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

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)

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:
(ÎUt + 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 + 1 , + A t) = — ^ Q * * (t + T | ) + 2 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

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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 -

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) + [ 1 • a ( t . At)]X(t)

(9)

where: c , < 1 - 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,<1, if X(t)=0 then rainfall index IM(t) will be
decreased; otherw ise IM(t) may be increased. It is able to be adaptable with changing law of
th e m oisture condition of the river basin. In this case, runoff coefficient h as been
determ ined as function of rainfall index and may be expressed by th e following sim ulated
non-linear expression:
a(t) = 1 - exp[-(IM(t)/C2)-]

(10)

where: C2 - p aram eter and Cji > 0.
The form ulation (10) leads to obtain the relations: when I M( t ) —
coefficient a(t) —> 1 and if IM(t) —» 0 then a(t) —> 0.

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 .

4. 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 _> min

F?

(12)


Luông Tuan Anh

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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

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

7 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 ra 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 ra 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 1.
Table 1: Effective coefficients of the model with different concentration times
River

x,=0h

I, = 3h

I, = 6h

ĩị = 9h

T ra 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 , ) ;

1

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 ra 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 ra 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 ạ = 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


6

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.

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.

Minimization Methods for Technique, Moscow (in Russian), 1981.

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
Prevention Systems Based on Risk Analysis o f Natural Disasters Related to Typhoons and
Heavy Rainfall, 1988.

8.

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

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