„4 - 
points within a specified maximum distance from an interpolated point will be used. 
It is then necessary also to use the same maximum distance for all interpolations, 
not to restrict the number of points within this distance, and to use a weight 
function that approaches tangentially to zero, or nearly so, when the distance to 
a reference point approaches the specified maximum distance. Several DTMs do not 
satisfy these requirements. 
However, in almost all cases the moving surface method is used only to 
compute terrain heights either at the nodes of a square grid or at regularly spaced 
points along a road axis and along cross sections. The heights of other points are 
then determined by surface or curve fitting between the computed points. In these 
cases the choice of a weight function is not very critical and in practice widely 
different weight functions are used. 
In the ECU system and in program HOELI the weight function 
w = (1-r)2/r2 (2a) 
is used, r being the ratio between the distance from interpolated point to refer- 
ence point and a maximum distance beyond which reference points are not used. The 
first two of the above German programs use 
w= 1/0" (2b) 
with n even and, at least in the second program, equal to two. The IBM-Germany 
program uses n=2 for the second degree surface and n=4 for the tilting plane. The 
Swedish National Road Service uses n=3. Torlegard notes that a program by IBM- 
Sweden allows a choice between n=1, 2, and 3. The Czechoslovakian DTM uses 
w=1- 0.9 r? (2c) 
It is also important to note that a sharp drop-off of the weight at small 
distances produces a representation of the terrain which fits well at the reference 
points. Too sharp a drop-off, especially if the reference surface is a plane, tends 
to result in isolated bumps and hollows at the extreme reference points and in wavy 
contour lines, with a period that reflects the spacing of the reference points. A 
slow drop-off can be used to produce smoothing. 
In experiments by the present writer, reported in [9]*, the function 
w = (1-7)3(1-22) 3/7" (2d) 
with n=1 and n=2 produced good results without smoothing. The exponential function 
w = exp(-ar?) (2e) 
with a=14 and a=20 produced slightly better results with some smoothing. A 
suitable function for very strong smoothing proved to be 
w=1-2r2; (rs0.5) 
w= 2(1-r)2; (r20.5) (2f) 
In the ECU system, smoothing or increased smoothing is produced by 
replacing » in the weight function by s if r<s, where an acceptable value for s is 
  
*A more elaborate version of this paper was prepared but somehow did not get 
published. This version, with more detailed results of experiments and with the 
less brief but more accurate title "Evaluation of some interpolation methods" 
can be obtained from the author.