V N U JOURNAL OF SCIENCE Nal Sci.& Tech., T.xx, N„3AP 2004 A N O N - L IN E A R R A IN F A L L - R U N O F F M O D E L L u o n g T u a n A nh Research C enter o f Hydrology a n d W ater Resources A bstract This paper introduces a Non-Linear Rainfall-Runoff Model based on non-linear storage curve for runoff routing processes and rainfall index for estimation of effective rainfall The components of the system are constrained with non-linear relationships The paper presents a model simulating rainfall-runoff formulation based on concept of system of an input-output relating model [1] The following processes of the river basin will be considered: Method for routing processes: Estimation of effective rainfall or excess of rainfall; Overland and underground (base) runoff; Method for determination of parameters Keywords: Rainfall Index, Non-linear Relationships Rainfall-Runoff M e th o d fo r r o u t i n g p ro c e s s e s The routing processes are based on non-linear storage curve equations which can be expressed in the form as follows [3, 7]: - Equation of continuity: R(t)-Q(t+T,) = dS(t+T,)/dt (1) - Equation of motion in the storage curve expression: S(t+T,) = KQ'Xt+x,) (2) where: R(t): Effective rainfall in cm/h; Q(t+T|): Runoff in consideration of concentration time tj in cm/h; S(t+Ti): W ater storage of th e river basin in cm ; K, P: P aram eters Equation (1) can be approxim ated in differential form as follows: (ỴUt + A tM Q it + Tii + Q tt + Tj + At))/2)At = S(t + Tj + At) - S(t + 1) ) (3) S ubstitution equation (2) in (3), gives: R(t + At) - Q(t + T| )— - Q(t + X, + At)— = KQ*’(t + T| + At) - KQp (t + T, ) (4) Equation (4) is a non-linear equation which can be solved w ith initial condition Q(t=0) and given effective rainfall R (t + A t) R ew rite equation (4) in th e following form: Q u + T, — Q p ( t + , + A t) = — ^ Q * * (t + T | ) + R ( t + A t ) - Q ( t + T j ) At At (5) Equation (5) can be solved by different methods, one of effective algorithm is Newton iteration procedure Luong Tuan Anh 2_ By arran g in g Q(t + a = ^ - ; x = Q(t + T| + At) At Equation (5) will be transform ed into: f(x)= a xp + X - b= (6) In equation (6), unknow n variable is X and will be found by th e following relationship: XK+I = XK - -7— * f ( x K) (7) where: f(xk) is derivative of function f(xk) It will be not difficult to dem onstrate th a t th e convergence condition of interative procedure (7) is b>0 E s t im a t io n o f e f f e c t iv e r a in fa ll Rainfall index used for estim ating effective rainfall, it may be expressed in the following form: I M ( t ) = a X ( t ) + a | X ( t - A t) + a X ( t - A t ) + + a n X ( t - n A t) where - IM(t) is th e rainfall index a t tim e t; X(t) is the average tim e t; a, is param eters satisfying th e condition: a„>a,>a (8 ) rainfall over th e ba The rainfall index determ ined by (8) implies the change of m oisture condition of the river basin, but it will be difficult to estim ate this index due to m any p aram eters (a„ i=0,n) It will be easy to determ ine th is index if we rearrange (8) by an o th e r approxim ated non linear form ulation [3]: IM(t) = C ,IM (t - At) + [ • a ( t At)]X(t) (9) where: c , < - P ara m e ter of th e model; a (t - At) - Runoff coefficient a t tim e t - At, R elationship (9) shows th a t when C, The form ulation (10) leads to obtain the relations: when I M( t ) — coefficient a(t) —> and if IM(t) —» then a(t) —> t he runoff O v e rla n d a n d u n d e r g r o u n d r u n o f f Underground runoff is sim ulated by using underground runoff coefficient determ ined by th e following non-linear relationship: A Non-linear rainfall • m n o U model a N(t) = C3exp(-R{t)/C4) (11) w here : a N(t)- underground runoff coefficient, a function of effective rainfall R(t) ; c „ c.| param eters satisfying conditions < C :l< and C4>0 R elationship (11) m eans th a t proportion between underground and overland runoff is inverted w ith excess of rainfall S tru ctu re of th e operation system of this input-output relating model is shown in figure F i g l : S tructure of th e non-linear rainfall-runoff model From th e above form ulations, it is not difficult to realizes th a t th e non-linear rainfallrunoff model based on rainfall index for estim ation of effective rainfall and non-linear storage curve for routing processes can be adaptable w ith monsoon clim ate conditions, w here runoff is usually determ ined by rainfall processes The model includes param eters: Cl, Cl, C;i, c.t : P aram eters for effective rainfall; K [, p , : P aram eters for overland runoff routing; K j , P : P ara m e ters for underground runoff routing M e th o d fo r e s tim a tio n o f m o d e l p a r a m e te r s T hese p aram eters are estim ated by simple method of optim ization created by Nelder I and M ead R (3, 4) This m ethod is effective for th e optim ization function w ithout derivatives Different types of objective function can be used for estim ating th e param eters of the model One objective function which may tak e place is: F(C„ Ci, c„ c„ K„ p„ K„ PJ = Í L _> F? (12) Luông Tuan Anh N where: c ,„ „ < c, < G,.„„ ; K * _ < K, < K ,„„; p, < _ Ï ? » J l Q i d - Qd)2: Ff =^(Qi(C1,C2.C3.C.|.K1,P1,K2,P2)-Qid)2; Q,(C|, c„ c.„0„ K|, p„K,, p.) is theriver flow calculated bv the model; Q„| is th e observed river flow; The m ean value of coefficients Pj, P2 in th e case of turbulence flow according to M anning equation is 0.6 [2] Effective coefficient of th e model can be determ ined by th e form ula of WMO as follows [8]: F02 = 100(1 - Ff A p p lic a tio n - F la s h F lood s i m u la tio n The model has been applied for estim ation of flash flood occurred on 27 Ju ly 1991 on Nam La river basin w ith draining area of 206,8 km ' Using hourly rainfall and discharge from l ,h hour of 26 Ju ly to 24lh of 27 July, 1991, param eters of th e model has been determ ined as follow: c ,= 0.949 ; c ,= 5.50 ; c ,= 0.496 ; c ,= 13.2; K|= 3.48 ; P|=0.609 ; K.= 203.5 ; p ,= 0.668 Effective coefficient of the model is about 97.2% M aximum discharge erro r is 7.2% Computed and observed hydrographs are illustrated in figure 800 600 400 200 10 13 16 19 22 25 28 31 34 37 40 43 46 49 Fig 2: Computed and observed hydrographs of flash flood on Nam La river basin A NiHi-lmciU' fiiinl'iill - Iunol'l model - F lood F o r e c a s tin g Flood forecast in consideration o f concentration time If the forecast lead tim e is sh o rter th a n the tim e of concentration th en stream-flow forecast can be m ade based on observed rainfall from a netw ork of in gauges whose data are able to tran sm it to th e forecast center [4j In these cases stream flow forecast can be based on rainfall-runoff model For estim ation of concentration tim e from equation (1) and (2), variation of T| cản be m ade and the results of com putation will show th e suitable lead tim e for each basin Exam ples of T Khuc (flood-1999) and Ve river basins (flood-1998) have been taken to dem onstrate forecast ability of the model T he re su lts of computation are shown in table Table 1: Effective coefficients of the model with different concentration times River x,=0h I, = 3h I, = 6h ĩị = 9h T Khuc 84,1 88,5 89,2 68,3 I 1=12h 47,6 Ve 89,4 92,1 89,8 70,9 50.9 Update forecasting error w ith First Order Auto-Regression Model AR ( 1): U pdating procedure using AR(1) will be made in following steps: - E stim ate forecasting erro r of the previous forecast: A Q (t-T 1) = Q „ ( t - f ) - Q f( t - T , ) ; where Q„, Q, a re observeved and forecast flows respectively ■ Produce new forecast based on collected rainfall inform ation: Q(t) - U pdate forecast by adding value QU|H|llU,(t)= Q (t)+ R {l)A Q (t-Tj) w here R (l) is first order regression coefficient, - C o m p u te d a i l y r u n o f f fr o m d a ily r a in fa ll The stream flow of'T Khuc (F=2740knr) and Ve (F=854krrr) river basins has been synthesized from th e rainfall data Using daily rainfall and runoff da ta of period 1997-1999 for th e T Khuc riv er basin, the p aram eters of th e model have been calibrated w ith the following: c , =0.962 c , =13.8 C -=0.385 c ,= 80.0 K, =19.8 p , =0.620 K; = 1062 p , =0.960 Effective coefficient of th e mode] is 93,1% For verification of th e model, d a ta of periods: 1980 -1982 and 1986-1988 have b ee r used and the effective coefficients of the model are 94,9% and 91,2% respectively For th e Ve river basin, daily rainfall-runoff data of 1997-1999 are selected for calibration of the model The param eters are: c , =0.963 = 14.1 C» =0.355 c , = 111.6 K, = 23.7 p, = 0.745 K ,= 354 p ,= 0.793 Effective coefficient is about 95,1% For investigation of stability of th e p aram eters of th e model, rainfall-runoff d a ta of periods 1981-1982 and 1986 have been used for Luong Tuan Anh verification., th e re su lts show th a t effective coefficients of th e model are 92,4% and 93,3% respectively Conclusion It may be concluded th a t the model for sim ulation of rainfall-runoff process based on non-linear storage curve for runoff routing, rainfall index for estim ation of effective rainfall and the com ponents of th e system are constrained by non-linear relationships can be adaptable w ith rainfall-runoff conditions of th e sm all and average river b asins in Vietnam REFERENCES Mathematical Models in Surface Hydrology, IBM Italy Pisa Scientific Center, Dooge 1978 Chow V.T., Maidment D.R and Mays L.W., Applied Hydrology, McGraw-Hill, New York, 1988 Luong Tuan Anh, A Model for Simulation o f Rainfall, Runoff Processes on Small and Average River Basins of Northern Vietnam Thesis for Ph.D Hanoi (in Vietnamese) 1996 Maidment D R., Handbook o f Hydrology, McGraw-Hill, INC, 1991 Minimization Methods for Technique, Moscow (in Russian), 1981 Nelder I., Mead R A., A Simplex Method for Function Minimization, Computer Journal No.7, 1969, pp 308-313 United Nations, Proceedings o f the Expert Group Meeting on the Improvement o f Disaster Prevention Systems Based on Risk Analysis o f Natural Disasters Related to Typhoons and Heavy Rainfall, 1988 WMO, Guide to Hydrological Practices, 1994, No 168 TẠP CHỈ khoa học ĐHQGHN, KHTN&CN T.xx, sỏ 3PT 2004 MỘTMƠHÌNHMƯA-DỊNGCHẢYPHI TUYẾNTÍNH LươngTuấnAnh Trung tà m nghiên cứu T hủy vãn N guồn lợi nước Bài báo giới th iệ u m ột mơ h ìn h phi tu y ến m ưa - dòng chảy dự a trê n sở áp dụng đường cong lượng t r ữ phi tu y ến đ ể diễn tốn dòng ch ả y v sơ’ m ưa đ ể tính lượng m ưa h iệu C ác th n h p h ầ n củ a h ệ thơng (mơ h ìn h ) k ế t nối bằn g qu an hệ phi tuyến ... estim ation of effective rainfall and the com ponents of th e system are constrained by non-linear relationships can be adaptable w ith rainfall- runoff conditions of th e sm all and average river... these cases stream flow forecast can be based on rainfall- runoff model For estim ation of concentration tim e from equation (1) and (2), variation of T| cản be m ade and the results of com putation... relationship: A Non-linear rainfall • m n o U model a N(t) = C3exp(-R{t)/C4) (11) w here : a N(t )- underground runoff coefficient, a function of effective rainfall R(t) ; c „ c.| param eters satisfying