Global Positioning System Reference
In-Depth Information
where
K
= 40.3 m
3
s
-2
, Ψ and Φ are the code and carrier-phase pseudoranges, and their
subscripts correspond to measured signals on
f
1
,
f
2
and
f
3
frequencies, (Δ
s
3
)
tr
is the third
order residual term and (Δ
s
TEC
)
tr
and (Δ
s
len
)
tr
are the residual terms due to TEC difference
and excess path length, respectively. The quantities Δ
TEC
bend
and
d
I
len
are the TEC and path
length differences between curved and LoS paths and and their subscripts correspond to
received signals on frequencies
f
1
,
f
2
and
f
3
. The (
RRE
)
tr
and (
RREgr
)
tr
are the total residual
range errors in the triple-frequency carrier-phase and code pseudorange combinations,
respectively.
However, as already mentioned, such a multiple frequency combination amplifies all
uncorrelated errors or noises (multipath and noise). Assuming the same measurement noise
on each signal, it can be shown that the noise will be amplified by a factor of 33.7 in the GPS
L1-L2-L5 combination (see Hoque & Jakowski, 2010a).
The Galileo system will transmit signals on four frequencies E2-L1-E1, E5a, E5b and E6
(1575.42, 1176.45, 1207.14 and 1278.75 MHz, respectively). Simultaneous reception of four
signals will allow quadruple-frequency combinations to eliminate the first, second and third
order ionospheric terms. Such a combination would theoretically eliminate higher order
ionospheric terms successfully from the range equation. However, the noise will be
amplified by a factor of about 626.13 in the E2L1E1-E5a-E5b-E6 combination (assuming the
same measurement noise on each signal) which is about two orders larger than a dual-
frequency factor. Therefore, a quadruple-frequency combination is barely pragmatic.
However, if the frequency separation is large (e.g., combinations between 4-8 GHz C band
and 1-2 GHz L band frequencies), the amplification factor will be small. In such cases,
measurements on four frequencies may be useful.
2.4.3 Geometry-free combination
When microwave signals are transmitted on two frequencies, all the nondispersive effects,
e.g., tropospheric delay, satellite and receiver clock offsets, antenna phase centre offsets and
variations etc., manipulate the signals on both frequencies in the same way - apart from the
ionosphere. Therefore, by differencing code / carrier-phase pseudoranges measured on two
frequencies, all non-dispersive terms including
ρ
will be cancelled out giving the estimate of
TEC along ray paths as (combining code / carrier-phase pseudoranges Eq. (12) / Eq. (13)
measured on
f
1
and
f
2
frequencies and substituting ρ by Eq. (7) and neglecting the second
and higher order terms)
22
12
ff
)
(
)
⎡
⎤
(27)
TEC
=
ΨΨ
−
+
noise
(
⎣
2
1
ΨΨ
−
⎦
2
2
2
1
Kf
−
f
1
2
22
12
ff
)
(
)
⎡
⎤
(28)
TEC
=
ΦΦ
−
+
B
+
noise
(
⎣
⎦
1
2
ambiguity
ΦΦ
−
2
2
1
2
Kf
−
f
1
2
in which
B
ambiguity
=
λ
2
N
2
-
λ
1
N
1
is the carrier-phase ambiguity constant where
λ
1
,
λ
2
are wave
lengths and
N
1
,
N
2
are integer ambiguities measured on
f
1
and
f
2
frequencies,
noise
Ψ−
and
2
1
noise
Φ−
are noises (e.g., thermal noise etc.) in code and carrier-phase combinations,
respectively. For simplicity different terms such as inter-frequency satellite and receiver
biases and multipath effects are not considered.
1
2
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