INTRODUCTION
Water supply scheming as well as design and management of water reservoirs
essentially relies on the competence of the process of hydrologic modeling
adopted. For hydrologist, flood protection engineer and reservoirs operator
who concern with surface water planning for an arid and semiarid region
like central and southeastern Iran, where water is regarded as the essence
of existence for human beings and their related economic and agricultural
activities, forming a mathematical model so as to understand and simulate
the relationship between rainfall and runoff events takes on all shades
of meaning. The socalled RainfallRunoff (RR) process comprises of motion
of rainfall through different terrestrial settings and its transformation
into the runoff in natural or artificial canals. In the context of hydrology,
immense variations of watershed attribute and precipitation patterns over
time and location as well as interaction of numerous variables involved,
render the RR relation a highly complicated phenomenon to be tackled and
therefore make the conduction of a simple, quick and credible RR modeling
program a challenge to the hydrologists. The literature contains numerous
papers and contributions investigating the applicability and potentials
of Artificial Neural Networks (ANN) modeling approaches in the area of
RR modeling and time series forecasting (Lachtermacher and Fuller, 1994;
Hsu et al., 1995; Smith and Eli, 1995; van den Boogard et al.,
1998; Tokar and Johnson, 1999; Zealand et al., 1999; Luk et
al., 2000; Coulibaly et al., 2000; Toth et al., 2000;
Deo and Thirumalaiah, 2000; Thirumalaiah and Deo, 2000; Zhang and Govindaraju,
2000; Sudheer et al., 2003; Wilby et al., 2003). A comprehensive
review on the application of ANNs to hydrology can be found in ASCE Task
Committee on Application of Artificial Neural Networks in Hydrology (Govindaraju,
2000 a, b) and in Maier and Dandy (2000). Having utilized as alternative
tools in constructing nonlinear system theoretic models of the hydrological
processes, ANNs act as fundamentally semiparametric regression estimators
and can almost approximate any measurable function up to an arbitrary
degree of accuracy (Hornik, 1991). Although ANNs have confirmed many promising
results in hydrologic modeling and water resources simulation, their utilization
needs for special attention in certain cases viz., when the data are noisy
and show a large scatter over measured ranges. In such cases the performance
of all neural network structures seems to be of equal quality and relatively
unsatisfactory. This thus poses definite limitations on their application
for an accurate and suitable RR modeling program. Besides, choice of optimal
network structure, transfer functions or network type and training algorithm,
which devours considerable amounts of data in order to find the patterns
embedded in the system, require relatively great computing efforts and
trial and error cycles as well, making the modeling procedure timeconsuming
and tedious. To eliminate these restrictions, a fuzzy methodology approach
seems to be practical and promising. Zadeh (1965) introduced the theory
of fuzzy sets as an extension for conventional set theory where the membership
of an object to a set is restricted to one or zero. A fuzzy set, however,
is defined by assigning a membership grade from the interval of [0, 1]
for every article that its belonging to the set is the subject of the
question. Such a definition is very suggestive because it allows the representation
of concepts of interest as qualitative; that is, fuzzy methodology is
capable of dealing with vague or imprecise inputs from designers and human
experts who describe each system variable with some linguistic terms,
such as high, low, nearly 10, etc. These linguistic terms can mathematically
be represented by designating suitable membership functions
and be manipulated on the basis of fuzzy set theory. Indeed, fuzzy sets
theory has led to evolution of two distinct basic territories in modern
mathematics; while fuzzy arithmetic deals with fuzzy numbers and their
generalized arithmetical operations such as fuzzy regression on the basis
of Zadeh`s extension principle (Zimmermann, 1996), fuzzy logic employs
fuzzy sets and their associated membership functions as foundations for
carrying out the process of approximate reasoning, resembling a human
expert`s manner. Fuzzy methodology has proved quite useful applications
in different disciplines such as decisionmaking problems (Zimmermann,
1996), mechanical design (Wood and Antonsson, 1989), control engineering
(Nguyen et al., 2003), etc.
Fuzzy models exhibit a number of advantages compared to global nonlinear
models, such as neural networks. The model structure is easy to comprehend
and is in some cases interpretable. Integration of
knowledge of diverse types including statistical data and empirical knowledge
can be managed within the model structure with ease. Moreover, fuzzy models
have effectively substantiated tolerance for input data being imprecise
i.e., represented as intervals (fuzzy numbers) or vague i.e., depicted
as fuzzy sets.
The present study investigates the ability of fuzzy regression and fuzzy
logic in modeling RR process for Hali Rud River watershed, central Iran.
The predicted results are in good agreement with measured data.
FUZZY SETS
Membership functions serve as the foundations for any fuzzy approach
program, demonstrating the degree of belonging of a given object or value
from the spectrum of possible objects or values over a universe of discourse
to an arbitrary fuzzy subset; the meticulous act of their selection is
considered to be application dependent and subjective. They can be specified
via either human expert`s knowledge or automated techniques such as data
mining tools and methodologies.
The membership degree of article x to fuzzy set A is designated
by A(x) or byμ for short. The closer the magnitude of A(x) to unity
(zero), the higher (less) membership degree of x in A. There exist
some standard forms of membership functions that are suggested for engineering
purposes. Of simple ones, the triangular membership function shown in
Fig. 1 is nothing more than an arrangement of three
points creating a triangle. An arbitrary fuzzy set with a triangular membership
function can then be denoted by Ã = T(p, q, r) where p, q and r
correspond to minimum, mean and maximum values of the parameter of interest,
respectively. Due to its smoothness, symmetry and owning nonzero values
for the input range, Gaussian membership function shown in Fig.
1 is also a popular operator for designating a fuzzy set and specified
by two parameters σ and c as follows:
where, c and σ are mean value and standard deviation of the Gaussian
function, respectively. A fuzzy set with a Gaussian membership function
can then be denoted as Ã = G(σ, c) .
As Fig. 1 shows, fuzzy sets do not own crisp and clearly
defined boundaries, keeping values with only a partial grade of membership.
If membership grades encompass a central value or range in which the membership grade reaches to unity the fuzzy set is referred to as a normal
fuzzy set. As an example consider a vague description of high cost. A
system experts picks out a membership grade to express to what degree
he believes a given value from all of the possible ones, for example the
cost of 3000 dollars, can be classified as high. Now, consider the inexact
phrase of around 14. Somewhat different from a vague concept modeling
situation where the central value(s) can be uncertain and the assignment
of membership grades is subjected to the expert`s judgment, the entire
range of a word number or a fuzzy number is determined by an interval
whose normal value is that central value, 14 here and the membership grades
are continuously decreased by approaching two sides of the interval where
memberships finally become zero. Basically, fuzzy numbers can be considered
as a special class of fuzzy sets showing some specific properties (Zimmermann,
1996). The uncertainty in determining the true and precise values of model
inputs or outputs can be casted as appropriate fuzzy numbers.
DATA AND AREA OF STUDY
Halil Rud River flows in Jazmoriyan basin, central Iran, where Jiroft
dam is located within the basin. Rainfall data used were the records of
five stations of the area named Soltani, Baft, Henjan, Cheshm Aros and
Meydan; Konarooey station data was used for discharging. Halil Rud River
watershed and location of rainfall and hydrometry stations are shown in
Fig. 2. Data base of the present study comprises of
34 floods data. The unit of all data is m^{3} sec^{1},
cms. In the following modeling steps, rainfall data were used as input
and runoff data as output.

Fig. 2: 
Halil Rud River watershed and location of rainfall and
hydrometry stations 
THE FUZZY RULEBASED MODEL
Organization of an RR simulation system can be delineated and evaluated
through rules derived from expert`s point of view corresponding to a specified
structure based on the measurement data. Compared to routine modeling,
fuzzy rulebased structures provide a robust tool which is directly able
to handle the semantic models of human interpretation of the system of
interest. Accordingly, some appropriate fuzzy expressions such as very
low (VL), low (L), medium low (ML), medium (M), medium high (MH), high
(H) and very high (VH), should firstly be appointed for RR modeling. The
modeling practice proceeds towards the mathematical presentation of those
fuzzy sets with some standard membership functions. As shown in Fig.
3 and 4, Gaussian shapes were chosen for the input
and output variables and then adjusted to finetune the modeling. The
standard deviation values for rainfall and runoff variables were fixed
on 9.48 and 15.1, respectively.
A generalized deductive reasoning scheme comprised of fuzzy ifthen rules
lends itself to model a highly nonlinear relationship that exists between
rainfall and runoff values. Every fuzzy rule is responsible for mapping
some part of the input space to suitable part of the output space. A fuzzy
rule base consists of fuzzy rules and accumulates what knowledge a system
expert has acquired through experience, experimental data and common sense.
The fuzzy rule base for the present model is formed by the following statements:

Fig. 3: 
Membership functions for rainfall variable 

Fig. 4: 
Membership functions for runoff variable 
Rule 1: If rainfall is VL, then runoff is VL
Rule 2: If rainfall is L, then runoff is L
Rule 3: If rainfall is ML, then runoff is ML
Rule 4: If rainfall is ML, then runoff is L
Rule 5: If rainfall is M, then runoff is L
Rule 6: If rainfall is M, then runoff is M
Rule 7: If rainfall is MH, then runoff is L
Rule 8: If rainfall is MH, then runoff is ML
Rule 9: If rainfall is H, then runoff is L
Rule 10: If rainfall is H, then runoff is ML
Rule 11: If rainfall is VH, then runoff is M
Rule 12: If rainfall is VH, then runoff is ML
Rule 13: If rainfall is VH, then runoff is L
As can be seen from the above rule base, this model contains some contradictory
rules showing the same antecedent and different consequence parts. This
conflict arises when data have largely scattered over their measured range.
These rules can also be included in the fuzzy model to make the simulation
action more effective and precise.
Ifthen rules can mathematically be interpreted by means of the socalled
Mamdani synthesis. The input variable is evaluated for each rainfall and
a truth value matched to the grade of membership of the input variable
in each predefined antecedent fuzzy set is computed. For every Mamdani
rule, minimum truth value of the antecedent then propagates through and
truncates the membership function for the consequent graphically. Since
the rules are disjunctive, every fired rule produces its own truncated
membership function in the consequent fuzzy sets. To make a single decision,
however, the rules must be integrated in some suitable style. Aggregation
is referred to the process by which the fuzzy sets that portray the outputs
of each rule are combined into a single fuzzy set. An aggregated membership
function comprised of the outer envelope of the individual truncated membership
shapes from each rule is created by taking maximum truth values of the
output results. Combination of mentioned logical connectors minimum and
maximum employed in the present fuzzy rulebased model is called maxmin
inference method. Generally, these connectors can also be replaced with
other standard functions such as product, probabilistic sum and other
agents in a similar way to the membership functions in order to improve
the performance of the model.
Completing the modeling procedure necessitates some mechanism to be capable
of converting the aggregated fuzzy set, which encompasses a range of output
values, to a single value. To do this, an appropriate defuzzification
method must be used. The adopted defuzzifier for the present model is
the bisector agent.

Fig. 5: 
Comparison of results of fuzzy rulebased model with
experimental data 
In this method a nonfuzzy value that divides the
area bounded by the final membership function curve into two equal segments
is yielded as the most typical crisp value of the union of all output
fuzzy sets. The results were acquired by making use of Fuzzy Logic Toolbox
integrated in MATLAB software. The membership functions, rules and mathematical
operators can be chosen step by step in the software. As can be shown
from Fig. 5, the predicted results and real data are
in good agreement with each other.
THE FUZZY REGRESSION MODELS
Fuzzy regression techniques are considered to be useful when it is known
that a causal relation exists, but only few data points are available
(Bardossy et al., 1990; Ozelkan and Duckstein, 2000). The correlated
equations for a fuzzy regression program can be written in the usual forms:
It should be pointed out that four parameters of the Eq.
2 can generally be fuzzy. When it is so, ~ symbol over the parameter
will hereafter represent the fuzzy number. In the context of modeling,
it is expected that neither the model and its parameters nor the data
are certain. Because the oversimplification of a modeling system has always
been considered as a great risk in the evaluation process the uncertainties
arisen from incomplete or imprecise information must be reflected in some
appropriate manner. The uncertainties in the model parameters, model inputs,
etc., can then be taken into account by fuzzy numbers with their shape
derived from experimental data or expert knowledge (Hanss, 1999). The
text continues on considering two different states, where in phase (1)
a, b and y and in phase (2) x (rainfall) and y (runoff), are taken as
fuzzy numbers. It is worthy of special mentioning that the developed fuzzy
regression models do not correspond to an ad hoc implementation and can be applied for any other hydrologic
modeling purposes, the only difference, however, will appear in the context
of measured data.
Possibility regression: Here, those models with fuzzy coefficients
of the regression are considered. The mathematical details and background
are well documented in Tanaka et al. (1982) and Yen et al.
(1999). Let the model be of the form:
where Ã_{0} and Ã_{1} correspond to the
fuzzy numbers of the following form:
Note that the subscripts 0 and 1 were ignored for the sake of simplicity
and the triangular fuzzy number can then be denoted by. Ã = T(aS^{L},
a, a,+S^{R}). S^{L} and S^{R} are called the left
and right spreads of the fuzzy number, respectively. Ã_{0}
and Ã_{1} must be determined in such a way that the fuzziness
in the output of fuzzy of all the observed responds becomes minimum. In
other words, it is desired that we could find the regression parameters
that have the smallest spreads around their central value. Mathematically,
it corresponds to the minimization action of the following target function:
which is subjected to the following constraints:
It is supposed that the regression parameters are symmetric fuzzy numbers,
that is
Also, note that there exists only one input of in the model. Thus the problem is:

Fig. 6: 
Possibility regression modeling of rainfall and runoff
variables 
which is subjected to the following refined constrains:
In the present study, QSB software was used and the desired parameters
viz. a_{0}, a_{1}, S_{0} and S_{1} obtained.
As can be shown from Fig. 6, the response of the possibility
regression modeling corresponds to a special region where nearly all data
are dispersed within. The lower and upper limits of this region depict
a fuzzy box i.e., the generalized concept of intervals for a twodimensional
problem. These limits possess the membership grade of zero. The line placed
in rainfallrunoff plane that possess the highest membership grade will
be given by this method. The projection of this line in rainfallrunoff
plane yields the average expected runoff values.
Least square regression with ordinary coefficients, but fuzzy data:
In this case, the regression model is written as the following from Diamond
(1998) and Ming et al. (1997):
where, the rainfall and runoff values are essentially considered as symmetric
triangular fuzzy numbers for simplicity and can then be represented by
respectively. For short, the triangular fuzzy numbers are now denoted
by the pair of (b, S)_{T} where b and S are meanvalue and spread of the membership function. Therefore, all
are taken the same and equal to S = 2.5. The values of a and b can be
obtained by minimizing,
Minimization action of the sum of square of distances between inputs
and estimated outputs according to the above equation leads to appearance
of the following equations:
Where:
andμ stands for membership grades of x_{i} and y_{i}. Indeed,
the inverse of triangular membership functions for lefthand and righthand
sides of rainfall and runoff variables are computed through Eq.
14ad. It is possible to find a^{+} and b^{+} from Eq.
13a, b and similarly a^{} and b^{} from Eq.
13c, d). It is obvious that b^{+}≥b^{} and to find the
feasible solutions, one has to check whether 0≤b^{}#b^{+}
or b^{}#b^{+}≤0. If b^{}#b^{+}≤0 then
the data are positively tight and thus the unique solution for the problem of
finding regression parameters in Eq.11 is the pair (a^{+},
b^{+}), otherwise the unique solution is (a^{}, b^{}).
The system of equations that appeared as Eq. 13ad
was solved and answers were obtained by making use of MAPLE package.
RESULTS AND DISCUSSION
In order to compare the obtained results from the different models considered
under the different assumptions, conditions and considerations, the Mean
of Relative Errors (MRE) for each model were computed. The predicted results
of fuzzy rulebase model show an MRE of 16.29%. This amount of error seems
to be acceptable because data has widely scattered. As mentioned before,
two fuzzy regression models were also developed and the results have been
expressed in the form of triangular numbers. Obtained fuzzy results for
the Possibility Regression (PR) and Fuzzy Least Squares Regression (FLSR)
are shown in Table 1.
In case of fuzzy regression models, different comparison criteria could
be considered according to operator`s point of view because those models
predict an interval of possible runoff values whose importance or degree of believeth in their occurrence would be determined by their
corresponding membership grades.
Table1: 
Predicted results with fuzzy regression models 

That is, fuzzy regression models provide
more informative results since they predict minimum, mean and maximum
expected runoff values and the operator can therefore obtain a wider perspective
over the situation. To have a consistent criterion, the average of predicted
runoff values was considered for computing MRE. The mean of relative errors
for PR and FLSR were 21.76 and 19. 38%, respectively. This means that
the results obtained by the use of fuzzy rulebase model show better correlation
with observed data.
CONCLUSIONS
With respect to importance of flood modeling in water resources management,
several models were developed by means of advanced fuzzy methodologies.
These novel models employ fuzzy logic and fuzzy regression techniques.
On the basis of data measured at the hydrometry stations, the models were
customized for Halil Rud River watershed. From the numerical results predicted
by the developed models, it can be pointed out that the adopted mathematical
approaches can effectively be used in flood simulation. While the fuzzy
rulebased model shows the lowest mean of relative errors, fuzzy regression
models are capable of predicting whole anticipated runoff values accompanied
by their respective grade of membership or possibility of occurrence on
the basis of observed data. The fuzzy rulebased model has the advantage
of flexibility and simplicity because the illdefined relation between
rainfall and runoff variables can be described in a semantic form. The
model is also considered as robust, that is, the performance do not depend
upon training phase and probable new input variables and rules can be
easily added. Fuzzy regression models are regarded more informative because
they can forecast the spectrum of possible results. The fuzzy methodology
succeeds in situations where data has broadly scattered. In such situations,
other global estimators like artificial neural networks seem to give results
of limited accuracy and precision.
ACKNOWLEDGMENTS
Authors would like to thank Dr. Ghezelayagh, department of mathematics
and statistics, Shahid Bahonar University of Kerman, Iran and Mr. Davoud
Oliaee for their continuous help in providing this manuscript. Suggestions
and comments from editors and anonymous reviewers are gratefully acknowledged.