ISPRS, Vol.34, Part 2W2, “Dynamic and Multi-Dimensional GIS”, Bangkok, May 23-25, 2001 
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Figure 1: Interpolation using the gravity model 
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Figure 4: Interplation using Sibson interpolation 
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Figure 2: Sibson interpolation 
The Sibson method, illustrated in Fig. 17, is based on the idea of 
inserting each grid point temporarily into the Voronoi diagram of 
the data points, and measuring the area stolen from each of a 
well-defined set of neighbours. These stolen areas are the 
weights used for the weighted average. The method is 
particularly appropriate for poor data distributions, as illustrated 
in Fig. 18, as the number of neighbours used is well defined, but 
dependent on the data distribution. 
Figure 3: Neighbour selection using Voronoi neighbours 
Fig. 19 shows the results. It behaves well, but is angular at 
ridges and valleys. Indeed, slopes are discontinuous at all data 
points (Sibson, 1980). One solution is to re-weight the weights, 
so that the contribution of any one data point not only becomes 
zero as the grid point approaches it, but the slope of the 
weighting function approaches zero also (Gold, 1989). 
Figure 5: Adding smoothing to Fig. 19 
Fig. 20 shows the effect of adding this smoothing function. While 
the surface is smooth, the surface contains undesirable “waves” 
- indeed, applying this function gives a surface with zero slope 
at each data point. This is sometimes called the “wedding-cake 
effect", and is also apparent in Fig. 16. 
Figure 6: Sibson interpolation using slopes at data points 
SLOPES - THE IGNORED FACTOR. 
This brings us to a subject often ignored in selecting a method 
for terrain modeling - the slope of the generated surface. In real 
applications, however, accuracy of slope is often more important 
than accuracy of elevation - for example in runoff modeling, 
erosion, insolation, etc. Clearly an assumption of zero slope, as 
above, is inappropriate. However, in our weighted average 
operation we can replace the height of a neighbouring data point 
by the value of a function defined at that data point - probably a 
planar function involving the data point height and local slopes. 
Thus at any grid node location we find the neighbouring points