Ionospheric delay

Where:

• 𝑃^𝑖 is the code pseudorange measurement in meters

• 𝛿^𝑖 is the carrier phase measurement in meters

• 𝜌^𝑖 is the geometrical distance between the receiver and the satellite 𝑖.

• 𝐼^𝑖 is the ionospheric propagation delay in seconds.

• 𝑇^𝑖 is the tropospheric propagation delay in seconds.

• 𝐷_{𝑚𝑢𝑙𝑣𝑖}^𝑖 is the code pseudorange measurement error induced by multipath propagation.

• 𝛿_{𝑚𝑢𝑙𝑣𝑖}^𝑖 is the phase measurement error due to multipath propagation.

• 𝜆 is the carrier wavelength.

• 𝑁^𝑖 is the carrier phase measurement ambiguity, constant over time as long as the PLL does not lose lock on the signal.

• 𝑛_𝑃^𝑖 is the code phase thermal noise error

• 𝑛_𝜑^𝑖 is the carrier phase thermal noise error

In the next sections we will present the error models that we used in this study. First, we will present the models that were used to model the pseudorange received by the receiver. Then, we will present two different pseudorange measurement error models that we used depending on the type of augmentation considered which are:

• Receiver Autonomous Integrity Monitoring (RAIM)

• Ground Based Augmentation System (GBAS)

In fact, as we have seen previously, RAIM does not provide correction for pseudorange measurements while GBAS provide differential corrections. Thus, pseudorange measurement error models considered for RAIM are the models corresponding to GPS standalone.

Additional augmentation models of SBAS and GALILEO GIC have been developed but we did not have the opportunity to use them in the frame of the current study. However, they will be used for future researches.

4.2 Pseudorange measurement model 4.2.1 Pseudorange measurement error models

The models presented in this section are used to generate the pseudorange at the input of the receiver.

4.2.1.1 Ionospheric delay

4.2.1.1.1 General considerations

The ionospheric delay model is used to represent the total error affecting pseudorange measurements due to the propagation of the signal through the ionosphere. Ionospheric error is the most severe error affecting GNSS measurements since it is very complex to model it and therefore it is difficult to

predict the delay so as to correct measurements. However, we will see further in this document that some properties of the ionosphere can be used to nearly cancel its effects on GNSS measurements when dual frequency measurements are available.

In this study, we focus on the nominal effects of the ionosphere on the group delay of GNSS signals.

Abnormal behaviour of the ionosphere is not considered and in particular ionospheric storms and ionospheric scintillations are not studied.

The ionosphere is a dispersive medium which is located between 60 km and 1000 km above the earth’s surface, in the atmosphere. In this area ultraviolet rays coming from the sun ionize a portion of gas molecules and thus, it releases free electrons. These electrons influence the propagation of the electromagnetic waves and thus, the GNSS signals [LEICK, 1995].

More precisely, the propagation velocity of GNSS signals through the ionosphere depends on their frequency and the total electron content (TEC, in e/m2) integrated along the LOS. It represents the number of free electrons in a 1 m² column along the LOS. It can be modelled as [LEICK, 1995]:

𝑇𝐸𝐶 = � 𝑁_𝑣. 𝑟𝑆

𝐿𝑂𝑆 𝑝𝑎𝑣ℎ

(4-3)

With 𝑁𝑣 the local electron density expressed in units of electron per cubic meters.

The group delay can be approximated at the first order as [LEICK, 1995]:

𝐼_𝑃≈40.3

𝑓² 𝑇𝐸𝐶 𝑚

(4-4)

For simulation purpose, it was necessary to model the total ionospheric delay so as to build the pseudorange. Different models have been identified and compared:

IRI-2001/IRI-2007:

In [RTCA, 2004], Appendix R, it is specified that to model ionosphere error, the model IRI-2001 should be used:

“Ionospheric error shall be modelled using the International Reference Ionosphere 2001 (IRI-2001) model. The IRI-2001 model was developed and validated by IRI, an international project sponsored by the Committee on Space Research (COSPAR) and the International Union of Radio Science (URSI). It models the ionospheric daily variation but does not model storms. Since statistical data is needed from a deterministic model, the model inputs will be randomized in the test procedures. IRI-2001 accounts for temporal and spatial correlation between satellite measurements. Other iono models may be used but they must be validated.”

The procedure that should be used to model storm conditions is the following:

“In addition to the ionospheric daily variation modelled by IRI, recorded storm data shall be used as the basis for the ionospheric component of pseudorange error in some trials.”

IRI-2007 is an updated version of IRI-2001. The main limitation of this model for us is that it is very complex to use and we do not have the necessary background to properly use it.

Moreover, in the frame of our study and since it is a deterministic model it would be necessary to randomize the input parameters so as to obtain a statistical model (as it is mentioned above).

Finally, the objective was to integrate a ionospheric error model in our simulator which would be very complex to do with the IRI-model and that’s we decided to consider other solutions.

GPS/GALILEO correction algorithms:

For single frequency receivers, ionospheric models can be used so as to correct a fraction of the ionospheric delay, using as inputs parameters broadcasted in the navigation message.

In the case of GPS, the model specified in [ARINC, 2004] is called the Klobuchar algorithm.

This model is reputed to allow removing at least 50% of the total ionospheric error. It is easy to find in the appropriate frame of the navigation message the input parameters of this model.

In the case of GALILEO, a more recent model should be used which is called the Nequick algorithm [ARBESSER, 2006]. It is supposed to have better performance than Klobuchar algorithm and is assumed to correct 70% of the ionospheric delay when operating on E5a, E5b, and E1 frequencies [GJU, 2010]. However, this model is a little more complex, and since GALILEO is not in service, there is not enough information available at the moment of the development of our software.

The final solution chosen in the frame of this study is to use Klobuchar model to generate ionospheric delays because of its simplicity and the availability of large amounts of data. We remind hereafter this algorithm.

4.2.1.1.2 Klobuchar Algorithm

Klobuchar algorithm can be found in [KLOBUCHAR, 1987] or [ARINC, 2004]. It is a thin shell model. Eight coefficients are transmitted to the users through the navigation message. The following notations are used to define the algorithm:

Satellite transmitted terms:

- 𝛼𝑛, 𝛽𝑛 are the satellite transmitted data words with 𝑛 = 0,1,2 𝑎𝑛𝑟 3 Receiver generated terms:

- 𝐸 the elevation angle between the user and satellite (semi-circles)

- 𝐴 the azimuth angle between the user and satellite, measured clockwise positive from the true north (semi-circles)

- 𝜙_𝑢 user geodetic latitude (semi-circles) WGS-84 - 𝜆_𝑢 user geodetic longitude (semi-circles) WGS-84 - 𝐺𝑃𝑆 𝑡𝑖𝑚𝑒 receiver computed system time

Computed terms:

- 𝑋 phase (radians)

- 𝐹 obliquity factor (dimensionless) - 𝑡 local time (sec)

- 𝜙𝑚 geomagnetic latitude of the earth projection of the ionospheric intersection point (mean ionospheric height assumed 350 km) (semi-circles)

- 𝜆𝑖 geodetic longitude of the earth projection of the ionospheric intersection point (semi-circles)

- 𝜙𝑖 geodetic latitude of the earth projection of the ionospheric intersection point (semi-circles) - 𝜓 earth's central angle between the user position and the earth projection of ionospheric

intersection point (semi-circles)

𝜓 = 0.0137

𝐸 + 0.11 − 0.022 (𝑖𝑒𝑚𝑖 − 𝑐𝑖𝑟𝑐𝑙𝑒𝑖) 𝜆_𝑖 = 𝜆_𝑢+𝜓 sin 𝐴

cos 𝜙_𝑖 (𝑖𝑒𝑚𝑖 − 𝑐𝑖𝑟𝑐𝑙𝑒𝑖)

𝜙_𝑖 = � 𝜙_𝑢+ 𝜓 cos 𝐴 , |𝜙𝑢| ≤ 0.416 𝑖𝑓 𝜙𝑖 > +0.416, 𝑡ℎ𝑒𝑛 𝜙𝑖 = +0.416

𝑖𝑓 𝜙_𝑖 < −0.416, 𝑡ℎ𝑒𝑛 𝜙_𝑖 = −0.416� (𝑖𝑒𝑚𝑖 − 𝑐𝑖𝑟𝑐𝑙𝑒𝑖)

𝜙_𝑚 = 𝜙_𝑖+ 0.064 cos(𝜆_𝑖− 1.617) (𝑖𝑒𝑚𝑖 − 𝑐𝑖𝑟𝑐𝑙𝑒𝑖)

𝑡 = � 4.32(10⁴)𝜆𝑖+ 𝐺𝑃𝑆 𝑇𝑖𝑚𝑒, 𝑖𝑓 0 ≤ 𝑡 < 86400 4.32(10⁴)𝜆𝑖+ 𝐺𝑃𝑆 𝑇𝑖𝑚𝑒 − 86400, 𝑖𝑓 𝑡 ≥ 86400 4.32(10⁴)𝜆_𝑖+ 𝐺𝑃𝑆 𝑇𝑖𝑚𝑒 + 86400, 𝑖𝑓 𝑡 < 0

� (𝑖𝑒𝑐)

𝐹 = 1.0 + 16.0[0.53 − 𝐸]³

𝑃𝐸𝑅 = � � 𝛽_𝑛𝜙_𝑚^𝑛

3

𝑛=0

, 𝑃𝐸𝑅 > 72000 𝑖𝑓 𝑃𝐸𝑅 < 72000, 𝑃𝐸𝑅 = 72000

� (𝑖𝑒𝑐)

𝑥 =2𝜋(𝑡 − 50400)

𝑃𝐸𝑅 (𝑟𝑎𝑟)

𝐴𝑀𝑃 = � � 𝛼^𝑛𝜙𝑚𝑛 3

𝑛=0

, 𝐴𝑀𝑃 > 0 𝑖𝑓 𝐴𝑀𝑃 < 0, 𝐴𝑀𝑃 = 0

� (𝑖𝑒𝑐)

We can then obtain the final ionospheric group delay as:

𝐼_𝐿1^𝑝(𝑡) = �𝑐𝐹 �5 × 10⁻⁹+ 𝐴𝑀𝑃 �1 −𝑥² 2 +

𝑥⁴

24�� , 𝑖𝑓 |𝑥| < 1.57 𝑐𝐹(5 × 10⁻⁹), 𝑖𝑓 |𝑥| > 1.57

(𝑚𝑒𝑡𝑒𝑟𝑖)

Table 5: Klobuchar algorithm [ARINC, 2004]