Global Positioning System Reference
In-Depth Information
effects is to compute the so called first order ionosphere-free combination of carrier-phase or
code pseudoranges measured on two frequencies. However, the second and third order
ionospheric terms and errors due to bending of the signal remain uncorrected in this
approach. Such a dual-frequency combination can be written in units of length as
(combining code / carrier-phase pseudoranges Eq. (12) / Eq. (13) measured on
f
1
and
f
2
frequencies and substituting
p
by Eq. (7), for details see Hoque & Jakowski, 2008)
2
2
f
f
1
2
Ψ
−
Ψ ρ Δ
= −
s
−
2
Δ
s
−
3
Δ
s
−
Δ
s
(14)
1
2
TEC
2
3
len
2
2
2
2
f
−
f
f
−
f
1
2
1
2
RRE
gr
2
2
f
f
1
2
Φ
−
Φ ρ Δ
= +
s
+
Δ
s
+
Δ
s
−
Δ
s
(15)
1
2
TEC
2
3
len
2
2
2
2
f
−
f
f
−
f
1
2
1
2
RRE
K TEC
(
−
TEC
)
K
(
Δ
TEC
−
Δ
TEC
)
2
1
bend
2
bend
1
Δ
s
=
=
(16)
(
)
(
)
TEC
2
2
2
2
f
−
f
f
−
f
1
2
1
2
∫
TEC
=
n ds
=
(
TEC
+
Δ
TEC
)
(17)
1,2
e
LoS
bend
1,2
q
(18)
Δ =
s
2
(
)
2
ff f
+
f
12
1
2
u
(19)
Δ =
s
3
22
12
3
ff
len
2
len
2
df df
−
22
11
(20)
Δ
s
=
(
)
len
2
2
f
−
f
1
2
where Ψ
1
, Ψ
2
and Φ
1
, Φ
1
are the measured code and carrier-phase pseudoranges on
f
1
and
f
2
frequencies, respectively, Δ
s
2
and Δ
s
3
are the dual-frequency second and third order residual
terms, respectively. The TEC along a
f
1
path will be different from that along a
f
2
path due to
ray path bending. Due to the same reason the excess path length will not be the same for
both signals. Therefore, they will not be cancelled out in the ionosphere-free solution. Thus,
the terms Δ
s
TEC
and Δ
s
len
in Eq. (14) and (15) refer to the dual-frequency residual errors due
to TEC difference and excess path length, respectively. Their expressions are given by Eqs.
(16) and (20). The quantities Δ
TEC
bend
1
and Δ
TEC
bend
2
are the differences between TECs along
curved and LoS paths and the quantities
le
d
are the differences between curved
and LoS path lengths for
f
1
and
f
2
signals, respectively. The RRE and RRE
gr
are the total
residual range errors in the carrier-phase and code combinations, respectively.
len
d
and
1
2
The disadvantages of such combinations (Eqs. 14, 15) are that i) the observation noise is
increased by a factor depending on frequencies involved in the combination, ii) the
ambiguity term of the carrier-phase combination is no more an integer value and iii) only
the first order term is eliminated, i.e., higher order terms remain uncorrected. Moreover, this
method cannot be applied to single-frequency receivers.
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