GENERATION OF RAINFALL INTENSITY-DURATION-FREQUENCY RELATIONSHIP FOR CENTRAL REGION IN BANGLADESH

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1 Proceedigs of the 3 rd Iteratioal Coferece o Civil Egieerig for Sustaiable Developmet (ICCESD 206), 2~4 February 206, KUET, Khula, Bagladesh (ISBN: ) GENERATION OF RAINFALL INTENSITY-DURATION-FREQUENCY RELATIONSHIP FOR CENTRAL REGION IN BANGLADESH Mushi Md. Rasel*, Md. Tashfique Uddi Chowdhury 2 ad Md. Mazharul Islam 3 Lecturer, Departmet of Civil Egieerig, Ahsaullah Uiversity of Sciece ad Techology, Bagladesh, mushimdrasel@gmail.com 2 URA, Departmet of Civil Egieerig, Ahsaullah Uiversity of Sciece ad Techology, Bagladesh, chowdhurytashfique@gmail.com 3 GRA, Departmet of Civil Egieerig, Ahsaullah Uiversity of Sciece ad Techology, Bagladesh, mdmazharulislamce@gmail.com ABSTRACT The objective of this research was to derive Raifall IDF relatioship for Cetral regio of Bagladesh. Two commo frequecy aalysis techiques Gumbel ad Log Pearso Type III (LPTIII) distributio were used to develop the IDF relatioship from raifall data of this regio. Yearly maximum raifall data for last 4 years ( ) from Bagladesh Meteorological Departmet (BMD) was used i this study. Idia Meteorological Departmet (IMD) empirical reductio formula was used to estimate the short duratio raifall itesity from yearly maximum raifall data. The results obtaied usig Gumbel method are slightly higher tha LPT III distributio method. Raifall itesities obtaied from these two methods showed good agreemet with results from previous studies o some parts of the study area. The chi-square goodess of fit test was used to determie the best fit probability distributio. The parameters of the IDF equatios ad coefficiet of correlatio for differet retur periods (2, 5, 0, 25, 50 ad 00 years) are calculated by usig oliear multiple regressio method. The results obtaied preseted that i all the cases the correlatio coefficiet is very high represetig the goodess of fit of the formulae to estimate IDF curves i the regio of iterest. It was foud that itesity of raifall decreases with icrease i raifall duratio. Further, a raifall of ay give duratio will have a larger itesity if its retur period is large. I other words, for a raifall of give duratio, raifalls of higher itesity i that duratio are rarer tha raifalls of smaller itesity. Keywords: Raifall itesity, raifall frequecy, Bagladesh Meteorological Departmet (BMD), Gumbel s Extreme Value distributio, LPTIII distributio. INTRODUCTION Raifall itesity-duratio-frequecy (IDF) curves are graphical exemplificatios of the amout of water that falls withi a give period of time i catchmet areas (Dupot ad Alle, 2000). IDF curves are used to aid the egieers while desigig urba draiage works. The establishmet of such relatioships was doe as early as 932 (Chow, 988 ad Dupot ad Alle, 2006). Sice the, may sets of relatioships have bee costructed for several parts of the globe. However, such relatioships have ot bee accurately costructed i may developig coutries (Koutsoyiais et al.,998). Koutsoyiais et al., 998; Koutsoyiais, 2003 cited that the IDF relatioship is a mathematical relatioship betwee the raifall itesity I, the duratio T d, ad the retur period T r. Ideed the IDF-curves allow for the estimatio of the retur period of a observed raifall evet or coversely of the raifall amout correspodig to a give retur period for differet aggregatio times. I Ketucky, for example IDF curves are used i cojuctio with ruoff estimatio formulae; e.g. the Ratioal Method, to predict the peak ruoff amouts from a particular watershed. The iformatio from the curves is the used i hydraulic desig to size culverts ad pipes (Dupot ad Alle, 2000). Further studies by (Iloa ad Fraces, 2002) performed raifall aalysis ad regioalizatio of IDF curves for differet regios. I Bagladesh water loggig ad flood is a commo problem durig Mosoo period because of iadequate draiage system. I order to solve this problem ew draiage desig is eeded where raifall data of differet duratio is eeded. But due to istrumetal limitatio these data were ot available. I the preset study, aual maximum raifall series is cosidered for Raifall Frequecy Aalysis (RFA). Raifall i ICCESD 206

2 a regio ca be characterized if the itesity, duratio ad frequecy of the diverse storms occurrig at that place are kow. The frequecy-data for raifalls of various duratios, so obtaied, ca be represeted by IDF curves, which give a plot of raifall itesity versus raifall duratio ad recurrece iterval. I recet studies, various authors attempted to relate IDF relatioship to the syoptic meteorological coditios i the area of hydrometric statios ( Dupot ad Alle, 2006). Mati et al.,984, i their study developed the IDF curve for North-East regio of Bagladesh ad also observed that the raifall data i this regio follow Gumbel s Extreme Value Distributio. Chowdhury et al.,2007, developed the short duratio raifall IDF curve for Sylhet with retur period of 2, 5, 0, 20, 50, ad 00 years. Kim et al. (2008) improved the accuracy of IDF curves by usig log ad short duratio separatio techique. They derived IDF curves by usig cumulative distributio fuctio (CDF) for the site uder cosideratio usig multi-objective geetic algorithm. Khaled et al. (20) applied L-momets ad geeralized least squares regressio methods for estimatio of desig raifall depths ad developmet of IDF relatioships. Rashid et al. (202) applied Pearso Type-III distributio for modellig of short duratio raifall ad developmet of IDF relatioships for sylhet city i Bagladesh. I probability theory, extreme value distributios amely Gumbel, Frechet ad Weibull are geerally cosidered for frequecy aalysis of meteorological variables. O the other had, Atomic Eergy Regulatory Board (AERB) guidelies described that the Order Statistics Approach (OSA) is the most appropriate method for determiatio of parameters of Gumbel ad Frechet distributios. I this preset study Gumbel s Extreme Value Distributio method is used to develop IDF curves ad equatios. I this cotext, a attempt has bee made to estimate the raifall for differet retur periods for differet duratios of such as 0-mi, 20-mi, 30-mi, 60-mi, 20-mi, 80-mi, 360- mi, 720-mi, 440-mi adoptig Gumbel distributios for developmet of IDF relatioships for seve divisios of Bagladesh. Model performace idicators (MPIs) such as correlatio coefficiet (R) was used to aalyze the performace of the developed IDF relatioships by Gumbel distributios for estimatio of raifall itesity for the statios uder study. 2. DATA COLLECTION AND METHODOLOGY For this study 24 hr daily raifall data from year 974 to 204 was collected from Bagladesh Meteorological Departmet (BMD) for Cetral regios. From the daily data maximum yearly raifall data was used i the aalysis. I Cetral regios there are six BMD statios (Dhaka, Comilla, M.Court, Mymesigh, Tagail, Chadpur) which are take ito cosideratio to develop IDF curve for cetral regio of Bagladesh. For accurate hydrologic aalyses, reliable raifall itesity estimates are ecessary. The IDF relatioship comprises the estimates of raifall itesities of differet duratios ad recurrece itervals. There are commoly used theoretical distributio fuctios that were applied i differet regios all over the world; (e.g. Geeralized Extreme Value Distributio (GEV), Gumbel, Pearso type III distributios) (Dupot ad Alle, 2000). Two commo frequecy aalysis techiques were used to develop the relatioship betwee raifall itesity, storm duratio, ad retur periods from raifall data for the regios uder study. These techiques are: Gumbel distributio ad LPT III distributio. 2. Estimatio of Short Duratio Raifall Idia Meteorological Departmet (IMD) use a empirical reductio formula (equatio ()) for estimatio of various duratio like -hr, 2-hr, 3-hr, 5-hr, 8-hr raifall values from aual maximum values. Chowdhury et al (2007), used Idia Meteorological Departmet (IMD) empirical reductio formula to estimate the short duratio raifall from daily raifall data i Sylhet city ad foud that this formula give the best estimatio of short duratio raifall. I this study this empirical formula (equatio ()) was used to estimate short duratio raifall of six statios of Cetral regio of Bagladesh. P t = P 24 () Where, P t is the required raifall depth i mm at t-hr duratio, P 24 is the daily raifall i mm ad t is the duratio of raifall for which the raifall depth is required i hr. 2.2 Gumbel Theory of Distributio For Gumbel distributio methodology was selected to perform the flood probability aalysis. The Gumbel theory of distributio is the most widely used distributio for IDF aalysis owig to its suitability for modellig maxima. It is relatively simple ad uses oly extreme evets (maximum values or peak raifalls). The Gumbel ICCESD 206 2

3 method calculates the 2, 5, 0, 25, 50 ad 00 year retur itervals for each duratio period ad requires several calculatios. Frequecy precipitatio P T (i mm) for each duratio with a specified retur period T (i year) is give by the followig equatio: P T = P ave + KS (2) Where K is Gumbel frequecy factor give by: K=- (3) Where P ave is the average of the maximum precipitatio correspodig to a specific duratio. I utilizig Gumbel s distributio, the arithmetic average i equatio (2) is used: P ave = (4) Where Pi is the idividual extreme value of raifall ad is the umber of evets or years of record. The stadard deviatio is calculated by equatio (5) computed usig the followig relatio: S= 2 ] /2 (5) Where S is the stadard deviatio of P data. The frequecy factor (K), which is a fuctio of the retur period ad sample size, whe multiplied by the stadard deviatio gives the departure of a desired retur period raifall from the average. The the raifall itesity, I T (i mm/h) for retur period T is obtaied from: I T = (6) Where T d is duratio i hours. The frequecy of the raifall is usually defied by referece to the aual maximum series, which cosists of the largest values observed i each year. A alterative data format for raifall frequecy studies is that based o the peak-over threshold cocept, which cosists of all precipitatio amouts above certai thresholds selected for differet duratios. Due to its simpler structure, the aual-maximum-series method is more popular i practice (Bogra, Vezzai ad Fotaa, 2005) 2.3 Log Pearso Type III The LPT III probability model is used to calculate the raifall itesity at differet raifall duratios ad retur periods to form the historical IDF curves for each statio. LPT III distributio ivolves logarithms of the measured values. The mea ad the stadard deviatio are determied usig the logarithmically trasformed data. I the same maer as with Gumbel method, the frequecy precipitatio is obtaied usig LPT III method. The simplified expressio for this latter distributio is give as follows: P * = log(pi) (7) PT* = Pave* + KTS * (8) Pave* = P* i = /2 2 (0) i= S* = ( P* Pave*) Where P T *, P ave *, S* are logarithmic extreme value of raifall, average of maximum precipitatio correspodig to a specific duratio, stadard deviatio of P* data respectively. K T is the Pearso frequecy factor which depeds o retur period (T) ad skewess coefficiet (Cs). The skewess coefficiet, Cs, is required to compute the frequecy factor for this distributio. The skewess coefficiet is computed by equatio (). (9) ICCESD 206 3

4 C S ( Pi * Pave*) i= = 3 ( )( 2)(S*) 3 () K T values ca be obtaied from tables i may hydrology refereces. By kowig the skewess coefficiet ad the recurrece iterval, the frequecy factor, K T for the LPT III distributio ca be extracted. The atilog of the solutio i equatio (7) will provide the estimated extreme value for the give retur period. 2.4 Derivatio of IDF Equatio The IDF formulae are the empirical equatios represetig a relatioship betwee maximum raifall itesity as a depedet variable ad other parameters of iterest; for example the raifall duratio ad frequecy as idepedet variables. There are several commoly used fuctios relatig those variables previously metioed foud i the literature of hydrology applicatios ( Chow, 988; Burke ad Burke, 2008). To derive a equatio for calculatig the raifall itesity (I) for the regios of iterest, there are some required steps for establishig a equatio to suit the calculatio of raifall itesity for a certai recurrece iterval ad specific raifall period which depeds maily o the results obtaied from the IDF curves. Two approaches were tried to estimate the equatio parameters. A. By applyig the logarithmic coversio, where it is possible to covert the equatio ito a liear equatio, thus to calculate all the parameters related to the equatio. The followig steps are followed: I. Covert the origial equatio i the form of power-law relatio (Chow, 988; Koutsoyiais et al., 998) as follows: CT I = (2) T m r e d By applyig the logarithmic fuctio to get log I = log K e log T d (3) Where K=CTr m (4) Ad e represets the slope of the straight lie. II. III. IV. Calculate the atural logarithm for (K) value foud from Gumbel method or from LPTIII method as well as the atural logarithmic for raifall period T d. Plot the values of (log I) o the y-axis ad the value of (log T d ) o the x axis for all the recurrece itervals for the two methods. From the graphs (or mathematically) fid the value of (e) for all recurrece itervals. The it was foud out the average value of e value, e ave, by usig the followig equatio e ave e = (5) Where represets recurrece itervals (years) value oted as Tr. V. From the graph, it was foud (logk) values for each recurrece iterval where (logk) represets the Y-itercept values as per Gumbel method or LPTIII method. The covert equatio (4) ito a liear equatio by applyig the atural logarithm to become: ICCESD 206 4

5 log K= logc + m logtr (6) VI. Plot the values of (logk) o the y-axis ad the values of (logtr) o the x-axis to fid out the values of parameters c ad m as per Gumberl method or LPTIII where (m) represets the slope of the straight lie ad (C) represets the (ati log) for the y itercept. B. Estimatio of the equatio parameters by usig oliear regressio aalysis: Usig the Solver fuctio of the ubiquitious spreadsheet programme Microsoft Excel, which employs a iterative least squares fittig routie to produce the optimal goodess of fit betwee data ad fuctio. The R 2 value calculated is desiged to give the user a estimate of goodess of fit of the fuctio to the data. 2.5 Goodess of Fit Test The aim of the test is to decide how good is a fit betwee the observed frequecy of occurrece i a sample ad the expected frequecies obtaied from the hypothesized distributios. A goodess of fit test betwee observed ad expected frequecies is based o the chi-square quatity, which is expressed as, χ k 2 2 = (Oi E i ) / Ei i= (7) 2 Where χ is a radom variable whose samplig distributio is approximated very closely by the chi-square distributio. The symbols Oi ad Ei represet the observed ad expected frequecies, respectively, for the i-th class iterval i the histogram. The symbol k represets the umber of class itervals. If the observed 2 frequecies are close to the correspodig expected frequecies, the χ value will be small, idicatig a good fit; otherwise, it is a poor fit. A good fit leads to the acceptace of ull hypothesis, whereas a poor fit leads to its rejectio. The critical regio will, therefore, fall i the right tail of the chi-square distributio. For a level of 2 sigificace equal to a, the critical value is foud from readily available chi-square tables ad χ > costitutes the critical regio. 3. RESULTS AND DISCUSSION The purpose of this study was to develop IDF curves ad derive a empirical formula to estimate the raifall itesity at Cetral regio i Bagladesh. The IDF curves are used as a aid whe desigig draiage structures for ay egieerig project. The curves allow the egieer to desig safe ad ecoomical flood cotrol measures. Raifall estimates i mm ad their itesities i mm/hr for various retur periods ad differet duratios were aalysed usig the two techiques: (Gumbel ad LPT III). Accordig to the IDF curves, raifall estimates are icreasig with icrease i the retur period ad the raifall itesities decrease with raifall duratio i all retur periods. Raifall itesities rise i parallel with the raifall retur periods. The results obtaied from the two methods have good cosistecy. Table : The parameters values used i derivig formula Regio Parameter Gumbel Log Pearso III Cetral Regio c m e ICCESD 206 5

6 Figure : IDF curve by average of Gumbel ad LPTIII methods at Cetral Regio Figure shows result of the IDF curves obtaied by average of Gumbel ad LPT III methods for Cetral regio. It was show that there were small differeces betwee the results obtaied from the two methods, where Gumbel method gives slightly higher results tha the results obtaied by Log Pearso III. This is show also from parameters of the derived equatio for calculatig the raifall itesity usig the two methods. Parameters of the selected IDF formula were adjusted by the method of miimum squares, where the goodess of fit is judged by the correlatio coefficiet. The results obtaied showed that i all the cases the correlatio coefficiet is very high, ad rages betwee ad 0.994, except few cases where it rages betwee 0.98 ad whe usig LPT III at 50 ad 00 years. This idicates the goodess of fit of the formulae to estimate IDF curves i the regio of iterest. For each regio the results are give as the mea value of the poits results. Table shows the parameters values obtaied by aalyzig the IDF data usig the two methods ad those are used i derivig formulae for the two regios. Also, goodess-of-fit tests were used to choose the best statistical distributio amog those techiques. Results of the chi- square goodess of fit test o aual series of raifall are show i Table 2. As it is see most of the data fit the distributios at the level of sigificace of α =0.05, which yields χ <3.84. Oly the data for 0 mi ad 20 mi do ot give good fit usig Gumbel method distributio. Also the data for 0 mi usig LPTIII method do ot give good fit. Table 2: Results of chi-square goodess of fit test o aual maximum raifall cal Regio Cetral Duratio i miutes Distributio Gumbel LPTIII CONCLUSIONS This research presets some isight ito the way i which the raifall is estimated i Bagladesh. Sice Bagladesh has differet climatic coditios from regio to regio, a relatio for each regio has to be obtaied to estimate raifall itesities for differet duratios ad retur periods ragig betwee 2 ad 00 years. This study has bee coducted for the formulatio ad costructio of IDF curves usig data from recordig statios by usig two distributio methods: Gumbel ad LPT III distributio. Gumbel method gave some larger raifall itesity estimates compared to LPT III distributio. I geeral, the results obtaied usig the two approaches are very close at most of the retur periods ad have the same tred, this agrees with the results obtaied by Rashid et al., 202. The results obtaied from that work are cosistet with the results from previous studies doe i some parts of the study area. It is cocluded that the differece observed betwee the results of this ICCESD 206 6

7 study ad the results doe before by Rashid et al. (202) are accepted ad i good agreemet, ad this ca be attributed to the record legths of the raifall data used for this study ad the studies before. The parameters of the desig storm itesity for a give period of recurrece were estimated for each regio. The results obtaied showed a good match betwee the raifall itesity computed by the methods used ad the values estimated by the calibrated formulae. The results showed that i all the cases data fitted the formula with a correlatio coefficiet greater tha This idicates the goodess of fit of the formulae to estimate IDF curves i the regio of iterest for duratios varyig from 0 to 440 mi ad retur periods from 2 to 00 years. The chisquare test was used o oe had to examie the combiatios or cotigecy of the observed ad theoretical frequecies, ad o the other had, to decide about the type of distributio which the available data set follows. The results of the chi-square test of goodess of fit showed that i all the duratios the ull hypothesis that the extreme raifall series have the Gumbel distributio is acceptable at the 5% level of sigificace. Oly few cases i which the fittig was ot good obtaied by usig the LPT III distributio. Although the chisquare values are appreciably below the critical regio usig Gumbel distributio ad few values are higher tha the critical regio usig LPT III distributio, it is difficult to say that oe distributio is superior to the other. Further studies are recommeded wheever there will be more data to verify the results obtaied or update the IDF curves. ACKNOWLEDGEMENTS Authors are idebted to the Bagladesh Meteorological Departmet (BMD) for providig the ecessary meteorological data. REFERENCES Bougadis, J., & Adamowski, K. (2006). Scalig model of a raifall itesity-duratio-frequecy relatioship. Hydrological processes, 20(7), Burke, C. B., & Burke, T. T. (2008). Stormwater Draiage Maual Burlado, P., & Rosso, R. (996). Scalig ad muitiscalig models of depth-duratio-frequecy curves for storm precipitatio. Joural of Hydrology, 87(), Chow, V.T., 988. Hadbook of Applied Hydrology. McGraw-Hill Book. Chowdhury, R. K., Alam, M. J. B., Das, P., & Alam, M. A. (2007). Short duratio raifall estimatio of Sylhet: IMD ad USWB method. JOURNAL-INDIAN WATERWORKS ASSOCIATION, 39(4), 285. Elsebaie, I. H. (202). Developig raifall itesity duratio frequecy relatioship for two regios i Saudi Arabia. Joural of Kig Saud Uiversity-Egieerig Scieces, 24(2), Iloa, V., & Fracés, F. (2002). Raifall aalysis ad regioalizatio computig itesity duratio frequecy curves. Uiversidad Politecica de Valecia Departameto de Igeieria Hidraulica Medio Ambiete APDO, Kim, T., Shi, J., Kim, K., & Heo, J. H. (2008, May). Improvig accuracy of IDF curves usig log-ad shortduratio separatio ad multi-objective geetic algorithm. I 2008 EWRI Cogress (pp. 2-6). Koutsoyiais, D. (2003, October). O the appropriateess of the Gumbel distributio for modellig extreme raifall. I Proceedigs of the ESF LESC Exploratory Workshop held at Bologa (pp ). Koutsoyiais, D., Kozois, D., & Maetas, A. (998). A mathematical framework for studyig raifall itesity-duratio-frequecy relatioships.joural of Hydrology, 206(), Mati, M. A., & Ahmed, S. M. U. (984). Raifall Itesity Duratio Frequecy Relatioship for the NE Regio of Bagladesh. Joural of Water Resource Research, 5(). Rashid, M. M., Faruque, S. B., & Alam, J. B. (202). Modelig of short duratio raifall itesity duratio frequecy (SDRIDF) equatio for Sylhet City i Bagladesh. ARPN Joural of Sciece ad Techology, 2(2), ICCESD 206 7