Global Positioning System Reference
In-Depth Information
=
1
− α
f
−
1
Φ
− λ
1
N
km,
1
+ λ
2
N
km,
2
I
km,
1
,P
p
km,I
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
(7.226)
Let
k
now denote the master reference station and
m
the other reference stations.
The master reference station generates its own observations and receives observations
from the other reference stations in real time. The Kalman filter, which runs at the
master reference station, generates the corrections
T
km
=
T
m
m(ϑ
m
)
and
I
km,
1
,P
at every epoch, for all reference stations and all satellites. These corrections
are used to predict the respective corrections at a roving receiver's location. Various
models are in use for computing these corrections. For example, the parameterization
could be in terms of latitude, longitude, and height and using different models for
T
km
and
I
km,
1
,P
to consider their characteristic spatial and temporal behavior. One of the
simplest location-dependent models is a plane
T
k
m(ϑ
k
)
−
[29
T
km
(t)
a
p
a
p
a
p
a
p
=
1
(t)
+
2
(t) n
m
+
3
(t) e
m
+
4
(t) u
m
(7.227)
I
km
1
,P
(t)
b
1
(t)
b
2
(t) n
m
+
b
3
(t) e
m
+
b
4
(t) u
m
=
+
(7.228)
Lin
—
1
——
No
PgE
The symbols
n
m
,e
m
, and
u
m
denote northing, easting, and up coordinates in the
geodetic horizon at the master reference station
k
. The symbol
m
varies to include
all other reference stations in the network. A set of coefficients
a
i
(t)
and
b
i
(t)
, also
called the network coefficients, are estimated by least-squares for every satellite
p
and, in principle, every epoch. Because of the high temporal correlation of the tropo-
sphere and ionosphere, one might model these coefficients over time, thus reducing
the amount of data to be transmitted. The master reference station
k
transmits its
ow
n carrier phase observations, or alternatively, the carrier phase corrections as de-
sc
ribed by (7.212) and the network coefficients
[29
over the network. A rover
n
applies the tropospheric and ionospheric corrections (7.227) and (7.228) for its
ap
proximate position, and determines its precise location by least-squares from the
se
ries of double-difference observations
{
a
i
,b
i
}
pq
kn,
IF
(t)
T
pq
pq
kn
(t)
+ β
f
λ
1
N
pq
kn,
1
− γ
f
λ
2
N
pq
d
pq
kn,I,
Φ
−
kn
(t)
= ρ
kn,
2
+
(7.229)
Φ
−
1
f
I
pq
pq
kn,I
(t)
= λ
1
N
pq
− λ
2
N
pq
d
pq
kn,
IF
,
Φ
Φ
− α
kn,
1
,P
(t)
+
(7.230)
kn,
1
kn,
2
us
ing the standard ambiguity fixing techniques.
Rather than transmitting network coefficients
a
i
,b
i
}
{
, one might consider trans-
T
km
,I
km,
1
,P
}
m
itting corrections
for a grid of points at known locations within the
ne
twork. The mobile user would interpolate the corrections for the rover's approxi-
m
ate location and apply them to the observations. Vollath et al. (2000) suggest the
us
e of virtual reference stations (VRSs) to avoid changing existing software that
do
uble-differences the original observations directly. The VRS concept requires that
the rover transmit its approximate location to the master reference station, which
computes the corrections
{
T
km
,I
km,
1
,P
}
for the rover's approximate location. In addi-
tion, the master reference station computes virtual observations for the approximate
{
