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A New Approach for Global DTM Modelling 
Rüdiger Brand !, Jochen Fróhlich ? 
! Chair for Photogrammetry and Remote Sensing 
Technical University Munich, Germany 
Phone: +49-89-2892 2673, Fax: +49-89-280 95 73 
E-Mail: ruediger@photo.verm.tu-muenchen.de 
? Konrad-Zuse Zentrum für Informationstechnik 
Berlin, Germany 
Phone: 4-49-30-89604-0, Fax: 4-49-30-89604-125 
E-Mail: froehlich@zib-berlin.de 
Commision IV, Working Group 4 
KEY WORDS: Mathematics, Global DEM/DTM Modelling, IDA, Adaptive Hierarchical Approximation 
ABSTRACT 
In Photogrammetry and Geodesy various methods for DTM-modelling exist. A surface, where the height coordinates 
z(z,y) are a function of the planimetric coordinates z and y, is usually described by an height matrix, but we can also 
do a Fourier series expansion. Since many processes in environmental modelling and climate research are described on 
the sphere, we have developed a method which uses spherical coordinates instead of planimetric coordinates. Starting 
with a set of scattered data the height of a surface point is calculated by analyzing adjacency relationships using locally 
supported basis functions. Organizing our algorithm in a hierarchical way with different scale functions errors in the 
data set can be detected. The method is triggered by B-spline interpolation and wavelet techniques. It is tested on data 
sets of Australia and the asteroid 243 IDA. 
1 INTRODUCTION 
Topographical information, especially digital terrain mod- 
els (DTM), are used in various applications, i.e. in Me- 
teorology, climate and environmental modelling and in 
Geodesy. In these applications often regional and global 
processes are simulated. One of the main requirements is 
the fast access of topographic data in different aggregation 
levels. The same aspects arise in computer graphics, vi- 
sualization and visual simulation (Gross, 1995). For this 
purpose efficient algorithms for coding and representation 
of very large data sets, in particular DTM, are needed. 
The variation of the surface elevation over a planar, spheri- 
cal or ellipsoidal area can be modelled in many ways. DTM 
can be represented either by mathematically defined sur- 
faces or by point and line methods. For global DTM the 
point methods often result in coarse regular grids, in which 
one grid cell represents a large area. Since the height is 
calculated as an average over this region, terrain features 
such as peaks, pits, passes, ridge lines, and stream courses 
cannot be represented using a coarse matrix. The advan- 
tage of these grid DTM is the fast data access nevertheless 
much storage capacity is needed in case of a global DTM. 
The mathematical methods of surface modelling rely on 
continuous 3-dimensional functions that are capable of 
representing complex forms with a high degree of smooth- 
ness. The surface can be split into patches of equal area 
and the associated heights are calculated from point obser- 
vations within the patches. Weighting factors are used to 
ensure that the surface patches match at the edges, though 
149 
the surface does not always need to be continuous in slope 
at the borders. For global DTM modelling Fourier series 
and interpolation with multiquadric polynomials are also 
used. In the latter case we have to solve a linear system 
which causes problems if the data amount is too high. The 
above methods are described in (Burrough, 1986; Gold, 
1989) in more detail. 
Wavelets have proven their efficiency for use in numeri- 
cal analysis and signal processing and there are also a few 
publications (Li, Shao, 1994; Zhou, Dorrer, 1994) in Pho- 
togrammetry. Their power lies in the fact that a small 
number of coefficients can represent general functions and 
large data sets quite accurately. The method proposed in 
this paper is similar to Fourier series expansions but gov- 
erned by ideas of B-spline functions and wavelet theory. 
After a short description of the mathematical fundamen- 
tals (Chapter 2) the new algorithm is given in Chapter 3. 
In Chapter 4 we present results of the first tests with a 
DTM of Australia. In a second example a data set derived 
from images of the asteroid 243 IDA is processed. Chapter 
5 concludes the paper with an outlook on future work. 
2 MATHEMATICAL FUNDAMENTALS 
Starting with the description of the well known “next 
neighbour” approximation we introduce a new approach 
for global DTM modelling on the sphere. In our descrip- 
tion we denote as reference points the given primary data, 
acquired from digitized maps, Photogrammetry (e.g. im- 
age matching) or GPS measurements. In the next neigh- 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996