Global Positioning System Reference
In-Depth Information
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rover location using its own observations and then corrects them for troposphere and
ionosphere, i.e.,
P
v,
1
P
v,
2
Φ
P
k,
1
P
k,
2
Φ
11
1
ρ
p
vk
T
vk
I
vk,
1
,P
+
α
f
=
+
(7.231)
p
v,
1
p
k,
1
1
−
1
p
v,
2
p
k,
2
1
α
f
Φ
Φ
and transmits the corrected, virtual observations to the rover. The rover merely has to
double-difference its own observations with those received from the master reference
station. No additional tropospheric or ionospheric corrections/interpolations are re-
quired at the rover because the effective, virtual baseline is very short, typically in the
range of meters corresponding to the rover's initial determination of its approximate
location from pseudoranges. In Equation (7.231) the subscript
v
in
[29
p
vk
indicates that
the distances are evaluated for the location of the virtual reference station
v
. The need
fo
r the rover to transmit data can be eliminated if the master reference station trans-
m
its corrected virtual observation to an evenly spaced grid of predetermined points
w
ithin the network. The rover can determine its position with respect to the nearest
vi
rtual grid point. The grid approach supports many mobile users, since they all use
th
e same data sent from the master reference station.
The multiple reference station techniques described above depend on the master
re
ference station operator's skill in modeling the spatial and temporal corrections
(7
.227) and (7.228). The success of fixing the ambiguities correctly at the rover di-
re
ctly depends on the validity of the tropospheric and ionospheric corrections. Raquet
(1
998), Lachapelle et al. (2000), and Fortes (2002) compute a covariance function
fro
m the double-difference carrier phase discrepancies of the known network base-
lin
es. They then use least-squares collocation to compute undifferenced corrections
fo
r each satellite at all reference stations and predict undifferenced corrections for a
gr
id of known locations. The conversion of corrections from the double-difference
do
main to the undifferenced domain is carried out based on covariance functions
as
sociated with spatial differential errors (for troposphere/orbits and ionosphere) and
as
signing the absolute errors equal to zero at a reference point normally located close
to
the center of the region covered by the network (considering that the user software
no
rmally implements the double-difference model, what matters is how the residual
er
rors change from one location to the other and not their actual absolute values).
Th
ese covariance functions are then used to compute covariance matrices to be ap-
pl
ied in the prediction of the errors at the user location using least-squares colloca-
tio
n. The master reference station transmits these corrections to the other reference
sta
tions, where they are applied to the undifferenced observations. These corrected
un
differenced observations are broadcast over the network (in addition to the pre-
dicted gridded corrections). Zebhauser et al. (2002) suggest transmitting the obser-
vation of the master reference station and the observation differences between pairs
of reference stations. The latter would be corrected for location, receiver clock, and
ambiguities, i.e.,
ρ
Lin
—
8.9
——
Lon
*PgE
[29
