Developement of Fuzzy Rainfall Runoff...

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Hydrological Sciences-Journal~des Sciences Hydrologiques, 46(3) June 2001

363

Development of a fuzzy logic-based rainfall-runoff

model

YESHEWATESFA HUNDECHA, ANDRAS BARDOSSY &

HANS-WERNER THEISEN

Institut fiir Wasserba u, UniversitdtStuttgart D-70550Stuttgart Germany

e-mail:[email protected] ;[email protected]

Abstract Rainfall-runoff models are used to describe the hydrological behaviour of a

r iver catchment. Many different models exist to simulate the physical processes of the

relationship between precipitat ion andrunoff. Some of them are based on simple and

easy-to-handle concepts, others on highly sophisticated physical and mathematical

approaches that require extreme effort in data input and handling. Recently,

mathematical methods using l inguist ic variables, rather than conventional numerical

variables applied extensively in other disciplines, are encroaching in hydrological

studies. Among these is the application of a fuzzy rule-based modell ing. In this paper

an attempt was made to develop fuzzy rule-based routines to simulate the different

processes involved in the generation of runoff from precipitat ion. These routines were

implemented within a conceptual , modular , and semi-distr ibuted modelthe HBV

model. The investigation involved determining which modules of this model could be

replaced by the new approach and the necessary input data were identif ied. A fuzzy

rule-based routine was then developed for each of the modules selected, and

application and validation of the model was don e on a rainfall-runoff analysis of the

Neckar River catchment, in southwest Germany.

Key words

rainfall-runoff mod elling; HBV mo del; fuzzy logic; River Neckar, Germany

Dveloppement d'un modle pluie-dbit base de logique floue

R sum Les modles pluie-dbit sont uti l iss pour dcrire le comportement

hydrologique d'un bassin versant. De nombreux modles existent pour simuler les

processus physiques dterminant la transformation de la pluie en dbit . Certains

d'entre eux sont bass sur des concept simples et aisment transposables, tandis que

d'autres s 'appuient sur des approches physiques et mathmatiques trs sophist iques

qui ncessitent beaucoup d'efforts au niveau de la pr ise en compte et du trai tement des

donnes . Depuis que lque temps, des mthodes mathmat iques manipulant auss i bien

des variables alphanumriques que les habituelles variables numriques, ont t

dveloppes en hydrologie. Parmi celles-ci se trouve la modlisation base de logique

floue. Dans cet ar t icle nous prsentons une tentative de dveloppement de routines

base de logique f loue pour simuler les diffrents p roce ssus mis en jeu dans la

production d 'coulem ent part ir des prcipitat ion s. Ces routines ont t implmen tes

au sein du modle HBV, conceptuel , modulaire et semi-distr ibu. L 'tude a ncessit

de dterminer quels modules du modle pouvaient tre remplacs par la nouvelle

approche d'une part et d' identif ier les donnes ncessaires d'autre part . Une routine

base de logique f loue a alors t dveloppe pour chacun des modules ainsi identif is,

et la mise en uvre et la validation du modle global ont t ralises avec les

donnes de pluie et de dbit du bassin versant de la r ivire Neckar , au Sud-Ouest de

l 'Al lemagne .

Motsclefs modles pluie-d bit; modle HBV ; logique floue; Rivire Neckar, Allemagne

INTRODUCTION

It is customary to establish a rainfall-runoff relationship in hydrological studies of a

river basin. Traditionally, this task has been accomplished using methods ranging from

Open for discussion until I Decem ber 2001
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


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364 Yeshewatesfa H undecha et al.

those that give explicit emphasis to the underlying physical processes involved in the

generation of runoff from rainfall to simple conceptual approaches that treat the

catchment system in a simple idealized way.

In using models that are based on descriptions of the physical processes, rigorous

mathematical equations are often needed to solve the problem at hand. Such models are

highly demanding in terms of their data requirement and it is often necessary to estimate

input parameters specific to the catchment being modelled. In many cases, a large number

of these parameters are involved and there is no way of estimating them uniquely. Instead,

they are determined subjectively based on the modeller's judgement and the effect is

normally manifested in the output of the model (Prakash, 1986). Hence, models which are

easy to handle and have a minimum data requirement are often sought to solve problems

where data availability is limited and the system is too complicated to be handled by

physical m odels.

Recently, mathematical methods using linguistic variables rather than conventional

numerical variables, which have been applied in other disciplines, are encroaching into

hydrological studies as well. Among these is the application of a fuzzy rule-based

approach in modelling processes involved in the hydrologie cycle. A fuzzy rule-based

modelling is a qualitative modelling scheme by which one describes system behaviour

using a natural language (Sugeno, 1993). In using a fuzzy logic-based approach in

modelling cause and effect, relationships are described verbally rather than using known

governing physical relationships. Some of the causes that are taken into account in the

physically-based models may be omitted. On the other hand, some of the causes that are

not considered in idealized types of models, because of the nature of generalization or

unavailability of known relationships, can be included in a fuzzy logic-based approach.

Establishing these relationships depends on observing trends between the cause and effect

and a detailed knowledge of the underlying processes is, therefore, not required. A few

attempts have been made so far to implement such an approach, in modelling processes

taking place in a river catchment, and promising results have been obtained. Brdossy &

Disse (1993) have already demonstrated the applicability of such an approach in modelling

infiltration. See & Openshaw (1999, 2000) have also worked on the application of entirely

Artificial Intelligent approaches and a hybrid model constructed by integrating conven

tional and Artificial Intelligent approaches in river level and flood forecasting. In the area

of meteorological data management, Abebe et al. (2000) have shown the applicability of

fuzzy rule-based models for reconstruction of missing precipitation events.

The purpose of this paper is to demonstrate the applicability of a fuzzy logic-based

approach to rainfall-runoff mo delling. An attempt was m ade to mod el the individual

processes involved in a watershed system, and their applicability was investigated by

incorporating them in a mod ular conceptual model already in use.

A

F U Z Z Y L O G I O B A S E D M O D E L L I N G A P P R O A C H

Quantitative rules pertaining to physical science are normally described by

mathematical functions which, for every element in the domain, assign a unique output

value. There are also certain classes of rules applied to linguistic variables, which do

not have unique numerical values. A very simple example that can demonstrate such

classes of rules could be drafted as:

If the air is warm, one has to consume m uch water.


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Developmentof a fuzzyrule-basedrainfall-runoff model 365

Here , the term wa rm is not a quantity that can be clearly defined. It can have

values w ithin an arbitrarily chosen range. But all temp erature values w ithin the defined

range may not be considered equally warm. Similarly, the consequence, i.e.

consum ption of m uch water is not a quantity that can be assigned a unique value. A

fuzzy logic-based modelling approach enables one to establish a one-to-one relation

ship between air temperature and water consumption in a way that is quite different

from a conve ntional functional form (Zadeh, 1973).

Fuzzy logic modelling is based on the theory of fuzzy sets (Zadeh, 1965). Unlike

in an ordinary binary set, in a fuzzy set the boundary is not clearly defined and partial

membership of elements is possible. Each element of the set is assigned a membership

value w hich can be betwe en 0 and 1 inclusively. The function that assigns this value is

referred to as the membership function associated with the fuzzy set. Fuzzy numbers

are special types of fuzzy sets defined on the set of real numbers. Fuzzy numbers are

usually defined by using membership functions that have triangular shapes and are

expressed as

(a.\,

02, a

3

)

T

such thata\ < a

2

aj a

2

a

3

04 x

Fig. 2 Membership function of a trapezoidal fuzzy number.

\


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366

Yeshewatesfa Hundecha et al.

given set of model variable values depends on the degree to which they fulfil the condi

tion of the rule. The truth value corresponding to the fulfilment of the conditions of a

rule for a given set of values of the arguments is referred to as the degree of fulfilment

(DOF) of the rule and has values in the interval

[0,1].

This value is determined based on

the membership value of each of the arguments and the logical connectors used (Brdossy

&Duckstein, 1995).

Normally, several rules are partially satisfied for a given set of model variables

and hence there are several associated fuzzy consequences, which are then combined

into an overall fuzzy consequence using different techniques of rule combination. The

combination method used in this paper is the maximum combination technique in

which the membership function of the overall consequence is determined as the maxi

mum of the product of the degree of fulfilment of each of the rules and the mem ber

ship function of their corresponding rule consequence. The combined rule consequence

1

lnput-1

Fuzzification

Xfj

Input-N

Fuzzification

Defuzzification

Crisp output

(y)

Fig. 3 Schematic representation of a fuzzy logic-based modelling procedure.


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Developmentof a fuzzyrule-basedrainfall-runoff model 367

is then conve rted into a crisp real num ber using defuzzification techniqu es. T he

defuzzification technique com mo nly used is the mean defuzzification in which the

centroid of the overall fuzzy consequence is taken as the crisp output of the fuzzy rule

system (Brdossy & Duckstein, 1995). Figure 3 shows the schematic representation of

the modelling procedure using a fuzzy rule-based approach.

DESCR I PTI ON OF M O DEL APPLI CATION AR EA

Application and validation of the model was carried out on the catchment of the River

Neckar in southwest Germany, most of which is in the state of Baden-Wurttemberg, It

has a total dr aina ge area of 13 957 km

2

and its complex topography ranges from

moderate hills to plains with erratic geological formations. The elevation of the land

surface varies from 100 to 1000 m a.s.l. A variety of land-use patterns ranging from

residential areas to forests characterize the basin. The outlet of the basin is at the point

of confluence of the River N eckar with the River R hine.

The entire basin was subdivided into 41 sub-basins so that each sub-basin could be

modelled separately. Each sub-basin was further subdivided into up to ten different

elevation classes. This was performed based on a 100 m elevation difference.

Daily time series of precipitation from 1980 to 1995 and daily time series of mean

temperature data from 1960 to 1998, observed at all meteorological stations within the

basin, were used to run the model. For each elevation zone in each sub-basin, the

corresponding daily time series of temperature and precipitation were computed using

geostatistical methods from the station data. Daily discharge values from 1980 to 1996

were obtained from different gauging stations on the River Neckar and many of its

tributaries. Fro m th ese, data from 15 stations were chosen to calibrate the m odel.

F O R M U L A T I O N O F M O D E L C O M P O N E N T S

Four different processes taking place in a watershed system were identified and a fuzzy

logic-based routine was formulated for each of these processes. The modules identified

are: snowmelt, vapotranspiration, runoff, and basin response. A fuzzy rule-based

routine was formulated for each of the modules independently of the others.

Snowmelt

The dynamics of snowmelt depends on the energy balance of the accumulated snow,

which in turn depends on temperature, net short-wave radiation, net long-wave

radiation, and additional heat energy input due to incoming rainfall. In utilizing a fuzzy

rule-based routine, air temperature, accumulated snow depth and magnitude of daily

precipitation were considered as factors that influence the amount of snowmelt. Since

the short-wave and long-wave radiations largely depend on the air temperature, they

were omitted. Although the effect of temperature on the amount of snowmelt is

apparent, the effect of the magnitude of precipitation and the amount of accumulated

snow is indirect. The magnitude of rainfall plays a role in increasing the amount of

snowmelt by providing additional energy input to the snowpack. This is especially true

if the temperature of the incoming rainfall is higher.


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368

Yeshewatesfa Hundecha

et al.

The depth of the accumulated snow also affects the amount of snowmelt. Only the

upper few centimetres of the snowpack are under the direct influence of the atmos

phere. Temperature of this layer depends on the atmospheric temperature. Snowmelt

only occurs from this layer when its temperature reaches 0C and the net energy

balance is positive. If the temperature of the lower layer is below 0C, part or all of the

snowmelt is refrozen. The heat released in the process is used to satisfy part or the

entire heat deficit of the lower snowpack. The amount of heat energy needed to raise

the temperature of the lower layer from some negative value to 0C depends on the

volume of this part of the snowpack. When the temperature of the lower snowpack

reaches 0C, any additional snowmelt is used to satisfy the free water holding capacity

of the snowpack. Only snowmelt in excess of this free water holding capacity becomes

available for infiltration and surface runoff (Anderson, 1968).

Two systems of rules were established to determine the proportion of precipitation

that is in the form of snow and the amount of snowmelt, respectively. The arguments

to be used in each sy stem of rules were identified and were div ided into different fuzzy

classes.

In the first system of rules where the proportion of precipitation in snow form is

determined, the only argument is temperature and the consequence is proportion of

solid precipitation. In the second system of rules, temperature, accumulated snow

depth, and magnitude of daily precipitation are used as arguments and the consequence

is the daily amount of snowmelt.

The temperature values were divided into five fuzzy classes. The magnitude of

precipitation and accumulated snow were also partitioned into three fuzzy classes. The

consequences of the rule systems were also fuzzified by dividing the amount of snow

melt into four fuzzy classes and the proportion of solid precipitation in the form of

snow into three fuzzy classes. These are summ arized in Tables 1-5.

Table 1Fuzzy classes of temperature for snowmelt computation.

Class of temperature

Cold

About zero

Cool

Warm

Table 2Fuzzy classes of accumulated snow depth.

Accumulated snow depth

Low

Moderate

High

Table3 Fuzzy classes of daily precipitation.

Daily precipitation

Low

Moderate

High

Fuzzy number representation

(C)

( _ o o , _ l , 0 ) j -

( - 1 , 0 . 1 . 2 ) ,

( 1 , 2 , 3 , 4 ) ,

(3 ,

4,

+

~ )

r


(mm water equivalent)

(0 ,

20,

35)

r

(20,35,

45)

T

(35,45,

+ ~ )

r


(mm)

(0 , 10, 15)j-

(10, 15 ,20)

r

(15 ,20 , + )j-


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Development

of a

fuzzy rule-based rainfall-runoff model

369

ble4Fuzzy classesofamountofsnowmelt.

Amountofsnowmelt


(mm day ')

Low

Moderate

High

Extreme

(0,

4, 8)

r

(4,8, 15)

7

-

(8,

15,

20)

r

(15,20,55)7-

ble5Fuzzy classesofpercentageofprecipitationinsnowform.

Proportionofprecipitationin

snow form


( )

Nosnow

Half

All

(0,25,5 0)

r

(25,50,

7 5)

T

(50,

75, 100)

r

ble6Asystemofrules describing proportionofprecipitationinsnowform.

Ruleno.

Argument (temperature)

Proportionofprecipitationinsnow form

1

2

3

4

Cold

About zero

Cool

Warm

All

Half

No

snow

Nosnow

ble7Asystemofrulesfor thedeterminationofsnowmelt.

Rule no.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

Argument:

Snow depth

L ow

L ow

L ow

L ow

L ow

L ow

L ow

Moderate

Moderate

Moderate

Moderate

Moderate

Moderate

Moderate

Moderate

Moderate

High

High

High

High

High

High

High

High

High

Tempe

About

About

About

Cool

Cool

Cool

Warm

About

About.

About.

Cool

Cool

Cool

Warm

Warm

Warm

About:

About;

About

:

Cool

Cool

Cool

Warm

Warm

Warm

rature

zero

zero

zero

zero

zero

zero

zero

zero

zero

Precipitation

High

Moderate

L ow

High

Moderate

Low

-

High

Moderate

Low

High

Moderate

L ow

High

Moderate

L ow

High

Moderate

L ow

High

Moderate

L ow

High

Moderate

L ow

Amount of snowmelt

High

Moderate

L o w

High

Moderate

L o w

Extreme

High

Moderate

L o w

High

Moderate

L o w

Extreme

Extreme

High

Moderate

L ow

L o w

High

Moderate

L ow

Extreme

Extreme

High


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370

Yeshewatesfa Hundechaet al.

In the first system of rules, for each fuzzy class of temperature values, a

corresponding fuzzy class of proportion of precipitation in snow form was assigned

(Table 6). The consequence of each rule was computed for a given temperature value

and the crisp output of the system of rules was computed by combining the fuzzy

consequences of each rule using the maximum combination technique and defuzzi-

fying the com bined fuzzy c onsequenc e using the mean defuzzification techniqu e. The

same techniques were used in all rule systems in the other modules as well.

In the second system of rules, the arguments, namely the fuzzy classes of

temperature, magnitude of precipitation, and accumulated snow, were connected using

the A N D operator. The complete system of rules is shown in Table 7. No te that the

increase in temperature and the magnitude of precipitation lead to the increase in the

amount of snowmelt. The consequence of increase in the amount of accumulated snow

is to decrease the potential for snowmelt as explained earlier in this section.

Evapotranspiration

The fuzzy logic routine for this process was formulated based on the long-term series

of observed mean monthly temperature and the corresponding mean monthly vapo

transpiration values. A set of rules was developed in which the incremental tempera

ture value above the long-term monthly mean value (T - T

m

) is used as an argument

and the ratio of the actual to the long-term mean monthly vapotranspiration value

(PEJPEm) is the consequence. The incremental temperature value above the long term

monthly mean value was divided into seven fuzzy classes and the corresponding

classification of the consequence was m ade. These are summ arized in Table 8.

Table 8Summary of the fuzzy rule system used to describe vapotranspiration.

r-r,(c) PE

a

/PE

m

(-2,0,2)

r

(0.8,1.0,1.2)7-

(1,3,5),- (1.1, 1.2, 1.3)

r

(3,5,7)

T

(1.2, 1.4,

1,

6)

r

(5,7,+oo)

r

(1.4, 1.6, 1.8)

r

( -5 , -3 , - l )

r

(0.3,0.6,0.8)7-

(-7,

-5 ,

- 3 )

r

(0.2,0.3,0.4 , 0.6)*

( -00, -7, -5 )7- (0 .1 ,0 ,2 ,0 .3)7-

Calculation of runoff

This mod ule com putes the proportion of rain or snowm elt that is converted to runoff at

a given soil moisture deficit. The proportion of rain or snowmelt that contributes to

runoff depends on the relative soil moisture. The relative soil moisture is the ratio of

the actual soil moisture to the field capacity of the soil. The proportion of rain or

snowmelt that contributes to runoff increases with the relative soil moisture. In the

HB V m odel, their relationship is expre ssed by an exponential function.

The fuzzy logic based routine was formulated by using the relative soil moisture as

the rule argument and the proportion of rain or snowmelt that contributes to runoff as

the consequence of the rule system. The field capacity and the permanent wilting


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Developmento f a fuzzyrule-basedrainfall-runoff model

371

Table 9A rule system for soil water accounting.

Relative soil moisture (ratio of actual to the

maximum soil moisture storage)

(0,0, 0.2, 0.3)

R

(0.2,0.3,0.4,0.5)*

(0.4,

0.5, 0.6, 0.7)

R

(0.6,0.7, 0.8, 0.9)

R

(0.8,0.9, 1.0, 1.0)

R

Percentage of rain or snow that goes to runoff

(0,2,4)

T

(10,

15,20)

r

(20,

30,4 0)

r

(60,

70 ,80)

T

(75, 90, 100, 100)R

points for the different zones were defined based on the type of soil in each zone. Five

fuzzy classes of the relative soil moisture values were established and the

corresponding classes of percentage of rain or snowmelt that contributes to runoff were

defined as shown in Table 9.

The basin response

This module describes the dynamics of the generated

runoff,

and thus its distribution in

time once the water balance is set by the modules for snowmelt and

runoff.

The

conceptual system used in the HBV model (Bergstrm, 1995) was adapted to use a

fuzzy logic-ba sed rou tine. The proce ss is conc eptua lized by a fictitious system of two

reservoirs arranged one over the other in which the outflows from the upper and lower

reservoirs simulate the direct runoff and the base flow respectively at the outlet of the

basin. These reservoirs are conceptually defined to simulate movement of the runoff

within the basin before reaching the outlet of the basin. The infiltration excess water

computed in the runoff module is input to the upper reservoir. The lower reservoir is

replenished by percolation from the upper reservoir.

In formulating the fuzzy logic-based routine of this module, three systems of fuzzy

rules were utilized to determine the outflows from the two reservoirs and the

percolation from the upper reservoir to the lower one respectively. The volume of

water in the upper reservoir was used as an argument in the rule systems established to

determine the outflow from the upper reservoir and the percolation from the upper to

the lower reservoir. In the rule system used to determine the outflow from the lower

reservoir, the volume of water was used as an argument. The volume of water in the

upper reservoir was classified into 10 fuzzy sets while that in the lower reservoir was

classified into 15 fuzzy sets to establish the rules.

Because of the erratic nature of the entire basin, the rules that apply for one part of

the basin would not apply to other parts of the basin. This would necessitate the

formulation of different sets of rules for each sub-basin, which would be difficult to

handle. For this reason, rules were established for three different types of upper and

three different types of lower reservoirs (Tables 10-12). For each sub-basin, a pair of

reservoir types was selected and the rules corresponding to the chosen pair were used.

RESULTS AND DI SCUSSI ON

Before testing the applicability of the fuzzy logic-based routines for the different

catchment processes, the HBV model was calibrated and used for the basin.


10/14

37 2 Yeshewatesfa Hundechaet al.

Table 10A rule system to predict outflow from the upper reservoir.

Volume of water in the

reservoir (mm)

( 0 , 1 ,

2)

T

( 1 , 2 , 4 , 8 ) *

(4 , 8, 10)

r

(8 ,

10, 15)

T

( 1 0 , 1 5 , 2 0 ) *

( 1 5 , 2 0 , 2 5 )

r

( 2 0 , 3 0 , 4 0 )

r

(30, 40, 45 )

T

(40, 45, 55)

r

(45 , 55, =o)j.

Reservoir outf low (mm

Res. Type 1

(0.0,0 .17,0.33)7-

(0.17,0.33,0.5)7-

(0.33,

0.5, 0 .67)

r

( 0 . 5 , 0 . 6 7 , 1 . 0 )

7

(0.67, 1.0,

1.33)

r

(1 .0 ,1 .33,2.0)7-

(3 .0 , 3 .33, 3 .67)T-

(3 .33 ,

3.67, 4.33)

T

(3.67, 6.67,

10)T-

(10.0, 13.33, 15.0)7-

day

1

):

Res. Type 2

(0.0, 0.25, 0.5)

r

(0 .25,0.83,1.5)7-

(0.83,1.5 ,2 .17)7-

( 1 .5 ,2 .17 , 3 .0 )

r

(2 .17,3.0 ,4 .33)7-

(3 . 0 , 4 . 3 3 ,

6 . 0 )

r

(5 .0 ,8 .33,11.67)7-

(8.33, 11.83, 15.0)

r

( 11 .83 ,15 .0 ,18 .5 ) 7

(15.0, 18.5, 21.67)

r

Res. Type 3

(0.0, 0.17, 0.33) ,-

(0 .17,0.33,0.67)7-

(0 .33 , 0 .67 .1 .33)

r

( 0 . 6 7 , 1 . 3 3 , 2 . 0 )

r

(1 .33 , 2.0, 2.67)7-

(2.0 , 3 .33, 5 .67)

r

(6.67,

9.0, 10.0)T-

(9 .33 ,

10.67, 12.33)7-

(11.67,12.67, 14.0)7-

(13.0, 15.0, 16.0)7-

Table 11A rule system to predict percolation from the upper to the lower reservoir.


reservoir (mm)

( 0 , l , 2 )

r

( 1 , 2 , 4 , 8 ) *

(4 , 8, 10)

r

(8 ,

10,

15)T-

(10, 15, 20)

r

(15,20,25)7-

(20, 30, 40)

r

(30, 40, 45)

r

(40, 45, 55)

7

(45 , 55, )

r

Percolation rate (mm

Res. Type 1

(0.0, 0 .28, 0 .56)

r

(0 .28 ,

0.56, 0.83)7-

( 0 . 5 6 , 0 . 8 3 ,

1,11)

T

( 0 . 8 3 , 0 . 9 5 , l . l l )

r

(0 .95,1.11,1.22)7-

(1 .11,

1.39,

1.67)

r

(1.38,

1.67,

1.95)

T

(1.67, 1.95, 2.22)

r

( 1 . 9 5 , 2 . 2 2 , 2 . 5 )

r

(2.22, 2.5, 2.78)T-

. d a y

1

) :

Res. Type 2

(0.0, 0 .43, 0 .86)

r

(0.43, 0.86, 1.24)T-

(0.86,

1.24, 1.67)T-

( 1 . 2 4 , 1 . 6 7 , 2 . 1 5 )

r

(1 .67, 2 .15, 2 .54)

r

( 2 . 1 5 , 2 . 5 4 , 3 . 2 7 )

r

(2.54, 3.27, 3.89)

r

( 3 .27 , 3 .89 ,4 .17)

r

(3 .89,4.17,4.72)7-

(4.17,4.72,5.17)7-

Res . Type 3

(0.0, 0 .3 , 0 .44)

r

(0 .3 ,

0 .5 , 0 .86)

r

( 0 . 5 , 0 . 8 6 ,

1.06)

r

(0.86 , 1.06, 1.3)

r

(1.06, 1.3,

1.47)

r

(1 .3 ,

1.58,

1.78)

r

(1 .58,1.78,2.03)7-

(1.8 ,2 .03,2.26)7-

(2 .03 , 2 .26 , 2 .6 )

r

(2.26, 2.6, 2.78)7-

Table 12A rule system to predict outflow from the lower reservoir.


reservoir (mm)

(0.0, 5.0, 10.0)7-

(5.0 ,10.0,20.0)7-

( 10 .0 ,20 .0 ,40 .0 ) r

(20.0, 40.0, 60.0)

r

(40.0, 60.0, 80.0)T-

(60.0, 80.0, 100.0)7-

(80.0, 100.0, 120.0)7-

(100.0, 120.0, 140.0)7-

(120.0, 140.0, 160.0)

r

(140.0, 160.0, 180.0)7-

(160.0, 180.0, 200.0)

r

(180.0, 200.0, 220,0)

r

(200.0, 220.0, 240.0)

r

(220.0, 240.0, 260.0)

r

(240.0, 260.0, ~ )

r

Outflow rate (mm day

1

)

Res . Type 1

(0.0,0 .05,0.1)7-

( 0 . 0 5 , 0 . 1 , 0 . 1 5 )

r

(0 .1 ,0 .15,0.2)7-

( 0 . 1 5 , 0 . 2 , 0 . 2 5 )

7

(0.2, 0.25, 0 .6)

r

(0 .25 , 0.3, 0 .35)

T

(0 .3 ,

0.35, 0.43)r

(0 .35 ,

0.43, 0.5)7-

(0.43, 0.5, 0.58)T-

(0 .5 , 0 .58 , 0 .65)

r

(0 .58 ,

0 .65 , 0 .73)

r

(0 .65 ,

0 .73 , 0 .80)

r

(0.70, 0.80, 0.90)7-

( 0 . 8 0 , 0 . 9 0 , 1 . 0 )

r

(0.90,

1.0, 1.10)T-

1:

Res. Type 2

(0.0,0 .08,0.15)7-

(0 .08 ,

0.15, 0.25)7-

(0.15,0.25,0.35)7-

(0 .25 ,

0.35, 0 .45)

T

(0 .35 , 0.45, 0 .53)

7

(0 .45 ,0.5 3, 0.60)j-

(0 .53 , 0.6, 0 .68)

r

(0.6,

0.68, 0 .75 )

r

(0 .68 , 0 .75 , 0 .83)

r

(0 .75 , 0 .83 , 0 .90)

r

(0 .83 ,

0.90, 0 .98)

r

( 0 . 9 0 , 0 . 9 8 ,

1.05)7-

( 0 .98 ,1 .05 ,1 .13) 7

(1 .05 ,

1.13,

1.2)7-

( l . l , 1 . 2 , 1 . 3 )

r

Res. Type 3

( 0 . 0 , 0 . 0 5 , 0 . 1 )

r

(0 .05 ,

0.10, 0 .15)

r

(0 .1 ,0 .15,0.20)7-

( 0 . 1 5 , 0 . 2 0 , 0 . 2 5 )

r

(0 .20, 0 .25, 0 .30)

7

(0 .25 , 0.30, 0.35)7-

(0.30, 0.35, 0.40)2-

(0 .35 ,

0 .40 , 0 .45)

r

(0.40, 0.48, 0.55)7-

(0 .48 , 0.55, 0.63)7-

(0 .55 ,

0 .63 , 0 .70)

r

(0 .63 ,

0 .70 , 0 .78)

r

(0 .70, 0 .78, 0 ,85)

T

(0 . 7 8 ,

0.85, 0 .90)

T

(0 .85 ,

0 .90 , 0 .95)

r


11/14

Development of a fuzzy rule-based rainfall-runoff model

373

Applicability of the fuzzy logic-based routines was investigated by incorporating only

one of the fuzzy rule-based modules at a time while retaining the HBV versions for the

other modules. Calibration of the modules was performed by using a multi-objective

calibration procedure (Gupta & Sorooshian, 1998). Three different measures of model

performance were used to calibrate the modules. These measures are the correlation

coefficient of the modelled and the observed discharges (RCOEF), the Nash-Sutcliffe

measure (NS), and the bias (the average of the difference between the modelled and the

observed discharges). The consequences of each rule system were manually changed

until acceptable values of all the three measures of model performance were obtained.

The model performance values were calculated based on simulation of daily discharge

for a period of 10 years, i.e. N =365 2. Fina lly, all the inde pen den tly ad justed fuzzy

logic-based routines were coupled together to model the catchment. All computations

were carried out for a time step of two hours.

Performance of each of the fuzzy rule-based modules was found to be nearly the

same as that of the corresponding HBV modules. Generally, the fuzzy rule-based

routine for snowmelt was observed to show the best performance (Table 13).

Estimation of the winter high flows by the HBV model was improved a little at some

observation points by introducing this module (Fig. 4). This is because, in the HBV

model, the snowmelt was represented by a simple degree-day method in which only

temperature is considered a driving force for snowmelt. In the fuzzy rule-based

routine, additional factors that were not considered in the HBV routine were

incorporated, leading to an improvement in the estimation of the amount of snowmelt.

On the other hand, the fuzzy rule-based routine for the basin response module was

found to be the most difficult to handle. Assignment of the combination of upper and

lower reservoir types to the sub-catchments was undertaken through a kind of trial and

160.00-]

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HBVModel

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ff 80.00-

a

g 60.00-

Date

Fig. 4 Com par ison be tween the HB V and the Sno w-F uzzy models a t one of the

observa t ion nodes .


12/14

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13/14

Development

of a fuzzy

rule-based

rainfall-runoff model

375

Feb/85 Feb/

85

Mar/

85

Apr/

85

Apr/85 May/85 Jim/85 Jun/85

Date

Fig. 5 Comparison between the HBV and the All-Fu zzy models at one of the

observation nodes.

error procedure and the associated large number of rules pertaining to the outflows

from the reservoirs resulted in difficulty in adjustment of the rales.

The entirely fuzzy logic-based model was found to reproduce the observed

discharges well. Generally, under low and normal flow conditions the model

performed well and no noticeable difference was observed between the HBV and the

fuzzy rule-based models. The fuzzy rule-based model was observed to overestimate

the peak flows (Fig. 5).

C O N C L U D I N G R E M A R K S

The applicability of a fuzzy logic-based approach to modelling catchment processes

has been illustrated. When using such an approach, a sound knowledge of the

underlying physical processes is not prerequisite.. Only knowledge of the factors that

influence a process and a qualitative relationship between the cause and effect is

required. Since the mathematical relationship between the cause and effect is not

necessary, quantities that are not explicitly included in other conceptual or physically

based models because of the nature of idealization of the process in the models or

unavailability of a known relationship can easily be included as rule arguments in a

fuzzy rule-based modelling.

Parameters are normally required to be determined by field measurements or

estimated through model calibration in other types of models. Each parameter can have

different values for different zones of the model area. No model parameters are


14/14

376

Yeshewatesfa H undecha et al.

considered in a fuzzy rule-based modelling approach. This makes the approach easier

and faster to work with.

REFERENCES

Abebe, A. I , Solomatine, D. P. & Venneker, R. G. W. (2000) Application of adaptive fuzzy rule-based models for

reconstruction of missing precipitation events.

Hydrol. Sci. J.

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Anderson, E. A. (1968) D evelopment and testing of snowpack energy balance equations.

Wat. Resour. Res.

4(1), 19-38.

Brdossy, A. & Disse, M. (1993) Fuzzy rule-based models for infiltration. Wat. Resour. Res. 29, 3 7 3 -3 8 2 .

Brdossy, A. & Duckstein, L. (1995)

Fuzzy Rule-Base d Modeling w ith Applications to Geoph ysical, Biological, an d

Engineering Systems.

CRC Press , Inc., Boca Raton, Florida, USA.

Bergstrom, S. (1995) The HBV model. In:

Computer Models of Watershed Hydrology

(ed. by V. P. Singh), 443^176.

Water Resources Publications, Litt leton, Colorado, USA .

Gupta, H. V. & Sorooshian S. (1998) Toward improved calibration of hydrologie models: multiple and noncomm ensurable

measures of information.

Wat. Resour. Res.

34(4), 751-763.

Prakash, A. (1986) Current state of hydrologie modeling and its limitations. In: Proc. Int. Symp. on Flood Frequency and

Risk Analysis

(14-1 7 May 1986). Louisiana State University, Baton Rouge, Louis iana /US A.

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Hydrol. Sci. J.

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778.

See,

L. & Openshaw, S. (2000) A hybrid multi-model approach to river level forecasting.

Hydrol. Sci. J. 45(4),

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Sugeno, M. & Y asukawa, T. (1993) A fuzzy-logic-based approach to qualitative modeling. IEEE Trans, on Fuzzy Systems

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7 - 3 1 .

Welstead, S. T. (1994)

Neural Network and Fuzzy Logic Applications in C/C++.

John Wiley & Sons Inc., New York,

USA.

Zadeh, L. A. (1965) Fuzzy sets.

Information and Control

8(3), 338-353 .

Zadeh, L. A. (1973) Outline of a new approach to the analysis of complex systems and decision processes.

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Systems, Man, and Cybernetics 3 , 28-44 .

Received

6 June 2000;

accep t ed

20 December 2000

Developement of Fuzzy Rainfall Runoff...

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