International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004 
  
Surface Creeted Usng Nearest Nelohbour interpolation on a 1m Grid 
no 
Pest ores 
  
x Coordnate 
Surface Crested Using Ecubic Interpolation on a tm Gad 
  
x coordinate 
Sutsce Crealed Using Bi^armmonic Spios interpolation on a tm God 
Height imedres) 
  
x coacdeute 
y coordnw e 
(d) 
Figure 2: Showing the (a)bilinear, (b) nearest neighbour, (c) 
bicubic, and (d) biharmonic spline surfaces created using the 
interpolation methods at 1m grid spacing. For information 
regarding the specifics of the four algorithms see Sandwell 
(1987), Watson (1992), and Smith et al (2003b). 
The differences in surface form between the DSMs shown in 
Figure 2 can clearly be seen. The nearest neighbour surface 
is blocky and 'stepped' in appearance, whilst the spline 
surface is much smoother, with many of the building and 
vegetation edges appearing curved. The quantified 
differences between the surface height predictions and the 
raw data are presented below. 
2.2 Comparing the Surfaces 
For the purposes of this investigation, the error ( € ) at each 
investigated point within the surfaces was considered to be 
  
998 
the difference between the raw data point, Zx)., and the 
interpolated value, Zi(x), for that location (see eq. 
below). 
&(x) 2 Z(x) - Zi(x) (1) 
where &(x) = measured error at location (X), Z height 
value, Zi = interpolated height value. 
This calculation was repeated across the surface to assess the 
success of the four interpolation algorithms at the 
investigated grid spacings. Model suitability was assessed in 
relation to how much error was introduced to the surface by 
each of the techniques - the most suitable model being the 
one which introduced the least error ( € ). 
In order to assess model suitability, some of the raw data 
points had to be omitted from the surfacing process. If all of 
the data points were used to interpolate a surface, then the 
goodness-of-fit of the surface could not be assessed with 
these same data points as this would yield an overly 
optimistic (low) prediction error. For this reason a standard 
validation procedure was employed which involved omitting 
some raw data from the interpolation procedure and assessing 
the success of the procedure to model in the absence of these 
known values. The method chosen is called the split-sample 
validation routine which is advocated by Declercq (1996). In 
this procedure part of the sample values are omitted, the 
interpolation is performed, and the difference between the 
predicted and the raw data values at these locations are 
calculated. This difference is then used as a measure of the 
success of the algorithm. The usefulness of this technique is 
increased when it is used iteratively, where the number of 
omitted points is progressively increased and the differences 
calculated. This can return useful information regarding the 
stability of the algorithm, and its ability to cope with 
differences in input point density. The  split-sample 
methodology was used in this way in Lloyd and Atkinson 
(2002). The authors used sample sizes of 95%, 50% and 25% 
of original points to measure the effects of interpolation for 
rural DEMs. The methodology outlined in Lloyd and 
Atkinson (2002) is adhered to in this study, and the same 
proportions of omitted data are used here for the comparison 
of interpolation methods. For the grid spacing investigation a 
random selection of 5% of the raw points were omitted and 
the surfaces produced at different resolutions. The success of 
each resolution was then assessed by calculating the 
difference between the omitted data and the surface 
predictions at these locations. The amount of omitted data 
points in each sample was not varied for this part of the 
investigation. 
In each investigation the omitted data points were chosen ina 
random selection process, and the tests run multiple times to 
ensure different points were selected each time and similar 
results obtained. This ensured the reliability of the 
investigation. 
3. RESULTS 
The model errors were calculated in accordance with 
Equation (1), and are recorded in Table 1 below. 
[nternatic 
Eds 
Table 1: " 
Assess th 
| 
'INTERPOLA 
195% Split sz 
{Bilinear 
iBicubic 
[Nearest Nei 
IBiharmonic : 
L———— 
L———— 
[450 Split s: 
|Bilinear 
iBicubic 
Nearest Nei 
|Biharmonic 
| 
i 
Bilinear 
[ficubic 
{Nearest Nei 
IBiharmonic 
Fam 
Table | 
method « 
there 1s : 
A discus 
Table 2 
Designec 
the Meth 
INTERPOLAT 
Bilinear 
Bicubic 
Nearest Neigl 
Biharmonic Sj 
31 Dis 
The res 
bicubic 
of all th 
spline n 
to be c: 
strong € 
unconst 
and mi 
general, 
highest 
across 
found | 
probabl 
there ar 
very s 
necessa 
In terr 
resoluti 
interpo 
errors, 
betwee 
was, he