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]