fiequ enc y : anjul - Gambia

400

Rgure 4RainfaIl / duration Ifiequ enc y : edan - Samatru


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200 -

150 -EEI2 100i

-a: T-2.18

T=24T =8

i50 -0 -

O 100 200 300 400minutes

50A A

1 AAX0 J

0 50 100 150 200 250Iminutes 300


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V L U e ou mtes hydrologiques - Orstorn -Septembre 1992 127Since the formula is based on he 60 minute 1O-year storm then his value is

correct for al1 the stations. With the possible exception of Banjul,he 60minutestorms for al1tropical stations re well represented, suggestinghatthe part oftheformula relating to different return periodss accurate. The 10-year storms fordifferent durations re also well represented.The area where the largest errors,in percentage terms, occurs for short intensity stormsor return periods otherthan ten years. It is also clear that the errors get larger for durations larger t han2 hours, which washe limit puton he formula by Bell. t is interesting that theaccuracy of the formula in Medan which inAsia, a continent not used inheoriginal analysis, shouldppear similar tothat for the African stations. he finalfigure, number6 , for Oxford in he United Kingdom indicateshat the formulamay not be applicable in temperate climates.The question of whetherhe results of the formulare accurate enough hasobe set gainst other factors which influencehe accuracy, andlsothe possibilityof differentmethods which mightgive better results.The estimation of rainfall ofhort duration is anotably difficulttask.Anormalautographic rain gauge as a single chartor a twenty four hour period. Readingoff values for durations shorterhan an hours hard to do accurately, particularlya s du r hgn intense storm thetrace may riserapidly making italmost impossibleto identify slight changesn the gradient ofthe line which inurn epresent changesin intensity. A steep gradient also makes it almost impossible to identify theduration of an ntense burst of rain. If the total of the values read iom the chartis different to hose fiom a check gauge then urther errors may be introducedin adjustinga id21for partsfthe &y. Other devices,or example tipping-bucketgauges, present similar difficulties.There is also the questionfwhich probability distributiono use. Most of theabove calculations used he Gumbel (type 1) distribution ut different resultswould be obtainediom different distributions. Theresalso the question ofwhichof the different distribution formulas most appropriateor the representation ofrainfall, particularly wheret is necessaxy to estimate rainfallorareturn periodsignificantly greaterhanthe length of theata series.To test the accuracy of the part of the formula dealing with return periodshevalues fiom the formula were compared with those obtained directly fromheGumbel method. Values of 60 minutes rainfall for Banjul were used. The

following table shows the results :


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Table 160 minute raiddl :Banjul -Gmbia

In he above table the a Min, N and a Max, IBmlumns refer t o the 95 96confidence lhits for the Gumbel distribution.The a Central I) column gives thecentralestimate from the fornulla and he


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E U l c ourneshydrologiques - Orstom -septembre 1992 129the 1 100 data items and the appropriate Gumbel actors ;he second was toca lcu late the ean and tandard deviation from theO to 5 items ofhe data set ;the thirdwas to use the Bell formula using thein 1O year values calculated asdescribed above. The return periods tested were 2, 10 and 50 years. The testshowed that fo r coefficient of variation of.35,he same as that implicitn theBell formula, herew a s little to choose betweenusinghe formula or the datawiththe Gum bel distribution.fthe coefficient ofva riationwas 0.22, he lowest inhedata sets described above, then it was better to use the mean and standarddeviation calculated from even10years of data only With 10years of data theformula gave aworse accuracy or around 70 %of the trials and with50 yearsof data it was worse n95%of therials. If the Coefficient of ariation was.43there was a slight advantage n using the ata rather than the formula ut not asmuch as in the previous test- t was worse for lower periods of returnut gavesimilar answ ers for return periods of 10 and 50 years. For the test with thecoefficient ofvariation of 0.35 the average absolute errors were aroundto 7 %for al1 ranges and both methods. Forcoefficient of variation of 0.22he errorswith 1O-year data sets were 6 % o 7%using the Gumbel distribution but upo17%usingthe formula. For the highest coefficient ofariationthe errors startedoff at around 10 % for a 10 year return period and droppedo around 5%for50 years but with the errors for 1 in 2 year storm being almostwiceas highusing the form ula.The basicersion of the Bell formula uses theO-year 60-minute storm as hebasis forxtrapolation.Bell however proposed another formula whichas basedon he 2-year 60-minute storm.This is :R[t,T] =(0.35 In(T) +O. 76)(0.54 P S 0.50)R[60y2] (7)

This form ula was considered less reliablehanthe formula usinghe ten yearstorm. Itoes however havehe advantage tha t if the atais normally distributedthen the 2-year storm ishe average. For the Gumbel distribution the 2.33 earStorm ishe average. By adjustingthe parametersfthe above equation differentversion of the ormula for the 2.33 storm was produced whichs :R[t, T] =(O. 34 ln@) +O. 712) (O. 54 P. - O.50)R[6OY .331 (8)The second test using the trial data setwas to use data for fiom to 1O0yearslength to estimate the mean and o use that value as the .33 year eturn periodstorm n the above formula. For coefficient ofvariation 0.35 here was littledifference inaccuracybetween the methods withslight improvementiom usingthe Gum bel distributionor more than20 year of data,ut even sothe differencew a s etween 7% or the Gumbel formula and%for equation 8. For a coefficientofvariationof0.22 similar resultsothose in theirst series of tes t were obtained,with the Gumbel distributiongiving better results afterears datawas available.In this case the rror for a 1 in50 storm was always around0%. For the 0.43


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coefficient of variation, the use of the Gumbel distribution &vasbeteer after10 years but the errorswere ofthe order of10%wthlitele variationin accuracyberneen the tws metho

From a visual inspection of the results pr es en td on the 6 figures the broadconclusions are that the Bell formula gives sable aesults for tropicalcountriesbut the aesults areworst for the one r a b gauge in a temperate clhate. Furtherlysis demonstrates19hata criticd factor is the fficient of variation of the

&ta set being studied. I f the fficient of variation oflow, around 0.22,thenit is better to use even a few s of data t o calculate the parmeters for theGumbel distribution.On he otherhand if the coefficient of ariation is close t othe implicit value in the Bell formula, 0.35, or even higher then some 10 to20 years of daka is needed before the s deviation cmbe estimatedwths uE c kn t aecuracy fo r theGumbeldist w ernaaccuracy.However,from ody a few years it wi I I not be possible to know with m y accuracywhether the esefficient ariation is %owor not. Thethesefore is :


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EUTjournes hydrologiques -Orstom-Septembre 1992 131BIBLIOGRAPHIE

BELL .C., 1969. Generalised rainfall-duration-fquency relationships.Proc ASCE, 95, HY1,3 11-327.BILHAM E.G.,936. Classification of heavy falls in short periods.ritishRainfallr 1935, 262-280.DHV, 198 1. Drenagem da Cidade o Mputo -Plan0 General.HOLLAND.J., 1967. Rain intensity fkequency relationships in Britain.British RainfallI961 , Part III, 43-51.NATIONALNVIRONMENTALESEARCH COUNCIL NERC,975. Flood StudiesReport, Volume II.SHAW . M., 1988. Hydrology in Practice., NR International.TOWNSEND, 1977. Technical report no.2, Analyses of intense rainfall.UNDP/WMO project RAF/74/074.WILD J.K., ND HALL J.K., 1982.Proc. Inst. Civil Eng., 73, 85-108.WILSON E.M.,983. Engineering Hydrology., MacMillan.

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