ISPRS, Vol.34, Part 2W2, “Dynamic and Multi-Dimensional GIS”, Bangkok, May 23-25, 2001 
70 
The discrepancy for spatial features can be mathematically 
expressed by a multiple integral. However, it is known that 
integrals may not have an exact solution. Moreover, there are no 
standard procedures of finding the exact solution of the integral. 
As a result, a traditional technique is unsatisfactory to solve this 
problem in GIS. 
Gaussian quadrature, a numerical integration technique, 
approximates the integral of a function by integrating the linear 
function that joins points of the function’s graph (Burden and 
Faires 1993). In other words, the integral Jg(x) dx is 
approximated by X Wi g(xj) where the nodes Xi, x 2 , .... x n and 
coefficients wi, w 2 , ..., w n are chosen to minimize the expected 
error between the integral and the approximation. This technique 
can be modified in a straightforward manner for use in the 
approximation of multiple integrals. This Gaussian quadrature is 
implemented to calculate the multiple integral in Eq. 5, Eq. 7, Eq. 
8 and Eq. 9. 
4. RESULTS AND DISCUSSION 
The analytical model for the uncertainty of spatial features is 
applied to the example data of Shi and Cheung (1999). The two 
expected nodes of the line segments are (0, 0, 0) and (1000, 0, 
0). a,, ¿>1, Ci, a 2 , th and c 2 are 100ft, 196ft, 148ft, 30ft, 78ft and 
90ft respectively. The covariance matrix, in the pdf of the 
multivariate normal distribution f, is a 6x6 diagonal matrix with 
nodes x1, yl, zl, x2, y2 and z2 and these nodal errors are 
assumed to be independent. The confidence coefficient (1 -or) is 
0.95. The expected discrepant area of the line segment is 
62145.0ft 2 (as shown in Table 1). 
Table 1. The expected discrepant area of spatial features 
Spatial 
feature 
The expected discrepancy 
Ratio 
Analytical model 
Simulation 
model 
Line 
segment 
62145.0ft 2 
59344.1ft 2 
1.047 
Linear 
85553.6ft 2 
89036.1ft 2 
0.961 
Areal 
15209871.4ft 3 
18570274.0ft 3 
0.819 
Volumetric 
37688986.2ft 3 
44339983.8ft 3 
0.849 
* Ratio - ana| y tical resu ' t 
simulated result 
For the polyline, the three expected nodes are (0, 0, 0), (500, 
500, 707.1) and (1500, 500, 707.1). a,, b u Ci, a 2 , tfc, c 2 , a 3 , ¿13 
and c 3 are 100ft, 196ft, 148ft, 30ft, 78ft, 90ft, 100ft, 196ft and 
148ft respectively. The covariance matrix in f is a 9x9 diagonal 
matrix. The confidence coefficient (1 -a) is 0.95. The expected 
discrepant area of the line feature is 85553.6ft 2 . Using the three 
nodes of the polyline for the areal feature results in an expected 
discrepant area equal to 5209871.4ft 3 . 
For the example of a volumetric feature, an additional node (to 
the existing three) is considered. This addition now specifies a 
volumetric feature. This additional node is (500, 707.1, 500), and 
its error ellipsoid has parameters a4 = 30ft, b 4 = 78ft and c 4 = 
90ft. The expected discrepant area of the volumetric feature is 
37688986.2ft 3 . 
In Table 1, the expected discrepant areas of the spatial features 
are recorded for both the numerical integration and the 
simulation techniques. The ratio from the analytical model to that 
from the simulation model is in the range of 0.8 to 1.1. In an 
ideal situation, this ratio should be 1. A ratio varying from 1 is 
due to the approximation of the expected discrepancy for both 
techniques (numerical integration and simulation techniques). 
In the simulation model, the accuracy of the result depends on 
the number of simulation. The larger the number of simulation, 
the higher the accuracy of the result. The accuracy of the 
approximation in the analytical model is related to the number of 
nodes chosen in the integral region. To a certain extent, the 
oscillatory nature of the integral function affects the 
approximation. In any case, the expected discrepant areas 
calculated from the numerical integration and from the simulation 
technique, are close to each other. Both techniques are valid. 
The above four examples considered the discrepancy of the 
spatial features when there is no correlation of nodal errors. 
5. CONCLUSIONS 
A newly developed analytical model to measure the uncertainty 
of a spatial feature in 3D GIS was presented in this paper. The 
uncertainty is determined by the ‘true’ location and the measured 
location of the spatial feature. The discrepant area (or volume) 
was used as a measure of the uncertainty. It was expressed as 
a mathematical function of which the measured location and the 
‘true’ location of the spatial feature were variables. Given the 
measured location of the spatial feature, the discrepant area (or 
volume) could be obtained and the derived result was the 
discrepant area for this measured location. In general, the 
measured location may be in the vicinity of the ‘true’ location. 
Based on the assumption of error of a spatial feature, a number 
of possible measured locations were considered in our proposed 
model rather than one measured location of the spatial feature. 
Therefore, the expected discrepant area was in the form of a 
multiple integral. Since this multiple integral could be solved 
analytically, the Gaussian quadrature, a numerical integration, 
was implemented to provide an approximate solution for the 
analytical model. The estimated expected discrepancy was 
finally compared to the simulated solution. 
In our previous uncertainty model for 3D spatial features, the 
uncertainty model of 3D spatial features was studied using the 
simulation technique. This simulation model was generated 
some possible measured locations of a spatial feature based on 
the same assumption of error of the spatial feature as stated in 
this paper, and computed the average discrepant area (or 
volume). In a mathematical point of view, there are a number of 
infinite points inside a region. However, the simulation model 
only sampled a certain number of the possible measured 
location of the spatial feature instead of all possible measured 
locations of the spatial feature. The accuracy of the expected 
discrepant area (or volume) is question although more 
simulations can provide a more precise result. Moreover, the 
simulation model is quite time-consuming. Thus, we proposed 
the analytical model by taking all possible measured locations of 
the spatial feature into account, in order to provide the expected 
discrepant area (or volume) with great accuracy in real time. 
In this paper, an analytical model was provided to validate the 
simulation model. The numerical results obtained from the 
analytical model and the simulation results given in our previous 
study can approximate a similar discrepant area (or volume). 
However, the calculation for the numerical solution is much 
faster than that for the simulated solution for the same problem. 
Therefore, the numerical integration technique is considered the 
preferred approach in studying the uncertainty of 3D spatial 
features. 
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Burrough, P.A. and R.A. McDonnell. 1998. Principles of 
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Caspary, W. and R. Scheuring. 1992. Error-band as measure of 
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Cassettari, S. 1993. Introduction to Integrated Geo-Information 
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