Hierarchical Terrain Representation
Description
Multiresolution digital elevation models are getting more and more importance
for a scale-sensitive modelling of elevation data. In principle, they allow the
representation of the data at a variety of scales. Of special concern
are adaptive methods, where the resolution does not to be uniform but may be
variable.
Adaptivity is very important for error measurement and control. In general,
equal distribution of points (or, for that matter, triangles) does not imply
equal distribution of error. The error is usually measured by some norm.
Popular examples are the L2-norm, which is the squared integral of
the difference between the original or the approximation, or the maximum norm,
which is the maximum difference. Applying adaptive refinement, the approximation
error can be controlled and distributed as uniformly as possible for a
given error norm.
However, some types of error cannot (or cannot easily) reflected by some norm.
For example, in hydrological modelling and simulation, changes in the topology
of the DEM may yield surprising and unwanted effects. Small changes in
elevation values can lead to large changes in catchment size and structure.
Luckily, adaptive refinement can control possible changes in topology as well.
The necessary computation involves the determination and quantification of
critical points (i.e. local minima, maxima and saddles).
We have developed a tool which allows a fast multiresolution construction of
very large digital elevation models with error control and preservation of
topology. Some examples are given below.
Examples
If below pictures appear scrambled, please increase the width of your web
browser. Click on the pictures to see an enlarged version.
Above we show successive approximations of a digital terrain model of the
North Sea area. The approximations are based on the L
1 norm.
From right to left, the error threshold increases. Although the approximations
are quite satisfying for mountainous areas, coastlines are not preserved very
well. In the Netherlands, for example, land is quite flat leading to a small
approximation error. Note that the land/sea topology is also not preserved at
all (e.g. in Northern Germany and Denmark).
Here we put a special constraint on coastlines forcing the error to be smaller
in these areas. The resulting terrain models are still continous, though (the
corresponding triangulations do not contain so-called hanging vertices). The
numbers of triangles are comparable to the previous case, however, here
coastlines are preserved much better and topological problems are less severe.
These images show adaptive grids for different error thresholds and
corresponding DEMs of a crater lake (Laacher See) in western Germany. Note that
in the leftmost image the lake in the foreground gets an opening at its upper
border which would be disastrous for hydrological simulations.
Here we show the corresponding images with topology preservation. Note that
the numbers of triangles is again not much higher than in the previous examples.
Here, all isolines retain their original topological structure (not only the
isoline surrounding the crater lake). This implies that the connectivity and
number of components of the isolines are identical to the original
fine-resolution elevation model. Their exact geometric position does not have
to be preserved. The geometrical error of the isolines can controlled by a
gradient-type norm.
References
-
T. Gerstner,
Ein adaptives
hierarchisches Verfahren zur Approximation und effizienten Visualisierung von
Funktionen und seine Anwendung auf digitale 3-D Höhenmodelle,
Diplomarbeit, Institut für Informatik, TU München, 1995.
-
T. Gerstner,
Adaptive hierarchical methods
for landscape visualization and analysis, in "Process Modelling and
Landform Evolution", S. Hergarten, H.J. Neugebauer (eds)., Lecture Notes in
Earth Sciences 78, pp. 75-92, Springer, 1999.
-
T. Gerstner, M. Hannappel,
Error
measurement in multiresolution digital elevation models, in Accuracy
2000 (Proc. 4th International Symposium on Spatial Accuracy Assessment in
Natural Resources and Environmental Sciences), pages 245-252, Delft
University Press, 2000.
-
T. Gerstner, H.-P. Helfrich, and A. Kunoth
Wavelet analysis of
geoscientific data, in "Dynamics of Multiscale Earth Systems"
H.-J. Neugebauer (ed.), Lecture Notes in Earth Sciences, Springer, 2001,
to appear.
-
T. Gerstner, M. Rumpf, U. Weikard,
Error indicators for
multilevel visualization and computing on nested grids,
Computers & Graphics, 24(3):363-373, 2000.
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