PROCESSING PSEUDORANGES AND CARRIER PHASES - GPS Satellite Surveying - page 244

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7.3 GEOMETRY-FREE SOLUTIONS

The undifferenced pseudorange (5.7) or (5.72) and carrier phase (5.10) equations
make up the epoch solution. Using the short notation, which neglects the subscript
for station k , the superscript for satellite p , and the epoch designator t , the geometry-
free epoch solution using dual-frequency pseudoranges and carrier phases is written
in the following form:

11 00
1

ρ + ∆
I 1 ,P
N 1
N 2

δ 1 ,P
δ 2 ,P
δ 1 , Φ
δ 2 , Φ

ε 1 ,P
ε 2 ,P
ε 1 , Φ
ε 2 , Φ

cT GD
P 2 − α f cT GD
Φ 1
Φ 2

P 1 −

α f 00

= + +

(7.19)

1 − 1 λ 1 0

1 −α f 0 λ 2

[24

cdt

∆ =− cd t + + T (7.20)

Lin
—
0.1
——
Nor
PgE

The carrier phases have been expressed in terms of distance values Φ 2 .

In the usual notation, the interfrequency code offset at the satellite is T GD , and

Φ 1 and

ρ
denotes the geometric topocentric distance from the receiver antenna to the satellite
at the instant of signal transmission. The auxiliary parameter ∆ combines the receiver

¯

clock correction d t , satellite clock correction d t , and the tropospheric delay T . Other

parameters are the ionospheric delay I 1 ,P , and the ambiguities N 1 and N 2 . The factor
α f is given in (5.13). The

terms represent the hardware delays at the receiver and
satellite and the signal multipath, and the epsilons are the noise. We can write (7.19)
in matrix notation,

δ

[24

b = Ax + δ + ε (7.21)

Because the A matrix contains constants that do not depend on the receiver-satellite
geometry, (7.19) is called the geometry-free model and is valid for static or moving
receivers. Whereas the parameters

and I 1 ,P change with time, the ambiguity
parameters are constant unless there are cycle slips. The parameters can be estimated
using least-squares or Kalman filtering. For example, Goad (1990) and Euler and
Goad (1991) use the geometry-free model to study optimal filtering for the combined
pseudorange and carrier phase observations for single and dual frequency. Neglecting
the

ρ + ∆

δ terms and the observational noise, the epoch solution for the parameters is

A − 1

x = b (7.22)

β f −γ f 0 0

−γ f γ f 0 0

A − 1

−λ − 1 β f + γ f 2 λ − 1 β f − γ f

= (7.23)

λ − 1
1

γ f 0

λ − 2 β f + γ f λ − 2 β f − γ f λ − 1

2

− 2 β f 0