Erosion Modeling in a Raster-Based GIS with Fuzzy Logic

Hans W. Guesgen∗

Computer Science DepartmentUniversity of AucklandAuckland, New Zealandhans@cs.auckland.ac.nz

Jochen Albrecht and Stefi GrundnerDepartment of Geography

University of AucklandAuckland, New Zealand

j.albrecht@auckland.ac.nz

Abstract

In recent years, fuzzy logic has been applied success-fully to a wide range of problems. This paper showshow it can be utilized in the area of spatial reason-ing, in particular geographic information systems. Theproblem that we address here is the prediction of ero-sion 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 givenas input to a fuzzy rule-based system, which producesan erosion map as output.

IntroductionErosion of the surface of the earth, caused by grav-ity, wind, water, and ice, is a natural geological phe-nomenon, which sculptures our land and forms newsedimentary rocks. In inhabited regions of the world,erosion processes are often caused by human influences,like modification to the vegetative cover by farm prac-tices. Erosion is a diffuse process and involves manydifferent parameters. It is the result of the interplaybetween environmental factors such as soil, topography,drainage, rainfall, and land use patterns. Due to thelarge number of parameters involved, erosion modelingcan be quite complex and therefore often requires sup-port by a computer system, which in many cases is ageographic information system (GIS).

Geographic information systems have been used forquite a while (Coppock & Rhind 1991), but the func-tionality of the systems has changed only little. In spiteof their name, geographic information systems haveso far been mostly geometric in nature, ignoring thethematic and temporal dimensions of geographic fea-tures (Molenaar 1996; Sinton 1978; Usery 1996). Var-ious attempts to overcome these limitations are doc-umented in a number of disciplines. (Frank 1992;Goodchild 1992; Gupta, Weymouth, & Jain 1991;Herring 1991; 1992; Raper & Maguire 1992) deal withextensions of the data model, while Allen’s work and

∗Partly supported by the University of Auckland Re-search Committee through various grants (most recentlythrough grant number XXX/9343/3414100).Copyright c© 2000, American Association for Artificial In-telligence (www.aaai.org). All rights reserved.

its derivatives (Allen 1983; Freksa 1990; Guesgen 1989;Hernandez 1991; Mukerjee & Joe 1990) form the ba-sis for numerous temporal logic endeavors that dealwith dynamic aspects of geographic information (Egen-hofer & Golledge 1997; Frank 1994; Peuquet 1994).Applications of fuzzy techniques are most commonlyfound in remote sensing literature but (Altmann 1994;Brimicombe 1997; Jiang & Eastman 2000; Molenaar1996; Papadias, Karacapilidis, & Arkoumanis 1999;Plewe 1997; Stefanakis, Vazirgiannis, & Sellis 1999;Woodcock & Gopal 2000) provide examples that theinherent fuzziness of geographic features becomes in-creasingly acknowledged in geographic information sci-ence as well.

A standard way in which a GIS performs spatial rea-soning, i.e., the extraction of new information fromstored spatial data, is through map overlap. New mapsare computed from existing ones by applying one of thefollowing operations:• Buffer operation, which increases the size of an object

by extending its boundary.• Set operations, such as union, intersection, and com-

plement.These operations are usually quantitative operations.On the other hand, humans often prefer an inexact,vague, uncertain, or qualitative analysis over a quan-titative one, as this is more adequate in many casesfrom the cognitive point of view (Clementini, Di Fe-lice, & Hernandez 1997). By using fuzzy logic, we areable to introduce vagueness into traditional GIS. Wewill discuss in the next section how erosion modelingcan benefit from such an introduction.

Erosion Factors and VaguenessThe division of natural continua such as soil, vegeta-tion, or climate into classes, and the assignment of newobservations to classes, is problematic, as the boundaryvalues that delimit the classes are usually sharply de-fined, which implies that all changes between classes oc-cur at the class boundaries and that there is no changewithin a class. The division into classes neglects thefact that spatial changes in the field may occur gradu-ally over distances that are substantially greater than

From: FLAIRS-00 Proceedings. Copyright © 2000, AAAI (www.aaai.org). All rights reserved.

Figure 1: Membership functions for the precipitationmap.

Fuzzy setFactor low medium high

Slope 0.7 0.3 0Precipitation 0.8 0.2 0Vegetation 0 0.1 0.9

Soil 0 0.5 0.5Rock 0.4 0.6 0

Table 1: Example of an assignment of membershipgrades.

the narrow zone covered by the thin boundary. By us-ing fuzzy sets rather than crisp sets to define classes, itis possible to have a vague or fuzzy definition of a class.

The factors from which we compute the fuzzy setsfor modeling erosion include slope, annual precipita-tion rate, rock, vegetation, and soil. These factors arerepresented as maps in a raster-based GIS. The pre-cipitation map, for example, is a grid in which the cellvalues range from 1353 to 2994 mm/year. From thesemaps, we compute various fuzzy classes, like the fuzzysets for high, medium, and low annual precipitation,by applying fuzzy membership functions. Such a func-tion maps the values given by a map to membershipgrades between 0 and 1. In Figure 1, the membershipfunctions for the precipitation map are shown. Similarfunctions are defined for the slope, vegetation, soil, androck maps. Since the factors are all derived from maplayers that have been compiled by domain experts, wechose the semantic import approach (Gupta, Knopf, &Nikiforuk 1988) for determining the appropriate mem-bership function. The membership functions are de-signed in such a way that a membership grade of zerois assigned to a cell of the map by at least one mem-bership for each factor. Table 1 shows an example of apossible assignment of membership grades. Since eachof the five erosion factors is associated with three mem-bership functions, we obtain a total of 3× 5 = 15 fuzzysets. These fuzzy sets constitute the input to our fuzzyinference system.

Fuzzy Inference and Defuzzification

Fuzzy inference uses fuzzy rules to deduct new infor-mation from given information. In our case, a typicalfuzzy rule looks as follows:

IF Slope = highPrecipitation = high. . .

THEN 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 membershipgrade for the THEN-part of a fuzzy rule. In our sys-tem, the minimum of the membership grades in IF-partof the rule is computed and taken as the membershipgrade of the THEN-part. For example, to determinethe degree to which a particular cell of the grid hasa high erosion potential (according to this particularfuzzy rule), we look up the the membership grades forthat cell in the fuzzy sets for high slope, high precipi-tation, etc. and compute the minimum of these grades.Since there might be other fuzzy rules with the sameTHEN-part, we have to repeat the computation for theother rules. Since the fuzzy rules are viewed in a dis-junctive way, the final value for the cell under consider-ation is the maximum of the membership grades fromthe individual rules. Figure 2 illustrates this inferenceprocess graphically.

After applying the fuzzy rules to the whole map or thepart of it that we are interested in, we have to translatethe fuzzy sets for the three erosion categories back toquantitative values. This process is called defuzzifica-tion. There are several defuzzification methods, whichare of different complexity and have different impact onthe results (Driankov, Hellendoorn, & Reinfrank 1996).The height defuzzification, which we use in our system,is a very simple and quick method. It computes theweighted sum of the membership grades in the threeerosion fuzzy sets, where the weights are given by thevalues that maximize the membership functions. Forthe erosion fuzzy sets low, medium, and high, thesevalues are 1, 2, and 3, respectively.

Implementation and Evaluation

We implemented our system in ARC/INFO-GRID, ageographic information system, which manipulates, an-alyzes, and displays geographic information using cell-based data. As study area, we chose the Hokianga re-gion in the north of New Zealand. There are two reasonsfor this choice:

1. Due to the high variability in elevation and the highprecipitation rates, the Hokianga region is particu-larly affected by erosion.

2. Various GIS data is available for that region, includ-ing topographic data as well as rainfall data, whichcan be transferred into grid format in a more or lessstraightforward way.

Figure 2: Applying fuzzy rules to compute the erosion fuzzy sets.

After applying the fuzzy rules, we converted the result-ing map into a map with six (rather than three) erosionclasses to make it comparable to erosion data that weobtained from the New Zealand Land Resource Inven-tories (NZLRI). The comparison of the maps showedsimilarities but also differences (see Figure 3).

The differences are particularly evident in the south-ern part of the Hokianga harbor, where one can recog-nize patterns emerging from the slope map. This is dueto the fact that the fuzzy rules put particular emphasison the slope factor—something that has to be adjustedin a revised version of the system.

Similarities between the two maps are obvious in thenorthern, the very west part, and the southeast regionsof the Hokianga area, which fall into the class of negli-gible erosion. Some similarities between the maps canbe detected in the southern and northwest part of thestudy area, which belong to the class of slight erosion.Large discrepancies can be found in areas of higher ero-sion potential, where our map categorizes more cellsas cells with extreme erosion potential than the ero-sion 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 verysurprising, as the methodologies applied for the designof both maps are based on very different concepts. The

erosion map from the NZLRI was created by survey-ing the landscape and incorporating the frequency andextent of visible erosion as well as background factorssuch as feasibility and cost of control and economic ef-fect. The erosion map developed with the method de-scribed in this paper, on the other hand, is based ondata that has significant influence on erosion to occur,such as slope, precipitation, soil, vegetation, and rocktype. Therefore, it can be considered as contributingto the potential of erosion, which does not necessar-ily mean that erosion is actually occurring at locationswith high potential. The dissimilarities in the mapswere to be expected, and therefore comparisons betweenthem have to be undertaken very carefully.

SummaryIn this paper, we described how fuzzy logic can be ap-plied to produce information about erosion from dataabout slope, annual precipitation, vegetation, rock, andsoil. The input information is stored in a raster-basedGIS as maps, each of which is converted into three fuzzysets. The resulting fuzzy sets are the input to a infer-ence system based on fuzzy rules, which produces threefuzzy sets corresponding to a low, medium, and highpotential of erosion, respectively. These are defuzzifiedin 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 fuzzylogic (bottom).

data.As our evaluation has shown, there are significant

similarities between the map that was produced by oursystem and a reference map taken from the NZLRI, butalso significant discrepancies. Some of these discrepan-cies have to be investigated further and might lead torevisions of the fuzzy rules that underly our system.

References

Allen, J. 1983. Maintaining knowledge about temporalintervals. Communications of the ACM 26:832–843.Altmann, D. 1994. Fuzzy set-theoretic approachesfor handling imprecision in spatial analysis. Inter-national Journal of Geographical Information Systems8(3):271–289.Brimicombe, A. 1997. A universal translator of linguis-tic hedges for the handling of uncertainty and fitness-of-use in GIS. In Kemp, Z., ed., Innovations in GIS4. London, England: Taylor and Francis. 115–126.Clementini, E.; Di Felice, P.; and Hernandez, D. 1997.Qualitative representation of positional information.Artificial Intelligence 95(2):317–356.Coppock, J., and Rhind, D. 1991. The history ofGIS. In Maguire, D.; Goodchild, M.; and Rhind,D., eds., Geographical Information Systems: Princi-ples and Applications. Essex, England: Longman Sci-entific & Technical. 21–43.Driankov, D.; Hellendoorn, H.; and Reinfrank, M.1996. An Introduction to Fuzzy Control. Berlin, Ger-many: Springer, 2nd edition.Egenhofer, M., and Golledge, R. 1997. Spatial andTemporal Reasoning in Geographic Information Sys-tems. New York: Oxford University Press.Frank, A. 1992. Spatial concepts, geometric data mod-els, and geometric data structures. Computers andGeosciences 18:409–417.Frank, A. 1994. Qualitative temporal reasoning inGIS-ordered time scales. In Proc. SDH-94, 410–431.Freksa, C. 1990. Qualitative spatial reasoning. InProc. Workshop RAUM, 21–36.Goodchild, M. 1992. Geographic data modeling. Com-puters and Geosciences 18:401–408.Guesgen, H. 1989. Spatial reasoning based on Allen’stemporal logic. Technical Report TR-89-049, ICSI,Berkeley, California.Gupta, M.; Knopf, G.; and Nikiforuk, P. 1988.Sinusoidal-based cognitive mapping functions. InGupta, M., and Yamakawa, T., eds., Fuzzy Logic andKnowledge Based Systems, Decision and Control. Am-sterdam, The Netherlands: North-Holland. 69–92.Gupta, A.; Weymouth, T.; and Jain, R. 1991. Anextended object-oriented data model for large imagebases. In Gunther, O., and Schek, H.-J., eds., Ad-vances in Spatial Databases, Lecture Notes in Com-puter Science 525. Berlin, Germany: Springer. 45–61.

Hernandez, D. 1991. Relative representation of spatialknowledge: The 2-D case. In Mark, D., and Frank, A.,eds., Cognitive and Linguistic Aspects of GeographicSpace. Dordrecht, The Netherlands: Kluwer. 373–385.Herring, J. 1991. The mathematical modeling of spa-tial and non-spatial information in geographic infor-mation systems. In Mark, D., and Frank, A., eds.,Cognitive and Linguistic Aspects of Geographic Space.Dordrecht, The Netherlands: Kluwer. 313–350.Herring, J. 1992. TIGRIS: A data model for an object-oriented geographic information system. Computersand Geosciences 18:443–452.Jiang, H., and Eastman, R. 2000. Application of fuzzymeasures in multi criteria evaluation in GIS. Inter-national Journal of Geographical Information Science14:173–184.Molenaar, M. 1996. The extensional uncertainty ofspatial objects. In Proc. SDH-96, 9B.1–13.Mukerjee, A., and Joe, G. 1990. A qualitative modelfor space. In Proc. AAAI-90, 721–727.Papadias, D.; Karacapilidis, N.; and Arkoumanis, D.1999. Processing fuzzy spatial queries: A configura-tion similarity approach. International Journal of Ge-ographical Information Science 13:93–118.Peuquet, D. 1994. It’s about time: A conceptualframework for the representation of temporal dynam-ics in geographic information systems. Annals of theAssociation of American Geographers 84(3):441–461.Plewe, B. 1997. A representation-oriented taxonomyof gradation. In Hirtle, S., and Frank, A., eds., Spa-tial Information Theory: A Theoretical Basis for GIS,Lecture Notes in Computer Science 1329. New York:Springer. 121–136.Raper, J., and Maguire, D. 1992. Design modelsand functionality in GIS. Computers and Geosciences18:387–394.Sinton, E. 1978. The inherent structure of informationas a constraint to analysis: Mapped thematic data asa case study. In Dutton, G., ed., Harvard Papers onGeographic Information Systems, volume 6. Reading,Massachusetts: Addison-Wesley.Stefanakis, E.; Vazirgiannis, M.; and Sellis, T. 1999.Incorporating fuzzy set methodologies in a DBMSrepository for the application domain of GIS. Inter-national Journal of Geographical Information Science13:657–676.Usery, L. 1996. A conceptual framework and fuzzy setimplementation for geographic features. In Burrough,P., and Frank, A., eds., Geographic Objects with In-determinate Boundaries, GISDATA Series No. 2. Lon-don, England: Taylor and Francis. 71–85.Woodcock, C., and Gopal, S. 2000. Fuzzy set theoryand thematic maps: Accuracy assessment and areaestimation. International Journal of Geographical In-formation Science 14:153–171.