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 phase19.
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+ c2 f2 + c3
f3 +. . . with the frequency derivative dηΦ
d f =−2c2 f3 − 3c3
f4 −. . . . (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+ c2
f2 + c3
f3 +· · · − 2c2 f2 − 3c3
f3 −. . . . (3.3)
Based on the formula of dispersion (Davies, 1990), defined with η =1− cNe
f2 , (3.4)
the approximation c2 = −40.3Ne can be found (Seeber, 2003). Under negligence of higher order termsO(c3), Eq. (3.3) can thus be reformulated as
ηP = 1+ c2 f2 − 2c2
f2 = 1+ 40.3Ne
f2 (3.5)
and it follows from Eq. (3.1), that ηΦ= 1− 40.3Ne
f2 . (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 f1and f2it follows for the group delay thatPr,1t < Pr,2t and for the phase advance thatλ1Φtr,1> λ2Φtr,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
Prt = ρtr+c
∆tr −∆tt
+∆tr,NTR+ f
∆tr,ION+c
br +bt +P
g
f (3.7)
and
Φtr = ρtr +c
∆tr −∆tt
+∆tr,NTR+ f
−∆tr,ION+CPBtr +Φ
g
f (3.8)
where the frequency dependent terms have been collected in cornered brackets. Concerning the fre-quency independent parameters, ρtr is the geometric LOS distance between the satellite transmittert and receiverr,∆tr and∆tt 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 andbt 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 ρtr or the bias parameters br, bt and CPBtr. 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 L1 (1 TECU corresponds to 0.163 m range error inC1). Therefore, the ionospheric delay can be related to the first frequency by consideration of the carrier phase frequency ratio
ξ = f12/f22. (3.9)
The Eqs. (3.7) and (3.8) can thus be modified to Prt = ρtr+c
∆tr −∆tt
+∆tr,NTR
+∆tr,ION+
c br+bt +P
f1
+ξ∆tr,ION+
c br +bt +P
f2
(3.10) for the code pseudoranges and
Φtr = ρtr +c
∆tr −∆tt
+∆tr,NTR
−∆tr,ION+ f
CPBtr +Φ
g
f1
−ξ∆tr,ION+ f
CPBtr +Φ
g
f2
(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)= f12(k)/f22(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 = a1λ1Φ1+a2λ2Φ2 (3.13) in case of dual-frequency measurements, where a1 and a2 are arbitrary numbers. With respect to Eq. (3.10) and Eq. (3.11), the selection of a1 = 1 and a2 = −1 allows to eliminate the geometry dependent terms, i.e., the non-dispersive parameters. The so-called geometry-free linear or ionosphere combination, denoted asP4andL4, is then defined with
Pr,4t = Pr,1t −Pr,2t = (1−ξ)∆tr,ION+DCB+P4 (3.14)
Ltr,4= λ1Φtr,1−λ2Φtr,2= (ξ−1)∆tr,ION+CPBt4,r +L4. (3.15) The remaining unknowns in this equations are the bias terms for code and phase measurements.
DCB=c(∆br+∆bt) is denoted as inter-frequency Differential Code Bias (DCB) and CPBt4,r contains the merged carrier phase biases, in particular the ambiguity differencesλ1N1−λ2N2. The ionospheric signal is contained in the target parameter∆tr,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∆tr,IONin 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
f2 and ηP = 1+ 40.3Ne
f2 . (3.17)
According to Fermat’s21 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 s0 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
ds0. (3.19)
Taking into account that the signal delay is mainly caused by ionospheric refraction, leads to∆s =∆tr,ION. Furthermore, the substitution of Eq. (3.17) into Eq. (3.19) yields
∆tr,ION = Z
1+ν40.3Ne
f2
! ds−
Z
ds0 (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 = s0, i.e., the integrals appearing in Eq. (3.20) can be merged in the reformulation
∆tr,ION = ν40.3 f2
Z
Neds0 (3.21)
which can also be written as
∆tr,ION = ν40.3
f2 STECtr. (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
f12 = 40.3f12− f22
f12f22 (3.23)
the geometry-free linear combinations for code and phase can be written as
Pr,4t = Pr,1t −Pr,2t =−αSTECtr +DCB+P4 (3.24) Ltr,4= λ1Φtr,1−λ2Φtr,2= αSTECtr +CPBt4,r +L4. (3.25) For GLONASS signal processing, this formulation yields accordingly
α= α(k)= (ξ(k)−1) 40.3
f12(k) =40.3f12(k)− f22(k)
f12(k)f22(k) . (3.26)
The Eqs. (3.24) and (3.25) can finally be solved as STECtr = α−1
Pr,2t −Pr,1t +DCB+P4 (code)
λ1Φtr,1−λ2Φtr,2−CPBt4,r +L4 (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
STECtr,2nd = α−1 − 7,527c 2f1f2(f1+ f2)
! 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 STECtr,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.