PROPAGATION OF DEM UNCERTAINTY: AN INTERVAL ARITHMETIC APPROACH. Gil Gonalves and Jorge Santos Departamento de MatemÆtica  Universidade de Coimbra, Apartado 3008, 3001-454 Coimbra, Portugal. Email: {gil,jmfs}@mat.uc.pt ABSTRACT In several GIS applications it is necessary to classify the topographic surface represented by a Digital Elevation Model (DEM) into specific slope classes. However, the computation of these classes from a DEM is influenced by the uncertainty in the elevations of those models. Monte-Carlo method has been used to study the effect of DEM uncertainty on topographic parameters. This method requires intensive computation, which turns the implementation very hard in most GIS software packages. Interval Arithmetic (IA) has been used as a successfully alternative to the Monte-Carlo method. Intervals of variation are used instead of a significant number of simulations. In this paper, we will show that the use of the IA constitutes a valid alternative to study the propagation of the uncertainty. Comparing the results from both methods, we conclude that the propagation of DEM uncertainty to the calculation of slope classes using IA is preferable to Monte-Carlo method, since it does not require so intensive computations, which turns the implementation easier in GIS analyses. INTRODUCTION The classification of the topographical surface in slope classes assumes an important role in several spatial analyses performed by GIS tools. The results of these analyses are frequently critical in decision making, which can affect many sectors in economic and social activities. Delimitation of ecological protection zones, identification of areas for urban development and the cartography of geohazard zones are some examples where slope is an important parameter. However, slope classification is affected by elevation uncertainty in DEM which is generated by errors in the acquisition of topographical data and in the interpolation methods used to build the elevation model. In this paper, we will go to use the word uncertainty to express the lack of knowledge about the true value of some quantity. The uncertainty includes errors or uncertainties due to imperfections of measurement systems and also the effect of the cartographic generalization which can not be avoid in cartographic modeling. In order that information quality about topography can be accessed by GIS users during spatial analysis, the following procedure can be executed:  build an elevation uncertainty model;  propagate that uncertainty to derived terrain features (slope, aspect, curvature);  specify appropriate methods for uncertainty evaluation, including visualization. The Monte-Carlo method is usually applied to study the uncertainty propagation from DEMs to derived slope classes ([10], [11], [11] e [13]). However, this method needs a large amount of computation efforts, which turns it very hard to be used in GIS applications. Interval arithmetic has been successfully used as and alternative to Monte-Carlo method ([17]). Instead to generate a certain number of simulations for some probability distribution which parameters has been estimated, interval arithmetic uses intervals of possible variation. Operations from interval arithmetic can be used to process that type of data ([15] e [16]). Therefore, interval arithmetic is an appropriated tool to deal with the uncertainty of spatial analysis with DEMs, giving intervals which express the propagated uncertainty. This work will show that interval arithmetic is a good alternative to study uncertainty propagation on this type of problems. First a common uncertainty model is used for the cartographic terrain representation. That model states that for a terrain representation based on contour lines and point elevations, 90% of control point elevations should have a mean squared error not bigger than half of the contour interval. It is used a specific interpolator (i.e., elastic grid), to generate a DEM from contours, to ensure the equivalence between the cartographic model and the digital model. After that, Monte-Carlo simulation and interval arithmetic are used do propagate DEM uncertainty to the derived Digital Slope Model (DSM), based on Horns method. Finally, slope classes are derived from DSM: I (0%-5%), II (5%-15%), III (15%-30%), IV (30%-40%) and V (>40%). Slopes classes uncertainty is also evaluated. This paper is organized in the following way: in section 2 the problem of DEMs uncertainty modeling is studied and the main uncertainty propagation methods are presented; in section 3 will be presented the methodology used to implement