ISPRS, Vol.34, Part 2W2, “Dynamic and Multi-Dimensional GIS”, Bangkok, May 23-25, 2001 
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where R^ and R z are the regions of the two ellipsoids expressed 
in Eq. 1 and Eq. 2 respectively; U is union of the two regions; 
and f[x 1 ,y1 ,z1 ,x2,y2,z2) is a probability density function (pdf) of 
normal distribution. The latter’s mathematical expression is 
shown below 
/(x1,y1,z1, x2, y2,z2) 
(2*r|: 
IV2 
*1 
yi-p„ 
x2-/j Ml 
Yt-Vn 
where ris the covariance matrix of x1’s, yl’s, zl’s, *2’s, y2's, 
and z2’s errors. 
True areal feature 
Figure 4 The discrepancy of an areal feature defined by area 
True areal feature 
Joining several line segments together yields a polyline, which is 
a broad linear feature. The reliability of the linear feature is 
measured by the discrepancy between the measured location 
and the ‘true’ location of the linear feature. In this instance, the 
linear feature only refers to an acyclic polyline. 
The discrepancy between the measured location and the ‘true’ 
location of the linear feature is shown in Fig. 3. The solid linear 
feature is the ‘true’ location of the linear feature and the dashed 
linear feature is the measured location. The ‘true’ location is 
connected by three ‘true’ nodes (h*\, Hy\, Ha), [Ha, Hy2, Hzz) and 
(//a. Hy3, Hz3)- The measured location is connected by three 
measured nodes (xl, yl, zl), (x2, y2, 22) and (x3, y3, 23). The 
shaded area in Fig. 3 represents the discrepancy between the 
measured location and the ‘true’ location. As a result, the 
expected discrepant area is presented in Eq. 7. 
E(discrepanc y) 
= jfxarea dz3dy3dx3 dz2dy2dx2dzfdyfdxf ( 7 ) 
R,UR,UR, 
where R u R 2 and ft 3 are regions of the three ellipsoids for three 
nodes of the linear feature; U is union of regions; f is a pdf of 
normal distribution; and area is the shaded area in Fig. 3. 
Figure 5. The discrepancy of an areal feature defined by volume 
Fig. 4 relates to the discrepancy of the areal feature based on 
area, and Fig. 5 relates to the discrepancy of the areal feature 
based on volume. The solid and the dashed areal features are 
the ‘true’ location and the measured location of the areal feature 
respectively. Both the area of the shaded area in Fig. 4 and the 
volume of the shaded area in Fig. 5 represent the discrepancy. 
Since discrepancy is the difference between reality and users’ 
representation of reality, using the volume of the shaded area to 
express the discrepancy of areal feature is satisfactory. For 
instance, the expected discrepant area of an areal feature 
containing three nodes can be given as follows. 
E (discrepane y) 
- jf x volume dz3dy3dx3dz2dy2dx2dzfdyfdxf ( 8 ) 
where R it R 2 and R 3 are regions of the three ellipsoids for three 
nodes of the areal feature; U is union of regions; f is a pdf of 
normal distribution; and volume is the volume of the shaded 
object in Fig. 5. 
True linear feature 
1 
(Px3. P/3, Prt) 
(Px 1, P/1, Pzl) 
Figure 3 The discrepancy of a linear feature 
2.3 DISCREPANCY OF A VOLUMETRIC FEATURE 
In a 3D GIS, another important element is volumetric features. 
The difference between the measured location and the ‘true’ 
location of a volumetric feature is a measure of the reliability of 
the volumetric feature. This discrepancy can be viewed as a 
union of surfaces’ discrepancies. For example, a volumetric 
feature contains four nodes and therefore four surfaces. For 
each surface, its corresponding discrepancy is computed. The 
discrepancy of the volumetric feature is computed by the union 
of all of the surfaces’ discrepancies. The expected discrepancy 
is illustrated in Eq. 9 whereby the volumetric feature consists of 
four nodes. 
2.2 DISCREPANCY OF AN AREAL FEATURE 
Areal features, as discussed in this paper, refer to polygons in 
the digital database sense. Though the reliability of an areal 
feature can be appraised by the discrepancy between the 
measured location and the ‘true’ location of the areal feature, its 
definition of discrepancy is distinct from that of linear features. 
The discrepancy of the areal feature refers to a volume of the 
area whose boundaries are the measured areal feature and the 
‘true’ areal feature. According to the definition for linear features, 
the discrepancy should be the surface area bounded by the 
measured linear feature and the ‘true’ linear feature. Fig. 4 and 
Fig. 5 illustrate this difference in conformity with respect to area 
and volume. 3 
E(discrepancy) 
- j f xvolumedz4dy4dx4dz3dy3dx3dz2dy2dx2dz\dyfdxf ^ 
r,\jr 2 . r, 
where R,, R 2 , R 3 and are regions of the four ellipsoids for four 
nodes of the volumetric feature; U is union of regions; f is a pdf 
of normal distribution; and volume is the union of the four 
surfaces’ discrepancies. 
This expected discrepancy might differ from that in the 
dependence case. This could indicate that the covariance matrix 
of f is not diagonal. 
3. NUMERICAL INTEGRATION