Bernese GPS Software Version 5.0 - Bernese GNSS Software

8. Initial Phase Ambiguities and Ambiguity Resolution
Let us inspect the effect of introducing the known (integer valued) ambiguities into the
normal equation system. In the case of the float solution we may write the observation
equations in matrix form as
(A1, A2) ·
p 1
p 2
− (l − Ψ(p 0 1,p 0 2)) = v (8.1)
y
(A1 and A2 are the parts of the first design matrix corresponding to the non-ambiguity
resp. ambiguity parameters). The corresponding system of normal equations is
N11 N12
N21 N22
·
p 1
p 2
=
A ⊤ 1Py
A ⊤ 2Py
(P is the weight matrix). Eliminating p 2 from Eqn. (8.2) we obtain
=
b1
b2
(8.2)
(N11 − N12N −1
22 N21) · p 1 = b1 − N12N −1
22 b2 . (8.3)
Assuming that the ambiguity parameters are known we may write
which gives
We may write
and therefore
A1p1 − (l − Ψ(p 0 1, ¯p
2))
y ′
= v ′ , (8.4)
N11p 1 = A ⊤ 1Py ′ = b ′ 1 . (8.5)
y − y ′ = Ψ(p 0 1 , ¯p 2) − Ψ(p 0 1 ,p0 2 ) = A2 · (¯p 2 − p 0 2 ) = A2 · dp 2
m
0.06
0.05
0.04
0.03
0.02
0.01
0.00
(8.6)
N11p 1 = A ⊤ 1Py − A ⊤ 1P A2 · dp 2 . (8.7)
Amb. fixed
Amb. free
0 2 4 6 8 10 12 14 16 18 20 22 24
Session Length (hours)
Figure 8.1: RMS of a 7-parameter Helmert Transformation with respect to the “true”
coordinate set.
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