can be determined by solving for the position 
unknowns (X,Y,Z) and the receiver clock error 
(dt). It has to be stressed that all parameters of 
equation 1 are double differenced as explained 
in the previous chapter. 
p=8(XYZyrodird, +e (1) 
The remaining formula descriptors are p which 
is the double differenced pseudorange 
observation, c the constant speed of light, d, 
the remaining atmospheric signal delay after 
double differencing and € which is the 
observation noise. s which implicitly contains 
the moving station coordinates, is the double 
difference range which describes the double 
differenced distance between the satellite- 
receiver combination. The noise of the 
pseudorange observations is the critical part in 
real-time mapping applications. The 
measurement accuracy which can be achieved 
with double-differenced C/A-Code observations 
is conventionally in the range of 3-5 m, resulting 
in a positioning accuracy of 5-10 m. Figure 2a 
shows the position differences of the projection 
center coordinates determined with GPS 
pseudorange observations and conventional 
aerial triangulation. From this figure it can be 
seen, that the positioning accuracy with C/A- 
Code pseudoranges is sufficient only for small 
scale mapping  («  1:50000). Recent 
developments to minimize the measurement 
noise, with narrow correlation techniques might 
improve the situation considerably. Observation 
accuracies for C/A-Code observations in the 
range of 5-30 centimeters have been reported by 
LACHAPELLE ET AL. [1992], resulting in 
positioning accuracies of a few decimeters. As 
soon as this technology is routinely available the 
simple data handling and data processing makes 
this observation type highly interesting for small 
and medium scale real-time mapping. 
The stringent accuracy requirements for large 
scale mapping makes the use of carrier phase 
observations necessary. In the case of carrier 
phase observations the incoming sine wave is 
measured against a reference wave which is 
generated by the GPS receiver. The problem 
with this measurement principle, is that the 
186 
correlation of the two signals is ambiguous, and 
only the fraction of the wavelength, by which 
the incoming signal is shifted, can be measured. 
Once the receiver locks onto the signal, the 
cycle counts are updated and a range difference 
With respect to the initial epoch can be 
measured. Equation 2 shows the observation 
equation for the carrier phase observation (À 
beeing the wavelength of the carrier phase 
observations). 
p = s (X,Y,Z)+cdt+d mo AN+e (2) 
If the initial ambiguity N can be successfully 
determined the inherent measurement accuracy 
(<2 mm) of the phase observations is sufficient 
for all map and image scales, resulting in a sub- 
decimeter position accuracy. Hence, especially 
for large scale mapping the initial ambiguity N 
has to be determined to exploit the high 
positioning accuracy with carrier phase 
observations. Further, it has to be kept in mind 
that the ambiguity has to be reinitialized as soon 
as the receiver does not update the cycle count 
correctly (cycle slips) or if the receiver can not 
track the satellite signal continuously (loss of 
lock e.g. due to signal obstructions). The need to 
determine the initial ambiguity complicates the 
use of carrier phase observations for real-time 
applications extremely. In principle, the initial 
ambiguities can be estimated and fixed at the 
beginning of a continuous sequence of 
observations in a static initialization, but due to 
banking angles in flight turns and the highly 
kinematic environment, losses of phase lock and 
cycle slips are frequent in airborne, real-time 
mapping applications. Hence, in most cases 
there is a need to re-initialize the ambiguities 
while the aircraft is moving. In recent 
publications several authors proposed methods 
for a real-time initialization of the ambiguities 
("ambiguity resolution on the fly"), based on 
statistical searching algorithms (e.g. HATCH 
[1990], FREI/BEUTLER [1990], SCHADE [1992]). 
The mentioned algorithms share some basic 
principles to distinguish between the correct 
cycle ambiguities and the incorrect ones. 
Usually, a n adjusted pseudorange position and 
its associated covariance information is used as 
a searching cube in which the potential solution 
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