Global Positioning System Reference
In-Depth Information
When the distance between the two receivers is lower than 10 kms, the atmospheric
propagation delays and the ephemeris errors can be irrelevant, allowing to achieve a
centimetrical accuracy. Over this distance, these errors grow up and can not be neglected.
Otherwise, these errors are very spatially correlated and can be spatially modelled
(Wübbena et al., 1996). However, to be able to predict and use in real-time these biases, three
conditions must be satisfied: the knowledge with a centimetric accuracy of the masters
positions, a control centre able to process in real-time data of all the stations, the continuous
carrier-phase ambiguity fixing also when inter-station distances reach 80-100 kms. This
concept is equal to bring to the left-hand side of (3), among the known terms, the first two
terms on the right-hand side, i.e.:
pq
pq
pq
pq
pq
pq
φ
()
i
−− = ++
ρ
λ
Ni
α
I
T
E
(4)
hk
hk
i
hk
i
hk
hk
hk
In this way, it is possible to model, not only between stations h and k , but also among all the
reference stations of the network, the residual ionospheric and tropospheric biases and the
ephemeris error. When these errors are modelled, they can be broadcasted to any rover
receiver.
3. From the concept to the implementation
First, it is possible to note that the (4) was written using two satellites. Otherwise, if a
network of GNSS reference stations is considered, the same satellites that are visible and
usable for the two or more master stations can not be necessarily visible from the rover
receiver. Therefore, it is better to move from ionospheric and tropospheric delays, which
depend by a couple of satellites, to something which depends only by a single satellite.
The network biases can be calculated in real-time using double differences, single differences
or non-differential equations. The achievement of a common value of ambiguities is required.
A theoretical proof of the equivalence between the non-differential and differential methods,
in particular, can be found in Schaffrin & Grafarend (1986). The use of a differential method
has pros and cons. The best advantage of the differential method is that the unknown
parameters are fewer. Otherwise, the main disadvantage is that there is a correlation problem.
Although the approaches are identical, in recent years the trend is to use a non-differential
approach. The network state parameters are evaluated by the use of a Kalman filter.
This methodology is obviously more complex, since both dispersive and non-dispersive
components of each station are considered as unknowns in the Kalman filter state vector.
This increased complexity is balanced by many advantages. The use of a Kalman filter
allows to increase the number of equations available at each epoch, including for example
measurements related to satellites that are not tracked by all the stations, in order to make
the network estimation more robust also when one or more permanent stations are not
available (e.g. transmission problems).
3.1 The non-differential model
p
p
φ ) and
Start in g from pseudorange and carrier-phase equations written for the L1 (
P
and
p
and p
L2 (
φ ) GPS frequencies considering only one receiver ( k ) and one satellite ( p ), it is
possible to separate the unknowns that depend on the receiver and the satellite:
P
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