Fuzzy Hydrologic Model in Tropical Watershed - Advances in Data, Methods, Models and Their Applications in Geoscience - page 291

The width of the transition zones was chosen based in the scale of the map, according to
indications provided by Lagacherie et al. (1996) and Burrough and McDonnell (1998).
Burrough and McDonnell (1998) exemplified that a sharp boundary drawn as a 0.2 mm-
thick line on a 1:25,000 scale map covers 50 m (25 m to the right and 25 m to the left from
drawn boundary) and a diffuse boundary at the same scale might extend over 500 m. In this
study, sharp boundaries were drawn as a 1 mm-thick line on a 1:100,000 scale map, so that
the width of the spatial transition zone centered over the drawn boundary location was 200
m. Only the uncertainty related to the drawn boundaries of the map was used, although the
transition zones verified in the field show larger extensions.
In order to model fuzzy transition zones, the computer model involved the following steps:
(i) separation of each soil unit (polygon boundary) in different map layers; (ii) isotropic
spread of the boundary of each polygon (inside and outside the polygon); (iii) application of
a membership function. Each soil unit was considered as a fuzzy set A = {x, µ A (x)}. In this
case, x denotes a point in geographic space that belongs to A, and µ A (x) is a number that
ranges from 0 to 1 and reflects the grade of membership of x in A. The fuzziness of the
boundary between soil units A and B were indicated by both distributions of grades of
membership, µ A (x) and µ B (x) (Fig. 5). Points located far enough from the boundary have
either µ A (x) = 1 and µ B (x) = 0 (if x is contained in A) or µ B (x) = 1 and µ A (x) = 0 (if x is
contained in B). Points located close to the boundary µ A (x) and µ B (x) have values between 0
and 1; (iv) the procedure was repeated for each soil unit, yielding a soil boundary fuzziness
map for each soil unit. The width of the transition zone can be defined by the user before the
computer program is run. Fig. 6 and Fig. 7 show the boundary of organic soil using the
fuzzy and boolean model, respectively.
Fig. 5. Linear membership function and illustration of the methodology employed in the
conversion of crispy soil data to fuzzy soil data