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Bernese GPS Software Version 5.0 - Bernese GNSS Software

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Using a Taylor series development we may rewrite the last equation as<br />

2.3 Observation Equations<br />

ψ i Fk(t) = φFk(t) − φ i F(t) + τ fF + n i Fk , (2.31)<br />

where fF is the frequency of the carrier. The difference<br />

φFk(t) − φ i F(t)<br />

is zero in the case of ideal oscillators and equal to<br />

(δk − δ i ) fF<br />

if the receiver clock error δk and the satellite clock error δ i are taken into account. The<br />

observation equation is then given by<br />

ψ i Fk(t) = (δk − δ i ) fF + τ fF + n i Fk . (2.32)<br />

Multiplying this equation by the wavelength λF we receive the phase observation L i Fk (in<br />

meters)<br />

L i Fk = ̺ i k + c δk − c δ i + λF n i Fk . (2.33)<br />

2.3.3 Measurement Biases<br />

The phase measurements and the code pseudoranges are affected by both, systematic and<br />

random errors. There are many sources of systematic errors: satellite orbits, satellite and<br />

receiver clocks, propagation medium, relativistic effects, and antenna phase center variations<br />

to name only a few. In the <strong>Bernese</strong> <strong>GPS</strong> <strong>Software</strong> all relevant systematic errors are carefully<br />

modeled. Here we discuss only two kinds of systematic errors, namely tropospheric and<br />

ionospheric refraction.<br />

∆̺ i k is the tropospheric refraction. It is the effect of the neutral (i.e. the non-ionized) part<br />

of the Earth’s atmosphere on signal propagation. Note that tropospheric refraction<br />

does not depend on the frequency and that the effect is the same for phase and code<br />

I i k<br />

measurements.<br />

is the ionospheric refraction. The ionosphere is a dispersive medium for microwave<br />

signals, which means that the refractive index for <strong>GPS</strong> signals is frequency-dependent.<br />

In a first (but excellent) approximation ionospheric refraction is proportional to<br />

1<br />

,<br />

f2 where f is the carrier frequency. In our notation the term Ii k is the effect of the ionosphere<br />

on the first carrier L1. Due to this frequency dependence the ionospheric refraction on the<br />

second carrier L2 may be written as<br />

f2 1<br />

f2 I<br />

2<br />

i k .<br />

<strong>Bernese</strong> <strong>GPS</strong> <strong>Software</strong> <strong>Version</strong> <strong>5.0</strong> Page 37

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