Geoscience Reference
In-Depth Information
where Φ
i
(
t
) is the measured carrier phase expressed in cycles referred to
station
i
and satellite
j
at epoch
t
. The time-independent phase ambiguity
N
i
is an integer number and, therefore, often called integer ambiguity or
integer unknown or simply ambiguity.
Relative positioning requires simultaneous observations at both the ref-
erence and the unknown point. This means that the observation time tags
for the two points must be the same. Assuming such observations (5-14) at
the two points
A
and
B
to satellite
j
and another satellite
k
simultaneously
at epoch
t
, the following measurement equations may be set up:
Φ
j
A
(
t
)=
1
λ
j
A
(
t
)+
fδ
A
(
t
)+
N
A
,
Φ
A
(
t
)=
1
λ
A
(
t
)+
fδ
A
(
t
)+
N
A
,
Φ
j
B
(
t
)=
1
(5-15)
λ
j
B
(
t
)+
fδ
B
(
t
)+
N
B
,
Φ
B
(
t
)=
1
λ
B
(
t
)+
fδ
B
(
t
)+
N
B
.
Introducing the short-hand notations
Φ
j
AB
(
t
)=Φ
B
(
t
)
Φ
j
B
(
t
)
Φ
A
(
t
)+Φ
j
A
(
t
)
,
−
−
j
AB
(
t
)=
B
(
t
)
j
B
(
t
)
A
(
t
)+
j
A
(
t
)
,
(5-16)
−
−
N
j
AB
N
B
+
N
A
,
=
N
B
N
A
−
−
we form the double-difference model which is defined as
Φ
j
AB
(
t
)=
1
λ
j
AB
(
t
)+
N
j
AB
.
(5-17)
Note that the receiver clock biases have canceled; this is the reason why
double-differences are preferably used. This cancellation resulted from the
assumptions of simultaneous observations and equal frequencies of the satel-
lite signals (which is justified for GPS).
Assuming
A
as reference station with known coordinates, the remain-
ing unknowns of the double-difference model are the desired coordinates
X
B
,Y
B
,Z
B
- which are comprised in
j
B
(
t
)and
B
(
t
) - and the ambiguities.
To solve for these unknowns, we need more satellites (to set up additional
double-differences) and also more epochs.
We do not consider linearization, possible redundant measurements, etc.
We just intended to demonstrate the principle. The desired result obtained
from (5-17) is the baseline vector
b
AB
with the components ∆
X
AB
,
∆
Y
AB
,
∆
Z
AB
or, finally, the GPS coordinates
X
B
,Y
B
,Z
B
derived from (5-10) via