2.2 GNSS
2.2.1 Principle
GNSS systems rely on the transmission of an electromagnetic signal between GNSS satellites and active receivers. For the purposes of this study, only ground receivers are here taken into account, but a number of space missions (e.g. Jason 1-2- 3, ICESat, GRACE, TerraSAR-X...) are equipped with GNSS positioning payloads in order to support precise orbit determination (POD). For GPS and GLONASS constellations, each satellite transmits on two carriers, L1 and L2, which are integer multiples of the main frequency provided by the accurate atomic clocks carried on- board GNSS satellites (at least two caesium and as many as two rubidium clocks in the case of GPS). The broadcast signal is then additionally modulated according to specific codes also derived from the fundamental frequency, Fig. 2.1.
Receivers are capable to measure the phase of the incoming signal and are pro- vided with an oscillator on the basis of which the broadcast information can be reproduced. Assuming that all the involved clocks are perfectly synchronized, the phase of the wave generated within the receiver at the time of recording t differs from the acquired phase by an amount which depends on the signal time of flight τ . Considering the finite velocity c of electromagnetic waves, in fact, measurements carried out at time t actually sense the signal emitted by the satellite at time t − τ . The basic observable ψi
F k(t), then, consists in the phase difference [Dach et al., 2007]: ψiF k(t) = φF k(t) − φiF(t − τ ) + niF k (2.15) where
ψi
F k(t) is the phase measurement (in cycle) at epoch t and frequency fF φF k(t) is the phase generated by the receiver k oscillator at the epoch t φi
F(t − τ ) is the phase of the signal emitted by the satellite at time t − τ ni
F k is the (unknown) integer number of cycles. A sketch representation is provided in Fig. 2.2.
a)
b)
c)
Figure 2.1 – Representation of GNSS carrier phase modulation on the basis of a superimposed code: (a) original carrier, (b) modulating code, (c) resulting signal. In the case of GPS two codes are employed. The C/A (clear acquisition) code has a frequency of 1/10 of the fundamental one maintained by the clocks and is modulated on L1 only; the P (precise) code, instead, provides a ten times higher accuracy and is superimposed to both L1 and L2. The navigation message, identifying the satellite that broadcast the signal, its health status and other operative information, is also modulated on both carriers.
a
b
t
t-τ
Figure 2.2 – Scheme of GNSS receivers working principle. The green dot indicates the phase of the signal at time t − τ , which is maintained by the wave broadcast at that epoch. The signal copy locally generated within the receiver, instead, continues to evolve during the flight time τ , so that, at the acquisition epoch t the cumulative phase difference consists of a certain number of complete oscillations (a) plus a subinteger discrepancy (b). Modulations are ignored in this sketch for the sake of clarity.
In operational conditions, Eq. 2.15 can be linearized with a first order Taylor series expansion
ψiF k(t) = φF k(t) − φiF(t) + τ fF + niF k. (2.16) For ideal, perfectly synchronized clocks, the difference φF k(t) − φiF(t) is equal to zero, but in operational conditions station- and satellite-clock errors, δk and δi re- spectively, must be taken into account so that
φF k(t) − φiF(t) = δk− δi fF. (2.17)
If the relation expressed in Eq. 2.17 is substituted in Eq. 2.15 and this last is multiplied by the carrier wavelength λF, the basic form of the phase observation equation in meters is obtained
LiF = ρik+ cδk− cδi+ λFniF k (2.18) where ρik = cτ represents the geometric distance between satellite i at time t − τ and the receiver k at time t. For users interested in positioning applications, ρi
k can be explicitly related to the 3-dimensional coordinates of the involved satellite-receiver pair through
ρik= REOP · Xk(t) − Xi(t − τ )
(2.19)
where
Xk is the coordinate vector of receiver k at the time of signal acquisition t expressed in a Terrestrial Reference Frame
Xiis the coordinate vector locating the satellite i at the time of signal emission t − τ in a Geocentric Celestial Reference Frame
REOP is the rotation matrix of Earth Orientation parameters necessary to transform coordinates from a Terrestrial to a Geocentric Celestial Reference Frame, see Sec. 1.3.1.1. The application of such operator allows expressing station and satellite positions in the same system, so that it is possible to differentiate the two vectors.
The electromagnetic signal broadcast by the satellite, however, interacts with the environment through which it propagates causing delays that need to be accounted for in the observation equation in order to achieve the best possible accuracy. In addition, systematic errors do also affect the measurements. Equation 2.15, then, should be expanded to
Lik = REOP · Xk(t) − Xi(t − τ ) + + λ · Nki + c · δtk− c · δti+
+ δρtrop+ δρion+ δρphas+ δρrel+ δρmult+ ik
(2.20)
where the added corrective terms accounts for
– ρtrop delays accumulated for the signal propagation through troposphere – ρion delays accumulated for the signal propagation through ionosphere
– ρphas corrections for instabilities/inaccurate characterization of phase center offsets and variations at satellite and receivers antennas
– ρrel corrections for relativistic effects affecting satellite clocks and orbits as well as signal propagation (Shapiro effect)
– ρmult multipath effects – i
k measurement errors
An observation of this type can be formulated for each of the two fundamental carriers and for any of their linear compositions. In GNSS analysis practice, several combinations are often employed according to the different parameters which need to be addressed. Solutions computed for this study, are obtained with the quasi ionosphere-free linear combination
L3 = 1 f2
1 − f22
(f12L1− f22L2)
for which the observation equation is, in first approximation, independent from iono- spheric path delays. Apart from second and higher order terms, in fact, ionospheric refraction is given by
Iki ≈ 1
f2 (2.21)
where f is the carrier frequency, and its relevant contributions at f1 and f2 cancel out in the L3 equation.
Additionally, differences of the original observation equation can be exploited in order to eliminate or, at least, reduce some biases. Results presented in this study derive from a double-difference adjustment between the data collected by pairs of receivers, kl, simultaneously tracking pairs of satellites, ij:
LijF kl= LiF kl− LjF kl (2.22) where
LiF kl= LiF k− Li
F l (2.23)
Equation 2.22 is independent from satellite and receiver clock errors and is therefore often used in GNSS data analysis.