ISPRS, Vol.34, Part 2W2, "Dynamic and Multi-Dimensional GIS", Bangkok, May 23-25, 2001 
COMPONENTS OF AN INTERPOLATION MODEL. 
On the basis of a sufficient set of data points, we now wanted to 
generate a terrain model with satisfactory elevations and slopes, 
as the basis of a valid rainfall runoff model. Our approach was to 
interpolate a height grid over the test area, and to view this with 
an appropriate terrain visualization tool - in this case Genesis II, 
available from www.qeomantics.com. We feel that 3D 
visualization has been under-utilized as a tool for testing terrain 
modeling algorithms, and the results are often more useful than 
a purely mathematical, or even statistical, approach. 
We have restricted ourselves to an evaluation of several 
weighted-average techniques, as there are a variety of 
techniques in common that can be compared. All of the methods 
were programmed by ourselves - which left out the very popular 
Kriging approach, as too complicated. Nevertheless, many 
aspects of this study apply to this method as well, since it is a 
weighted-average method, with the same problems of neighbour 
selection, etc., as the methods we attempted. 
Figure 1: Interpolation from nearest point 
In general, we may ask about three components of a weighted- 
average interpolation method. Firstly: what is the weighting 
process used? Secondly: what is the set of neighbours used to 
obtain the average? Thirdly: what is the elevation function being 
averaged? (Often it is the data point elevation alone, but 
sometimes it is a plane through the data point incorporating 
slope information as well.) 
Figure 2: Interpolation from Delaunay triangulation 
The simplest possible technique, useful on occasion, is merely 
to give each grid node (if a grid is being created) the height of 
the nearest data point. While trivial, it is valuable for a variety of 
applications, such as image rectification, rainfall estimation, and 
others. All grid cells falling within the Voronoi cell of a particular 
data point are assigned its elevation. Fig. 12 shows the result for 
our contour data set: the skeleton can be seen to separate each 
plateau around a contour. 
Figure 3: Adding skeleton points to Fig. 14 
The next most simple weighted-average model is the 
triangulation, using the Delaunay triangulation described 
previously. Fig. 13 shows the result, including the skeleton 
draped over the flat triangles. Fig. 14 shows the improvement 
when estimated skeleton points are added. 
Figure 4: Selecting neighbours using a counting circle 
The other weighted average models that were tested were the 
traditional gravity model, and the more recent “area-stealing” or 
"natural neighbour” or perhaps more properly “Sibson” 
interpolation methods (Sibson 1980, Watson and Philip 1987, 
Gold 1989). Here the number of neighbours used may vary. In 
the case of the gravity model the weighting of each data point 
used is inversely proportional to the square of the distance from 
the data point to the grid node being estimated, although other 
exponents have been used. There is no obvious set of data 
points to use, so one of a variety of forms of “counting circle” is 
used, as in Fig. 15. When the data distribution is highly 
anisotropic there is considerable difficulty in finding a valid 
counting circle radius. 
Figure 16 shows the resulting surface for a radius of 5 (about a 
quarter of the map). Data points form bumps or hollows. If the 
radius is reduced there may be holes in the surface where no 
data is found within the circle. If the radius is increased the 
surface becomes somewhat flattened, but the bumps remain. 
The result depends on the radius, and other selection properties, 
being used. Clearly, in addition, estimates of slope would be 
very poor, and very variable.