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5.3.2 Fuzzy land cover map
The fuzzy feature of the LPI map (land cover map) was calculated by the membership
functions illustrated in Fig. 4. Field observations allowed the identification of transition
zones on the vegetation cover (forest and pasture). The transition zones are covered by
brushwood, as well as by degraded forest with grass fields. The diffuse boundaries
observed in field were identified on the LPI map, so allowing proper membership function
parameters to be used. This procedure generated four fuzzy maps: forest and pasture fuzzy
maps using linear and bell-shaped membership functions.
5.3.3 Fuzzy rule-based modeling
In the fuzzy rule-based modeling, the relationships between variables are represented by
means of fuzzy
if-then
rules that assume the form:
If
x is A
then
y is B (6)
where x and y are linguistic variables, A and B are linguistic constants. The if-part of the
rule “x is A” is named the antecedent, while the then-part of the rule “y is B” is named the
consequent.
In this study, the Sugeno's method of fuzzy inference (Sugeno, 1985) was used to calculate
the CN of all cells in the watershed map. In this method the antecedent is a fuzzy
proposition and the consequent is a crisp function. Two typical fuzzy rules used in a Sugeno
fuzzy model will be demonstrated as an example:
If
x
1
is A
11
and
x
2
is A
12
then
y is B
1
(7)
If
x
1
is A
21
and
x
2
is A
22
then
y is B
2
(8)
where x
i
(i = 1, 2) is an input variable (e.g. soil, vegetation), y is an output variable (e.g. CN
parameter), A
ij
(i = 1, 2 and j = 1, 2) is a fuzzy set (e.g. high infiltration capacity, forest), and
B
i
is a number that represents the consequent of the rule.
If x
1
0
and x
2
0
are values assumed by x
1
and x
2
and A
ij
(x
i
0
) the grade of pertinence then the
consequent value (crisp function) is W
1
and W
2
:
W
1
= min(A
11
(x
1
0
), A
12
(x
2
0
))
(9)
W
2
= min(A
21
(x
1
0
), A
22
(x
2
0
))
(10)
where “min” denotes “minimum value of”. The global output y
0
, that can be the CN
parameter, is calculated by equation (11) (Kruse et al, 1994; Burrough, 1998):
y
0
= (W
1
B
1
+ W
2
B
2
)/(W
1
+ W
2
) (11)
The fuzzy inference system of the Fuzzy SCSCN model was accomplished through the
following steps: (i) transformation of the input data in a fuzzy set; (ii) application of the
fuzzy rules (Table 3); (iii) computation of the information associated to transition zones on
different soil and vegetation map units, using the Sugeno's method; (iv) generation of CN
raster maps with the CN values of all pixels of the studied watershed (Fig. 8); and (v) runoff
calculation.
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