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
1. 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)



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= 0

(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.
2. 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 0 X ( t ) + a | X ( t - A t) + a 2 X ( t - 2 A t ) + + a n X ( t -



t he runoff

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 runoff is sim ulated by using underground runoff coefficient determ ined
by th e following non-linear relationship:


3

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 0 < C :l< 1 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 1.

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 8 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 .

follows [8]:
F02 = 100(1 -

Ff

5. 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 2.

800

600

400

200
0

.

1

4


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 , ) ;

1

6. 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,

1.

Dooge
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.

Maidment D. R., Handbook o f Hydrology, McGraw-Hill, INC, 1991.

5.