Global Positioning System Reference
In-Depth Information
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superscripts
p
and
r
denote the respective base satellites. Following this notation, the
single-difference observations for a GPS and a GLONASS satellite, respectively, can
then be written as
f
1
c
ρ
ϕ
km,
1
,
GPS
=
q
km
N
km,
1
,
GPS
+
+
d
km,
1
,
GPS
−
f
1
d
t
km
(7.115)
f
1
c
ϕ
km,
1
,
GLO
=
s
km
N
km,
1
,
GLO
+
f
1
d
t
km
ρ
+
d
km,
1
,
GLO
−
(7.116)
These equations are based on the assumption that the receiver clock errors
d
t
km
are
the same for both types of observations, GPS and GLONASS. The receiver hardware
delays
d
km,
1
,
GPS
and
d
km,
1
,
GLO
, on the contrary, are dealt with separately. We have
neglected the signal multipath terms.
In case of GPS-only processing, one can combine the receiver delay
d
km,
1
,
GPS
and clock
f
1
d
t
km
term into a new receiver-dependent term
[27
ξ
km
that is estimated
every epoch. The station coordinates, the single-difference ambiguities, and the epoch
parameter
Lin
—
0.8
——
Nor
*PgE
ξ
km
can then be estimated from observations to several satellites over a
number of epochs using either least-squares or Kalman filtering. The usual ambiguity
fixing techniques can be applied. In the case of combined processing of GPS and
GLONASS single differences one uses the satellite-dependent parameterization,
q
km,
GPS
N
km,
1
,
GPS
+
ξ
=
d
km,
1
,
GPS
(7.117)
s
km,
GLO
N
km,
1
,
GLO
+
ξ
=
d
km,
1
,
GLO
(7.118)
[27
ξ
W
e note that
parameters are constants in time but are not integers because of
th
e receiver hardware delays. Using these auxiliary parameters the single-difference
eq
uations become
f
1
c
ρ
ϕ
km,
1
,
GPS
=
q
km
q
km,
1
,
GPS
−
+ ξ
f
1
d
t
km
(7.119)
f
1
ϕ
km,
1
,
GLO
=
km
km,
1
,
GLO
−
f
1
d
t
km
c
ρ
+ ξ
(7.120)
which allow us to estimate the station coordinates, the
constants, and the epoch
clock parameters, again, using classical least-squares or Kalman filtering formulation.
Unfortunately, the usual ambiguity fixing techniques cannot be directly applied to this
single-difference formulation because the
ξ
parameters are not integers. However, one
can still fix the double-difference ambiguities.
An immediate outcome of using (7.119) and (7.120) are the estimates
ξ
ξ
q
km,
1
,
GPS
,
s
km,
1
,
GLO
and their respective variance-covariance matrix, denoted by the symbol
C
ξ
. Using a matrix
D
containing elements 1,
ξ
1, and 0 at appropriate places, the
estimated double-difference ambiguities with respect to the GPS reference satellite
p
and GLONASS reference satellite
r
are
−
