The GNSS integer ambiguities: estimation and validation

The single difference models can now be presented as follows. The full-rank geometryfree
model is given by:
⎛⎡
⎤ ⎞
µ
e2f
E{y} = ⊗ Im ρ + ⎝⎣−µ
⎦ ⊗ Im⎠
I
0
1
⎛⎡
⎤
0
I2f
+ ⊗ em dtqr + ⎝⎣Λ⎦
0
⊗
0
Im−1
0
⎞
(2.45)
⎠ a
The observation vector is given by (2.44), the range vector ρ by (2.40), and the parameter
vectors I, dtqr, and a by equations (2.32), (2.42) and (2.43) respectively.
The ionosphere-weighted, troposphere-float, geometry-based model is obtained as:
⎛⎡
⎤ ⎞
µ
e2f
e2f
E{y} = ⊗ G ∆rqr + ⊗ Ψ T + ⎝⎣−µ
⎦ ⊗ Im⎠
I
0
0
1
⎛⎡
⎤
0
I2f
+ ⊗ em dtqr + ⎝⎣Λ⎦
0
⊗
0
Im−1
0
⎞
(2.46)
⎠ a
2.3.3 Double difference models
The number of parameters in the observation equations can be even further reduced
by also taking differences between observations of different satellites. In this double
difference (DD) approach, one reference satellite t is chosen and the single difference
observations of this satellite are subtracted from the corresponding single difference
observations of all other satellites. For one satellite-pair the geometry-free observation
equations become then:
p ts
qr,j = p s qr,j − p t qr,j = ρ ts
qr + T ts
qr + µjI ts
qr + e s qr,j
φ ts
qr,j = φ s qr,j − φ t qr,j = ρ ts
qr + T ts
qr − µjI ts
qr + λjN ts
qr,j + ε ts
qr,j
(2.47)
The instrumental delays and clock errors of the receivers have now also cancelled from
the observation equations, as well as the initial phases in the receivers. This leaves only
the integer carrier phase ambiguity as extra parameter in the phase observations. Only
if this ambiguity is resolved, the phase observations can be considered as very precise
pseudorange measurements, so that high precision positioning solutions can be obtained.
Therefore, the integer nature of the ambiguities must be exploited. Integer parameter
resolution is, however, a non-trivial problem, which will be the topic of the remaining
chapters.
If the first satellite is chosen as the reference satellite, the following transformation matrix
can be used to arrive at the double difference geometry-free model:
D T =
−em−1 Im−1
(2.48)
18 GNSS observation model and quality control