From: FLAIRS-00 Proceedings. Copyright © 2000, AAAI (www.aaai.org). All rights reserved.
Erosion Modeling in a Raster-Based GIS with Fuzzy Logic
Hans W. Guesgen∗
Jochen Albrecht and Stefi Grundner
Computer Science Department
University of Auckland
Auckland, New Zealand
hans@cs.auckland.ac.nz
Department of Geography
University of Auckland
Auckland, New Zealand
j.albrecht@auckland.ac.nz
Abstract
In recent years, fuzzy logic has been applied successfully to a wide range of problems. This paper shows
how it can be utilized in the area of spatial reasoning, in particular geographic information systems. The
problem that we address here is the prediction of erosion from factors such as slope, annual precipitation,
etc. These factors are represented as maps in a GIS.
The maps are converted into fuzzy sets and then given
as input to a fuzzy rule-based system, which produces
an erosion map as output.
Introduction
Erosion of the surface of the earth, caused by gravity, wind, water, and ice, is a natural geological phenomenon, which sculptures our land and forms new
sedimentary rocks. In inhabited regions of the world,
erosion processes are often caused by human influences,
like modification to the vegetative cover by farm practices. Erosion is a diffuse process and involves many
different parameters. It is the result of the interplay
between environmental factors such as soil, topography,
drainage, rainfall, and land use patterns. Due to the
large number of parameters involved, erosion modeling
can be quite complex and therefore often requires support by a computer system, which in many cases is a
geographic information system (GIS).
Geographic information systems have been used for
quite a while (Coppock & Rhind 1991), but the functionality of the systems has changed only little. In spite
of their name, geographic information systems have
so far been mostly geometric in nature, ignoring the
thematic and temporal dimensions of geographic features (Molenaar 1996; Sinton 1978; Usery 1996). Various attempts to overcome these limitations are documented in a number of disciplines. (Frank 1992;
Goodchild 1992; Gupta, Weymouth, & Jain 1991;
Herring 1991; 1992; Raper & Maguire 1992) deal with
extensions of the data model, while Allen’s work and
∗
Partly supported by the University of Auckland Research Committee through various grants (most recently
through grant number XXX/9343/3414100).
c 2000, American Association for Artificial InCopyright telligence (www.aaai.org). All rights reserved.
its derivatives (Allen 1983; Freksa 1990; Guesgen 1989;
Hernández 1991; Mukerjee & Joe 1990) form the basis for numerous temporal logic endeavors that deal
with dynamic aspects of geographic information (Egenhofer & Golledge 1997; Frank 1994; Peuquet 1994).
Applications of fuzzy techniques are most commonly
found in remote sensing literature but (Altmann 1994;
Brimicombe 1997; Jiang & Eastman 2000; Molenaar
1996; Papadias, Karacapilidis, & Arkoumanis 1999;
Plewe 1997; Stefanakis, Vazirgiannis, & Sellis 1999;
Woodcock & Gopal 2000) provide examples that the
inherent fuzziness of geographic features becomes increasingly acknowledged in geographic information science as well.
A standard way in which a GIS performs spatial reasoning, i.e., the extraction of new information from
stored spatial data, is through map overlap. New maps
are computed from existing ones by applying one of the
following operations:
• Buffer operation, which increases the size of an object
by extending its boundary.
• Set operations, such as union, intersection, and complement.
These operations are usually quantitative operations.
On the other hand, humans often prefer an inexact,
vague, uncertain, or qualitative analysis over a quantitative one, as this is more adequate in many cases
from the cognitive point of view (Clementini, Di Felice, & Hernández 1997). By using fuzzy logic, we are
able to introduce vagueness into traditional GIS. We
will discuss in the next section how erosion modeling
can benefit from such an introduction.
Erosion Factors and Vagueness
The division of natural continua such as soil, vegetation, or climate into classes, and the assignment of new
observations to classes, is problematic, as the boundary
values that delimit the classes are usually sharply defined, which implies that all changes between classes occur at the class boundaries and that there is no change
within a class. The division into classes neglects the
fact that spatial changes in the field may occur gradually over distances that are substantially greater than
Fuzzy Inference and Defuzzification
Fuzzy inference uses fuzzy rules to deduct new information from given information. In our case, a typical
fuzzy rule looks as follows:
IF
THEN
Figure 1: Membership functions for the precipitation
map.
Factor
Slope
Precipitation
Vegetation
Soil
Rock
low
0.7
0.8
0
0
0.4
Fuzzy set
medium high
0.3
0
0.2
0
0.1
0.9
0.5
0.5
0.6
0
Table 1: Example of an assignment of membership
grades.
the narrow zone covered by the thin boundary. By using fuzzy sets rather than crisp sets to define classes, it
is possible to have a vague or fuzzy definition of a class.
The factors from which we compute the fuzzy sets
for modeling erosion include slope, annual precipitation rate, rock, vegetation, and soil. These factors are
represented as maps in a raster-based GIS. The precipitation map, for example, is a grid in which the cell
values range from 1353 to 2994 mm/year. From these
maps, we compute various fuzzy classes, like the fuzzy
sets for high, medium, and low annual precipitation,
by applying fuzzy membership functions. Such a function maps the values given by a map to membership
grades between 0 and 1. In Figure 1, the membership
functions for the precipitation map are shown. Similar
functions are defined for the slope, vegetation, soil, and
rock maps. Since the factors are all derived from map
layers that have been compiled by domain experts, we
chose the semantic import approach (Gupta, Knopf, &
Nikiforuk 1988) for determining the appropriate membership function. The membership functions are designed in such a way that a membership grade of zero
is assigned to a cell of the map by at least one membership for each factor. Table 1 shows an example of a
possible assignment of membership grades. Since each
of the five erosion factors is associated with three membership functions, we obtain a total of 3 × 5 = 15 fuzzy
sets. These fuzzy sets constitute the input to our fuzzy
inference system.
Slope = high
Precipitation = high
...
Erosion = high
The IF-part of a fuzzy rule is interpreted as conjunction,
i.e., as intersection of the underlying fuzzy sets.
There are several ways of computing the membership
grade for the THEN-part of a fuzzy rule. In our system, the minimum of the membership grades in IF-part
of the rule is computed and taken as the membership
grade of the THEN-part. For example, to determine
the degree to which a particular cell of the grid has
a high erosion potential (according to this particular
fuzzy rule), we look up the the membership grades for
that cell in the fuzzy sets for high slope, high precipitation, etc. and compute the minimum of these grades.
Since there might be other fuzzy rules with the same
THEN-part, we have to repeat the computation for the
other rules. Since the fuzzy rules are viewed in a disjunctive way, the final value for the cell under consideration is the maximum of the membership grades from
the individual rules. Figure 2 illustrates this inference
process graphically.
After applying the fuzzy rules to the whole map or the
part of it that we are interested in, we have to translate
the fuzzy sets for the three erosion categories back to
quantitative values. This process is called defuzzification. There are several defuzzification methods, which
are of different complexity and have different impact on
the results (Driankov, Hellendoorn, & Reinfrank 1996).
The height defuzzification, which we use in our system,
is a very simple and quick method. It computes the
weighted sum of the membership grades in the three
erosion fuzzy sets, where the weights are given by the
values that maximize the membership functions. For
the erosion fuzzy sets low, medium, and high, these
values are 1, 2, and 3, respectively.
Implementation and Evaluation
We implemented our system in ARC/INFO-GRID, a
geographic information system, which manipulates, analyzes, and displays geographic information using cellbased data. As study area, we chose the Hokianga region in the north of New Zealand. There are two reasons
for this choice:
1. Due to the high variability in elevation and the high
precipitation rates, the Hokianga region is particularly affected by erosion.
2. Various GIS data is available for that region, including topographic data as well as rainfall data, which
can be transferred into grid format in a more or less
straightforward way.
Figure 2: Applying fuzzy rules to compute the erosion fuzzy sets.
After applying the fuzzy rules, we converted the resulting map into a map with six (rather than three) erosion
classes to make it comparable to erosion data that we
obtained from the New Zealand Land Resource Inventories (NZLRI). The comparison of the maps showed
similarities but also differences (see Figure 3).
The differences are particularly evident in the southern part of the Hokianga harbor, where one can recognize patterns emerging from the slope map. This is due
to the fact that the fuzzy rules put particular emphasis
on the slope factor—something that has to be adjusted
in a revised version of the system.
Similarities between the two maps are obvious in the
northern, the very west part, and the southeast regions
of the Hokianga area, which fall into the class of negligible erosion. Some similarities between the maps can
be detected in the southern and northwest part of the
study area, which belong to the class of slight erosion.
Large discrepancies can be found in areas of higher erosion potential, where our map categorizes more cells
as cells with extreme erosion potential than the erosion map of the NZLRI—albeit much less localized, i.e.,
smaller areas spread throughout the study region.
The discrepancies between the two maps are not very
surprising, as the methodologies applied for the design
of both maps are based on very different concepts. The
erosion map from the NZLRI was created by surveying the landscape and incorporating the frequency and
extent of visible erosion as well as background factors
such as feasibility and cost of control and economic effect. The erosion map developed with the method described in this paper, on the other hand, is based on
data that has significant influence on erosion to occur,
such as slope, precipitation, soil, vegetation, and rock
type. Therefore, it can be considered as contributing
to the potential of erosion, which does not necessarily mean that erosion is actually occurring at locations
with high potential. The dissimilarities in the maps
were to be expected, and therefore comparisons between
them have to be undertaken very carefully.
Summary
In this paper, we described how fuzzy logic can be applied to produce information about erosion from data
about slope, annual precipitation, vegetation, rock, and
soil. The input information is stored in a raster-based
GIS as maps, each of which is converted into three fuzzy
sets. The resulting fuzzy sets are the input to a inference system based on fuzzy rules, which produces three
fuzzy sets corresponding to a low, medium, and high
potential of erosion, respectively. These are defuzzified
in order to obtain an erosion map with quantitative
Figure 3: Erosion severity map provided by the NZLRI (top) and erosion potential map derived by applying fuzzy
logic (bottom).
data.
As our evaluation has shown, there are significant
similarities between the map that was produced by our
system and a reference map taken from the NZLRI, but
also significant discrepancies. Some of these discrepancies have to be investigated further and might lead to
revisions of the fuzzy rules that underly our system.
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