DSpace at VNU: A non-linear rainfall - runoff model = Một mô hình mưa-Dòng chảy phi tuyến tính

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 co

A N O N - L I N E A R R A I N 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 b stract 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

1 M e th o d fo r r o u t i n g p r o c e s s e s

T he routing processes are based on non-linear storage curve e q u a tio n s which can be expressed in the form a s follows [3, 7]:

- Equation of continuity:

- Equation of motion in the storage curve expression:

where: R(t): Effective rainfall in cm/h; Q(t+T|): Runoff in consideration of co n c en tra tio n tim e

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 a s follows:

(ÎUt + A tM Q it + Tii + Q tt + Tj + At))/2)At = S(t + Tj + At) - S(t + 1) ) (3)

S u b stitu tio n 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 in itia l condition Q(t=0) an d given effective rainfall R (t + A t )

R ew rite equation (4) in th e following form:

Q u + T, — Q p ( t + 1 , + A t) = — ^ Q * * (t + T | ) + 2 R ( t + A t ) - Q ( t + T j ) (5)

Equation (5) can be solved by different methods, one of effective alg o rith m is Newton

By arra n g in g Q(t +

a = ^ - ; x = Q(t + T| + At)

At

E quation (5) will be transform ed into:

In equation (6), unknow n variable is X and will be found by th e following relationship:

* f ( x K) 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 in tera tiv e procedure (7) is b>0

2 E s t i m a t i o n o f e f f e c t i v e r a i n f a l l

Rainfall index used for estim atin g effective rainfall, it m ay be expressed in the following form:

I M ( t ) = a 0 X ( t ) + a | X ( t - A t) + a 2 X ( t - 2 A t ) + + a n X ( t - n A t ) (8 )

where - IM(t) is th e rainfall index a t tim e t; X(t) is the average rainfall over th e basin at tim e t; a, is param eters satisfying th e condition: a„>a,>a

The rainfall index determ ined by (8) implies the change of m oisture condition of the river basin, b u t it will be difficult to estim ate th is index due to m any p a ram eters (a„ i=0,n)

It will be easy to d ete rm in e th is index if we re arran g e (8) by an o th e r approxim ated non­ linear form ulation [3]:

where: c , < 1 - P a ra m e te r of th e model; a ( t - At) - Runoff coefficient a t tim e t - At,

R elationship (9) show s t h a t when C,<1, if X(t)=0 th en rainfall index IM(t) will be decreased; otherw ise IM(t) m ay be increased It is able to be ad a p tab le with changing law of

th e m oisture condition of the river basin In th is case, runoff coefficient h a s been determ ined as function of ra in fall index and m ay be expressed by th e following sim ulated non-linear expression:

where: C2 - p a ram eter an d Cji > 0

The form ulation (10) leads to obtain the relations: w hen I M( t ) — t he runoff coefficient a(t) —> 1 an d if IM (t) —» 0 th en a(t) —> 0

3 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 ru n o ff is sim u la te d by using underground runoff coefficient determ ined

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 0 < C :l< 1 and C4>0

R elationship (11) m eans th a t proportion between underground an d overland runoff is inverted w ith excess of rainfall

S tru c tu re of th e operation system of this in p u t-o u tp u t re la tin g model is shown in figure 1

F i g l : S tru ctu re 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 rainfall- runoff model based on rainfall index for estim ation of effective rainfall and non-linear storage curve for ro u tin g processes can be adaptable w ith monsoon clim ate conditions,

w here runoff is usu ally determ ined by rainfall processes

The model includes 8 param eters:

Cl, Cl, C;i, c.t : P ara m e ters for effective rainfall;

K [, p , : P ara m e ters for overland runoff routing;

K j , P : P a ra m e te rs for underground runoff routing

4 M e th o d f o r e s t i m a t i o n o f m o d e l p a r a m e t e r s

T hese p aram eters are estim ated by sim ple m ethod of optim ization created by Nelder

I an d M ead R (3, 4) T his m ethod is effective for th e optim ization function w ithout derivatives Different types of objective function can be used for e stim atin g th e param eters

of the model One objective function w hich m ay ta k e place is:

F(C„ Ci, c„ c„ K„ p„ K„ PJ = Í L _> min (12)

F?

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 the river

flow calculated bv the model; Q„| is th e observed riv er flow;

The m ean value of coefficients Pj, P2 in th e case of tu rb u len c e 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

5 A p p lic a tio n

- F la s h F lo o d s i m u l a ti o n

The model has been applied for estim ation of flash flood occurred on 27 Ju ly 1991 on Nam La riv er b asin w ith d ra in in g a rea of 206,8 k m ' Using hourly rain fall an d discharge from l ,h hour of 26 J u ly to 24lh of 27 July, 1991, param eters of th e model has been determ ined a s 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 axim um discharge e rro r is 7.2% Computed and observed hydrographs are illustrated in figure 2

800

600

400

200

0

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

- F lo o d 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 e n stream -flow forecast can be m ade based on observed rainfall from a netw ork of ra in gauges whose data

a re able to tra n sm it to th e forecast ce n ter [4j In th ese cases strea m 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 an d the re su lts of com putation will show th e suitable lead tim e for each basin Exam ples of T ra Khuc (flood-1999) an d Ve riv er b asins (flood-1998) have been tak e n to d em o n stra te forecast ability of the model T he re su lts of com putation

a re shown in tab le 1

Table 1: Effective coefficients of the model with different concentration times

Update forecasting error w ith First Order Auto-Regression M odel A R ( 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 - f1) - Q f ( t - T , ) ;

w here Q„, Q, a r e 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-T j) 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 i l y r a i n fa l l

The strea m flow of'T ra Khuc (F=2740knr) and Ve (F=854krrr) river b asins h as been synthesized from th e rainfall data Using daily rainfall an d runoff d a ta of period 1997-1999 for th e T ra Khuc riv e r basin, the p aram eters of th e model have been ca lib rated 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 an d 1986-1988 have b e e r used and the effective coefficients of the model are 94,9% an d 91,2% respectively

For th e Ve riv er basin, daily rainfall-runoff d a ta of 1997-1999 are selected for calibration of the model The param eters are:

c , =0.963 ạ = 14.1 C» =0.355 c , = 111.6

K, = 23.7 p, = 0.745 K ,= 354 p ,= 0.793

Effective coefficient is ab out 95,1% For investigation of stab ility of th e p a ram eters of

6 Luong Tuan Anh verification., th e re su lts show th a t effective coefficients of th e model a re 92,4% an d 93,3% respectively

6 Conclusion

It m ay be concluded t h a t the model for sim ulation of rain fall-ru n o ff process based on non-linear storage curve for runoff routing, rainfall index for estim atio n of effective rainfall and the com ponents of th e system are constrained by n o n -lin ea r re la tio n sh ip s can be adaptable w ith ra infall-runoff conditions of th e sm all and average riv er b asin s in Vietnam

REFERENCES

1 Dooge Mathematical Models in Surface Hydrology, IBM Italy Pisa Scientific Center,

1978

2 Chow V.T., Maidment D.R and Mays L.W., Applied Hydrology, McGraw-Hill, New York,

1988

3 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

4 M aidment D R., Handbook o f Hydrology, McGraw-Hill, INC, 1991.

6 Nelder I., Mead R A., A Simplex Method for Function Minimization, Computer Journal

No.7, 1969, pp 308-313

7 United Nations, Proceedings o f the Expert Group Meeting on the Improvement o f Disaster Heavy Rainfall, 1988.

8 WMO, Guide to Hydrological Practices, 1994, No 168

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