Establishment of Intensity-Duration-Frequency Curves for Precipitation in the Monsoon Area of Vietnam

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1 京都大学防災研究所年報 第 49 号 B 平成 8 年 4 月 Annuls of Diss. Prev. Res. nst., Kyoto Univ., No. 49 B, 2006 Estblishment of ntensity-durtion-frequency Curves for Precipittion in the Monsoon Are of Vietnm Le MNH NHA*, Ysuto ACHKAWA, n Koru AKARA *Grute school of Urbn n Environment Engineering, Kyoto University Synopsis he rinfll ntensity-durtion-frequency (DF) reltionship is one of the most commonly use tools in wter resources engineering, either for plnning, esigning n operting of wter resource projects, or for vrious engineering projects ginst floos. he estblishment of such reltionships ws one s erly s in 932 (Bernr). Since then, mny sets of reltionships hve been constructe for severl prts of the globe. However, such mp with rinfll intensity contours hs not been constructe in mny eveloping countries, incluing Vietnm. here is high nee for DF curves in the monsoon region of Vietnm. his reserch is to construct DF curves for seven sttions in the monsoon re of Vietnm n to propose generlize DF formul using bse rinfll epth, n bse return perio for Re River Delt (RRD) of Vietnm. Keywors: Rinfll intensity, Design rinfll, ntensity-durtion-frequency reltionship (DF), Vietnm.. ntrouction he rinfll ntensity-durtion-frequency (DF) reltionship is one of the most commonly use tools in wter resources engineering, either for plnning, esigning n operting of wter resource projects, or for vrious engineering projects ginst floos. he estblishment of such reltionships ws one s erly s in 932 (Bernr, 932). Since then, mny sets of reltionships hve been constructe for severl prts of the globe. However, such mp with rinfll intensity contours hs not been constructe in mny eveloping countries, incluing Vietnm. here is high nee for DF curves in the monsoon region of Vietnm but unfortuntely the equte long historicl t sets re frequently not vilble. A rinfll intensity-urtion-frequency (DF) reltionship is commonly require for esigning of the wter resource projects. here hs consierble ttention n reserch on the DF reltionship: Hershfiel (96) evelope vrious rinfll contour mps to provie the esign rin epths for vrious return perios n urtions. Bell (969) propose generlize DF formul using the one hour, 0 yers rinfll epths; P 0, s n inex. Chen (983) further evelope generlize DF formul for ny loction in the Unite Sttes using three bse rinfll epths: P 0, P 0 24, P 00, which escribe the geogrphicl vrition of rinfll. Kouthyri n Gre (992) presente reltionship between rinfll intensity n P 2 24 for ni. Koutsoyinnis et l. (998) cite tht the DF reltionship is mthemticl reltionship between the rinfll intensity i, the urtion, n the return perio (or, equivlently, the nnul frequency of exceence, typiclly referre to s frequency only). his pper proposes the pproch to the formultion n construction of DF curves using

2 t form recoring sttion by using empiricl equtions, n comprison the equtions, choosing wht eqution cn be use in the monsoon re of Vietnm. Normlly, rinfll intensity-urtionfrequency reltionship is erive from the point rin guges, the network of ily rinfll recoring rin guges in Vietnm hs higher ensity thn short urtion (hourly or minutes) rin guges. he regionl DF formul prmeters re generte for unguge res to estimte rinfll intensity for vrious return perio n rinfll urtion. he metho propose in this stuy is resonbly pplicble to unguge rinfll loctions, which is conclue from the verifiction of itionl rin guges. More specificlly, this reserch is to stuy the generlize DF formul using some bse rinfll epth n bse return perio. wo min proceures re presente in this stuy. he first prouces the set of DF curves t 7 sttions by using empiricl functions. he secon prouces generlize DF eqution for loction re. he pper is orgnize in five sections, the first being this introuction. n Section 2 we give the tritionl methos to estblish DF curves using empiricl equtions n regionl prmeters of equtions. Section 3 els with generliztion rinfll intensity urtion frequency formuls. Section 4 emonstrtes the propose proceures with pplictions using rel worl t (Re River Delt in Vietnm). Conclusions re rwn in Section 5. Cumultive Distribution Function F(x) CDF F(x) Rinfll ntensity (mm/hr) CDF of 0 minute 2. Methos to estblish intensity urtion frequency curves for precipittion For mny hyrologic nlyses, plnning or esign problems, relible rinfll intensity estimtes re necessry. Rinfll intensity urtion frequency reltionship comprises the estimtes of rinfll intensities of ifferent urtions n recurrence intervls. he typicl technique for estblishment the DF curves of precipittion is conucte vi three steps. he first step is to fit Probbility Distribution Function (PDF) or Cumultive Distribution Function (CDF) to ech group comprise of the t vlues for specific urtion. t is possible to relte the mximum rinfll intensity for ech time intervl with the corresponing return perio from the cumultive istribution function. Given return perio, its corresponing cumultive frequency F will be: F or () F Once cumultive frequency is known, the mximum rinfll intensity is etermine using chosen theoreticl istribution function (e.g. GEV, Gumbel, Person type istributions). he Person type istribution tht is commonly use in Vietnm for frequency nlysis is utilize in this stuy. CDF of 60 minute 0.5 CDF of 440 minute DF ( 00 yer return) 0 minute 60 minute ( hr) Rinfll urtion (minute) DF ( 2 yer return) DF ( 2 yer return) 440 minute ( y) Fig.. he trnsformtion of the CDF into the DF curves.

3 n the secon step, the rinfll intensities for ech urtions n set of selecte return perios (e.g. 5, 0, 20, 50,00 yers, etc.) re clculte. his is one by using the probbility istribution functions erive in the first step. he figure show the trnsformtion of the CDF into the DF curves. n the thir step, the empiricl formuls (Section 2.) re use to construct the rinfll DF curves. he lest-squre metho is pplie to etermine the prmeters of the empiricl DF eqution tht is use to represent intensity-urtion reltionships. 2. Empiricl DF formuls he DF formuls re the empiricl equtions representing reltionship mong mximum rinfll intensity (s epennt vrible) n other prmeters of interest such s rinfll urtion n frequency (s inepenent vribles). here re severl commonly use functions foun in the literture of hyrology pplictions (Chow et l., 988), four bsic forms of equtions use to escribe the rinfll intensity urtion reltionship re summrize s follows: lbot eqution: i (2) b Bernr eqution: i (3) e Kimijim eqution: i (4) e b Shermn eqution: i e ( b) (5) where i is the rinfll intensity (mm/hour); is the urtion (minutes);, b n e re the constnt prmeters relte to the metrologicl conitions. hese empiricl equtions show rinfll intensity ecreses with rinfll urtion for given return perio. All functions hve been wiely use for hyrology prcticl pplictions. he lest-squre metho is pplie to etermine the prmeters of the four empiricl DF equtions tht re use to represent intensity-urtion reltionships. he vlue of prmeters in the rinfll DF equtions were chosen on minimum of Root Men Squre Error (RMSE) between the DF reltionships prouce by the frequency nlysis n simulte by the DF eqution. 2.2 Regionliztion of the prmeter of rinfll intensity urtion frequency equtions he rinfll DF curves re erive from the point rin guges; only sets of DF curves t point re estblishe. However, we nee the DF curves t ny point, s the network of ily rinfll recoring rin guges in Vietnm hs higher ensity thn recoring rin guges. he regionl DF formul prmeters re generte for unguge res to estimte rinfll intensity for vrious return perio n rinfll urtion. he metho propose in this stuy h resonble ppliction to unguge rinfll loction, which ws conclue from the verifiction of itionl rin guges. After etermining the prmeters of DF formul such s prmeters, b n prmeter e, for the sme return perio, using Arc view/gs interpolting the prmeter contour mps, ht mp cn generte for the prmeters which cn then be use for unguge rinfll with return perios. For tht mp, it is possible to estimte the prmeter set of ny point in this re, the rinfll DF curves cn be constructe by using these prmeters mp. 3. Generlize rinfll intensity urtion frequency formul A set of ntensity-durtion-frequency (DF) curves constitutes reltion between the intensity (more precisely, the men intensity) of precipittion (mesure in mm/h), the urtion or the ggregtion time of the rinfll (in min) n the return perio of the event. he return perio of n event (here the rinfll intensity or epth) inictes how rte/how frequent this event is n is efine by the inverse of the nnul exceence probbility. Denote by i the rinfll intensity (mm/h), the urtion of the rinfll

4 (min) n the return perio (yers). he DF reltion is mthemticlly s follows: i f (, ) (6) he rinfll intensity is function of the vribles n. Koutsoyinnis et l. (998) cite tht the DF reltionship is mthemticl reltionship between the rinfll intensity i, the urtion, n the return perio (or, equivlently, the nnul frequency of exceence, typiclly referre to s frequency only). he typicl DF reltionship for specific return perio is specil cse of the generlize formul s given in eqution (7) i e ( (7) b) where, b, e n re non-negtive coefficients. hus, the eqution tht is more generl: with = n e= it will be lbot eqution; = n b=0 is Shermn; e= is Kimijim eqution n = is Shermn. his expression is n empiricl formul tht encpsultes the experience from severl stuies. An numericl stuy shows if ssume =, the corresponing error is much less thn the typicl estimtion errors which results eqution (8) i (8) e ( b) Bell (969) propose generlize DF formul using the one hour, 0 yers rinfll epths; P 0, s n inex. Cheng-lung Chen (983) further evelope generlize DF formul for ny loction in the Unite Sttes using three bse rinfll epths: -hour rinfll epth, 0- yer returns P 0 ; 24-hours rinfll epth, 0-yers returns P 0 24, n 24-hours rinfll epth, 00-yers returns P 00, which escribe the geogrphicl vrition of rinfll. Bell evelope generlize DF reltionships for high intensity shorturtion rinfll. Bell estblishe two generl reltionships: P (5<<20 min) (9) P 60 P 0.2ln 0.52 (2 00 yers) (0) 0 P he DF reltionship prouce by frequency nlysis t ech recoring rin guge ws fitte to the following eqution suggeste by Bell (969) n Chen (983) my consier expressions of the type: f( ) f 2 ( ) () ' ' where is the return perio (yer), the rinfll urtion; is constnt return perio (yer) s the bse vlue; constnt rinfll urtion s the bse vlue. is the rinfll intensity with yer return perio n minute rinfll urtion. ' ' is the rinfll intensity with bse yer return perio n bse minute rinfll urtion. f () is function of only return perio n ssume to be the rtio of ' to. Here the function oes not epen on the urtion. f 2 () is function of only urtion n ssume to be the rtio of is the rtio of to '. Here the function oes not epen on the return perio. Bell (969), Chen (983) n Koutsoyinnis et l. (998) propose the function of the return perio f () is the rtio of f ' to propose by : ' ) c ln (2) ' ( ' ' An f 2 () is the rtio of ' 2 ( ) ' ' ' t to t ' n is function of the rinfll urtion f (3) e ( b) After combining Eqution (), (2) n (3), the generlize formul of rinfll intensity frequency cn be written s ' ' ( c ln ) (4) e ( b) he eqution 4 is generlize formul of rinfll intensity frequency formul using bse on rinfll intensity with -min rinfll urtion, n -yer return perio.

5 4. Appliction Bse on the bove methoology, we present the rel prt of Vietnm. he Re River n hi Binh River systems in the North hve bsin re of 69,000 km 2. he Re River Delt re is 5,540 km 2. Annul rinfll strongly vries over the Re river re in rnge mm/yer. ble. he nnul mximum series for vrious rinfll urtions, i.e. 0 min, 20 min, 30 min, 45 min, h, 2h 24h, were tken from the Vietnm nstitute of Meteorology n Hyrology (VNMH). 4. Estblishment of DF curves n comprison equtions Frequency nlysis techniques re use to evelop the reltionship between the rinfll intensity, storm urtion, n return perios from rinfll t. Anlysis of istribution for rinfll frequency is bse on the Person ype istribution, which is commonly use in Vietnm for this kin of nlysis. he Person ype istribution is written s: x x0 x x0 f ( x) exp (5) Fig. 2 Loction of Re River Delt. A 30 yers recor (from 956 to 985) of the seven sttions: Hnoi (Lng), Bcging, Hiuong, Nminh, Ninhbinh, hibinh, Vnly locte t the Re River Delt in Vietnm (Figure 2) ws use. he length of recor for recoring rin guges is list in where x 0 is the loction prmeter, is the scle prmeter, is the shpe prmeter. he Person ype probbility moel is use to clculte the rinfll intensity t ifferent rinfll urtions n return perios to form the historicl DF curves for ech sttion. Figure 3 using this frequency istribution functions, the mximum rinfll intensity for consiere urtions n 2, 5, 0, 20, 50,00 n 200 yers return perios, hve been etermine. he results re shown in Figure 3b t Hnoi sttion. he reltionship between the mximum rinfll intensities n the urtions for every return perios re etermine by fitting empiricl functions. ble. List of recoring rin guges use in the nlysis. No Nme of Sttion Longitue (E) Ltitue (N) Elevtion No. of yer recor Hnoi (Lng) Bcging Hiuong Nminh Ninhbinh hibinh Vnly Source: Vienm nstitute of Meteorology n Hyrology (VNMH)

6 he DF curves for seven sttions were constructions by using equtions (2) to (5): lbot, Bernr, Kimijim n Shermn. Lest squre metho is pplie to etermine the prmeter of four empiricl DF equtions use to represent intensityurtion reltionships. he vlue of prmeter in the Rinfll DF equtions were chosen on the minimum of Root Men Squre Error (RMSE) between the DF reltionship prouce by the frequency nlysis n tht simulte by the DF equtions. he RMSE (men squre error) ws efine s RMSE m j n k k k * ij ij mn 2 (6) where m is the number of vrious rinfll urtions (m=4, from 0 minutes to 24 hours), n is the number of vrious return perios (n=8, from 2 yer to 200 yer return perio), k ij is the rinfll intensity erive by Person type istribution for j hour urtion, k rinfll intensity estimte by Eqution, for j hour urtion, k yer return perio t the i sttion. At the Hnoi sttion, the prmeters of four empiricl equtions were etermine. he DF curves for the Hnoi sttion ws constructe with the Kimijim eqution re shown in Figure 4. he prmeters re etermine, presente in ble 2. ble 2. he prmeters of Kimijim equtions s DF curves Return perios (yers) b e yer return perio t the sttion, n k*ij is the ntensity(mm/hr) Return perios (yers) ) b) Fig. 3. ) Distribution rinfll intensity (60 minutes) nlysis (Person ype ) t Hnoi sttion. b) Mximum rinfll intensities for ifferent time intervls n return perios obtine from the cumultive ensity function Person type.

7 ntensity(mm/hr) yer s 00yer s 50yer s 20yer s 0yer s 5yer s 3yer s ntensity(mm/hr) yer s 00yer s 50yer s 20yer s 0yer s 5yer s 3yer s 2yer s 2yer s Durtion (min) Durtion (min) ) b) ntensity(mm/hr) yer s 00yer s 50yer s 20yer s 0yer s 5yer s 3yer s ntensity(mm/hr) yer s 00yer s 50yer s 20yer s 0yer s 5yer s 3yer s 2yer s Durtion (min) 2yer s Durtion (min) c) ) Fig. 4 Rinfll ntensity Durtion Frequency (DF) curves t the Hnoi Sttion ) lbot eqution b)bernr eqution c) Kimijim Eqution ) Shermn Eqution. ble 3 Constnt prmeters with 4 empiricl equtions t the Hnoi with 00 yers return perio. Function b e RMSE R lbot Bernr Kimijim Shermn Comprison of the results for the four empiricl methos n seven sttions: ble 3 n Figure 5 shows tht Kimijim n Shermn equtions my fit well t the Hnoi sttion tht hs Root men squre error (RMSE) only 3.2 to 4.7 mm/hour n its reltive coefficient R is pproximte he results re tht the Kimijim n Shermn equtions re cceptble fit to the DF reltionship in Vietnm. he root men squre error with Shermn n Kimijim re less thn 5 mm/ hour. he empiricl DF equtions likes Kimijim n Shermn cn be use for monsoon re of Vietnm. wo equtions re cceptble fit to the DF reltionship in Vietnm. M en Squre E rror ( m m /hr) 7 6 lbot Bernr Kimijim Shermn Return Perios ( yers) Fig. 5 Comprison Root Men Squre Error with four equtions.

8 ) b) ntensity(mm/hr) i b e Durtion(min) Durtion (min) c) ) Fig. 6 Contour mp of prmeter of Kimijim eqution with 00 yers returns perio n DF curves t un - guge point. ) Prmeter contour mp b) Prmeter b contour mp c) Prmeter e contour mp n ) Rinfll DF curves t Hungyen (unguge loction) using prmeter contour mps. 4.2 Regionliztion the prmeter of rinfll intensity urtion frequency equtions After etermining the prmeters of DF formul such s prmeters, b n prmeter e, for the sme return perio, using Arc view/gs interpolting the prmeter contour mps, tht mp cn generte for the prmeters which cn then be use for unguge rinfll with return perios. he regionl DF formul prmeters re generte for unguge res to estimte rinfll intensity for vrious return perio n rinfll urtion. he metho propose in this stuy h resonble ppliction to unguge rinfll loction, which ws conclue from the verifiction of

9 itionl rin guges. he prmeters contours mp for Kimijim eqution crete, s shown in Figure 6. Rinfll intensity urtion frequency t Hung yen (unguge loction) cn etermine. Prmeters set: =9500, b=20 n e=0.83. he DF curve t Hung yen cn be follow eqution for 00 yer return perio: / '00 yers for vrious urtions n return perios re given in ble 4. he rtios show little vrition with urtion, n re function of perio. ble 4 Averge reltionship between rinfll b e i (7) intensity n urtion (Rtio of sme urtion t Hnoi sttion. 00 yers / t ' ) t he rinfll DF curves for Hung yen by using tht mp estblishe t figure 6. with 00-yer return. Yer Return Rtios f() Generlize rinfll intensity urtion frequency formul t Hnoi sttion () ntensity-frequency Rtios he function f () is the rtio of ' to ' n is function of the return perio (eqution ). he Hnoi sttion is use to illustrte how to efine the generlize DF formul. For this exmple: =00 yers s the bse return perio. he rtio of he prmeter is slope vlue of liner regression reltionship between the log-trnsforme vlues of return perios () n the rtios of rinfll intensity: f ( ) log (8) 00 yr he prmeter =0.36 n c=0.272 with correltion coefficient vlue r=0.99. ble 5. he Rtio of (,) to (,h) t Hnoi Sttion. Return perio(yers) Durtion (min) 200yr 00yr 50yr 20yr 0yr 5yr 3yr 2yr Ave vlue StD Coeff. vrition

10 (2) ntensity-durtion Rtios he intensity-urtion rtios (or epth-urtion rtios) re clculte for ech vilble t. he clcultions re me in orer to obtin the verge vlue of the rtios ech consiere urtions. ble 5 shows rtios 60-minute rinfll intensity n urtion (Rtio of / t t' 60' ) for sme return perio. he rtios show little vrition with return perios, n re function of the rinfll urtion. he rtio f2 ws fitte by Shermn eqution: f 2 ( ) (9) ( 76.3) ' ' ' ' he prmeter =88.93, b=76.3 n e=0.945 with correltion coefficient vlue r=0.99 n RMSE=3.56 (mm/hr). Combining eqution (8), (9) the generlize ntensity Durtion Frequency formul t Hnoi (Hnoi), with rinfll intensity in 60 minutes n 00 yers return is mm/hr, gives: ( ln ) ( 76.3) 0.94 Generlize rinfll intensity urtion frequency formul t Hnoi sttion s: ln 0.94 ( 76.3) (20) he rinfll intensity cn clculte from (20) eqution for ny urtion () n return perios () t Hnoi sttion. 5. Conclusions his stuy hs been conucte to the formultion n construction of DF curves using t form recoring sttion by using empiricl equtions, four empiricl functions use to represent ntensity- Durtion-Frequency reltionship for Re River Delt (Vietnm). n generl, the 3 prmeters functions (Kimijim n Shermn) showe cceptble fitting to the rinfll intensity qurtiles. he regionliztion of the prmeters of rinfll intensity-urtion-frequency equtions were generte for unguge res to estimte rinfll intensity for vrious return perio n rinfll urtion. he prmeter contour mps were me to estimte unguge rinfll with return perios. More specificlly, this reserch is to generlize DF formul using some bse rinfll epth n bse return perio. n fct, DF curves give the rinfll intensity t point. Storm sptil chrcteristics re importnt for lrger ctchments. ntensity-durtion-are- Frequency curve (DAF) is stuie for the evlution of esign storms using scling pproch. References Bell, F.C. (969). Generlize rinfll urtion frequency reltionships. Journl of Hyrulic Div., ASCE, 95(), Chen, C.L. (983). Rinfll intensity-urtion - frequency formuls, Journl of Hyrulic Engineering, ASCE, 09(2), Chow, V.. (964). Hnbook of Applie Hyrology, McGrw-Hill, New York, Chow, V.., Miment, D.R. & Mys, L.W. (988). Applie Hyrology, McGrw-Hill. Dvi M. Hershfie (96). Estimting the Probble Mximum Precipittion, Journl of the Hyrulic Division, Proceeing of the ASCE, HY5, 99-6 Kothyri, U.C. n Gre, R.J. (992). Rinfll intensity urtion frequency formul for ni, J. Hyr. Engrg., ASCE, 8(2), Koutsoyinnis, D., Mnets, A. (998). A mthemticl frmework for stuying rinfll intensity-urtion-frequency reltionships, Journl of Hyrology, 206, Nottion he following symbols re use in this pper: = rinfll intensity for t-min urtion n -yer return perios; 60 = rinfll intensity for 60-min urtion n -yer return perios; 00 = rinfll intensity for t-min urtion n 00-yer return perios; i = rinfll intensity (mm/hr) P = rinfll epth (mm) = return perio (yers); n = rinfll urtion (min).

11 rinfll ntensity-durtion-frequency (DF) reltionship DFBernr(932) 7DFDF Re River Delt (RRD) DF