ISSN: (Print) (Online) Journal homepage:

Transcription

1 Hydrologcal Scences Journal ISSN: (Prnt) (Onlne) Journal homepage: Dervaton of varable IUH correspondng to tme-varyng ranfall ntensty durng storms / Dérvaton d'un HUI varable correspondant à l'évoluton temporelle de l'ntensté pluvométrque durant les averses KWAN TUN LEE, NAI-CHIN CHEN & YI-RU CHUNG To cte ths artcle: KWAN TUN LEE, NAI-CHIN CHEN & YI-RU CHUNG (28) Dervaton of varable IUH correspondng to tme-varyng ranfall ntensty durng storms / Dérvaton d'un HUI varable correspondant à l'évoluton temporelle de l'ntensté pluvométrque durant les averses, Hydrologcal Scences Journal, 53:2, , DOI:.623/hysj To lnk to ths artcle: Publshed onlne: 8 Jan 2. Submt your artcle to ths journal Artcle vews: 346 Vew related artcles Ctng artcles: Vew ctng artcles Full Terms & Condtons of access and use can be found at Download by: [ ] Date: 8 November 27, At: 7:57

2 Hydrologcal Scences Journal des Scences Hydrologques, 53(2) Aprl Dervaton of varable IUH correspondng to tme-varyng ranfall ntensty durng storms KWAN TUN LEE, NAI-CHIN CHEN & YI-RU CHUNG Department of Rver and Harbor Engneerng, Natonal Tawan Ocean Unversty, Keelung, Tawan 22, R.O.C. Downloaded by [ ] at 7:57 8 November 27 Abstract The nstantaneous unt hydrograph (IUH) of a watershed s the result of one nstantaneous unt of ranfall excess dstrbuted unformly over the watershed. Although the geomorphologcal characterstcs of the basn reman relatvely constant, the varable characterstcs of storms cause varatons n the shape of the resultng hydrographs. It s, therefore, nadequate to use one typcal IUH to represent the hydrologcal response generated from any specfc storm. In ths study, a varable IUH was derved that drectly reflects the tme-varyng ranfall ntensty durng storms. The ranfall ntensty used to generate the varable IUH at tme t s the mean ranfall ntensty occurrng from the tme t T c to t n whch T c s the watershed tme of concentraton. Hydrologcal records from three watersheds n Tawan were used to demonstrate the applcablty of the proposed model. The results show that better smulatons can be obtaned by usng the proposed model than by usng the conventonal unt hydrograph method, especally for concentrated ranstorm cases. Key words varable IUH; temporal ranfall ntensty; runoff modellng; runoff travel tme Dérvaton d un HUI varable correspondant à l évoluton temporelle de l ntensté pluvométrque durant les averses Résumé L hydrogramme untare nstantané (HUI) d un bassn versant et la résultante d une unté nstantanée de plue nette dstrbuée unformément dans l espace du bassn versant. Ben que les caractérstques géomorphologques du bassn versant restent relatvement constantes, les caractérstques varables des averses génèrent des varatons dans la forme des hydrogrammes résultants. Il est par conséquent nadéquat d utlser un HUI typque pour représenter la réponse hydrologque générée par n mporte quelle averse partculère. Dans cette étude, un HUI varable a été dérvé, qu reflète drectement l ntensté pluvométrque évolutve au cours du temps durant les averses. L ntensté pluvométrque utlsée pour générer l HUI varable au temps t est l ntensté pluvométrque moyenne entre les temps t T c et t, où T c est le temps de concentraton du bassn versant. Les enregstrements hydrologques de tros bassns versants de Tawan ont été utlsés pour démontrer l applcablté du modèle proposé. Les résultats montrent que de melleures smulatons peuvent être obtenues en utlsant le modèle proposé plutôt que la méthode conventonnelle de l hydrogramme untare, en partculer pour les averses concentrées. Mots clefs HUI varable; ntensté pluvométrque temporelle; modélsaton de l écoulement; temps de parcours de l écoulement INTRODUCTION The nature of streamflow n a regon s related to the temporal and spatal dstrbuton of the ranfall, watershed geomorphology, vegetaton and evaporatve demand. The geomorphologcal characterstcs are the channel network and the surroundng landscapes that transform the ranfall nput nto an output hydrograph at the outlet of the watershed. A smple approach to ranfall runoff modellng was proposed by Sherman (932) as the unt graph method (now called the unt hydrograph, UH). The unque hydrologcal response functon for a watershed s a practcal hypothess that has been used n many scentfc and engneerng felds. The conventonal UH method s based on an assumpton that the UH s not storm dependent. It mples that the hydrodynamc characterstcs n the converson of ranfall to runoff reman constant for dfferent szes of storms. Several researchers have questoned the assumpton of the tme-nvarant relatonshp between ranfall and runoff. Through examnng watershed hydrologcal records, Mnshall (96) showed that the UH s a functon of ranstorm characterstcs and, hence, that there should be a set of UHs for a watershed nstead of only one. Mandevlle & O Donnell (973) derved varable response functons usng the lnear channel and lnear reservor conceptual models. By usng a generalzed Muskngum model, Dng (974) derved a twoparameter varable IUH showng that the hydrologcal response functon of a watershed was Open for dscusson untl October 28 Copyrght 28 IAHS Press

3 324 Kwan Tun Lee et al. Downloaded by [ ] at 7:57 8 November 27 correlated wth effectve ranfall ntensty. Chen & Sngh (986) proposed an alternatve procedure for the dervaton of the varable IUH form proposed by Dng (974) to yeld a second-order nonlnear kernel. Ahsan & O Connor (994) developed a smple nonlnear model to ncorporate a varable gan factor dependng on the prevalng wetness ndex of the watershed. By usng a tme area concept, Saghafan et al. (22) derved tme-varable sochrones to respond to temporal changes n the ranfall ntensty. Rodrguez-Iturbe & Valdes (979) proposed the geomorphologcal nstantaneous unt hydrograph model (GIUH). In the GIUH approach, the IUH of a watershed can be derved by only usng the nformaton obtanable from a topographc map or from a dgtal elevaton data set. The proposed GIUH model s a tme-varyng IUH of a basn that depends on the runoff velocty. Instead of arbtrarly assgnng a runoff velocty or adoptng an emprcal equaton for travel tme estmaton n the GIUH model, Lee & Yen (997) estmated the tme of concentraton based on a knematc-wave approxmaton by gnorng the nertal and frcton forces to obtan a knematc wave-based GIUH model (KW-GIUH). Consequently, the resultant IUH has been explctly expressed as a functon of the ranfall ntensty. In applyng the KW-GIUH model, temporal ranfall ntensty was used to generate the tmevaryng IUH, and then the IUH was multpled by the ranfall depth to obtan a component hydrograph for superposton to produce the total hydrograph. Recently, the present authors found that although the KW-GIUH model obtaned good smulaton results n most storm events, t nevertheless resulted n an overestmaton of the peak dscharge, especally for concentrated ranstorm cases. The objectve of ths study s to develop a tme-varyng IUH by ntroducng a tme delay concept for selecton of the representatve ranfall ntensty. Conventonal unt hydrographs, whch were derved usng hydrologcal records, were used to demonstrate the necessty of havng a varable UH/IUH model. Hydrograph smulatons usng the prevous KW- GIUH and the proposed varable IUH methods were compared n detal. Furthermore, the assumptons of the th-order channel wdth and roughness coeffcent n the orgnal KW-GIUH model (Lee & Yen, 997; Yen & Lee, 997) were also revsed. VARIABLE IUH MODEL The hydrograph of outflow from a watershed s the sum of the component hydrographs from all the sub-areas of the watershed modfed by the effect of transt tme through the overland flow planes, stream network and storage n the channels. Snce the geomorphologcal characterstcs of the watershed, such as shape, sze and slope, are constant, one may expect consderable smlarty n the shape of hydrographs from storms of smlar ranfall characterstcs, whch s the essental concept proposed by Sherman (932). If the watershed ranfall runoff process can be approxmated by a lnear system concept, the rate of drect runoff at tme t s gven by: t Q( t) = ( τ) u( t τ) dτ () e where e (τ) s the effectve ranfall ntensty over the block of ranfall at tme τ; u(t τ) s the ordnate of the nstantaneous unt hydrograph (IUH) at tme t τ, and τ s the dummy tme varable of ntegraton. Nevertheless, t would be wrong to mply that one typcal hydrograph would suffce for any storm because varable characterstcs of the storms cause varatons n the shape of the resultng hydrographs. Instead of usng u(t τ) to represent the IUH n equaton (), the representaton of IUH should be modfed to u( e, t τ) to nclude the nfluence of ranstorm characterstcs. Dscharge observed at the watershed outlet at tme t results from the ranfall that occurred between the tme t and a short perod before t representng tme of concentraton (T c ). Therefore, the ranfall ntensty requred to generate the tme-varyng IUH at tme t can be approxmated as mean of the ranfall ntensty between tme t T c and t as: Copyrght 28 IAHS Press

4 Dervaton of varable IUH correspondng to tme-varyng ranfall ntensty durng storms 325 () t = ( ) d = [ () t + ( t Δ t) + ( t 2 Δ t) + L + ( t nδt) ] (2) t e - e τ τ tt e e e e T c c n where () e t s the mean ranfall ntensty between t T c and t; Δ t s the tme nterval of the measured ranfall record where the ranfall ntensty s averaged over each nterval, and n = T c /Δt. Schematc descrpton of equaton (2) can be shown as n Fg.. Consequently, nstead of usng equaton () for runoff estmaton, the dscharge at the watershed outlet at tme t s gven by: t Q( t) = ( τ) u(, t τ) dτ (3) e e where e (τ) s the ranfall excess at tme τ; and u ( e, t ) s the ordnate of the nstantaneous unt hydrograph correspondng to a mean ranfall ntensty of e () t, whch s calculated by usng equaton (2). e (6) Downloaded by [ ] at 7:57 8 November 27 e T c = 3 hr t Fg. Transformed hyetograph used to generate the varable IUH. Basc structure of the varable IUH model e e(6) = e(4) + e(5) + e(6) 3 [ ] t The IUH of a watershed results from one nstantaneous unt of effectve ranfall unformly dstrbuted over the watershed. Based on Strahler s orderng scheme, a watershed of order Ω can be dvded nto dfferent states, such as x o for the th-order overland flow regons, and x for the th-order channels, n whch =, 2,..., Ω. The IUH of the watershed can be expressed as (Rodrguez-Iturbe & Valdes, 979): u() t = fx () t f () () () ( ) o x t f x t f j x t L P w (4) Ω w W where W s the flow path space gven as W = x o, x, x j,..., x Ω ; f x o () t s the travel-tme probablty densty functon n state x o wth a mean value of T ; f () t s the travel-tme probablty densty functon n state x j wth a mean value of T x ; denotes a convoluton ntegral; j P(w) s the probablty of a drop of effectve ranfall adoptng path w; and the relatonshp of the and j means the subsequent state of the flow from an th-order channel to a jth-order channel. The travel-tme probablty densty functon, both for the overland-flow and channel-flow states, s assumed to follow an exponental dstrbuton as: t fx () t = k exp( ) T T (5) xk xk where T x k s the mean travel tme n state x k. In the Rodrguez-Iturbe & Valdes (979) approach, the stream network structure was embedded nto equaton (4). The hydrodynamc effect was smulated by assgnng dfferent mean w xo x j Copyrght 28 IAHS Press

5 326 Kwan Tun Lee et al. flow veloctes nto the runoff travel tme estmaton, thereby mplctly reflectng the storm sze over the watershed. By usng the knematc-wave approxmaton, Lee & Yen (997) derved travel tme equatons for dfferent orders of overland areas and streams. The ranfall excess ntensty at tme t was substtuted nto the travel tme equatons to generate the component IUH at tme t. Lnear superposton was then appled to combne the component hydrographs of temporal effectve ranfall ntensty to produce a complete drect runoff hydrograph. In consderng the tme delay to respond to the effectve ranfall to the watershed outlet, the quantty of the ranfall ntensty used to generate the tme-varyng IUH at tme t s the mean value of the ranfall ntensty between tme t T c and t, whch has been shown n equaton (2). Consequently, the runoff travel tme for the th-order overland-flow path s (Henderson & Woodng, 964): Downloaded by [ ] at 7:57 8 November 27 T xo nl o o = 2 m So e m where e s the mean effectve ranfall ntensty calculated usng equaton (2); S o s the mean thorder overland-flow slope; n o s the roughness coeffcent for the overland planes; and m s a constant whch can be recognzed as 5/3 from Mannng s equaton; o (6) L s the mean th-order overland-flow length. The mean length of the th-order V-shaped overland-flow planes s (Lee & Yen, 997): L o APOA = (7) 2N Lc where A s the total area of the watershed; P OA s the rato of the th-order overland area to the total watershed area; N s the number of the th-order channels; and L c s the mean channel length of the th-order subbasns. As the flow n the th-order overland planes gradually ncreases wth tme to reach equlbrum, an addtonal tme T s needed for the th-order channel to reach ts x equlbrum. If the overland plane equlbrum state s reached, the lateral overland dscharge contrbutng to the channel can be assumed as 2e L o n order to smplfy the analytcal soluton of the travel tme equaton. Consequently, the source term for the channel flow contnuty equaton s the excess ranfall on the overland flow plane. The travel tme for the th-order channel s (Lee & Yen, 997): m B 2 o c n e c L L m Tx = h co h 2 co 2L + e o Sc B where L c s the mean channel length of the th-order subbasn; B s the th-order channel wdth; S c s the mean th-order channel slope; nc s the roughness coeffcent for the th-order channel flow; and h co s the nflow depth of the th-order channel due to water transported from upstream reaches. Equatons (6) and (8) are bascally the same as those n the prevous KW-GIUH model derved by Lee & Yen (997), except that the temporal ranfall ntensty e n the orgnal dervaton of the equatons s replaced by the mean ranfall ntensty, e, between t T c and t as shown n equaton (2). In applyng equatons (6) and (8), the mean ranfall ntensty, e, s obtaned by averagng the ranfall quanttes between t T c and t. The watershed tme of concentraton can be estmated (8) Copyrght 28 IAHS Press

6 Dervaton of varable IUH correspondng to tme-varyng ranfall ntensty durng storms 327 Downloaded by [ ] at 7:57 8 November 27 by combnng equatons (6) and (8) as (Lee et al., 26): Ω T T T = + c oc cc = = + + S S B m Ω m nl 2 o c o o B n e c L L m 2 hco h 2 co m o 2L (9) = o e e c where T c s the watershed tme of concentraton; s the mean runoff travel tme on the frstorder overland planes; and Tcc Toc s the mean runoff travel tme n the th-order channels. Snce the above equaton contans the e factor, a successve substtuton method s performed to obtan the T c value. Consequently, the IUH expressed n equaton (4) can then be modfed as: ( e, ) = x () x () x () x () ( ) L o j Ω () w u t f t f t f t f t P w W where u ( e, t) shows the IUH of a watershed beng functons of e and t. Hence, the hydrodynamc effect shown on the flow hydrograph has been explctly presented usng the average of the temporal ranfall ntensty over the perod T c. Channel geometry and channel bed resstance As shown n equatons (6) and (8), although a number of geomorphc factors are requred to estmate the runoff travel tmes on overland areas and n channels, the travel tmes can be obtaned from topographc maps or by the applcaton of a dgtal elevaton model (DEM). The only excepton s the nformaton for channel wdth and roughness coeffcent, whch can only be collected n feld nvestgatons. To smplfy the feld nvestgaton work, Lee & Yen (997) proposed a lnear varaton for channel wdth for dfferent orders of streams, and assumed the channel roughness coeffcent to be a constant value for all the streams. If the classcal study for natural streams reported by Leopold & Maddock (953) s followed, the channel wdth s correlated to the stream dscharge wth an exponent of.5, and the peak dscharge of a watershed s proportonal to the watershed sze (Leopold et al., 964; Goodrch et al., 997). Consequently, the relatonshp between channel wdth and watershed area s gven by:.5 A B = BΩ () A where Ω s the order of the watershed stream network; B Ω s the channel wdth at the watershed outlet; A s the mean of the dranage area of order ; and A s the watershed area. Snce the exponent of.5 n equaton () has been verfed by Chen et al. (996) usng the data from 62 watersheds n Tawan, equaton () s consdered acceptable for channel wdth estmaton. Thus, the channel wdth at the watershed outlet s the only geometrc feature that needs to be measured n the feld nvestgaton. In general, the sze reducton of the bedload grans n natural rvers accompanes decreasng gradents from upstream to downstream (Mangelsdorf et al., 99). As the gradent decreases, energy losses dmnsh due to the reducton n wake turbulence and the formaton of localzed hydraulc jumps downstream from boulders (Jarrett, 992). By usng emprcal data collected n mountan rvers, Jarrett (987) related the channel roughness to the channel-bed slope and hydraulc radus as:.38.6 n =.32S R (2) c where S s the energy gradent, and R s the hydraulc radus n m. If the energy slope can be approxmated by the channel-bed slope and the nfluence of the hydraulc radus can be neglected because of ts small exponent, then the varablty of the channel roughness coeffcents for dfferent order of streams can be expressed as: w Copyrght 28 IAHS Press

7 328 Kwan Tun Lee et al. where.38 S c n = c n c (3) Ω ScΩ nc s the roughness coeffcent of the th-order channel; nc Ω and Sc Ω are the channel roughness coeffcent and slope at the watershed outlet, respectvely. Consequently, nstead of adoptng a constant channel roughness coeffcent for the entre channel network, a spatal varaton of the roughness coeffcent accordng to stream order s assumed and then appled n the present study. Downloaded by [ ] at 7:57 8 November 27 CONVENTIONAL UNIT HYDROGRAPH METHOD For comparson purpose, conventonal one-hour UHs for dfferent storms were derved. The convoluton of the effectve ranfalls and the conventonal UH can be expressed as a matrx form as follows: IU e where: I = Q (4) [( + ) ] e m n n 2 M M 2 M m M L = m M 2 M M M m U [ n ] u u 2 = M un Q [( m+ n ) ] q q 2 = M qm+ n n whch I e s the effectve ranfall matrx wth (m + n ) rows and n columns; U s the UH matrx wth n rows and one column; Q s the drect runoff matrx wth (m + n ) rows and one column. Snce the hydrologcal records of I e and Q may suffer from nstrumental and observatonal errors, and the lnearty assumpton adopted for the ranstorm runoff relatonshp, an exact soluton for U does not exst. By usng a lnear programmng method, we can derve the UH by renderng: mn W [Q R I e U] (6) where Q R s the recorded drect runoff hydrograph, and W s the weghtng factor expressed as: qk Wk = (7) m+ n q = n whch W k s the weghtng factor at tme k, and q k s the recorded drect runoff dscharge at tme k. The objectve functon shown n equaton (6) expresses the mnmum of the dfferences between the recorded and model estmated drect runoff hydrograph, whch dffers from the sum of square error. Sngh (976) appled the lnear programmng method to derve the one-hour UH, but dd not nclude the weghtng factor shown n equaton (7). We ntroduced the weghtng factor so that the derved UHs would have better runoff smulatons, especally for concentrated storm events n Tawan. To evaluate the sutablty of the proposed varable IUH model and the conventonal UH model for the basn of nterest, three crtera were chosen to analyse the degree of goodness of ft. These crtera can be defned as follows: (5) Copyrght 28 IAHS Press

8 Dervaton of varable IUH correspondng to tme-varyng ranfall ntensty durng storms 329 Downloaded by [ ] at 7:57 8 November 27 (a) The coeffcent of effcency (Nash & Sutclffe, 97) s defned as: n [ Qrec ( t) Qsm ( t) ] = CE = (8) 2 n [ Qrec ( t) Q rec ] = 2 where Q rec (t) s the recorded dscharge at tme t, Q sm (t) s the smulated dscharge at tme t, Q rec s the average recorded dscharge durng the storm event, and n s the number of dscharge records durng the storm event. The better the ft, the closer CE s to unty. (b) The error of peak dscharge s defned as: ( Q p ) sm ( Q p ) rec EQ p (%) = (9) ( Q ) p rec where (Q p ) sm s the peak dscharge of the smulated hydrograph, and (Q p ) rec s the recorded peak dscharge. (c) The error of the tme to peak dscharge s defned smply as: ET = ( T ) ( T (2) p p sm p ) rec where (T p ) sm s the smulated tme to peak dscharge, and (T p ) rec s the recorded tme to peak dscharge. MODEL APPLICATION Descrpton of study watersheds To demonstrate the capablty of the proposed varable IUH model for runoff smulaton, hydrologcal records from three watersheds, namely, P-An-Chao (PAC), Po-Chao (PC), and San-Hsa (SH) n the mountan regon of Tawan (as shown n Fg. 2) were used to perform the storm hydrograph smulatons. Most of the mountan regons n Tawan are composed of sedmentary and metamorphc rocks, whch are fragle and hghly weathered. The mean annual ranfall n Tawan reaches 25 mm, whch s 2.5 tmes the world s average. The PAC watershed s located n central Tawan, and the PC and SH watersheds are located n northern Tawan. The watersheds are upland forest watersheds wth heavy brush cover; the channels are flled wth large boulders and have brush banks. The stream networks for these three watersheds are of 4th order. The szes of the PAC, PC and SH watersheds are.73,.85 and km 2, respectvely. Table shows the detals of the geomorphologcal factors, whch were obtaned by applyng a DEM developed by the frst author (Lee, 998) usng a 4-m resoluton raster elevaton data set, and equatons (7), (), and (3) were used to calculate the mean overland-flow lengths, the channel wdths, and the channel roughness coeffcents for dfferent orders of streams. Hydrologcal records show that hgh ranfall ntensty, due to typhoons and thunderstorms, occurs manly between May and October n Tawan. The rato of the baseflow to the total runoff of the storms s comparatvely small. The major component of the dscharge results from rapd drect runoff, whch results not only from the hgh ranfall ntensty, but also the steep slope of the watershed. In performng the smulatons, the effectve ranfall hyetograph was determned by deductng the abstractons from the ranfall usng the Horton nfltraton equaton (939). More than sx storm events from each watershed were chosen to smulate drect runoff hydrographs wth the recorded data usng the proposed varable IUH model and the conventonal UH model. Copyrght 28 IAHS Press

9 33 Kwan Tun Lee et al. Po-Chao Shh-Tng San-Hsa Ta-Po Downloaded by [ ] at 7:57 8 November 27 5 km Fg. 2 Locaton map of the study watersheds. P-An-Chao Ta-Hu Flow gaugng staton Ran gaugng staton Applcaton of the conventonal UH and the varable IUH model 6 km The derved conventonal one-hour UHs for dfferent szes of storms from the study watersheds are presented n Fg. 3. In ths fgure, the numbers n parentheses are the mean ranfall ntensty wthn a tme nterval of T c durng the heavest ranfall perod of the ranstorm, whch s denoted as ( e) n Table 2 to reflect the storm sze. Fgure 3 shows that hgher peaks and shorter tmes-topeak are observed n the one-hour UHs derved from the large-storm records. In contrast, the UH p wth a smaller peak and a longer tme-to-peak s generated from the mld-storm records. The tmeto-peak dscharge for a mld storm does not show an obvous lag, except n the case of the PAC watershed, because hourly ranfall and flow data were used n these analyses. Snce the conventonal UH dervaton dd not provde a drect lnkage between the UH and storm sze, t does not facltate the development of a set of UHs to ncorporate wth the hydrodynamc effect due to dfferent storm szes. As mentoned earler, the proposed varable IUH model produces a set of IUHs dependng on the gven tme-varyng ranfall ntensty nstead of just a sngle IUH for a watershed. Examples of the varable IUHs for the three watersheds are shown n Fg. 4, n whch an exponental dstrbuton was appled for the travel tme dstrbuton for overland-flow areas and channels. The overland-flow roughness coeffcent was set equal to.6; the channel-flow roughness coeffcent at the watershed outlets was equal to.6, and the ranfall ntenstes of the IUHs were set equal to, 2 and 3 mm/h, respectvely. The peak of the IUH s seen to ncrease wth ncreasng e, whereas the tme to peak decreases wth ncreasng e. Fgure 4 shows the sgnfcant nfluence of the effectve ranfall ntensty on the IUHs. Ths varaton of the IUHs shows a smlar tendency to that of the conventonal UHs shown n Fg. 3, whch were derved from usng dfferent szes of Copyrght 28 IAHS Press

10 Dervaton of varable IUH correspondng to tme-varyng ranfall ntensty durng storms 33 Downloaded by [ ] at 7:57 8 November 27 Table Geomorphologcal factors of the study watersheds. Factor / Watershed Order PAC PC SH Channel number N Watershed area A (km 2 ) Rato P OA Channel length L (km) c Overland length L (km) o Channel slope S (m/m) c Overland slope S (m/m) o Channel wdth B (m) Channel coeffcent n c Number of th-order channels contrbutng to jth-order channels N,j N, N, N, N 2,3 8 8 N 2,4 5 8 N 3, storm records. The varaton of the IUH drectly lnked to ranfall ntensty s consdered the major mert of the proposed varable IUH model. In practcal applcatons for runoff smulaton usng the proposed varable IUH model, temporally non-unform ranfall ntensty s used to generate the tme-varyng IUH. Then the ranfall depth s multpled by the IUH to obtan the component hydrograph. Consequently, the sum of the component hydrographs generated at dfferent tme steps s the complete drect runoff hydrograph. To demonstrate the ablty of the varable IUH model to produce runoff hydrographs of gven ran events on watersheds, and to show the valdty of the model through comparson wth observed data, ranstorm records on the study watersheds were studed. Copyrght 28 IAHS Press

11 332 Kwan Tun Lee et al PAC 25/6/987 (7.3 mm/h) 7/8/99 (9.86 mm/h) 2/3/997 (3.66 mm/h) Downloaded by [ ] at 7:57 8 November Tme (h) PC 8/8/99 (22.34 mm/h) 22/8/2 (4.25 mm/h) 27/7/987 (.77 mm/h) 2 3 Tme (h) SH 9/7/98 (2.43 mm/h) 7/8/984 (2.2 mm/h) 27/7/987 (5. mm/h) Tme (h) Fg. 3 One-hour UHs for dfferent storm szes from the study watersheds. Copyrght 28 IAHS Press

12 Dervaton of varable IUH correspondng to tme-varyng ranfall ntensty durng storms 333 Table 2 Smulated results of storm events n the study watersheds. Downloaded by [ ] at 7:57 8 November 27 IUH (h - ) Ranfall duraton (h) Storm records One-hour UH model Varable IUH model Total ( ) Qp e p ranfall (m 3 Tp CE EQp ETp CE EQp ETp /s) (h) (%) (h) (%) (h) depth (mm/h) (mm) PAC 22/8/ /6/ /8/ // /5/ /3/ PC 27/7/ /9/ /8/ /6/ /8/ /8/ SH 9/7/ /8/ /7/ /8/ // /8/ Note: ( ) s the mean ranfall ntensty wthn a tme nterval of T c durng the heavest ranfall perod of the ranstorm e p PAC e (mm/h) IUH (h - ) PC e (mm/h) 3 2 IUH (h - ).2.5. SH e (mm/h) Tme (h) Fg. 4 Examples of the varable IUHs for the study watersheds. Copyrght 28 IAHS Press

13 334 Kwan Tun Lee et al. RESULTS AND DISCUSSION To demonstrate the nherent hydrodynamc varablty n the study watersheds, hydrologcal records from three study watersheds were used to show the adaptablty of the varable IUH for varous szes of storms. Conventonal one-hour UHs for dfferent storms were also developed for comparson purposes. Comparsons of the conventonal UH and the varable IUH model Table 2 shows the smulated results generated by usng the conventonal UH and the proposed varable IUH model for storm events n the study watersheds. In these smulatons, tme-varyng IUHs were used n applyng the varable IUH model. A representatve UH for each watershed, Downloaded by [ ] at 7:57 8 November 27 e (mm) e (mm) (a) (b) e (mm) PAC 25 June 987 Varable IUH One-hour UH PC 8 September 987 Varable IUH One-hour UH SH 29 August 99 Varable IUH One-hour UH e (mm) e (mm) e (mm) PAC 7 August 99 Varable IUH One-hour UH SH 3 9 October Varable IUH 4 One-hour UH 3 2 PC 8 August 99 Varable IUH One-hour UH (c) Tme (h) Tme (h) Fg. 5 Hydrographs smulaton usng varable IUH model and one-hour UH model n: (a) the PAC watershed; (b) the PC watershed; and (c) the SH watershed. Copyrght 28 IAHS Press

14 Dervaton of varable IUH correspondng to tme-varyng ranfall ntensty durng storms 335 whch was obtaned by averagng the UHs derved from dfferent storm events, was used n applyng the conventonal UH model. The smulated results show that the values of the effcency coeffcent and error of the tme to peak dscharge are wthn the same range for these two models, but the peak dscharge error for the varable IUH model s generally lower than that for the conventonal UH model. Moreover, snce the representatve UH s derved by averagng the UHs obtaned from dfferent storm events, sgnfcant error of the peak dscharge for performng the conventonal UH model s found, especally for the smallest or largest storm events. For the sake of brevty, only two storm hydrographs from each watershed are gven here. As shown n Fg. 5, the recorded and smulated hydrographs generated by usng the varable IUH model are n good agreement for the storm events, especally n smulatng the crest segment of the hydrographs. Downloaded by [ ] at 7:57 8 November 27 e (mm) (a) e (mm) (b) e (mm) PAC 9 October 994 Method Method PC 8 August 99 Method Method 2 SH 7 August 997 Method Method 2 e (mm) e (mm) e (mm) PAC 2 March 997 Method Method PC 27 July 987 Method Method 2 SH 27 July 987 Method Method (c) Tme (h) Tme (h) Fg. 6 Hydrographs smulaton usng dfferent temporal ranfall ntenstes as nput n: (a) the PAC watershed; (b) the PC watershed; and (c) the SH watershed (Method denotes the prevous approach; Method 2 denotes the proposed approach). Copyrght 28 IAHS Press

15 336 Kwan Tun Lee et al. Downloaded by [ ] at 7:57 8 November 27 Hydrograph varaton accordng to the nput of dfferent tme-varyng ranfall ntenstes Prevous researchers (Dng, 974; Chen & Sngh, 986; Lee & Yen, 997; Yen & Lee, 997) adopted temporal ranfall ntensty e (t) to generate the varable IUH at tme t. In ths study, the quantty of the temporal ranfall ntensty used to generate the tme-varyng IUH at tme t s the mean ranfall ntensty between tme t T c and t, as shown n equaton (2) and Fg.. Fgure 6, shows a comparson of the hydrographs generated usng the temporal ranfall ntensty and the method proposed n ths study. In ths fgure, Method denotes Lee & Yen s (997) approach usng the storm temporal ranfall ntensty as nput, and Method 2 denotes the proposed approach usng the mean ranfall ntensty between tme t T c and t. As shown n the left-hand panels of Fg. 6, smlar smulaton results can be found by usng both methods. The estmaton error for the peak dscharge s only about 7% for the PAC watershed, 6% for the PC watershed, and 3% for the SH watershed. The maxmum hourly ranfall ntenstes n the study storm events are 5, 3 and 8 mm/h for the PAC, PC and SH watersheds, respectvely. As shown n the rght-hand panels of Fg. 6, three other storms were consdered n the study watersheds to confrm the capablty of usng methods and 2 for runoff smulaton. In these smulatons, the smulated peak dscharges of the storms are found to be always overestmated by usng Method and the smulated hydrographs are shfted to the left compared to the recorded hydrographs, but good smulaton results can be obtaned by usng Method 2. Comparson of the ranfall hyetographs of the storms reveal that the ranfall hyetographs shown n the left-hand panels of Fg. 6 are more unformly dstrbuted than those n the rght-hand panels, whch are concentrated storm events. For unformly dstrbuted ranstorms, the dstrbuton of the transformed hyetograph generated by usng equaton (2) s smlar to that of the orgnal hyetograph, so the smulated hydrographs look smlar for the applcaton of both methods (as shown n Fg. 6 (left)). If Method s appled for concentrated ranstorms as shown n Fg. 6 (rght), hgh ntensty of the ranfall wll result n an IUH wth a very hgh peak, and consequently an overestmate of the dscharge. Nevertheless, f equaton (2) s appled, the hgh ntensty ranfall s replaced by adoptng the mean ranfall ntensty over the perod of T c. The smulated hydrographs are then close to the measured ones even for the concentrated ranstorms shown n Fg. 6 (rght). Ths demonstrates that the proposed method s applcable to both mld and concentrated storm cases. CONCLUSIONS The hydrologcal response of a watershed s assumed to be unque and tme-nvarant when usng a conventonal hydrologcal approach, although the UH derved from hydrologcal records does show storm dependency. In ths study, a varable IUH model lnked explctly to the mean ranfall ntensty occurrng between tme t T c and t has been proposed to allow for tme delay effect for runoff transport n the watershed. The varable IUH shows a hgher peak and a shorter tme-topeak for a hgh ranfall ntensty condton. Addtonally, t shows a lower peak and a longer tmeto-peak for a low ntensty ranfall condton. Better smulaton s found n the hydrograph smulatons usng the proposed varable IUH model than the conventonal UH method, especally for concentrated storm events n the three study watersheds. Determnaton of the channel wdths and channel roughness coeffcents of dfferent order streams n the KW-GIUH model s also modfed, correspondng to the fluval geomorphology. The proposed varable IUH model warrants consderaton for ranfall runoff modellng n ungauged catchments that are nfluenced by hgh ntensty ranfall. Acknowledgements Ths research was supported by the Councl of Agrculture and the Natonal Scence Councl, Tawan, Chna, under grants CA94-..-B() and NSC E Fnancal support provded by the Councl of Agrculture and Natonal Scence Councl s Copyrght 28 IAHS Press

16 Dervaton of varable IUH correspondng to tme-varyng ranfall ntensty durng storms 337 gratefully acknowledged. The comments and suggestons made by the anonymous revewer were greatly apprecated. Downloaded by [ ] at 7:57 8 November 27 REFERENCES Ahsan, M. & O Connor, K. M. (994) A smple non-lnear ranfall runoff model wth a varable gan factor. J. Hydrol. 55, Chen, S. J. & Sngh, V. P. (986) Dervaton of a new varable nstantaneous unt hydrograph. J. Hydrol. 88, Chen, Y. P., Yang, S. Y. & Chen, K. H. (996) Rver tranng and utlzaton of flood plan between levees n Tawan. In: Proc. RIVERTECH96, Frst Int. Conf. on New/Emergng Concepts for Rvers, Internatonal Water Resources Assocaton, (Chcago, Illnos, USA), Dng, J. Y. (974) Varable unt hydrograph. J. Hydrol. 22, Goodrch, D. C., Lane, L. J., Shllto, R. M. & Mller, S. N. (997) Lnearty of basn response as a functon of scale n a semard watershed. Water Resour. Res. 33(2), Henderson, F. M. & Woodng, R. A. (964) Overland flow and groundwater flow from a steady ranfall of fnte duraton. J. Geophys. Res. 69(8), Horton, R. E. (939) Analyss of runoff plot experments wth varyng nfltraton capacty. Trans. Am. Geophys. Unon 2, Jarrett, R. D. (987) Errors n slope-area computatons of peak dscharges n mountan streams. J. Hydrol. 96, Jarrett, R. D. (992) Hydraulcs of mountan rvers. In: Channel Flow Resstance: Centennal of Mannng s Formula (ed. by B. C. Yen), , Water Resources Publcatons, Lttleton, Colorado, USA. Lee, K. T. & Yen, B. C. (997) Geomorphology and knematc-wave based Hydrograph dervaton. J. Hydraul. Engng ASCE 23, Lee, K. T. (998) Generatng desgn hydrographs by DEM asssted geomorphc runoff smulaton: a case study. J. Am. Water Resour. Assoc. 34(2), Lee, K. T., Chung, Y.-R., Lau, C.-C., Meng, C.-C. & Chang, S. (26) A wndows-based nqury system for desgn dscharge based on geomorphc runoff modellng. Comput. Geosc. 32(2), Leopold, L. B. & Maddock, T. (953) The hydraulc geometry of stream channels and some physographc mplcaton. US Geol. Survey Prof. Paper 252. Leopold, L. B., Wolman, M. G. & Mller, J. P. (964) Fluval Process n Geomorphology. W. H. Freeman, New York, USA. Mandevlle, A. N. & O Donnell, T. (973) Introducton of tme varance to lnear conceptual catchment models. Water Resour. Res. 9(2), Mangelsdorf, J., Scheurmann, K. & Wess, F.-H. (99) Rver Morphology a Gude for Geoscentsts and Engneers. Sprnger-Verlag, Berln, Germany. Mnshall, N. E. (96) Predctng storm runoff on small expermental watersheds. J. Hydraul. Engng ASCE 86(HY8), Nash, J. E. & Sutclffe, J. V. (97) Rver flow forecastng through conceptual models, Part I: a dscusson of prncples. J. Hydrol. (3), Rodrguez-Iturbe, I. & Valdes, J. B. (979) The geomorphologc structure of hydrologc response. Water Resour. Res. 5(6), Saghafan, B., Julen, P. Y. & Rajae, H. (22) Runoff hydrograph smulaton based on tme varable sochrone technque. J. Hydrol. 26, Sherman, L. K. (932) Stream-flow from ranfall by the unt-graph method. Engng News Rec. 8, Sngh, K. P. (976) Unt hydrographs a comparatve study. Water Resour. Bull. 2(2), Yen, B. C. & Lee, K. T. (997) Unt hydrograph dervaton for ungauged watersheds by stream order laws. J. Hydrol. Engng ASCE 2(), 9. Receved 22 March 26; accepted 3 November 27 Copyrght 28 IAHS Press