an outcome, based on the available data. Typically uncertainty 
is modeled by probability theory. Imprecision is a feature of the 
data itself. It refers to data expressed as a range of possible 
values. Other descriptions of data are accurate and exact, 
approximate, and ambiguous. Geographical features, such as 
spatial relations, may contain many kinds of uncertainties 
which are often caused by the fuzzily defined concepts and 
linguistics, the presence of varying shapes and features of 
complicated spatial objects, and the imprecise measurements of 
spatial data. In following section, we will briefly introduce 
these uncertainties of spatial relations. 
2.1. Conceptual Uncertainties 
Conceptual uncertainties of spatial relations in GIS are mainly 
caused by the fuzzy linguistic and conceptual variables. For 
example, we often use the fuzzy concepts (such as near, far, 
almost west, middle east, and et. al.) to deriving metric 
relations. In this case, the metric relations are derived by a set 
of metric values together with their fuzzy memberships. The 
fuzzy membership is generally a real number on [0,1], where 0 
indicates no membership and 1 indicates complete membership. 
Similarly, the fuzzy concepts of weak connected, strong 
connected, almost same, big different, and et. al., are caused the 
uncertainties of topologic and ordering relations. It should be 
emphasized that topologic relations are generally independent 
on the geometry, but topologic relations usually are derived 
from geometric descriptions, so this is also necessary when 
taking the uncertainty of the geometric entities into 
consideration which of cause leads to derived quantities of 
uncertainty of the topologic relations. Some related examples 
are shown in Fig.1 (a)-(c). 
  
  
  
Fig.1(a). Fuzzy distance describing near, middle and far 
  
  
| 
3d 
Fig.1(b). Topologically met objects with the weak 
and strong connections 
   
  
  
  
  
    
  
  
  
Fig.1(c). Directionally ordered objects with the 
almost same and big different orientations 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
  
  
  
  
  
  
  
  
  
  
Y e x 
    
Ó du «d, Sd, Ö du Man M 
Fig.2(c). Ordering between points and objects 
  
  
  
  
  
2.2. Object Feature Uncertainties 
Object feature uncertainties of spatial relations in GIS are 
mainly caused by the ambiguous variables of object shapes, 
sizes and distributions. A metric relation between two 
arbitrarily-shaped objects is a fuzzy concept and is thus often 
dependent on human interpretation [Fig.2(a)-(b)].The simplest 
way to calculate the distance or direction between two objects is 
to convert the object calculations to the represented point 
calculations, such as using the distance between two capitals to 
represent the distance between two countries. But this method 
will cause many problems when the country is big and its shape 
is complicated. As the descriptions in Chen and et. al [1995, 
1996], the rigorous method to calculate metric relations 
between two arbitrarily-shaped objects is to calculate the 
Hausdorff distances and directions between their sub-sets of 
objects and their fuzzy memberships. The general Hausdorff 
metric between two objects is just the special case of its fuzzy 
membership value equals to 1. Similarly, the object shape and 
distribution may cause the uncertainties of ordering relations 
[Fig.2(c)], but don't influence topologic relations. The ordering 
relations between objects can be derived by the method of 
partial-ordered segmentation of objects [Chen and et. at., 1995, 
1996]. 
2.3. Data Uncertainties 
Data uncertainties of spatial relations in GIS are mainly caused 
by imprecise measurements of spatial data. Generally, the 
locations of objects in spatial databases are not error-free, they 
may contain many kinds of errors, such as the errors of scanning, 
digitizing, selecting, projection, overlaying, and et.al. 
[Goodchild and et.al., 1989]. For describing the uncertainty in 
the positions of spatial objects (such as lines and areas), we can 
use the error model of. € -band developed by Chrisman [1982], 
in which the positional uncertainty of" a spatial object K, can 
106 
  
  
  
  
  
  
  
  
  
  
  
   
  
  
  
  
  
    
  
   
  
   
   
   
   
   
   
   
  
   
  
  
  
  
  
   
   
  
  
    
   
   
  
  
  
  
  
   
   
   
    
  
Fig.3(a 
  
Fig.3(b 
  
a 
fete 
Fig.3(c 
  
be represen 
distance o 
membershif 
spatial me 
error € -ban 
uncertaintie 
3.1. 9-Inter 
For driving 
et al., (199. 
usual conce 
in which th 
K, and K 
interior ( K 
interior ( K 
matrix 3, , 
By conside 
equation [. 
topological 
realized \ 
in IR^ [Eg 
The beauty 
the fact the 
by using tk 
digital con