Page 216 - Geochemical Anomaly and Mineral Prospectivity Mapping in GIS

218 Chapter 7

modeling. We now turn to the stage of logical integration of fuzzy evidence with the aid
of an inference network and appropriate fuzzy set operations (Fig. 7-8).
As in classical or Boolean set theory, set-theoretic operations can be applied or
performed on fuzzy sets or fuzzified evidential maps. There are several types of fuzzy
operators for combining fuzzy sets (Zadeh, 1965, 1973, 1983; Thole et al., 1979;
Zimmerman, 1991). Individual fuzzy operators, all of which have meanings analogous to
operators for combining classical or crisp sets, portray relationships between fuzzy sets
(e.g., equality, containment, union, intersection, etc.). There are five fuzzy operators,
which are in fact arithmetic operators, that are useful for combining fuzzy sets
representing spatial evidence of mineral prospectivity, namely, the fuzzy AND, fuzzy
OR, fuzzy algebraic product, fuzzy algebraic sum and fuzzy gamma (γ) (An et al. (1991;
Bonham-Carter, 1994). These fuzzy operators are useful in mineral prospectivity
mapping in the sense that each of them or a combination of any of them can portray
relationships between sets of spatial evidence emulating the conceptualised model of
inter-play of geological processes involved in mineralisation. Thus, the choice of fuzzy
operators to be used in combining fuzzy sets of spatial evidence of mineral prospectivity
must be consistent with the defined conceptual model of mineral prospectivity (see Fig.
1-3).
The fuzzy AND (hereafter denoted as FA) operator, which is equivalent to the
Boolean AND operator in classical theory, is defined as

μ FA = MIN (μ 1 ,μ 2 ,! ,μ n ) (7.9)

where μ FA is the output fuzzy score and μ 1, μ 2,…, μ n are, respectively, the input fuzzy
evidential scores at a location in evidence map 1, evidence map 2,…, evidence map n.
The MIN is an arithmetic function that selects the smallest value among a number of
input values. The output of the FA operator is, therefore, controlled by the lowest fuzzy
score at every location (Fig. 7-13). So, for example, if in one evidence map the fuzzy
score at a location is 0 even though in at least one of the other evidence maps the fuzzy
score is 1, the output fuzzy score for that location is still zero (Fig. 7-13A). Clearly, the
FA operator is appropriate in combining complementary sets of evidence, meaning that
the pieces of evidence to be combined via this operator are deemed all necessary to
support the proposition of mineral prospectivity at every location.
The fuzzy OR (hereafter denoted as FO) operator, which is equivalent to the Boolean
OR operator in classical theory, is defined as

μ FO = MAX (μ 1 ,μ 2 ,! ,μ n ) (7.10)

where μ FO is the output fuzzy score and μ 1, μ 2,…, μ n are, respectively, the input fuzzy
evidential scores at a location in evidence map 1, evidence map 2,…, evidence map n.
The MAX is an arithmetic function that selects the largest value among a number of input
values. The output of the FO operator is, therefore, controlled by the highest fuzzy
scores at every location (Fig. 7-13). So, for example, if in one evidence map the fuzzy