ISPRS, Vol.34, Part 2W2, “Dynamic and Multi-Dimensional GIS”, Bangkok, May 23-25, 2001 
68 
2.1 DISCREPANCY OF A LINEAR FEATURE 
A line segment has two nodes. Its discrepancy is defined as the 
area whose boundaries are the measured linear feature and the 
‘true’ linear feature. This area is shaded in Fig. 1. 
(xi. У1. *i) 
Figure 1. The discrepancy of a line segment 
In Fig. 1, the solid line segment is the ‘true’ line segment joining 
the ‘true’ locations of the two nodes (pPp.Pa) and (//*2, p^, 
Pa). The dashed line segment is the measured line segment 
linking the two measured nodes (x1, yl, zl) and (x2, y2, z2). a b 
bi and C1 are parameters in Eq. 1, which is a mathematical 
expression of the error ellipsoid for the node on the left-hand 
side. Similarly, a 2 , 62 and c 2 are parameters in Eq. 2, which is a 
mathematical expression of the error ellipsoid for the node on 
the right-hand side. The error ellipsoid is a confidence region for 
the expected location of the node with a confidence coefficient of 
(1-a) where 0< a< 1. 
Error ellipsoid for node on the left-hand side 
\ 2 y.,1 „ \ 2 <*■ 
*-Mz\ 
1 = ( Axt 
bi 
+ 2 
+ 2 
+ 2 
Px\z\Py\A Px1yl| xl —p^ Y y1 - Py\ ^ 
^ Л Sjrl Л S /1 
Px\y‘\Py\z\ ~ Px\z\ If Xl — Pp Y Zi — Pp 
к Л ®x1 д S z1 
Px\y'\Px\z\ ~ Py\z\ ^ ~ Py\ Zl — Pp 
T 
л 
(1) 
where k — (-2 In (or)) x (1 - px\y\ - Paa ■ Py\z\ + ^PapPaaPpa)- 
pxiyi is the correlation coefficient of xTs error and y1’s error. 
Similarly, ppp and p^A are the correlation coefficients of yl’s 
error and zl’s error, and x1’s error and zl’s error respectively; 
and Sxi, Syi and Sa are standard derivations of x1’s, y1’s and 
zl’s errors. 
Error ellipsoid for node on the right-hand side 
1 = 
( *2-p x2 1 
2 
+ 
У 2 - Руг 
cip 
К г ) 
I J 
1 ° 2 J 
+ 2 
Px2zZPy2z2 - Px2y2 | X2 - p^ 
к I S v o 
V y2-Ay2 A 
2 Px2y2Py2z2 Px2z2 
~~k 
Y *2 - Px2 Y 
A S * 2 Л 
öy 2 
^2-Az2 
+ 2 
Px2y2Px2z2 ~ Py2z2 У У2 ~ Ay2 У Z2- р л 
k \ 
Sy 2 
Sz2 
(2) 
where k — (-2 In (or)) x (1 - p*^ 2 - p*^ 2 - /A222 2 + 2-P&yzpx2a.pp.£)- 
Px2>2 is the correlation coefficient of x2’s error and y2’s error. 
Similarly, P## and p*^ are the correlation coefficients of y2’s 
error and z2's error, and x2’s error and z2’s error respectively; 
and Sx2, and are standard derivations of x2’s error, y2’s 
error and z2’s error. 
The shaded area in Fig. 1 is determined differently depending on 
its case. Three possible cases exist: (a) the measured locations 
of all nodes and their corresponding ‘true’ locations are on a flat 
plane but the measured and the ‘true’ line segments do not 
intersect; (b) the measured and the ‘true’ line segments 
intersect; (c) neither case (a) or case (b) is a possibility. 
In the first case, it is assumed that the measured locations and 
the ‘true’ locations of all nodes should be on a flat plane and that 
their two corresponding line segments should not intersect. In 
such a situation, the shaded area can be denoted by area_quad 
(xl, yl, zl, x2, y2, z2), which is a function of (xl, yl, zl) and 
(x2, y2, z2), as shown in Eq. 3. 
area quad 
/o\ 
= 0.5* (the magnitude of (Ax 0)+the magnitude of (CxD)) ' ' 
where A = (xl -p*i, y1 -pp, zl -p*) 
B = (px2-p*i, Pyz-Pp, Pur Pa) 
C = (x1 -px2, y1-Py2, Al-p^) 
D = (x2-p^, y2-py2, Z2-P22) and 
Ax B is a vector product of A and B 
and so on. 
The second case is illustrated in Fig. 2 whereby the discrepancy 
is shaded. The shaded area consists of two triangles whose 
vertices are (x1, yl, zl), (pa, Pp, Pa) and (x12, y12, z12) for the 
triangle on the left-hand side, and (x2, y2, z2), (p^, p^, p^) and 
(x12, y12, z12) for the triangle on the right-hand side. (x12, y12, 
z12) is an intersecting point on the ‘true’ line segment and the 
measured line segment. The discrepancy is denoted by 
area_triangle which is a function of (xl, yl, zl), (x2, y2, z2), and 
(x12,3x12, z12). This shaded area is expressed in Eq. 4. 
area_ triangle 
= 0.5* (the magnitude of (A’xS‘)+the magnitude of (CxCf)) 
where A' = (x1 -p x 1, yl -pyi, zl -p A ) 
B = (xl 2-Pa , yl 2-pp, z12-p*) 
C = (x12-pxz, yl 2-pp, z12-p22) and 
£7 = (x2-px2, y2-py2, z2-p 22 ). 
(xi, Vi, zi) 
Measured line segment ^ Jx2 ‘ iJ/2, ^ 
1 _iXi2, У12, Z12)/ 
T 
True line segment 
(Pxi. Up, Un) 
(x 2 , Уг. z 2 ) 
Figure 2. The discrepancy of a line segment forming two triangles 
It is also possible that both the measured and the ‘true’ nodes 
are neither on a ‘flat’ plane nor intersect. Under these 
circumstances the shaded area cannot be obtained exactly. The 
obscurity of the equation formed by the measured and ‘true’ 
nodes affects the discrepancy; the discrepancy cannot be readily 
calculated. To simplify and quantify such a case, the 
approximate area of the shaded region in Fig. 1 is described by 
Eq. 3. Due to errors of the spatial feature’s nodes, the expected 
discrepant area of the line segment is computed as per Eq. 5. 
E(discrepancy) 
= J f (xl, y1, zl, x2, y2, z2) area _ quad dz2dy2dx2dz1dy1dx1 
RU ^ (5) 
+ J f(x1, yl, zl, x2, y2, z2) area _ triangle dz2dy2dx2dz1dy1dx1 
R,UR,