Geoscience Reference
In-Depth Information
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d
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d
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d
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(a)
(b)
FIGURE 12.4
Two simple examples of fuzzy membership functions: (a) linear and (b) trapezoidal. (From
Fisher, P.F., Fuzzy modelling, in:
GeoComputation
, Openshaw, S. and Abrahart, R., eds., Taylor & Francis
Group, London, U.K., 2000, pp. 161-186.)
Examples of DA of fuzzy memberships are shown in Figure 12.5. Figure 12.5a shows major
roads and the distance away from them in an area close to Leicester (data from Langford, 1993).
Figure 12.5c shows one possible fuzzy membership function that has been used to transform
these explicit distances into a fuzzy membership map of proximity to major roads (Figure 12.5b).
Likewise, Figure 12.5e is a map of slopes based on a rather poor DEM with many ghost contours.
Figure 12.5d presents a fuzzy membership function that has been used to transform these elevations
into a map of
gentle
slopes (Figure 12.5f).
A widespread classification, in particular, of European and North American landscapes is the
distinction between urban and rural, but where does
rural
end and
suburban or even urban
begin?
This could be defined in a number of different ways but it is easy to see that any definition is
arbitrary and susceptible to a
sorites
argument. Ban and Ahlqvist (2009) use the DA of fuzzy
memberships to geographical and census data to generate a fuzzy view of the membership of
raster grid cells in the set of exurban areas (i.e. a region lying beyond the suburbs of a city). They
examined published definitions of exurbia and present six different definitions for the identifica-
tion of the set of census areas which are in an exurbia class. These use eight different membership
functions which combine a mixture of socio-economic and spatial variables including population
density, distance from the metropolitan statistical area (MSA), ethnicity, commuting time to the
MSA and housing density. Simple mathematical functions were then used to assign membership
values to grid cells, and the different memberships were then fused through a weighted linear
combination (which they call a convex combination after Burrough, 1989), giving a mapping of
exurban areas of Ohio. Of course, the extent of the MSA itself is a
iat
definition by the US Census
Bureau. A similar approach to selecting the membership function based on
the literature
is used
by Zeng and Zhou (2001).
The choice of membership function can also be based on the opinions of experts through
direct interviews (DeGenst et al., 2001). Even when membership functions are available, having
been computed or assigned during an analysis operation as part of a GIS, the threshold values
(
d
1
,
d
2
, etc.) and the form of the function must still be decided by experts perhaps in consultation
with technical experts or directly decided in an automated geospatial environment (Yanar and
Akyurek, 2006).
Many researchers have argued that the extent of a soil is poorly defined (Burrough, 1989;
McBratney and De Gruijter, 1992) by either location or attribute. Soils change continuously in
space, forming gradations, and only rarely are those gradations abrupt. Thus, the concept of a
soil map as a set of well-defined regions separated by distinct boundaries does not match the real-
world occurrence of soil characteristics. Furthermore, the soil frequently changes at a high spatial
frequency and so is poorly captured by the crisp map. Lagacherie et al. (1996) and Wang and Hall
