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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004
range of scales for many types of surfaces. Studies of fractals
in elevation surfaces have shown how they can be used to
model bare earth terrain (Batty and Longley, 1994) at
different resolutions on the basis of their self-repeating
properties. However, this same argument is not directly
transferable to the modelling of large scale urban surface
form (say at the level of modelling buildings) due to the
inherent complexity and lack of self-repeating structures at
this level in urban areas. If urban surfaces cannot be
modelled with fractals at these large scales, then it follows
that we do not understand how the characteristics of these
surfaces may alter at higher resolutions. In effect, this means
that we do not understand how the pattern of interpolation
errors may change for different grid resolutions.
However, several global characteristics of the various
surfaces created by different interpolation methods have been
noted previously in the literature. Zinger et al (2002)
commented that linear interpolation will tend to overly
smooth and deform building edges. However, such general
characteristics reveal little about the exact spatial pattern of
error within a surface model. Lloyd and Atkinson (2002)
further investigated the quantification of error within
interpolated surfaces. The authors focused on a comparison
of Inverse Distance Weighting (IDW) and kriging
interpolation, and quantified the inaccuracy in each surface.
Such measures are useful general indicators of error within
surface models, again however they do not reveal anything
about the spatial pattern of errors across the surface. One of
the most relevant comparisons of interpolation algorithms
was that of Rees (2000) who investigated the interpolation of
gridded DEMs to higher resolutions - whilst this study did
not look at the interpolation of irregularly spaced data onto a
regular grid (the subject of this paper) many of Rees' (2000)
conclusions are nevertheless relevant. Rees (2000) concluded
that simple bilincar and bicubic interpolations are adequate
for most elevation model requirements in non-urban areas.
Rees (2000) conclusions are tested within this study for urban
areas.
1.2.2 Grid Spacing
Whilst the effect of different interpolation methods on the
form of the surface has been investigated in the past (eg.
Zinger et al, 2002; Morgan and Habib, 2002; Lloyd and
Atkinson, 2002; Smith et al, 2003a) there has been little
research into the effect of changing grid size in the
interpolation stage save for that of Behan (2000). Behan
(2000) quantified error within models produced from
different interpolation algorithms. It was found that the most
accurate surfaces were created using grids which had a
similar spacing to the original points. Behan's (2000) study
looked at global or average error differences between two
interpolation methods.
1.3 Aim of the Investigation
The investigation presented here quantifies the amount of
model error introduced during the interpolation process, and
specifically examines the pattern of errors created when
modelling at different spatial resolutions. From previous
literature it is known that the interpolation method and scale
will influence the derived urban DSM, however we do not
know to what extent.
997
2. METHODOLOGY
2.1 Creating the Surfaces
2.1.1 The Data and the Study Area
DSMs were created from a subset of a first return laser
scanning dataset, supplied by the Environment Agency. The
data were captured from an airborne sensor, at a point
density of -2m. The area used for modelling is shown in
Figure 1, which shows that this sample region comprises a
complex roof structure (church), some bare earth, a flat roof
and a variety of vegetation. Despite being a small area (1315
points over a 80m by 50m region), the surface was
considered to be representative of the typical types of
structure found in the wider region. In addition, the
investigation has been conducted over 2 more study areas to
ensure the reliability of the results.
Figure 1: Orthorectified photograph of the corresponding
area. Aerial photography reproduced with permission of
Ordnance Survey € Cc Ordnance Survey. All rights
reserved.
2.1.2 The Choice of Interpolation Methods
There are many routines for spatial interpolation available
and these have been widely documented in the past (Watson,
1992). However, not all of the methods are suitable for
elevation modelling from LiDAR data. In particular, for the
urban surface environment where there are frequent
discontinuities, local interpolation rather than global or fitted
function methods are preferable in order that the local
complexity in the surface be retained as much as possible.
For this reason, this study compared only local deterministic
interpolation techniques.
2.4.3 Performing the Interpolation
The raw points were first resampled onto a regular Im, 2m
and 4m grid, using four interpolation methods: bilinear,
bicubic, biharmonic splining, and nearest neighbour, and the
resultant surface forms produced are shown below in Figure
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