ISPRS, Vol.34, Part 2W2, “Dynamic and Multi-Dimensional GIS”, Bangkok, May 23-25, 2001 
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MEASURING UNCERTAINTY IN SPATIAL FEATURES IN A THREE-DIMENSIONAL GEOGRAPHICAL 
INFORMATION SYSTEM 
Chui Kwan CHEUNG and Wenzhong SHI 
Dept, of LSGJ, the Hong Kong Polytechnic University 
tracv.cheunq@honqkong.com 
Keywords: geographical information systems, uncertainty, positional error, numerical integration 
ABSTRACT 
In this paper, uncertainty of spatial features including linear features, areal features and volumetric features in a three-dimensional (3D) 
vector-based geographical information system (GIS) is studied. Existing uncertainty models for 3D spatial features are divided into two 
directions: (a) a confidence region model derived from a statistical approach and (b) a reliability model based on a simulation technique. 
The confidence region model for a spatial feature could provide an area inside which the ‘true’ location of the spatial feature is with a 
predefined probability. In the reliability model, uncertainty was measured by a discrepant area, which is formed by the measured 
location and the ‘true’ location of a spatial feature and hence was determined in terms of the measured location and the ‘true’ location of 
the spatial feature in a mathematical expression. Based on an assumption of error of the spatial feature, uncertainty of the spatial 
feature could be simulated repeatedly and the average discrepant area would be obtained. However, it is known that simulation is quite 
time-consuming and cannot provide a precise solution. Hence, this study further proposes the development an analytical model on the 
reliability model on spatial features in a 3D GIS. The expected value of the discrepant area of the spatial feature is expressed as a 
multiple integral by a statistical approach. The authors proposed to solve the multiple integral based on a numerical integration, resulting 
in an approximate solution of the expected discrepant area. This uncertainty model is also compared with an earlier simulation model. 
1. INTRODUCTION 
A GIS is defined as a software package, which provides users 
with a tool to input, store, analyze, retrieve and transform 
geographical data (Cassettari 1993). It is now widely applied in 
many different areas including military applications, 
environmental studies and geological exploration. However, 
geographical data in GIS is not error-free (Heuvelink 1998). The 
market of GIS will be affected by evaluating uncertainty in GIS to 
a certain extent. 
Uncertainty in GIS may arise from data collection and input in 
the first step to spatial analyses. Hence, there are many different 
types of uncertainty in GIS (Burrough and McDonnell 1998). It is 
virtually impossible to represent the world completely due to the 
complexity of the geographical world. Some of the man-made 
utilities such as water pipes and road networks can be 
represented by points, lines and polygons while most natural 
phenomena cannot (Burrough 1986). Differences between the 
database contents and the phenomena they represent are 
mainly due to the characteristic of phenomena. In addition, 
measurement errors are introduced during data collection and 
input and propagated through GIS operations. 
There are different approaches to describe uncertainty of linear 
features in two-dimensional GIS (Caspary and Scheuring 1992; 
Dutton 1992; Stanfel and Stanfel 1993,1994; Shi 1994; Easa 
1995; Shi and Liu 2000). However, little research exists in the 
modeling of uncertainty in higher dimensional spatial features. 
Shi (1997, 1998) derived a confidence region model for 3D and 
N-dimensional linear features from strictly statistical approaches. 
Later on, the reliability of 3D spatial features, including linear 
features, areal features and volumetric features were studied by 
a simulation technique (Shi and Cheung 1999). 
Shi and Cheung (1999) earlier assessed the reliability of a 3D 
spatial feature by calculating the discrepant area between the 
measured location and the ‘true’ location of this spatial feature. It 
was first assumed that nodal error was normally distributed. In 
such simulation, the measured nodes of the spatial feature were 
generated and the discrepant area was calculated. After the 
simulation was repeated many times, the expected value and 
the variance of the discrepant area for the linear feature (or the 
discrepancy volume for the area or volumetric feature) were 
calculated. On the other hand, Stanfel (1996) suggested that a 
stochastic method could be used to calculate the expected 
discrepant area in 2D GIS, mainly due to weakness of the 
simulation technique such as time-consuming and unstable 
results. 
An analytical expression for the expected discrepant area (or 
volume) is derived mathematically in this study. Theoretically, its 
exact solution will be obtained automatically in GIS given the 
measured location and the ‘true’ location of the spatial feature. 
As a result GIS users will aware of the uncertainty of the spatial 
feature from this indicator. However the analytical expression will 
be expressed in terms of a multiple integral based on a 
probability theory. This multiple integral is unable to be solved 
analytically. This paper thus presents an analytical method with 
a numerical solution to describe uncertainty of three-dimensional 
spatial features in GIS. 
In this paper, we focus on uncertainty model for a 3D spatial 
feature in a vector-based GIS. Uncertainty of this spatial feature 
will be determined by discrepancy between the measured 
location and the ‘true’ location of the spatial feature. In Session 
2, we will explain the discrepancy of spatial features including 
linear features, areal features and volumetric features. Due to 
the weakness of simulation technique (as stated above), we will 
withdraw this technique in this paper and express the expected 
discrepant value analytically and this mathematical expression is 
also given in Session 2. A numerical integration given in Session 
3 will be implemented in order to find the approximate solution 
for the expected discrepant value. Finally, some experimental 
studies will be conducted and their numerical results will be 
compared with the simulation result from the previous simulation 
model. 
2. DISCREPANCY OF 3D SPATIAL FEATURES 
Uncertainty of spatial features is measured by a discrepancy, 
which is the difference between the measured location and the 
‘true’ location of spatial features. From a statistical point of view, 
this indicates that a mean of any variable X is close to its actual 
value. Thus, in this study, the ‘true’ locations of the nodes of 
spatial features refer to their mean locations. 
For this study, the following two assumptions are made. First, 
positional error of a node is within an error ellipsoid whose 
center corresponds to the ‘true’ location (Stanfel and Stanfel 
1994; Easa 1995). Second, the positional error of the nodes is 
assumed to follow a normal distribution inherent to specific 
measurement technologies (Stanfel and Stanfel 1993). The 
authors will therefore model the positional error of spatial 
features based on positional nodal error of the spatial features.