Geoscience Reference
In-Depth Information
(a)
(b)
(c)
(d)
FIGURE 12.6
Three realisations of the geomorphometric extent of peaks in the vicinity of Helvellyn shown
in solid white against a hillshaded background. Analytical windows of (a) 11 × 11, (b) 21 × 21 and (c) 31 × 31
are shown together with (d) the sum of 20 different such approximations of the extent of peaks giving a fuzzy
representation of the extent of Helvellyn and other peaks. (After Fisher, P.F. et al.,
Trans. Inst. Br. Geogr.,
29,
10 6, 20 0 4.)
of Ben Nevis (Fisher et al., 2007a), the highest mountain in the British Isles. In this work, they
employed geomorphometric analysis to identify the locations of mountain peaks at multiple res-
olutions, deriving a number of different extents (Figure 12.6a through c). Averaging the results
of analysis yielded a fuzzy footprint for the peak (Figure 12.6d). Fisher et al. (2007b) used the
same approach to analyse the changes in crests of a coastal dune field being mapped as the
union
(Figure 12.7b) of fuzzy ridges and peaks.
12.6 FUZZY LOGIC
Having defined the memberships of fuzzy sets, just as with crisp sets, it is possible to execute logical
set operations directly comparable with the
union
,
intersect
and
inverse
of crisp sets (Figure 12.1).
These operators were first proposed by Zadeh (1965) in his original development of fuzzy set theory.
Although these
original
operators are often still used, many others have been developed since. For
an exhaustive list of operators, both parameterised and non-parameterised, the reader is referred to
Zimmermann's (2010) review of fuzzy set theory.
For two fuzzy sets
A
and
B
(Figure 12.7a), the fuzzy
union
is simply a maximum operation
(Equation 12.5; Figure 12.7b), taking the maximum value of μ for every measured value. From
