Michal Gallay, Christopher D. Lloyd, Jennifer McKinley: Exploring DEM error with geographically weighted regression (poster), 9th International Symposium GIS Ostrava, VŠB – Technical Univerzity of Ostrava, from 23rd to 25th January 2012

1. Exploring DEM error with geographically weighted regression
Michal Gallay 1 - Christopher D. Lloyd 2 - Jennifer McKinley 2
1 - Institute of Geography, University of Pavol Jozef Šafárik in Košice, Slovakia
2 - School of Geography, Archaeology and Palaeoecology, Queen’s University Belfast, UK
Area ratio Fisher’s K
Adapted from Hobson (1972), McKean and Roering (2004)
Input data
DEM error
DEM surface
roughness
Exploring
relationship
0.6680.4560.1840.001InSAR DSM vs log(FK)
0.6380.4050.2020.002InSAR DTM vs log(FK)
0.6650.4270.2010.001Photog.DTM vs log(FK)
0.6830.4190.1870.033Cont. DTM vs log(FK)
0.7110.4660.2340.014Cont. DTM vs AR
0.6920.4660.2350.016Photog.DTM vs AR
0.6420.4020.1850.030InSAR DTM vs AR
0.6760.4970.2730.024InSAR DSM vs AR
3rd QrtMedian1st Qrt
GWR R2
OLS R2
DEM residuals against DEM roughness
* unobstructed and obstructed flat land, Intermap (2002),
** GeoPerspectives (2006),
*** with respect to the contour interval 10 m Ordnance Survey (2001)
5 ***3.045Contour OS DTM
1.5 **2.765Photog. DTM
1 , 2.5 *3.675InSAR DTM
1 , 2.5 *3.665InSAR DSM
Units in metres
Stated RMSERMSE maskCell sizeDEM origin
The methodology of geographically weighted regression (GWR) is
described in detail by its developers in Fotheringham et al. (2002). Carlisle
(2005) used ordinary least squares (OLS) for DEM error modelling noting
that GWR could improve the predictions if a larger reference dataset is
used. Erdogan (2010) conducted such GWR modelling of error of
interpolated DEM from levelling data. The findings of the OLS regression
reported in the scatter plots and the table below show no apparent linear
relationship between surface roughness and the DEM errors. However,
locally the relationship is evident and geographically weighted approach
increases the cooefficient of determination (R2) on average. For details on
the presented analysis consult Gallay et al. (2010) and Gallay (2010).
This case study compares five different commercial high-resolution digital
elevation models (DEMs) originated from airborne light detection and ranging
(LiDAR), interferometric SAR (InSAR), photogrammetric acquisition and
contour maps digitizing. The LiDAR DEM derived from last return points was
considered as the reference DEM (Gallay et al. 2008). The aim was to analyse
the statistical and spatial distribution of the DEM error and its relationship with
the DEM surface roughness (Desmet 1997, Wise 2011).
■ – observed value, InSAR, Photog.,
Contour DEMs
● – more accurately measured reference
value LiDAR last return point elevations
(ca. 2 m sampling interval)
O – interpolated reference DEM value,
cell size = 5 m
Z■ – ZO = DEM error
Surface roughness measured as area ratio and Fisher’s K (inverted
vector strength) were used to parameterise the DEM surface (Hobson
1972, Mckean and Roering 2004, Grohmann 2004). Useful as proxies of
slope angle and slope aspect as it avoids circularity. Area ratio roughness
is the ratio of real surface area to the area of its orthogonal projection.
Fisher’s K defines the dispersion of unit vectors normal to the surface.
While the first measure is sensitive to the local variation of the slope angle
the latter is sensitive to the variation of the slope aspect.
The results show that globally no linear relationship exists between the sur-
face roughness and DEM residuals but it was found to be very diverse locally
(Anselin 2003, Erdogan, 2010). High elevation errors occurred along DEM arte-
facts and sharply defined landforms. The applied roughness parameters were
found to be useful predictors of such features and could be used for their iden-
tification (Gallay 2010). The findings also suggest that the assumption of statio-
narity and Gaussian distribution of the DEM error field is questionable (Hunter
and Goodchild 1997, Wechsler and Kroll 2006).
Great Langdale
the Lake District
England
LiDAR data (c) Environment Agency, InSAR NextMap data (c) Intermap, Photogrammetric data (c) GeoPerspectives, Contour DTM data: OS LandForm Profile DTM (c) Crown Copyright Ordnance Survey.
Shaded relief calculated in LandSerf (c) Jo Wood, sun azimuth angle 134°, sun elevation angle 10°, resoluti on of the DEMs 5 metres.
DEM error calculated as ‘DEM – Reference DEM‘, cell size = 5 metres, the reference DEM calculated from last return LiDAR points with IDW in Geostat.Analyst ArcGIS 9.0, points = 12, power=2, cell size 5 metres.
Contour interval 20 metres, contour lines (c) Crown Copyright Ordnance Survey.
Surface roughness as area ratio calculated for a 5x5 moving window by r.roughness.vector (Grohmann 2004), GRASS GIS 6.2.2, cell sizes of the DEMs = 5 metres
Contour interval 20 metres, contour lines (c) Crown Copyright Ordnance Survey.
Surface roughness as Fisher’s K (inverted vector strength) calculated for a 5x5 moving window by r.roughness.vector (Grohmann 2004), GRASS GIS 6.2.2, cell sizes of the DEMs = 5 metres
Contour interval 20 metres, contour lines (c) Crown Copyright Ordnance Survey.
GWR coefficient of determination (R.sq). GWR between DEM error and area ratio roughness of the corresponding DEMs cell-size = 5 metres, bandwidth = 15 metres, gwr.weights: exp(-0.5(dist/bw)^2).
Calculated by spgwr package (Bivand and Yu, 2008). Contour interval 20 metres, contour lines (c) Crown Copyright Ordnance Survey.
GWR coefficient of determination (R.sq). GWR between DEM error and logarithm of Fisher’s K roughness of the corresponding DEMs cell-size = 5 metres, bandwidth = 15 metres, gwr.weights: exp(-0.5(dist/bw)^2).
Calculated by spgwr package (Bivand and Yu, 2008). Contour interval 20 metres, contour lines (c) Crown Copyright Ordnance Survey.
Four different DEMs were compared with
respect to an LiDAR interpolated reference
DEM. The rationale for the interpolation of
the relatively dense field of LiDAR points
was to estimate the elevation exactly on
locations of the observed values which
were the centres of grid cells of the DEMs.
The reference DEM was subtracted from
the other four DEMs. Thus, the difference
surfaces of the same spatial resolution
were generated. The DEM residuals were
masked on locations where above-ground
surface objects were present to avoid bias.
References Acknowledgements
European Social Fund
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