Pseudorange measurement errors

As introduced, the pseudorange measurement model is affected by different independent sources of errors all grouped in the 𝑒𝑟𝑟𝑜𝑟𝑠𝜌,𝑢𝑖 (or 𝑒𝑟𝑟𝑜𝑟𝑠𝜑,𝑢𝑖 ) term. In this section, a list of the main sources of errors is given and a measurement model is defined. Orders of magnitude of these errors are discussed as a conclusion.

2.1.3.1 Different sources of errors

A precise description and modeling of each error type can be found in the literature, for example in [Kaplan and Hegarty, 2006]. A brief overview of each error is proposed in the following:

- the receiver noise, - ionospheric effect, - tropospheric effect, - multipath,

- clock and ephemeris inaccuracies.

The satellite payload, the satellite antenna, the receiver antenna and the receiver processing channel are also sources of pseudorange errors. These errors are induced by signal delays and signal distortions generated at satellite level and at receiver level as detailed in next chapters.

38 2.1.3.1.1 Receiver noise

The pseudorange estimation is based on the synchronization of the receiver with the digitized incoming signal (thus providing the receiver with the capability to estimate the time of arrival of a specific part of the incoming signal). This synchronization mechanism, also called tracking, will be presented later. The noise affecting a digitized incoming signal has a direct impact on this synchronization and consequently on the pseudorange measurement. The repercussion on the pseudorange can be modeled as an additive Gaussian noise affecting the measurement.

The magnitude of this noise depends upon the setting of the tracking loop, the received signal strength, the antenna as well as upon the modulation of the signal of interest.

2.1.3.1.2 Ionospheric effect

The ionosphere is a dispersive medium located approximatively between 70 km and 1000 km above the earth’s surface. Because of free electrons which create an electric field, electromagnetic wave does not travel at the vacuum speed of the light as they cross this region. The signal group delay and by consequence the code pseudorange measurement are delayed in proportion to the number of electrons encountered, referred to as the Total Electron Content (TEC), whereas the carrier phase measurement is advanced by the same amount. The ionospheric delay is frequency-dependent. The delay on the code pseudorange measurement 𝑖, 𝜀_{𝐼𝑜𝑛𝑜}^𝑖 is modeled by:

where

- 𝑓 is the carrier frequency of the signal in hertz.

GNSS receivers systematically try to mitigate the ionospheric effect applying corrections. Different algorithms exist to estimate ionospheric delays:

- GPS receivers apply the Klobuchar ionospheric model and Galileo receivers the NeQuick ionospheric model. These model parameters are embedded in the navigation message.

- SBAS provides users with its own ionospheric delay correction model by the means of an ionospheric grid.

- Another method consists of combining pseudorange measurements from the same satellite but on two different frequencies, exploiting the fact that the ionosphere is a dispersive medium, meaning that the ionospheric delay is frequency-dependent. This method called dual-frequency iono-free combination removes the first order ionospheric delay.

- The use of differential measurements is also a means to compensate the ionospheric effect as presented in 2.3.1.1.

2.1.3.1.3 Tropospheric effect

The troposphere is a non-dispersive medium (for frequencies up to 15 GHz) located between about 40 km and the earth’s surface. Within this medium, the group and the carrier phase delays are delayed by the same amount compared to free space propagation. This delay, which leads to a pseudorange measurement bias, is function of the tropospheric refractive index, which is dependent upon the local temperature, pressure, and relative humidity.

𝜀_{𝐼𝑜𝑛𝑜}^𝑖 (𝑓) ≈ −40.3 × 𝑇𝐸𝐶

𝑓² (2-4)

2.1 PVT Computation using GNSS core constellations

39

GNSS receivers can estimate and correct their own tropospheric delay according to different models, usually fairly accurately. Civil aviation, for instance, recommends the UNB3 tropospheric model to be applied by airborne receivers [RTCA, 2006].

The use of differential measurements is also a means to compensate tropospheric effect as presented in 2.3.1.1.

2.1.3.1.4 Multipath error

Multipath are GNSS signal replicas induced by the reflection and/or the diffraction of GNSS signals on obstacles encountered during the signal propagation. This phenomenon is environment-dependent.

At receiver level, interferences exist between the Line of Sight (LoS – the only signal corresponding to the true satellite/receiver distance) and reflected/diffracted signals. Thus, the receiver sees multiple versions of the GNSS signal, each with different times of arrival, signal levels and carrier phases. The consequence is that multipath induces an error on the receiver synchronization with the LoS signal of interest. There are usually three types of methods used to mitigate the multipath at different stages of the receiver signal processing:

- at the antenna level, by carefully choosing antenna characteristics and location, in order to limit the power of the multipath entering the receiver,

- at the signal processing level by discriminating the LoS from the multipath. Tens of techniques were developed as the MRDLL [Laxton and DeVilbiss, 1997], the MEDLL [Townsend et al., 2000], the deconvolution technique [Dragunas and Borre, 2011], etc.,

- at the PVT computation level by trying to detect and exclude measurement with biases (high residual values).

2.1.3.1.5 Clock and ephemeris inaccuracies

Satellites clocks are highly stable but cannot remain fully synchronized with the constellation time. This is the reason why the navigation data message contains a clock correction field. Despite the satellite clock correction, some residual errors can affect the receiver. In the same way, ephemeris transmitted in the navigation data message, which contains the information of satellites positions, can be affected by some imprecisions because of the difficulty to forecast changes of satellites orbit. These inaccuracies entail equivalent residual error on pseudorange measurements. Assuming that the distribution of the clock and ephemeris inaccuracies (projected on the vector between the satellite and the user) is a zero mean Gaussian random variable, the standard deviation of the error budget attributed to the clock and ephemeris together is called URA (User Range Accuracy) for GPS and SISA (Signal-In-Space Accuracy) for Galileo. These parameters are broadcast in the navigation message.

2.1.3.2 Measurement error models

The error term which appears in the pseudorange measurements definition gathers the pseudorange measurement bias induced by different sources:

where

𝑒𝑟𝑟𝑜𝑟𝑠_𝜌^𝑖 = 𝜀_{𝜌,𝑁𝑜𝑖𝑠𝑒}^𝑖 + 𝜀_{𝑇𝑟𝑜𝑝𝑜}^𝑖 + 𝜀_{𝐼𝑜𝑛𝑜}^𝑖 + 𝜀_{𝜌,𝑀𝑃}^𝑖 + 𝜀𝑐𝑙𝑜𝑐𝑘&𝑒𝑝ℎ𝑖 (2-5) 𝑒𝑟𝑟𝑜𝑟𝑠𝜑𝑖 = 𝜀_{𝜑,𝑁𝑜𝑖𝑠𝑒}^𝑖 + 𝜀_{𝑇𝑟𝑜𝑝𝑜}^𝑖 − 𝜀_{𝐼𝑜𝑛𝑜}^𝑖 + 𝜀_{𝜑,𝑀𝑃}^𝑖 + 𝜀𝑐𝑙𝑜𝑐𝑘&𝑒𝑝ℎ𝑖 (2-6)

40

- 𝜀_{𝜌,𝑁𝑜𝑖𝑠𝑒}^𝑖 (𝜀_{𝜑,𝑁𝑜𝑖𝑠𝑒}^𝑖 ) is the error induced by the receiver noise on the code (respectively the carrier phase) pseudorange measurement.

- 𝜀_{𝑇𝑟𝑜𝑝𝑜}^𝑖 is the error induced by the tropospheric delay (after applying UNB3 tropospheric model) on the pseudorange measurement.

- 𝜀_{𝐼𝑜𝑛𝑜}^𝑖 is the error induced by the ionospheric delay (after applying ionospheric models such as Klobuchar and NeQuick etc.) on the pseudorange measurement.

- 𝜀_{𝜌,𝑀𝑃}^𝑖 (𝜀_{𝜑,𝑀𝑃}^𝑖 ) is the error induced by the multipath (after applying multipath mitigation techniques) on the code (respectively on the carrier phase) pseudorange measurement.

- 𝜀𝑐𝑙𝑜𝑐𝑘&𝑒𝑝ℎ𝑖 is the error induced by the satellite clock and ephemeris inaccuracies on the pseudorange measurement.

It is usually assumed that components of 𝑒𝑟𝑟𝑜𝑟𝑠𝜌𝑖 and 𝑒𝑟𝑟𝑜𝑟𝑠𝜑𝑖 are independent and can be modeled by zero-mean normal distributions that overbound the real error distributions. The total error induced on the pseudorange measurement model, also called User Equivalent Range Error (UERE), has a variance equal to:

where

- 𝜎_{𝑈𝐸𝑅𝐸}² is the variance of all residual errors affecting the pseudorange measurement.

- 𝜎_{𝑁𝑜𝑖𝑠𝑒}² is the variance of the receiver noise affecting the pseudorange measurement. 𝜎_{𝑁𝑜𝑖𝑠𝑒}² is lower on carrier phase than on code measurement.

- 𝜎_{𝑇𝑟𝑜𝑝𝑜}² is the variance of the tropospheric delay affecting the pseudorange measurement.

- 𝜎_{𝐼𝑜𝑛𝑜}² is the variance of the ionospheric delay affecting the pseudorange measurement.

- 𝜎_𝑀𝑃² is the variance of the error entailed by multipath affecting the pseudorange measurement.

- 𝜎𝑐𝑙𝑜𝑐𝑘&𝑒𝑝ℎ2 is the variance of the satellite clock error plus the ephemeris error affecting the pseudorange measurement.

2.1.3.3 Code pseudorange measurement error order of magnitude

Order of magnitude for these five components are given as an example for a single frequency receiver not using any augmentation system. All values given in Table 2-2 are dependent upon several parameters and characteristics of the GNSS receiver. Values given in Table 2-2 have to be considered as order of magnitude and represent the impact of the different errors on a typical receiver.

- 𝜎_{𝑇𝑟𝑜𝑝𝑜} is evaluated for a satellite elevation angle 𝜃 equal to 5° (highest value) and equal to 75° (lowest value). It corresponds to the residual tropospheric model after applying the UNB3 model. Values are estimated from formulas defined in [RTCA, 2006].

- 𝜎_{𝐼𝑜𝑛𝑜} is evaluated for a receiver latitude 𝛾 equal to 45° and a satellite elevation angle 𝜃 equal to 5° (highest value) and equal to 75° (lowest value). GPS results are provided after applying the Klobuchar ionospheric model, whereas Galileo results are provided after applying the NeQuick ionospheric model [Montloin, 2014]. Even if in general 𝜎𝐼𝑜𝑛𝑜 is lower after applying the NeQuick model than the Klobuchar model, at this latitude, the two models reach same performance.

𝜎_{𝑈𝐸𝑅𝐸}² = 𝜎_{𝑁𝑜𝑖𝑠𝑒}² + 𝜎_{𝑇𝑟𝑜𝑝𝑜}² + 𝜎_{𝐼𝑜𝑛𝑜}² + 𝜎_𝑀𝑃² + 𝜎𝑐𝑙𝑜𝑐𝑘&𝑒𝑝ℎ2 (2-7)