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3. ISOLINES AFFECTED BY ACCIDENTAL ERRORS 
Knowing the behaviour of isolines in analytic surfaces 
helps greately the interpretation of their behaviour in 
natural surfaces. Isolines in natural surfaces are affected 
by accidental errors originating, on the one hand, in the 
roughness of the terrain, and on the other, in errors of 
data acquisition. Next, contour lines are going to be 
analyzed, followed by an analysis of lines of equal 
slope. 
3.1 Contour Lines Affected by Accidental Errors 
In figure 5, contour lines with an interval of 10 m are 
shown, as derived from a digital elevation model (DEM). 
This area does not contain any horizontal parts, and 
therefore the derivation of contour lines is without 
problems. 
There are numerous ways to express the height 
accuracy g, of such contours (e.g. Li, 1982, Tempfli, 
1980). As we are in this paper primarily interested not 
in the height accuracy of the contours but rather in their 
accuracy in the XY plane, we are going to apply a rather 
simple formula to express height accuracy. This simple 
formula has proven to be quite adequate for describing 
the accuracy of contour lines derived from a DEM (e.g. 
Ackermann, 1978, Kraus, 1987): 
0,-ac-b*Z (2) 
a ... 0.15 thousandth of the flying height 
b... 150 um at image scale 
Z' — tan a ... maximum terrain slope 
Considering formulea (1) and (2), the accuracy of 
contour lines in the XY plane becomes: 
Gu 78/2 tb (3) 
For contour lines in figure 5, estimates based on the 
flying height and the focal length of the camera yield: 
a, [ml] = (0.4/ Z^) + 3 (4) 
The accuracy in the XY plane, Opp + becomes in areas 
with terrain slope of 100% 3.4 m, and in areas with 
20% 5.0 m. For the entire area of interest, 0,, is 
visualised as pixel graphics in figure 6. The error in the 
XY plane is in inverse ratio to the terrain slope. The 
values of the terrain slope Z' for each pixel have been 
inquired from the digital slope model (DSM, to be 
treated below); the values Z' represent the inverse value 
of the distance AZ, between neighbouring contour lines. 
3.2 Lines of Equal Slope Affected by Accidental Errors 
In a DEM it is possible to derive the components of the 
normal vector to the terrain surface in each point of the 
raster, and at every intersection of a break line with the 
raster. The component of the normal vector along the 
maximum slope is used as the function value of the 
DSM. From the DSM lines of equal slope can be 
derived. Figure 7 shows the lines of equal terrain slope 
with an interval of 1096 and, by thick lines, the break 
lines of the DEM. A comparision with the contour lines 
(figure 5) enables us to note: 
- At break lines there occur displacements of varying 
size in the lines of equal terrain slope. 
- The distance AS, of neighbouring lines of equal 
terrain slope becomes very large in areas with 
regular terrain slope. 
743 
The area of the terrain with elevations below some 
1450 m shows a fairly regular slope of 7096. In this 
area the slope lines are hardly defined. The position of 
the isolines of slope in this area is more or less 
arbitrary?. Eyes trained by contents of chapter 2 can 
easily detect a high degree of uncertainty in these lines. 
By the way, a raster image with the threshold 7096 
would yield in this area a so-called "salt-and-pepper" 
pattern. 
The uncertainty in lines of equal terrain slope can be 
well visualised applying slope zones. Figure 8 is such an 
image. It represents the slope zones 19-21%, 29-31%, 
39-41%, etc. Overlaying these zones and the vector 
graphics (figure 7) is of great advantage to GIS users. 
This accuracy overlay can be made even more 
impressive, as shown in figure 9. In it, the level of 
probability of the position of the lines in the XY plane is 
represented in corespondence with the Gaussian 
distribution. 
There are few theoretical investigations on the accuracy 
of terrain slope values as derived from DEMs. The 
following is based upon formula (2): 
g.=C+d*272" (5) 
c ... 1.5% as derived from empirical studies 
(Kraus, 1991) 
d ... To derive this value, there is an empirical 
study in process (for purposes of this paper, 
this parameter is of little importance). 
Z" ... Maximum slope of the surface as defined by 
the lines of equal slope. 
Considering formula (1) the accuracy of lines of equal 
slope in the XY plane can be expressed as: 
Og, =C/2" +d (6) 
sp 
This value for the lines of equal terrain slope in figure 7 
can be estimated as: 
g, lm] — 1.5 / Z" (7) 
E.g. when Z" = 0.1[m"], the distance AS_ between 
neighbouring lines of equal terrain slope is TOO m and 
their accuracy in the XY plane 0, is 1.5/0.1 — 15 m. 
For the entire area of interest, figure 10 contains a 
visualisation of 0. p 8S pixel graphics. In areas with very 
little curvature of the surface the corresponding errors 
exceed even the accuracy limit of 15 m represented on 
the graphics in black. 
4. CONCLUSION 
GISs have to be extended so to contain, in addition to 
models representing geographical data, corresponding 
models to describe their accuracy. Inquiries into the 
information system should yield both the value of the 
function and its accuracy. Accessing multiple models is 
characteristic for deriving complex results. Much work 
has yet to be done to enable the simultaneous 
derivation of the corresponding accuracies for such 
derived products. GIS users have to be informed about 
the accuracy of all direct and derived products of the 
system. Information on the accuracy model has to be 
given attractive visualisation. This way the misuse of 
geographical data yielded by GIS can be considerably 
reduced. 
  
3In the paper (Killian, Kraus, 1992) this topic will be 
treated in detail.