Higher Order Vagueness in a Dynamic Landscape: Multi-Resolution Morphometric Analysis of a Coastal Dunefield

Abstract

Many environmental phenomena can be conceptualised as vague and so may be suitable for storage as fuzzy sets and analysis by fuzzy logic. Fuzzy sets directly address the vagueness in the information, but many consider that any statement about a vague phenomenon must itself be vague. This is known as higher order vagueness, and is handled in fuzzy set theory by type-2 and, by extension, type-n fuzzy sets. In this paper we use the recognition of, and change in, a system of coastal sand dunes as an environmental example in which to explore the use of type-2 fuzzy sets. The crests and troughs of the dunes are identified as fuzzy sets from geomorphometric analysis of high resolution digital elevation models from two years (1998 and 2000). By varying the parameters of the morphometric extraction, multiple instances of type-1 fuzzy sets can be defined, and these can be summarised to yield type-2 fuzzy sets. The logic of change analysis is presented, and two alternative approaches to change analysis of type-2 fuzzy sets implemented. In one approach changes in the multiple instances of type-1 fuzzy sets are analysed and summarised as type-2 sets. The second approach directly examines change in the parameters of type-2 fuzzy sets, viewing the results for the different parameters as separate instances of change. All analyses produce satisfactory results which, although they arc hard to verify, make sense, yielding a range of possible but small degrees of fuzzy change. In a rather sedentary dune system (as is usual in a coastal location in mid-latitudes) this is to be expected. Change analysis in most applications of type-2 fuzzy sets, which might be based on expert advice for defining memberships, would have to rely on this second approach, and it is therefore interesting to note that this yields the largest range of possible change results.