Bernese GPS Software Version 5.0 - Bernese GNSS Software

2.3 Observation Equations
Using double-difference observations from two different epochs t1 and t2, the triple-difference
may be formed. In the Bernese GPS Software, the triple-differences of the phase measurements
are used in the data pre-processing (see Section 6.5).
L ij
1kℓ (t2) − L ij
1kℓ (t1) = ̺ ij
kℓ (t2) − ̺ ij
kℓ (t1) −
L ij
2kℓ (t2) − L ij
2kℓ (t1) = ̺ ij
kℓ (t2) − ̺ ij
kℓ (t1) − f2 1
f 2 2
I ij
kℓ (t2) − I ij
kℓ (t1)
I ij
kℓ (t2) − I ij
kℓ (t1)
(2.38a)
(2.38b)
In the above equations, we assumed that the unknown ambiguity parameters n ij
1kℓ ,nij
2kℓ remained
the same within the time interval [t1,t2] and that, therefore, the phase ambiguities
are eliminated (the main advantage of the triple-differences). This is indeed true if the receivers
did not loose lock within this time interval and if no cycle slip occurred. Tropospheric
refraction usually does not change rapidly with time and is thus considerably reduced on the
triple-difference level. This is not true, however, for the ionospheric refraction, which may
show very rapid variations in time, particularly in high northern and southern latitudes.
2.3.5 Receiver Clocks
We saw in Section 2.3.4 that the term c δk in Eqns. (2.29) and (2.33) may be eliminated by
forming the differences of the measurements to two satellites (the term c δ i may be eliminated
using the differences between two receivers). This does not mean, however, that the
receiver clock error δk is completely eliminated in the differences. By looking at Eqns. (2.23)
and (2.25), it becomes clear, that in order to compute the geometric distance between satellite
and receiver at time t (in GPS time scale) the receiver clock error δk has to be known
to correct the reading of the receiver clock tk
By taking the time derivative of this equation, we obtain
̺ i k(t) = ̺ i k(tk − δk) . (2.39)
d ̺ i k = − ˙̺i k dδk , (2.40)
where ˙̺ i k is the radial velocity of the satellite with respect to the receiver. This velocity is
zero if the satellite is at the point of closest approach and may reach values up to 900 m/s
may be interpreted as the error in the distance
for zenith distances z ≈ 80o . The term d ̺i k
̺i k we make, when assuming an error −dδk in the receiver clock synchronization with GPS
time.We conclude that the error |d ̺i k | in the geometric distance ̺i k induced by a receiver
clock error |dδk| will be smaller than 1 mm if the receiver clock error |d δk| is smaller than
1 µs.
2.3.6 Linear Combinations of Observations
It is often useful to form particular linear combinations of the basic carrier phase and/or code
measurements. The linear combinations used in the Bernese GPS Software are discussed
in this section. We form the linear combinations using either zero- or double-difference
measurements. L1, L2 represent the phase observables (zero- or double-differences), P1, P2
represent the code observables, both in units of meters.
Bernese GPS Software Version 5.0 Page 39