Bernese GPS Software Version 5.0 - Bernese GNSS Software

2. Fundamentals
Ionospheric refraction delays the GPS code measurements and advances the carrier phases.
The effect has the same absolute value for code and phase measurements, but with opposite
signs.
Taking into account these systematic errors, we may refine the observation equations (2.29)
and (2.33) for both frequencies, yielding:
P i 1k = ̺i k + c δk − c δ i + I i k + ∆̺i k (2.34a)
P i 2k = ̺ i k + c δk − c δ i + f2 1
f2 I
2
i k + ∆̺ i k
(2.34b)
L i 1k = ̺ik + c δk − c δ i − I i k + ∆̺ik + λ1 n i 1k (2.34c)
L i 2k = ̺i k + c δk − c δ i − f2 1
f 2 2
We use the same notation for the geometrical distance ̺ i k
implicitly contain tropospheric and ionospheric delays.
I i k + ∆̺i k + λ2 n i 2k
. (2.34d)
although Eqns. (2.33) and (2.29)
The reader has to be be aware of the fact that the consideration of further bias terms in
Eqns. (2.34) is requisite in some cases. For example, so-called “differential code biases”
should be considered in case of analyzing the difference P i 1k − P i 2k for ionosphere mapping
(see Chapter 12).
2.3.4 Forming Differences
Differences of the original observations allow to eliminate or reduce some biases. Let us
define the single-difference (between a pair of receivers k and ℓ) by
L i Fkℓ = L i Fk − L i Fℓ
(2.35)
and the double-difference (between a pair of receivers kℓ and between a pair of satellites ij)
by
L ij
Fkℓ = LiFkℓ − Lj
Fkℓ . (2.36)
Double-differences are the basic observables in the Bernese GPS Software. The corresponding
observation equations are
P ij
1kℓ
P ij
2kℓ
L ij
1kℓ
L ij
2kℓ
= ̺ij
kℓ + Iij
kℓ + ∆̺ij
kℓ
= ̺ij
kℓ + f2 1
f2 I
2
ij
kℓ + ∆̺ij
kℓ
= ̺ij
kℓ − Iij
kℓ + ∆̺ij
kℓ + λ1 n ij
1kℓ
= ̺ij
kℓ − f2 1
f 2 2
I ij
kℓ + ∆̺ij
kℓ + λ2 n ij
2kℓ
(2.37a)
(2.37b)
(2.37c)
(2.37d)
By forming the double-difference observations, receiver and satellite clock errors are eliminated
(assuming that the receiver clock errors are known accurately enough to compute the
distances ̺ correctly – see Section 2.3.5).
Page 38 AIUB