A number of GIS packages include special facilities for identifying, selecting and analysing boundaries. These may be the borders of distinct zones or areas, within which values on one or more attributes are relatively homogeneous, and distinct from those in adjacent areas, or they may be zones of rapid spatial change in more continuously varying attribute data. Amongst the tools available for boundary determination are raster-based GIS products, like Idrisi, and the more specialised Terraseer family of spatial analysis tools. Indeed Terraseer has a specific product, Boundaryseer, for boundary detection and analysis, using techniques such as overlap and sub-boundary statistical analysis. The problem these products seek to address is that of identifying where and how boundaries between different zones should be drawn and interpreted when the underlying data varies continuously, often without sharp breaks. For example, soil types may be classified by the proportion of sand, clay, organic matter and mineral content. Each of these variables will occur in a mix across a study region and may gradually merge from one recognised type to another.

In order to define boundaries between areas with, for example, different soil types, clear polygon-like boundaries are inappropriate. In such cases a widely used procedure is to allocate all grid cells to a set of k zones or classes based on fuzzy membership rather than binary or 0/1 membership. The idea here is to assign a number between 0 and 1 to indicate how much membership of the zone or class the cell is to be allocated, where 0 means not a member and 0.5 means the cell’s grade of membership is 50% (but this is not a probabilistic measure). If a zone is mapped with a fuzzy boundary, the 50% or 0.5 set of cells is sometime regarded as the equivalent to the boundary or delimiter (cross-over point) in a crisp, discrete model situation. Using this notion, crisp polygon boundaries or identified zone edges in raster modelled data may be replaced with a band based on the degree of zone membership. Alternatively grid datasets may be subjected to fuzzy classification procedures whereby each grid cell acquires a membership value for one of k fuzzy classes, and a series of membership maps are generated, one for each class (see further, the discussion on soft classifiers in Section 4.2.12.4). Subsequent processing on these maps may then be used to locate boundaries.

Several fuzzy membership functions (MFs) have been developed to enable the assignment process to be automated and to reflect expert opinion on the nature and extent of transitional zones. Burrough and McDonnell (1998, Ch.11) provides an excellent discussion of fuzzy sets and their applications in spatial classification and boundary determination problems. The GIS product Idrisi, which utilises some of Burrough’s work, provides four principal fuzzy MFs:

Sigmoidal or (double) s-shaped functions, which are produced by a combination of linear and cos²() functions in Idrisi’s case, or as an algebraic expression of the form
m=1/(1+a(z‑c)²) where a is a parameter that varies the width of the function, z is the property being modelled (e.g. proportion of clay) and c is the value of this proportion at the function midpoint (see Figure 4‑30; here membership values of >0.5 are regarded as being definitely members of the set A)

J-shaped functions, which are rather like the sigmoidal MFs but with the rounded top sliced off flat over some distance, x (if x=0 then the two sides of the J meet at a point). The equation used in this case is of the form:
m=1/(1+((z‑a)/(a‑c))²)

Linear functions, which are like the J-shaped function but with linear sides, like the slope of a pitched roof, and are thus simple to calculate and have a fixed and well-defined extent, and

User-defined functions, which are self-explanatory. In most applications MFs are symmetric, although monotonic increasing or decreasing options are provided in Idrisi.

An alternative to using MFs as spreading functions is to apply a two-stage process: first, use a (fuzzy) classification procedure to assign a membership value (m_ik=0.0‑1.0) to each grid cell, i, for each of k classes. This value is based on how similar the cell attributes are to each of k (pre-selected) attribute types or clusters. These may then be separately mapped, as discussed earlier.

Having generated the membership assignments boundaries are then generated by a second algorithm. Boundaryseer provides three alternative procedures:

·         Wombling (a family of procedures named after the author, W H Womble, 1951)

·         Confusion Index (CI) measures, which are almost self-explanatory, and

·         Classification Entropy (CE) based on information theoretic ideas

Wombling involves examining the gradient of the surface or surfaces under consideration in the immediate neighbourhood of each cell or point. Typically, with raster datasets, this process examines the four cells in Rook’s or Bishop’s move position relative to the target cell or point — boundaries are identified based on the rate of change of a linear surface fitted through these four points, with high or sudden rates of change being the most significant. Wombling methods can be applied to vector and image datasets as well as raster data.

The Confusion Index (CI) works on the presumption that if we compute the ratio of the second highest membership value of a cell i, m_i2, to the highest, m_i1, then any values close to 1 indicate that the cell could realistically be assigned to either class, and hence is likely to be on the boundary between the two classes.

Finally, Classification Entropy, devised by Brown (1998), creates a similar value to the CI measure, again in the range [0,1], but this time using the normalised entropy measure:

where the summation extends over all the k-classes and the measure applies to each of the j cells or locations. This latter kind of measure is used in a number of spatial analysis applications (see further, Section 4.6).

Fuzzy procedures are by no means the only methods for detecting and mapping boundaries. Many other techniques and options exist. For example, the allocation of cells to one of k-classes can be undertaken using purely deterministic (spatially constrained) clustering procedures or probabilistic methods, including Bayesian and belief-based systems. In the latter methods cells are assigned probabilities of membership of the various classes. A fuller description of such methods is provided in the documentation for the Boundaryseer and Idrisi packages. As is apparent from this discussion, boundary detection is not only a complex and subtle process, but is also closely related to general-purpose classification and clustering methods, spatially-constrained clustering methods and techniques applied in image processing.

As noted earlier, boundaries may also be subject to analysis, for example to test hypotheses such as: “is the overlap observed between these two boundaries significant, and if so in what way, or could it have occurred at random?”; “is the form of a boundary unusual or related to those of neighbouring areas?” Tools to answer such questions are provided in Boundaryseer but are otherwise not generally available.