Positioning and Navigation Using the Russian Satellite System
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The combined solution converges even earlier than the GLONASS solution and remains approximately 40 mm from the true position. So even with neglecting the GLONASS inter-channel biases and floating ambiguities, in combined GPS/GLONASS carrier phase processing sub-decimeter accuracy can be achieved. Figure 8.12 shows the computed single difference receiver inter-system hardware delay ∆δt U R,HW . It can be clearly seen that this bias is not constant. Its absolute value increases with time, until it obviously reaches some kind of saturation after approximately fifteen minutes. Figure 8.13 and 8.14 show the computed double difference floating ambiguities for some of the par- ticipating satellites in the GPS only and the GLONASS only solutions. To display the ambiguities, a constant value has been added to them such that the resulting ambiguities take values near zero. Once the filter has converged, the ambiguities vary only slightly. In the GPS only solution, this variation is in the range of a cycle or less. This corresponds to one wavelength of approximately 190 mm or less. In the GLONASS only solution, this variation is in the range of 2000 cycles or less. With a wavelength of approximate 65 µm of the common frequency, these 2000 cycles correspond to a variation of around 130 mm, which is comparable to the variation of the floating ambiguities in the GPS only processing. 8.5 Ionospheric Correction 8.5.1 Single Frequency Ionospheric Correction The observation equation for pseudorange observations from an observer R to a GPS or GLONASS satellite S (8.1.4) contains the delay of the satellite signal in the ionosphere δt S,Iono R . Strictly spoken, this ionospheric delay is unknown. It varies with satellite and may vary with time. Therefore, the ionospheric delay yields one unknown per satellite and epoch in addition to the unknowns that are already to be solved for, i.e. user position and receiver clock offset. With single frequency observations, having exactly 8.5 Ionospheric Correction 121 -60 -50 -40 -30 -20 -10 0 10 141000 141500 142000 142500 143000 143500 Hardw are Bias [ns] GPS Time [s] Figure 8.12: GPS/GLONASS double difference carrier phase inter-system hardware delay. one observation per satellite and epoch, the number of unknowns then is greater than the number of observations, rendering the system of observation equations unsolvable. To avoid this, GPS introduced a model to estimate the ionospheric path delay for single frequency users. Having determined the ionospheric delay employing this so-called Klobuchar model, the delay now is known and can be shifted to the left hand side of Eq. (8.1.8). Of course, with some minor modifications regarding the signal carrier frequency, this model may as well be applied to single frequency GLONASS pseudorange measurements. However, since GLONASS grants full access to the L 2 frequency, it enables the properly equipped user to deal with the ionosphere problem in a better way, by using dual-frequency ionospheric corrections. These will be treated in Section 8.5.2. The Klobuchar model is an empirical approach, known to eliminate only about half the actual iono- spheric path delay. Still it is used as the recommended model for computation of ionospheric delays for single frequency GPS users. The way this model is to be applied is defined in the GPS Interface Control Document (ICD-GPS, 1991). It is briefly described in the following. Given the geodetic coordinates of the user position λ R , ϕ R (in half circles), the elevation and azimuth E S , A S (in half circles) of the observed satellite and the broadcast coefficients α j , β j of the ionospheric correction model, the ionospheric time delay at the receiver computed GPS system time t R is obtained following the algorithm: Compute Earth’s central angle: ψ = 0.0137 E S + 0.11 − 0.022 Compute latitude of ionospheric intersection point: ϕ I = ϕ R + ψ cos A S Note: Absolute value of ϕ I is to be limited to 0.416. Compute longitude of ionospheric intersection point: λ I = λ R + ψ sin A S cos ϕ I Compute geomagnetic latitude of ionospheric intersection point: ϕ I m = ϕ I + 0.064 cos(λ I − 1.617) Compute local time at ionospheric intersection point: t I = t R + λ I · 43200 s Note: t I is to be limited to the range of 1 day: 0 ≤ t I < 86400 s. 122 8 OBSERVATIONS AND POSITION DETERMINATION -50 -40 -30 -20 -10 0 10 20 30 141000 141500 142000 142500 143000 143500 Am biguities [cycles] GPS Time [s] GPS 1 GPS 15 GPS 21 GPS 23 GPS 28 Figure 8.13: GPS double difference carrier phase floating ambiguities. -4000 -2000 0 2000 4000 141000 141500 142000 142500 143000 143500 Am biguities [cycles] GPS Time [s] GLO 2 A A A GLO 4 GLO 17 GLO 18 A A GLO 19 Figure 8.14: GLONASS double difference carrier phase floating ambiguities. 8.5 Ionospheric Correction 123 Compute amplitude of ionospheric delay: A I = 3 j=0 α j (ϕ I m ) j Note: Amplitude is to be limited to values ≥ 0. Compute period of ionospheric delay: P I = 3 j=0 β j (ϕ I m ) j Note: Period is to be limited to values ≥ 72000. Compute phase of ionospheric delay: X I = 2π · (t I − 50400 s) P I [rad] Compute obliquity factor: F = 1 + 16 · (0.53 − E S ) 3 Compute ionospheric time delay: δt S,Iono R, f L1, GP S = 5 · 10 −9 + A I · 1 − X 2 I 2 + X 4 I 24 · F , |X I | < 1.57 5 · 10 −9 · F , |X I | ≥ 1.57 (8.5.1) The ionospheric path delay is dependent on the frequency of the signal; it is inversely proportional to the square of the carrier frequency, cf. e.g. (Hofmann-Wellenhof et al., 1993). The algorithm introduced above is designed to compute an estimate of the ionospheric delay of a signal at GPS L 1 frequency (1575.42 MHz). Therefore, for a GLONASS satellite S transmitting on frequency f S , an additional frequency factor is required to compensate for this: δt S,Iono R, f S = f L 1 , GP S f S 2 δt S,Iono R, f L1, GP S (8.5.2) with δt S,Iono R, f L1, GP S the ionospheric path delay as calculated for satellite S according to Eq. (8.5.1). The ionospheric path delay from Eq. (8.5.2) for each single satellite is to be inserted in the system of observation equations Eq. (8.1.8) on computation of the receiver position. 8.5.2 Dual Frequency Ionospheric Correction The ionospheric path delay of a GPS or GLONASS satellite signal depends on the electron content of the ionosphere, the frequency of the signal and the distance that the signal travels through the ionosphere, which in turn depends on the satellite elevation. (Hofmann-Wellenhof et al., 1993) e.g. gives the equation c · δt S,Iono R = 1 cos z 40.3 m 3 s 2 f 2 T EC (8.5.3) with z Zenith distance of signal at ionospheric piercing point f Frequency of carrier signal T EC Total electron content of ionosphere The total electron content in this equation is measured in units of electrons per m 2 . Determination of the Ionospheric Group Delay Having dual frequency measurements to a satellite available, one can make use of this frequency dependence of the ionospheric path delay to estimate the actual delay without having to rely on a model that may be inaccurate. Denoting the measured pseudorange from receiver R to satellite S on L 1 with P R S R,L 1 and that on L 2 with P R S R,L 2 , we can rewrite Eq. (8.1.4) as: P R S R,L 1 = S R + c · δt R − c · δt S + c · δt S,T rop R + c · δt S,Iono R (f L 1 ) + ε S R,L 1 (8.5.4) = P R S R,0 + c · δt S,Iono R (f L 1 ) + ε S R,L 1 P R S R,L 2 = S R + c · δt R − c · δt S + c · δt S,T rop R + c · δt S,Iono R (f L 2 ) + ε S R,L 2 (8.5.5) = P R S R,0 + c · δt S,Iono R (f L 2 ) + ε S R,L 2 124 8 OBSERVATIONS AND POSITION DETERMINATION where the frequency independent parts of the pseudorange are summarized in P R S R,0 . Differencing these two measurements yields P R S R,L 2 − P R S R,L 1 = c · δt S,Iono R (f L 2 ) − c · δt S,Iono R (f L Download 5.01 Kb. Do'stlaringiz bilan baham: 
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