ribed using
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the satellite
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GPS station
S is given in
jd for some
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should be
avoid any
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jnearity
matic errors
systematics
coordinate
ree different
from small
e visible. An
to missing
e) is again
in the GPS
pplied using
eters (time
1e projection
GPS Seconds ri [sec]
Figure 2: Strip 23 of photo flight Vechtel: Difference in
northing, easting and height for kinematic GPS station
with and without PRN13
The linearity of GPS systematics has been investigated
using data from ‘Vechtel'. Generally, systematic effects
and discontinuities in GPS trajectories are not visible in
the absolute positions, because they cannot be separated
from the actual aircraft flight behavior (turbulence, etc.).
An adequate tool for detailed investigation is the
comparison of different GPS evaluations (e.g.
independent stations) or artificially manipulated data sets
(e.g. falsifying ambiguities). Then systematic errors
magnify and show up.
To investigate the GPS systematics the ambiguity of one
satellite was falsified by 1 cycle and compared to the
initial solution. After fixing as much ambiguities as
possible, the remaining were forced to integers and the
GPS positions were estimated individually for each strip
without any cycle slip or change of satellite constellation
using reference station ‘Meppen’.
25 j
cm —-
34
14
1] DW
À ris PILAM
T .
1d ud u.
+ ue FERIIS IAE Tr —
0 Ya a Min = e
— T——
si ento MM
Ma ore «wn
de ut ae =
:- /
1 /Á
onn h
1] /
'
3 Ÿ
I
i E A a Wu hurt I T T
219100 219200
GPS week seconds [sec]
Figure 3: Strip 21: Residuals after linear regression of
coordinate differences (PRN26 +1 cycle)
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
= -2 -1
de fd
-—
-
T
219800
GPS week seconds [sec]
T
210400
Figure 4: Strip 22: Residuals after line regression of
coordinate differences (PRN26 *1 cycle)
Applying a linear regression to the coordinate differences
between falsified and best solution gives an idea of the
not modeled behavior in the GPS supported block
adjustment. Figure 3 and Figure 4 show the residuals
from a linear regression in northing, easting and height.
Significant deviations from linearity can be observed,
which amounts to some decimeters. They correspond to .
remaining errors, which will not be modeled in the block
adjustment and therefore add uncertainties to the GPS
positions.
4. NEW APPROACH USING GPS SATELLITE
GEOMETRY
Comparisons of GPS kinematic trajectories with the
projection centers derived from a conventional block
adjustment show agreement at about 5 cm (Seeber et al.
1995). Obviously, the bundle block adjustment, even with
a reduced number of control points, can serve as a
constraint in the determination of ambiguities in the
combined adjustment.
Instead of modeling the systematic errors introduced by
false ambiguity resolution by a linear regression, which
does not suit the GPS model, a new approach to the
combined adjustment is proposed. Actual ambiguity
terms are improved in the combined bundle adjustment
using geometric GPS information and information on
ambiguities from the GPS processing.
The GPS and the combined block adjustment are
performed in the general way of intermediate
adjustments. The observation equation for the GPS
coordinates in the combined adjustment reads
= AB+v (2)
It can be extended by an additional ambiguity term,
which will be estimated in the combined adjustment
I+A-N = A+v. (3)
This requires, that the design matrix contains the
complete geometric information from the GPS
adjustment. Considering the normal equation
xl- Qn - (AT PA) (AT PI), (4)
357
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