Bernese GPS Software Version 5.0 - Bernese GNSS Software

Using a Taylor series development we may rewrite the last equation as
2.3 Observation Equations
ψ i Fk(t) = φFk(t) − φ i F(t) + τ fF + n i Fk , (2.31)
where fF is the frequency of the carrier. The difference
φFk(t) − φ i F(t)
is zero in the case of ideal oscillators and equal to
(δk − δ i ) fF
if the receiver clock error δk and the satellite clock error δ i are taken into account. The
observation equation is then given by
ψ i Fk(t) = (δk − δ i ) fF + τ fF + n i Fk . (2.32)
Multiplying this equation by the wavelength λF we receive the phase observation L i Fk (in
meters)
L i Fk = ̺ i k + c δk − c δ i + λF n i Fk . (2.33)
2.3.3 Measurement Biases
The phase measurements and the code pseudoranges are affected by both, systematic and
random errors. There are many sources of systematic errors: satellite orbits, satellite and
receiver clocks, propagation medium, relativistic effects, and antenna phase center variations
to name only a few. In the Bernese GPS Software all relevant systematic errors are carefully
modeled. Here we discuss only two kinds of systematic errors, namely tropospheric and
ionospheric refraction.
∆̺ i k is the tropospheric refraction. It is the effect of the neutral (i.e. the non-ionized) part
of the Earth’s atmosphere on signal propagation. Note that tropospheric refraction
does not depend on the frequency and that the effect is the same for phase and code
I i k
measurements.
is the ionospheric refraction. The ionosphere is a dispersive medium for microwave
signals, which means that the refractive index for GPS signals is frequency-dependent.
In a first (but excellent) approximation ionospheric refraction is proportional to
1
,
f2 where f is the carrier frequency. In our notation the term Ii k is the effect of the ionosphere
on the first carrier L1. Due to this frequency dependence the ionospheric refraction on the
second carrier L2 may be written as
f2 1
f2 I
2
i k .
Bernese GPS Software Version 5.0 Page 37