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.