ANALYSIS OF GEOGRAPHICAL DATA AND VISUALISATION OF THEIR QUALITY 
K. Kraus 
Institute of Photogrammetry and Remote Sensing, The Vienna University of Technology, Vienna, Austria 
ISPRS Commission IV 
Abstract: 
Geographical information such as elevations or immission, is most often represented either as vectors, e.g. isolines, 
Or as raster, e.g. density slicing. Here, the precision of the representation is dealt with. Special attention is paid to 
the accuracy of secondary forms of information such as terrain slopes as derived from DEMs. For a mathematical 
analysis, synthetic surfaces are employed. As a conclusion, new ways are shown to visualize the quality of 
geographical data. 
Key words: Accuracy, Data Quality, DEM, Theory, Visualisation 
1. PRELIMINARY REMARKS 
This paper is dealing only with geographical data such 
to be represented by some function of land coordinates. 
In other words, the value of the function, Z, is a 
function of X and Y, i.e. Z(XY). The mathematical form 
of function Z(XY) is not regarded here. The function 
must be continuous. The first derivatives Z',(XY) and 
Z' (XY) may, however, show discontinuity. This means 
thát the surface represented by Z(XY) contains break 
lines. 
Function Z(XY) may describe immissions of some 
polluting substance, terrain elevations, etc. Without 
limiting generality, let us consider the special case of 
function Z(XY) describing the terrain surface. 
In our time, geographical data are stored in geographical 
information systems (GIS). Until recently, little attention 
has been paid to the quality of data and of derived 
products. A first step in the right direction is to store, 
alongside with the function values, their accuracy - i.e. 
introducing into the GIS standard deviations o,. 
Many GISs possess the capability to derive isolines of 
the function Z(XY), i.e. Z(XY) =const. Isolines are then 
intersected with other data such as cadastral 
boundaries. Sometimes, isolines of different functions 
are intersected with each other. In these cases, 
accuracy characteristics in form of c, are of no use; 
what is needed is rather the accuracy of isolines in the 
XY plane, i.e. Gon: This accuracy can be obtained as: 
  
Applying such accuracy characteristics it becomes 
possible to derive the accuracy of areas as deduced, 
using formulae in (Prisley et al., 1989). 
741 
Op = 0,/tana = 0, * AZ, / AZ (1) 
tan a Maximum terrain slope at the point in 
question 
AZ ... )lsoline interval (usually constant over the 
entire area of interest) 
Distance to the neighbouring isolines in 
the XY-plane (AZ, varies along the 
isoline). 
AZ, 
We have to give careful consideration to the distance 
AZ, of neighbouring isolines. It may become very large. 
As a consequence, the standard deviation c,, of isolines 
may become very large, as well. And, therefore, 
processes of intersection with isolines may yield bad or 
even unacceptable results. It has to be emphasized at 
this point that the above conclusion is valid for applying 
isoline information, i.e. Z(XY) = const, in any form, both 
vector or raster (a way similar to density slicing). 
The above problem will be examined here from various 
points of view. We start in chapter 2 with considering 
the behaviour of distances between neighbouring 
isolines of analytical surfaces. In this, dealing with 
derived surfaces is of special interest. In chapter 3, 
function Z(XY) and its derivatives become stochastic 
variables; this means taking into account accidental 
errors 0,, and creating in GIS accuracy models in parallel 
to the corresponding functional ones. Visualising the 
quality of geographical data is prevalent in chapter 3.