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2022 | OriginalPaper | Buchkapitel

# 5. GNSS Observation Models

verfasst von: Clement A. Ogaja

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## Abstract

This chapter summarizes the necessary information to understand GNSS measurement and observation models. Section 5.1 presents the underlying observation equations for GNSS signals, while signal propagation errors and correction methods are introduced in Sects. 5.2 and 5.3. Specifically, receiver and satellite antenna phase center modeling is summarized in Sect. 5.3.2.3, and other topics such as phase wind-up and Earth deformation effects on station coordinates are also, respectively, discussed in Sects. 5.3.2.4 and 5.3.2.6. Such discussions are provided with the assumption that the reader is primarily interested in understanding the concepts applicable in achieving high precision for geodesy and similar applications. Pertinent references are provided as necessary for further background understanding.
Fußnoten
1
A GNSS receiver determines signal travel time Δt by correlating the received code from satellite with a replica of this code generated in the receiver, so this replica moves in time (Δt) until the maximum correlation is obtained.

2
Antenna’s mean electromagnetic reference point, also called electrical antenna phase center (APC), is the transmission or reception point of a carrier wave from satellite to receiver. APCs are unique to each antenna (hardware dependent) and are defined by the electromagnetic properties of the antennas. For example, for one antenna, the L1 APC, L2 APC, and L5 APC are all different.

3
Some GNSS navigation codes are encrypted for security reasons. Thus, a direct tracking of these code types is not possible for unauthorized users. However, several methods (receiver tracking modes) have been developed to circumvent the problem. These modes may lead to different signal hardware delays in the receiver.

4
“For a receiver with fixed coordinates, the wind-up is due to the satellite orbital motion. As the satellite moves along its orbital path, it must perform a rotation to keep its solar panels pointing to the Sun direction in order to obtain the maximum energy while the satellite antenna keeps pointing to the Earth’s center. This rotation causes a phase variation that the receiver misunderstands as a range variation.” [30]

5
The density of the atmospheric layers is not homogeneous. This causes spatial and temporal variations in the refractive index. Snell’s law states that n 1 ⋅ φ 1 = n 2 ⋅ φ 2 if an electromagnetic wave travels from a medium with refractive index n 1 to a second one with refractive index n 2 and crosses the distance between them at an angle φ 1 and deviates (leaves) at an angle φ 2.

6
Since multipath error depends on the receiver’s environment, to reduce its effects, (1) good data collection site(s) is necessary to avoid reflective environments, (2) use a good quality antenna that is multipath-resistant and/or can internally digitally filter out the multipath disturbance, and (3) avoid, de-weight, or minimize the use of low elevation satellite data.

7
The satellite orbit information is generated from the tracking data collected by the GNSS Control Segment (see Fig. 2.​1 in Chap. 2). The Control Segment updates satellite positions on a regular basis, calculates their predicted paths, and uploads this information to the satellites. The uploaded information includes the almanac and ephemeris data, which contains the predicted positions of the satellites. These are downloaded when GNSS receivers track the satellite signals.

8
The noise of a single, double, and triple difference is a factor of $$\sqrt {2}$$, $$\sqrt {4}$$, and $$\sqrt {8}$$ higher compared to the undifferenced observations. All differenced observations are mathematically correlated (see, e.g., [13]).

9
Estimating/fixing correct ambiguity depends on the bias term being introduced as additional parameter. If ignored, they are absorbed into ambiguity estimates making it difficult to fix them to correct integers. See Sect. 7.​3 of Chap. 7.

10
The GLONASS inter-frequency bias is commonly defined as the difference of bias at frequency number k with respect to the bias at frequency number 0 [29].

11
Applies to CDMA-based satellites.

12
Code and phase biases (B, b) caused by receiver and satellite hardware delays of signals (see Eqs. (5.3) and (5.4)).

13
The precise GPS and GLONASS satellite clocks provided by IGS are accurate to the order of 0.1 ns or better (1 ns of error ≈30 cm in range) [30].

14
GLONASS satellites transmit Δ rel within the satellite clock corrections $$\widetilde {\delta } t^{sat}$$ [30].

15
The rate of advance of two identical clocks, one on the satellite and the other on ground, will differ due to the difference in gravitational potential (general relativity) and the relative speed between them (special relativity).

16
The clock on the satellite appears to run faster by ≈ 38 μs∕day than on ground (since Δff = ΔTT). This effect is corrected (in the factory) by decreasing the oscillating frequency of the satellite by the amount 4.57 ⋅ 10−3 Hz.

17
The ionosphere is dispersive for GNSS signals, which means that the signal delays differ depending on the carrier frequency employed.

18
GNSS carrier waves propagate with the phase velocity, whereas code measurements are considered to propagate with group velocity (i.e., pseudoranges obtained from the codes modulated in the carriers); the carrier waves speed up in the ionosphere, affected by what is known as the phase delay, while code measurements appear to be delayed or slowed by what is known as the group delay.

19
TEC is the number of free electrons expressed in TEC units (TECUs), where 1 TECU = 1016 electrons per m2.

20
The water vapor part only covers the lowest layer of the troposphere below 13 km above the surface of the Earth, while the dry part extends to about 45 km above the surface of the Earth [9].

21
The electrical phase center of the antenna is the point to which all measurements derived from GNSS signals are referred. This point cannot be accessed (e.g., by physical measurement). Therefore, a geometrical point on the antenna denoted as antenna reference point (ARP) is introduced for the purpose of defining a phase center offset (PCO)—the difference between ARP and the mean electrical phase center [13].

22
GNSS baseline lengths of tens of kilometers or less.

23
GNSS “satellites transmit right circularly polarized (RCP) radio waves and therefore, the observed carrier phase depends on the mutual orientation of the satellite and receiver antennas. A rotation of either receiver or satellite antenna around its bore (vertical) axis will change the carrier-phase”[16]. The change in phase can be up to one cycle (one wavelength), which corresponds to one complete revolution of the antenna. Mostly, the receiving antenna is fixed (unless mobile), and satellite antennas undergo slow rotations as their solar panels are being oriented toward the Sun, and thus the station-satellite geometry changes causing the phase wind-up.

24
A unit vector pointing from satellite’s CoM to the geocenter is $${\mathbf {r}}^{s_{CoM}} / ||{\mathbf {r}}^{s_{CoM}}||$$.

25
The station position vector r P is computed by subtracting the ARP offset vector (Δ ARP), defining ARP positioning relative to the station point P (monument marker), and the APC offset vector (Δ APC ≡ PCO in Fig. 5.5), defining the APC position relative to the ARP, from the receiver’s APC position r. Thus r P = r −Δ ARP −Δ APC. It is also noted that if ARP is the same as the monument marker position, then r P = r −Δ APC.

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