Antti Lehtinen Doppler Positioning with GPS - Matematiikan laitos

geometry can be reduced to a single real number. The geometrical dilution of
precision value (GDOP) is defined as
H
T −1
GDOP = trace H (3.12)
where trace means the summation alongthe main diagonal of the matrix. The
GDOP value can be used for estimatingthe magnitude of the positioningerror,
as follows:
(3.13)
σn =GDOP× σρ
where σn is a combined standard deviation of the solution x, y, z and ctu components,
while σρ is the pseudorange measurement standard error. Besides of
the GDOP, there are other useful values that can be computed from the GDOP
matrix. These include position DOP, horizontal DOP, vertical DOP and time
DOP [Parkinson & Spilker, p. 414].
The pseudorange positioning error with a particular satellite geometry can be
estimated with the formula 3.13. Accordingto [Kaplan, p. 262], the standard
error σρ in pseudorange measurements is about 8 meters. According to [Kaplan,
p. 270], the GDOP value is on average below 3. The equation 3.13 now shows
that the positioningerror is usually below 25 meters. Thus, the positioning
performance with the GPS pseudorange positioning is far better than with the
Transit Doppler positioningsystem.
3.5 Limitations of the GPS
In the previous section it was stated that the GPS has a very good positioning
performance. The theory for the pseudorange positioning is well developed and
relatively easy. Still, the GPS has its limitations. One of the most severe is the
satellite availability. The standard GPS receivers need to measure pseudoranges
to at least four satellites simultaneously, in order to solve for the position estimate.
In theory, at least four satellites should be available nearly always nearly everywhere
[Kaplan, p. 287]. In practise, however, there are problems. The problems
usually arise in weak signal conditions. Sometimes the receiver should have
many satellites in sight but there is some barrier that prevents the GPS signal
from reachingthe receiver. This occurs frequently in urban areas and indoors
[Syrjärinne, p. 22]. Usually the signal is received, but it is too much attenuated
for the pseudorange positioning purposes.
Recently, some effort has been put to resolve the satellite availability problem.
Novel methods have been developed for enhancingthe positioningperformance
14