aligned to UTC with a tolerance of 100 ns.

Signals

As for GPS, Galileo and GLONASS, also BeiDou transmits right-hand circularly polarized signals.

Three frequency bands indicated as B1, B2 and B3 serve as carrier signals with modulations based on
Quadrature Phase Shifted Keying (QPSK) or BPSK. Similar to GPS, BeiDou differentiates between
the SPS as an open, free of charge service and an authorize service satisfying in particular the demand
for a high reliability. It should however be noticed, that the BeiDou signal transmission concept is
related to the current deployment stage and not all technical aspects are defined for the final FOC
phase^{19}.

### 3.2 Acquiring the ionospheric refraction from dual-frequency microwave signals

For electromagnetic waves, the ionosphere is a dispersive medium, meaning that the ionospheric delay on the signal depends on its frequency. After Seeber (2003) and Hofmann-Wellenhof et al. (2008), the phase refraction index for a given radio signal frequency can be approximated by the power series

ηΦ= 1+ c_{2}
f^{2} + c_{3}

f^{3} +. . . with the frequency derivative dηΦ

d f =−2c_{2}
f^{3} − 3c_{3}

f^{4} −. . . . (3.1)
The series coefficientscfidepend on the electron density along the signal propagation path.
Further-more, the relation between the group (P) and phase (Φ) refractive index as a function of the frequency
have been found many years ago by Rayleigh and Lindsay (1945)^{20} and can be expressed as

ηP = ηΦ+ f dηΦ

d f . (3.2)

Substituting Eq. (3.1) into Eq. (3.2) yields
ηP = 1+ c_{2}

f^{2} + c_{3}

f^{3} +· · · − 2c_{2}
f^{2} − 3c_{3}

f^{3} −. . . . (3.3)

Based on the formula of dispersion (Davies, 1990), defined with η =1− cNe

f^{2} , (3.4)

the approximation c_{2} = −40.3Ne can be found (Seeber, 2003). Under negligence of higher order
termsO(c_{3}), Eq. (3.3) can thus be reformulated as

ηP = 1+ c_{2}
f^{2} − 2c_{2}

f^{2} = 1+ 40.3Ne

f^{2} (3.5)

and it follows from Eq. (3.1), that ηΦ= 1− 40.3Ne

f^{2} . (3.6)

For a radio signal passing a planetary atmosphere, the refractive index can generally also be obtained fromη =c/vwherecis the speed of light in vacuum andvthe medium dependent propagation speed.

19BeiDou signal structure:http://www.navipedia.net/index.php/BeiDou_Signal_Plan 20John William Strutt, 3rd Baron Rayleigh (12.11.1842 - 30.06.1919) was an English physicist.

54

Chapter 3. Satellite techniques
for observing the total electron content
Therefore, it can be concluded that the ionosphere causes withηP > ηΦandvP < vΦa group delay
and phase advance of the signal. From the Eqs. (3.6) and (3.5) it can further be interpreted that the
following applies: The higher the frequency number, the smaller the ionospheric impact. For the GPS
carrier frequencies f_{1}and f_{2}it follows for the group delay thatP_{r,1}^{t} < P_{r,2}^{t} and for the phase advance
thatλ_{1}Φ^{t}_{r,1}> λ_{2}Φ^{t}_{r,2}.

The basic observation equations for code pseudorange and carrier phase measurements between a transmittert ∈ {1, ...,T}and a receiverr ∈ {1, ...,R}are given by

P_{r}^{t} = ρ^{t}_{r}+c

∆tr −∆t^{t}

+∆^{t}_{r,NTR}+ f

∆^{t}_{r,ION}+c

br +b^{t}
+P

g

f (3.7)

and

Φ^{t}_{r} = ρ^{t}_{r} +c

∆tr −∆t^{t}

+∆^{t}_{r,NTR}+ f

−∆^{t}_{r,ION}+CPB^{t}_{r} +Φ

g

f (3.8)

where the frequency dependent terms have been collected in cornered brackets. Concerning the
fre-quency independent parameters, ρ^{t}_{r} is the geometric LOS distance between the satellite transmittert
and receiverr,∆tr and∆t^{t} are clock offsets and∆_{NTR} identifies the impact of the neutral atmosphere
(mainly the troposphere).

The frequency dependent terms are defined by the ionospheric signal delay ∆_{ION}, hardware specific
code biases or instrumental delaysbr andb^{t} for receiver and satellite as well as a Carrier Phase Bias
(CPB). Additional frequency dependent effects may be induced by multipath, Phase Center Offsets
(PCOs) or Phase Center Variations (PCVs). Code multipath can cause pseudorange errors in the
decimeter level whereas the phase impact is usually less than a few centimeters (Yang et al., 2004).

Impacts due to PCVs can reach up to few centimeters (El-Rabbany, 2006). Such terms have not been
explicitly considered here but it can be expected that the errors are partially absorbed in ρ^{t}_{r} or the bias
parameters br, b^{t} and CPB^{t}_{r}. The CPB parameter contains in particular the carrier phase ambiguity
λN and phase variations due to circular polarization of the electromagnetic signal, known as Phase
Wind-Up (PWU), caused by continuous rotation of the satellite relatively to the receiver for aligning
its solar panels towards the sun. N designates a frequency dependent ambiguity parameter in cycles
that remains constant during a satellite pass where a pass is defined as an uninterrupted, continuous
data arc linked between a specific satellite and a specific receiver. P and Φ are unmodeled
mea-surement noise terms. It shall further be noted, that ρmay contain additional effects such as antenna
environment influences and relativistic effects on the satellite orbit due to the Earth’s gravity field and
the satellite’s velocity.

The ionospheric information is contained in the signal delay∆_{ION} where the sign indicates the group
delay and phase advance according to Eq. (3.5) and Eq. (3.6). A common procedure is the
determina-tion of the ionospheric impact for the correcdetermina-tion of single frequency receivers working only with L_{1}
(1 TECU corresponds to 0.163 m range error inC_{1}). Therefore, the ionospheric delay can be related
to the first frequency by consideration of the carrier phase frequency ratio

ξ = f_{1}^{2}/f_{2}^{2}. (3.9)

The Eqs. (3.7) and (3.8) can thus be modified to
P_{r}^{t} = ρ^{t}_{r}+c

∆tr −∆t^{t}

+∆^{t}_{r,NTR}

+∆^{t}_{r,ION}+

c br+b^{t} +P

f_{1}

+ξ∆^{t}_{r,ION}+

c br +b^{t} +P

f_{2}

(3.10) for the code pseudoranges and

Φ^{t}_{r} = ρ^{t}_{r} +c

∆tr −∆t^{t}

+∆^{t}_{r,NTR}

−∆^{t}_{r,ION}+ f

CPB^{t}_{r} +Φ

g

f_{1}

−ξ∆^{t}_{r,ION}+ f

CPB^{t}_{r} +Φ

g

f_{2}

(3.11)

3.2. Acquiring the ionospheric refraction from dual-frequency microwave signals 55 for the carrier phases. At this point, it is important to note, that the formula with fixed carrier frequen-cies on both signal links is valid for GPS, Galileo and also BeiDou, but requires a PRN or channel k dependent modification by

ξ = ξ(k)= f_{1}^{2}(k)/f_{2}^{2}(k) (3.12)

for the processing of GLONASS signals. The channel distribution, based on the status of 28 Novem-ber 2014, is provided by Table 3.3.

The availability of multi-frequency signals plays an important role allowing to compute linear data combinations in the form of

PLC = a1P1+a2P2 or ΦLC = a1λ_{1}Φ_{1}+a2λ_{2}Φ_{2} (3.13)
in case of dual-frequency measurements, where a_{1} and a_{2} are arbitrary numbers. With respect to
Eq. (3.10) and Eq. (3.11), the selection of a_{1} = 1 and a_{2} = −1 allows to eliminate the geometry
dependent terms, i.e., the non-dispersive parameters. The so-called geometry-free linear or ionosphere
combination, denoted asP_{4}andL_{4}, is then defined with

P_{r,4}^{t} = P_{r,1}^{t} −P_{r,2}^{t} = (1−ξ)∆^{t}_{r,ION}+DCB+P_{4} (3.14)

L^{t}_{r,4}= λ_{1}Φ^{t}_{r,1}−λ_{2}Φ^{t}_{r,2}= (ξ−1)∆^{t}_{r,ION}+CPB^{t}_{4,r} +L_{4}. (3.15)
The remaining unknowns in this equations are the bias terms for code and phase measurements.

DCB=c(∆br+∆b^{t}) is denoted as inter-frequency Differential Code Bias (DCB) and CPB^{t}_{4,r} contains
the merged carrier phase biases, in particular the ambiguity differencesλ1N1−λ2N2. The ionospheric
signal is contained in the target parameter∆^{t}_{r,ION} which is defined here in range units.

In Section 1.5, the STEC [TECU] has been introduced as a characteristic ionospheric parameter that is defined by the integral

STEC= Z

Neds (3.16)

over the electron density along the signal propagation path. The conversion between the ionospheric
delay∆^{t}_{r,ION}in range units to TECU will be derived in the following.

It has been previously shown by means of Eq. (3.5) and Eq. (3.6), that the refraction indices for phase ηΦand groupηPof first order can be expressed as

ηΦ= 1− 40.3Ne

f^{2} and ηP = 1+ 40.3Ne

f^{2} . (3.17)

According to Fermat’s^{21} principle a measured rangescan be described with
s=Z

ηds (3.18)

where the refractive indexη is integrated along the signal path. The LOS distance s_{0} can therefore
be obtained by settingη = 1 and consequently the signal delay as the difference between curved and
straight path can be computed from

∆s = Z

ηds− Z

ds_{0}. (3.19)

Taking into account that the signal delay is mainly caused by ionospheric refraction, leads to∆s =∆^{t}_{r,ION}.
Furthermore, the substitution of Eq. (3.17) into Eq. (3.19) yields

∆^{t}_{r,ION} =
Z

1+ν40.3Ne

f^{2}

! ds−

Z

ds_{0} (3.20)

21Pierre de Fermat (second half of 1607 - 12.01.1665) was a French mathematician and lawyer.

56

Chapter 3. Satellite techniques
for observing the total electron content
withν =1 for group andν =−1 for phase signals. A simplification is introduced at this point, where
the integration is performed along the LOS instead of the curvature signal. It follows thats = s_{0}, i.e.,
the integrals appearing in Eq. (3.20) can be merged in the reformulation

∆^{t}_{r,ION} = ν40.3
f^{2}

Z

Neds_{0} (3.21)

which can also be written as

∆^{t}_{r,ION} = ν40.3

f^{2} STEC^{t}_{r}. (3.22)

Equation (3.22) can now be considered in Eq. (3.15) to obtain a formulation including the ionospheric refraction on the first carrier signal in TECU. With the frequency dependent factors summarized in

α= (ξ−1) 40.3

f_{1}^{2} = 40.3f_{1}^{2}− f_{2}^{2}

f_{1}^{2}f_{2}^{2} (3.23)

the geometry-free linear combinations for code and phase can be written as

P_{r,4}^{t} = P_{r,1}^{t} −P_{r,2}^{t} =−αSTEC^{t}_{r} +DCB+P_{4} (3.24)
L^{t}_{r,4}= λ_{1}Φ^{t}_{r,1}−λ_{2}Φ^{t}_{r,2}= αSTEC^{t}_{r} +CPB^{t}_{4,r} +L_{4}. (3.25)
For GLONASS signal processing, this formulation yields accordingly

α= α(k)= (ξ(k)−1) 40.3

f_{1}^{2}(k) =40.3f_{1}^{2}(k)− f_{2}^{2}(k)

f_{1}^{2}(k)f_{2}^{2}(k) . (3.26)

The Eqs. (3.24) and (3.25) can finally be solved as
STEC^{t}_{r} = α^{−1}

P_{r,2}^{t} −P_{r,1}^{t} +DCB+P_{4} (code)

λ_{1}Φ^{t}_{r,1}−λ_{2}Φ^{t}_{r,2}−CPB^{t}_{4,r} +L_{4} (phase) (3.27)
for the determination of STEC.

Further details about ionospheric influences on GNSS signals can for instance be found in Hofmann-Wellenhof et al. (2008), Xu (2010) or Hoque and Jakowski (2012) and many other literature.

According to Eq. (3.17), this formulation considers only ionospheric influences of first order. Second order terms can be expressed through

STEC^{t}_{r,2nd} = α^{−1} − 7,527c
2f_{1}f_{2}(f_{1}+ f_{2})

! Z

NeBcosΘds (3.28)

taking the Earth’s magnetic field B and the angle Θ between the signal path and B into account
(Hernández-Pajares et al., 2007). The consideration of STEC^{t}_{r,2nd} is mainly relevant for clock and
orbit modeling but contributes less than 0.1% to the total ionospheric delay in GNSS signals why it is
mostly neglected.

Carrier phase leveling and code smoothing

Pseudorange measurements are noisy but unambiguous while the carrier phase data is precise but biased. For this reason, algorithms for smoothing the code or leveling the phase data, respectively, are applied to reduce the noise (roughly by a factor of √

N, whereN is the number of data samples of the data arc) while maintaining the precision of the phase measurements.