nd, in order 
control is 
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lat must not 
ulation this 
ervations at 
are well 
er hand this 
on since the 
are almost 
id the flight 
1 points are 
| (Alobaida, 
e of strip 
loys GPS 
er with the 
nd. In this 
g the linear 
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impossible 
ct physical 
]: 
ints are 
the 
these 
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image 
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verview of 
A detailed 
jangulation 
s for both 
ns 4 and 5, 
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sumed that 
sure times 
sition the 
additional 
  
Xs x dx 
= * 
xb uds SAU (1) 
Z dz 
G o 
with: 
XY Observed coordinates of 
GG G 
the GPS antenna, 
X.X452 unknown coordinates of 
o 0 o 
the perspective center, 
R unknown rotation matrix 
associated with the image, 
dx, dy, dz GPS antenna eccentricity 
from the perspective 
center with respect to the 
image coordinate system. 
This model takes into account the eccentricity 
of the. GPS antenna with respect to the perspective 
center of the aerial camera. This offset must be 
determined by pre-flight calibration using terrestrial 
techniques. We could treat these shift parameters as 
constants or as observations; the latter approach allows 
us to consider the accuracy of the calibration results. If 
they are treated as constants, we have regular 
Observation equations, if they are considered 
observations, we have to deal with condition equations 
with parameters. 
Additionally, linear "drift" parameters can be 
introduced as unknowns for each of the coordinates 
(Friess, 1992): 
Xs X dxV [ax\ [bx 
x = Y +R*( dy )+( ay ) +( by ]*dt (2) 
© A dz az bz 
with: 
ax, ay, az shift parameters, 
bx, by, bz time dependent parameters, 
dt is the time interval. 
These parameters are very effective for 
correcting the datum transformation and cycle slips. 
They may be block-, strip-, or sub-strip invariant. If the 
GPS observations are continuous for the whole flight 
mission, one set of "drift" parameters is sufficient. If the 
satellite signal is interrupted, a new set of drift 
parameters has to be introduced. These parameters cause 
correlations and may lead to singularities. Therefore, a 
minimum of four ground control points and cross strips 
at the beginnings and ends of the strips are needed to 
produce accurate results (Gruen et al., 1993; Ackermann 
et al., 1993). 
For our particular application all control is 
available in WGS84, both the GPS observations at the 
aircraft and those along the linear feature. Therefore, 
we do not have to worry about the datum transformation 
problem. 
3. Strip Triangulation with a Known Linear 
Feature on the Ground 
For this triangulation procedure we assume that 
GPS observations are available in the aircraft at the time 
of exposure. These observations are handled in the 
bundle adjustment according to the mathematical model 
shown in formula (1) As mentioned earlier, GPS 
controlled strip triangulation cannot be carried out 
without ground control, because all exposure stations are 
along a single flight line. Instead of ground control 
points, however, we utilize GPS observations along a 
linear feature on the ground which is approximately 
parallel to the flight line. These ground coordinates can 
be collected from a mobile mapping platform, e.g. the 
GPSVan (The Center For Mapping, 1991), that travels 
along the linear feature. The major problem of 
implementing features instead of points as observations 
for strip triangulation lies in the association of 
corresponding features on the ground and in the image. 
Since the collinearity equations are based on a point to 
point correspondence between image and object space, 
feature observations cannot be directly introduced in 
bundle adjustment. An extended model has to be 
developed to include these observations. It is explained 
below. 
As a first step the coordinates of arbitrary 
points are measured along the linear feature in the 
image. These points are selected individually in each 
image. They need not to be common in successive 
images, because they are not used as tie points. An 
analytical function, usually a low order polynomial, is fit 
to the measured image coordinates. It represents the 
linear feature in that particular image. Each image has 
its own function which is of the form (3). The order of 
the polynomial can be chosen either by the operator or 
automatically. The polynomial coefficients and their 
variance-covariance matrix are determined by a least 
squares adjustment in a separate step for each image 
individually. 
f; y1,,0,,,4, 02 js 8p )=0 (3) 
with: 
205