ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002 
  
differences between adjacent strips measured at these tie 
points. The distance between the tie points in a profile is 
usually approximately 1 km. The distance between two 
successive GPS observations is approximately 100 m. 
Therefore, it can be assumed that the nugget can be related to 
the variance of the GPS observations corresponding with the 
random GPS errors. For each strip overlap in a laser dataset 
(block) the nugget can be determined for two profiles (see 
figure 2). 
Overlap of strips 44 and 45 
  
| nugget 
sill 
Covariance (cr) 
= 
range 
  
  
  
  
  
5 10 15 20 
Distance (km) 
  
Figure 4. Characteristic parameters of a covariance function. 
4.3 Error per strip 
The introduced parameters, sill and range, of the fitted 
covariance function (see figure 4) describe systematic effects 
per strip in along-track direction. This could, for example, be 
a periodic effect of several kilometres length or an along- 
track tilt of the entire strip. Figure 5 shows the principle of 
such a long-time periodic error affecting groups of GPS 
observations (or generally spoken parts of strips). 
The depicted behaviour of the signal due to GPS-noise (see 
figure 5) is an approximation and simplification of reality 
because each laser position is determined by interpolation 
between at least two GPS measurements and, moreover, we 
have to deal with two interfering signals (from the two 
overlapping) strips. But actually we found systematic effects 
in the strip overlaps at distances much smaller than the 
sample spacing of 1 km (forming the nugget in figure 3). 
el distance of one 
| GPS observation 
long-time periodic error point error 
Figure 5. Height error € as function of along-track distance d 
For each strip overlap in a laser dataset two values (from two 
profiles) for the sill and for the range of the corresponding 
covariance function are determined as well. These values 
represent the errors per strip in along-track direction. 
Note that vertical offsets and tilts in across-track direction are 
not manifest in the covariance function. Vertical offsets are, 
however, described by the fourth error component as well. 
There are alternative methods to detect and visualise tilts in 
across-track direction (like hill shades and profiles). The 
errors per strip can be decreased with suitable strip 
adjustment procedures. 
4.4 Error per block 
One of the results from the strip adjustment described in 
Crombaghs et al. [2000] is a vertical offset correction for 
every laser strip. The size of these offsets are a measure for 
the height deviations with respect to the national height 
system. Assuming that the laser scanning companies already 
executed a strip adjustment, these offsets must nearly be zero 
when executing a further strip adjustment for controlling 
purposes at the Survey Department. 
Besides, the standard deviations of these offsets (a further 
output of the strip adjustment) indicate the height precision 
of the offset parameters and are therefore measures for the 
height precision of each individual strip. The estimated 
precision of the height offsets depends on the configuration 
of the block: position and quantity of strips, cross strips and 
control points. 
It is common practice to use height differences with the 
available ground control points to say something about the 
quality of a laser DEM. In our opinion, it is, however, by far 
preferable to use the estimated strip offsets themselves and 
the corresponding standard deviations to get a thorough 
insight intp the height precision of each strip and the entire 
block. 
The used observations in the strip adjustment are height 
differences between strips (*measured' at tie points), between 
strips and ground control points, and between strips and cross 
strips. In order to get realistic values for the standard 
deviations of the strip offsets, it is necessary to use realistic 
values for the covariance matrix of the observations in the 
strip adjustment This can be achieved by using realistic 
values for error components 2 and 3. These errors influence 
the precision of the *measured' height differences. This has to 
be taken into account in the stochastic model of the 
observations. The elements at the main diagonal of the 
observations covariance matrix originally comprised values 
for point noise and uncertainties due to interpolation errors. 
These main diagonal elements were increased for error 
component 2 and for the sill of error component 3. Because 
of error component 3, there are also correlations between 
neighbouring height differences (at distance s). They are 
modelled in the off-diagonal elements with a Gaussian 
function and the parameters from a representative covariance 
function derived from several datasets: 
1 s[km] | (3) 
cov [cm?1= sill[cm?] :e rengel Pa]