428 REMOTE SENSING/GIS Figure A simple map algebraic example, in which spatial changes in groundwater levels, over a fixed time period (between 1973 and 1986) are derived by layer subtraction Table Example of a pairwise comparison matrix, used to assess relative factor significance and to calculate factor weights (for landslide hazard assessment in this case) DEM Distance (from roads) 1/5 1/3 Slope Aspect FS 1 1/3 1/3 1/6 1/4 1/6 1/4 Distance (from drainage) Soil moisture Iron oxides Clay content Slope Aspect Factor of safety (FS) DEM Distance (from roads) Distance (from drainage) Soil moisture Iron oxides Clay content 1/7 1/7 1/4 1/5 1/5 1/5 1/7 1/5 1/5 1/7 1/3 1/3 1/5 1 1/2 1/2 1/2 1 1/2 1 1 Factor weights 0.278 0.278 0.151 0.037 0.086 0.034 0.057 0.041 0.039 necessary within the analysis Bayesian probability theory, Dempster–Shafer theory, and fuzzy set membership are commonly used to incorporate uncertainty in GIS analysis in geological applications, for example, landslide hazard assessment and mineral prospectivity mapping Considerable research has been carried out into the extraction of mineral prospectivity maps from multisource datasets In many cases, identification of the very lowest and very highest prospectivity has not been in dispute but uncertainty arises in definition of the regions of intermediate prospectivity, which then require further analysis and interpretation Suitability or prospectivity must be treated as a continuous phenomenon representing a measure of confidence in the outcome A fuzzy set describes a continuous membership function where represents a non-member and a value of represents a member, and the values between them represent the increasing possibility of membership The fuzzy set membership function can be most readily appreciated with reference to a simple linear function (Figure 7) The fuzzy function, may in fact be linear, sigmoidal or ‘J’ shaped, and monotonic or symmetric The threshold values, which define it, will depend on the phenomenon and desired outcome