Page 218 - Geochemical Anomaly and Mineral Prospectivity Mapping in GIS

220 Chapter 7

13A). Clearly, like the FA operator, the FAP is appropriate in combining complementary
sets of evidence, meaning that all input fuzzy scores at a location must contribute to the
output to support the proposition of mineral prospectivity, except in the case when at
least one of the input fuzzy scores is 0 (Fig. 7-13A). In contrast to the FA operator, the
FAP has a ‘decreasive’ effect (Figs. 7-13B and 7-13C), meaning that the presence of
very low but non-zero fuzzy scores tend to deflate or under-estimate the overall support
for the proposition.
The fuzzy algebraic sum (hereafter denoted as FAS) is defined as

n
−
μ FAS = 1 ∏ 1 ( μ i ) (7.12)
−
1 = i

where μ FAS is the output fuzzy score and μ i represents the input fuzzy evidential scores at
a location in i (=1, 2,…, n) evidence maps. The FAS is, by definition, not actually an
algebraic sum, whereas the FAP is consistent with its definition. The output of the FAS
is greater than or equal to the highest fuzzy score at every location (Fig. 7-13). So, for
example, if in one evidence map the fuzzy score at a location is 1 even though in at least
one of the other evidence maps the fuzzy score is 0, the output fuzzy score for that
location is still 1 (Fig. 7-13A). Clearly, like the FO operator, the FAS is appropriate in
combining supplementary sets of evidence, meaning that all input fuzzy scores at a
location must contribute to the output to support the proposition of mineral prospectivity,
except in the case when at least one of the input fuzzy scores is 1 (Figs. 7-13A and 7-
13B). In contrast to the FO, the FAS has an ‘increasive’ effect (Fig. 7-13C), meaning
that the presence of very high fuzzy scores (but not equal to 1) tend to inflate or over-
estimate the overall support for the proposition.
In order to (a) regulate the ‘decreasive’ effect of the FAP and the ‘increasive’ effect
of the FAS, meaning to constrain the range of the output values to the range of the input
values, or (b) make use of the ‘decreasive’ effect of the FAP or the ‘increasive’ effect of
the FAS, as is needed in order to derive a desirable output that is more-or-less consistent
with the conceptual model of mineral prospectivity, the fuzzy γ (hereafter denoted as
FG) operator can be applied. The FG operator is defined as (Zimmerman and Zysno,
1980)

n n
μ FG =( ∏μ ) 1 −Ȗ − 1 ( ∏ − ))1( μ i Ȗ (7.13)
i
= i 1 = i 1

where μ FG is the output fuzzy score and μ i represents the fuzzy evidential scores at a
location in i (=1, 2,…, n) evidence maps. The value of γ varies in the range [0,1]. If γ =
0, then FG = FAP. If γ = 1, then FG = FAS. Finding a value of γ that contains the output
fuzzy scores in the range of the input fuzzy scores, as illustrated in Fig. 7-14, entails
some trials. The example shown in Fig. 7-14 is based on only two input fuzzy scores. In
practise, however, the input fuzzy scores could come from two or more fuzzy evidence