ISPRS Commission III, Vol.34, Part 3A ,Photogrammetric Computer Vision“, Graz, 2002 
laser point in these areas is prognosticated by interpolating 
the heights of the neighbouring points (unweighted mean 
interpolation). Comparison of the interpolated height with the 
originally measured height yields a height difference. The 
standard deviation of all differences dH; in the area is a 
suitable measure for the laser point noise: 
T point noise = 
  
(1) 
with: 
n — number of laser points in 50m x 50m flat area 
dH, = cross correlated height difference 
In each laser dataset a large number of flat areas (about 100) 
have to be analyzed in order to eliminate the influence of the 
terrain roughness. 
4.2 Error per GPS observation 
Flat areas of 50m x 50m are also used for the computation of 
the amplitude of this error type. In this case, the areas are 
selected in strip overlaps (see figure 2). Note that these areas 
are used as tie points for the strip adjustment as well. For 
every tie point the height difference between both concerned 
strips is calculated in the following way. At the location of 
every laser point in the 50m x 50m area in strip 1, an 
interpolated value is computed by interpolation of the 
neighbouring laser points in strip 2. The measured height of 
the laser point in strip 1 is subtracted from the interpolated 
value of strip 2. The mean value of all height differences in a 
50m x 50m area is calculated yielding the height difference 
between adjacent strips for this tie point. Due to the 
averaging process, the error per laser point (point noise) is 
almost absent in this height difference. 
Profile 
  
       
tie point 
9 (50m x 50m 
flat area) 
Figure 2. Horizontal, flat areas (tie points) in strip overlap. 
The height differences calculated at all tie points along a strip 
overlap yield a profile (see figure 2). These profiles are 
analyzed for systematic effects. This can be done with 
empirical covariance functions, a tool from spatial statistics 
which is closely related to the (semi)-variogram from 
geostatistics. The empirical covariance function C(s) 
describes the correlation between the signal of sample points 
as a function of the distance s between these points in time or 
in space. 
n, 
Le 
C(s)=C(0)-—} (à TER (2) 
2n, 
izl 
with: 
C(0) — variance 
Ns = number of used point pairs for distance s 
Xi = signal at point ; (in this case: x; = A; ) 
Il 
point belonging by point i at distance s 
All possible point pairs lying at a distance s apart from each 
other are selected, and the covariance for the signal of these 
point pairs is calculated. This is repeated for increasing 
distances up to the largest chosen distance s,,,. Usually, the 
signals of sample points at larger distances from each other 
are less correlated than the signals of sample points close to 
each other. The covariance values can be plotted in a graph. 
An example is given in figure 3. The numbers in the graph 
indicate the number of point pairs used to produce the 
covariance value. In this example s,,, equals 15 km. 
Overlap of strips 44 and 45 
  
  
  
  
  
  
30 
qi 
25 1 
a J 
= 5754 
m % 
S NS 85 J 
© S 
E ko e 
& 5 NR J 1 
0 e Per 54 ais 
y A J Be 55 ul 
e 68 / bo 
ar % 53 7 
10 : : 2 
0 5 10 15 20 
Distance (km) 
Figure 3. Empirical covariance function with fitted curve. 
Usually, an analytical function is fitted through the 
empirically determined data points. In our case, a Gaussian 
function is chosen because it is representative for the 
behaviour of the calculated empirical covariance functions. 
From this curve three characteristic parameters can be 
determined (figure 4): 
e The nugget. This is the variance of measurement 
errors combined with that from spatial variation at 
distances shorter than the sample spacing. 
e The sill. This parameter gives the largest occurring 
covariance at smallest possible distance between 
sample points (here about 1 km). 
e The range (also called correlation distance). The 
signal values from sample points at this distance 
and farther away from each other are no longer 
correlated. 
In our height precision assessment the sample points are the 
tie points along a profile. The signal consists of the height