Page 212 - Geochemical Anomaly and Mineral Prospectivity Mapping in GIS

214 Chapter 7

beyond 2 km of FI to be considerably less prospective than locations within 2 km of FI.
The fuzzy membership function in equation (7.6) or Fig. 7-11B is, thus, apparently
consistent with the conceptual knowledge-based representation of spatial evidence
illustrated in Fig. 7-8.
Another example of a universe of discourse Y in mineral prospectivity mapping is a
variety of geochemical anomalies. A set of fuzzy evidence from this Y is multi-element
stream sediment anomalies defined by, say, catchment basin analysis (hereafter denoted
as ANOMALY). Based on the results of analysis of spatial association between
ANOMALY and epithermal Au deposit occurrences in the case study area (see Figs. 6-
12E and 6-12F), the following membership function may be defined for the fuzzy set
‘favourable ANOMALY’ (g):

0
­ 0 for y < 14
.
.
μ ( = ) y ° ) . 0 ( 34 − 14.0 ) for . 0 14 ≤y ≤ 34 (7.7)
® − 14.0(y
0
g
°
.
0
¯ 1 for y > 34

where y represents values of ANOMALY scores. The graph and generic form of this
function are illustrated in Fig. 7-12A. The parameters of the function (i.e., 0.14 and 0.34,
which are α and γ, respectively, in Fig. 7-12A) represent (a) the maximum of the range
of ANOMALY scores (e.g., 0.14; see Figs. 6-12E and 6-12F) considered to be
completely non-significant and (b) the minimum of the range to ANOMALY scores with
optimum positive spatial association with the epithermal Au deposit occurrences (i.e.,
0.34; see Figs. 6-12E and 6-12F). The fuzzy membership function in equation (7.7) or
Fig. 7-12A is linear and, thus, inconsistent with the conceptual knowledge-based
representation of spatial evidence illustrated in Fig. 7-8. Alternatively, the following
membership function may be defined for the set ‘favourable ANOMALY’ (g):

­ 0 for y < 14
.
0
° [ − 14.0(x ) . 0 ( 34 − 14 ] ) 2 for . 0 14 ≤y ≤ β
0
.
®
μ ( = ) y ° 2 . (7.8)
g
0
.
0
.
° 1 [ − 0( . 34 − ) x . 0 ( 34 − 14 ] ) for β ≤y ≤ 34
0
.
0
− )(max
.
¯ [ ° .0 1 (max y − 34 ) ]+ 9.0 for y > 34

The graph and generic form of this function are illustrated in Fig. 7-12B. The function in
equation (7.8) consists of a linear part (i.e., the last condition) and a continuous nonlinear
part (i.e., the first three conditions). The former is called a right-shoulder S function
+
(denoted as S in Fig. 7-12B). The parameters of the fuzzy membership function in
+
equation (7.8) are the same as those of the function in equation (7.7). However, the S
function in equation (7.8) requires another parameter, β, which is a value of y that forces
the function to equal the cross-over point (i.e., fuzzy membership equal to 0.5; Fig. 7-
12B). Choosing a suitable value of y to represent β requires expert judgment. The
median of the range of values between α and γ could, for example, be chosen for β. So,