In recent years, many simulation tools emerged to model the communication of connected vehicles. Thereby, the focus was put on channel modelling, applications or protocols while the localisation due to satellite navigation systems was treated as perfect. The effect of inaccurate positioning, however, was neglected so far. This paper presents an approach to extend an existing simulation framework for radio networks to estimate the localisation accuracy by navigation systems like GPS, GLONASS or Galileo. Therefore the error due multipath components is calculated by ray optical path loss predictions (ray tracing) considering 3D building data together with a well-established model for the ionospheric error.

When modelling the communication of connected vehicles, most simulation tools assume a precise positioning of transmitter (TX) and receiver (RX). A vehicle's position is the basis of the decision-making process whether sending a message or not. The localisation of a vehicle is mainly based on Global Navigation Satellite Systems (GNSS), because they are widely available on the earth surface. But GNSS-systems are based on several assumptions to work correctly. Therefore simulating the effect of the GNSS-System on localisation is prerequisite for further simulations of connected vehicles.

For a 3D localization, the signal of at least four satellites must be decodable, each enabling to estimate the distance between the receiver (RX) and the satellite (TX).
To determine the distance, the receiver assumes the signal to propagate along the direct path and to travel at constant speed of light.
However, due to the long distance between the TX and RX, signal propagation is affected by external effects.
Especially in urban environments, a direct path is not always guaranteed.
Ray optical path loss predictions enable to model the propagation by determining the true geometric distance between the TX and RX.
Furthermore, considering all valid paths from the satellite to the device yields the channel gain, making it possible to compute the corresponding received power.
Multipath modelling for GNSS using a ray tracing approach is already presented in

In addition, as specific nature of satellite-to-earth links, that signal has to pass the atmosphere of the earth.
As a consequence the different layers of the atmosphere like troposphere and ionosphere have to be considered in modelling propagation as well.
The ionosphere is a dispersive medium resulting in a frequency-related slowing-down effect on the signal depending on the geometry between the TX and RX and the time-variant state of the sun.
In contrast, the troposphere is a non-dispersive medium and according to the ITU Recommendation ITU-P.531-13

In contrast to all these publications, this research focus not only on multipath or ionosphere modelling, but to combine both and integrate it into an existing simulation tool with a focus on localisation accuracy. Furthermore this paper demonstrates how to extend this simulation tool usually used for mobile networks on earth with the ability to propagate satellite-to-earth links.

Models for multipath propagation and the ionosphere were integrated in the Simulator for Mobile Networks (SiMoNe) framework developed at the Institute for Communications Technology, Braunschweig, making it possible to evaluate positions along a trace with different focal points: on the one hand, visible and receivable satellites were evaluated to identify dead spots. On the other hand, determining localisation accuracies can lead to conclusions about areas with a high probability of insufficient positioning.

This paper is structured as follows: A guide for preparing the simulation data is proposed in Sect.

To simulate satellites by using ray optical pathloss predictions a valid scenario is required.
Information about buildings in 3D can be modelled based on open street maps

Satellite positions are described by their orbit and a timestamp.
Any GPS-Receiver receives these information via the GPS-Signal.
For our simulations we use publicly available orbit data, the almanach

For use in simulation, geocentric coordinates have to be harmonized with geodetic coordinates of buildings and subscriber positions. Usually, buildings and subscriber positions are in a projected coordinate system based on an assumption of the earth shape like the Universal Transversal Mercator (UTM) or Gauß-Krüger (GK) system. The ray tracer used for this paper is based on cartesian coordinates and can not work with angular coordinate systems, like the World-Geodetic System 84 (WGS84). As a consequence there are two options to choose from: Transform buildings and subscriber cordinates into ECEF or transform the satellite positions into UTM/GK.

Using a common projection algorithm to project WGS84 coordinates into UTM/GK leads to distortions up to 40 cm/km starting at meridian.
Therefore following solution is used:
First, a position on earth ground is defined as reference.
This reference position is transformed from UTM/GK to WGS84 and converted to ECEF coordinates.
Secondly, the reference position is defined as origin of its local coordinate system in a Earth-North-Up (ENU) orientation.
The relationship between a global ECEF- and a local ENU coordinate system is shown in Fig.

East North Up (ENU) local coordinate system in relation to Earth Centered Earth Fixed (ECEF) global coordinate system.

As a result the satellite position is a non valid GK/UTM position in the context of using transformation algorithms. But it works under the assumption that most of propagation effects occur near the position on the earth ground. For larger scenarios this solution needs to be repeated for every receiver position to reduce projection distortions.

The section of Propagation Modelling is divided in two parts. The first part is about ray optical propagation predictions in the context of satellites and methods to accelerate checking whether a link exist. The second part is about modelling the ionosphere representing the effect by the atmosphere. From the point of view by ray optical propagation modelling, a ray travelling through the ionosphere is assumed as a straight line. As a consequence, only a slowing effect on the propagation is modelled.

The simulation framework for this work is the Simulator for Mobile Networks (SiMoNe), developed at the Institute for Communications Technology at TU Braunschweig

In contrast to these use cases, a satellite to earth link differ from mobile communication in the field of distance and propagation effects.
GPS-Satellites travel on a height of 20 200

A link budget, including transmission power, attenuation by multipath components and antenna masking, is an enabler to identify visible and decodable satellites of the specific user.
Also the path of the channel components can be used for modelling the error by multipath.
GNSS-Receiver can use two techniques to estimate the signal duration: Code-Phase (CoP) or Carrier-Phase (CaP) observation.
CoP observation uses the coded signal for correlation and CaP observation uses the original modulated signal.
The technique of observing the CoP is not as distinct as the CaP and can lead to errors of tenth of meters due to multipath error

For calculating the multipath error for Code-Phase Observation

The variables mentioned in both equations can be derived from the results of ray tracing.

Before starting ray optical path loss predictions, determining which satellites are visible in the context of a receiver position reduces overhead. Therefore each elevation angle between the receiver and satellite position needs to be calculated. An positive elevation angle represents a visible satellite above the horizon.

Instead of calculating elevation angles and transforming all coordinates of the buildings for each receiver position, an assumption can be made. If the scenario is assumed to be small in comparison to the distance between satellite and ground level and in comparison to the appearing projection errors, we can use one receiver position as a reference position for the whole scenario. Usually the centre point of all buildings in the scenario is chosen as a reference position.

A common speed up algorithm to filter buildings of interest for ray tracing is using an ellipse with TX and RX in the focal points and a configurable extra path length. This filter is not usable here. Since the great distance in combination with a wide variety of elevation angles leads to a ellipse either excluding or including all buildings. The ellipse approach functions well, if obstacles are equally distributed between the transmitter and receiver. But any possible obstacle obstructing the signal path in satellite to earth links is located next to the receiver. Therefore an obstacle filter should concentrate on the receiver position. A rectangle or a circle around the receiver position are feasible forms for filtering.

The ionosphere is a dispersive medium. As mentioned before, the ionosphere has a negligible effect on the path geometry but on the velocity of the electromagnetic wave. The slowing-down effect depends on the number of electrons in the ionosphere layer which can be described by the Total Electron Count (TEC). The sun activity is responsible for ionizing the atmospheric layer. So the number of electrons are depending on the sun activity. An indicator for the sun activity is the number of sun spots. The more sun spots exists, the higher is the electron density in the ionosphere. Sun spots are described within a sun cycle which changes over several years. Another factor related to the sun is the time of day. By night the effect of the sun is less present than during the day. Therefore the ionospheric effect is related to the simulation time and need to be modelled by the sun spot number and time of day.

Based on ITU-P.531-13 (

NeQuick2 models the Slant Total Electron Count (STEC), same as the TEC along a ray path

Following Eq. (

Pseudoranges describe the distance between satellite and receiver based on measuring the signal delay by the receiver, which includes all errors. As a consequence in the simulation the errors by multipath propagation and ionosphere delay have to be added to the true distance. Calculating the pseudorange is carried out only for visible and receivable satellites based on ray-optical predictions.

For a 3D position a receiver needs at least four visible and decodable satellites available. The process of determining a position based on pseudoranges is called multilateration.

Multilateration describes the algorithm of finding the intersection of all spheres with the radius of the pseudoranges.
The relations between pseudoranges and the searched position for each satellite

At least four satellites are needed to determine the four variables in this equation system.
There exist several algorithms to solve this equation system by an iterative approach or in a closed-form algorithm.
For our simulations a closed-form algorithm, called the Bancroft-Algorithm

If the receiver can decode more than four satellites, the equation system is overdetermined. There are two options to handle this: on the one hand the receiver can choose four satellites based on the constellation geometry and exclude all other satellites. On the other hand all visible and decodable satellites are integrated and used for determining a position. Both options were implemented.

To assess the accuracy of a satellite-based localisation system, at least three indicators exists: The Dilution of Precision (DOP), comparing a simulated absolute position with a true position, and a deviation range within the true position shall exist with a given probability.

The constellation of the satellites can be described by DOP.
DOP is an qualitative measure to evaluate the constellation quality by the volume of the satellites span.
In Fig.

Geometrical Interpretation of Dilution of Precision adapted from

With the help of multilateration algorithm, an absolut position can be determined. The effect of the error sources are represented in the difference between the calculated and the true position and gives a statement about the localisation accuracy.

An alternative to an absolute positional difference for evaluating localisation accuracy is using an statistical measure like CE90.
CE90 indicates a radius of circle that covers the RX position by a chance of 90 %.
The Eq. (

For evaluation of the proposed methods a measurement was made.
The first method to prove is the coordinate transformations described in Sect.

The core at the measurement setup is a application board with a ZED-F9P module by u-blox functioning as a GNSS-Receiver.
It allows to record and export GPS-Logs in a NMEA-Format.
From these messages, the date, time, position in degrees minutes, DOP Values, elevation and azimuth angles and CNDR can be extracted.
CNDR stands for Carrier-to-Noise-Density-Ratio, expressed in

The receiver is capable of combining information from different satellite localisation systems like GPS, GLONASS, and Galileo. It can also be used as a dual frequency receiver and works with additional correction information from earth stations. Any feature going beyond a single-frequency-receiver without correctional parameters is turned off to reduce uncertainties in comparison with the simulation.

Map of Braunschweig (Germany) including modelled buildings and route of measurement based on © OpenStreetMap contributors 2021. Distributed under the Open Data Commons Open Database License (ODbL) v1.0.

The measurement was performed in northern Braunschweig at 09:15:00 UTC on 12 January 2021.
In Fig.

First the choosen configuration for transmiter and receiver in the simulation will be explained.
It is assumed that the satellites transmit with a power of 47

CNDR values.

In the GPS-Logs the power is given as Carrier-to-Noise-Density-Ratio (CNDR).
System parameters to calculate the CNDR based on a received power are presented in Table

In the following the configuration of the ray tracer is presented.
Most common frequency band used for GPS is the L1/L1C frequency.
Therefore the frequency is set to 1575.42

A satellite constellation diagram is shown exemplary for one timestep in Fig.

Satellite Constellation Diagram at 09:15:01 UTC 12 January 2021.

CNDR for satellite PRN13 from 09:15:00 to 09:18:10 UTC on 12 January 2021 along a measured path.

Figure

Position deviation between Simulation (Carrier-Phase) and Measurement from 09:16:10 to 09:18:00 UTC on 12 January 2021 along a measured path

Position deviation between Simulation (Code-Phase) and Measurement from 09:16:10 to 09:18:00 UTC on 12 January 2021 along a measured path

In Figs.

Analysing CE90 values for this measured trace results in similar outcome.

The transformation of coordination results in similar satellite constellations as measured. So the approach of integrating satellites in a non-spheric coordinate, but projected coordinate system, is applicable.

The received power calculated by the ray tracer depends on its configuration with regard of implemented multipath effects, level of detail in modelling obstacles, buildings and receiver equipment. Here the results of LOS links matches the measurement rather good. In the case of NLOS-scenarios there are unmodelled effects, especially in cases with only a diffracted link. Different reasons can lead to this behaviour: first other propagation paths were not modelled, like diffraction on sides of buildings. Secondly, the model of diffraction are not feasible for this scenario. Therefore further development needs to be invested in the ray tracer.

The main problem of comparing the simulation and the measurement is the post-processing made by the GNSS-Receiver. In the simulation, the positions are individual determined. In contrast to that, the positions resulting from the receiver are filtered and CE90 and position offset does not have any information about previous positions, that might be available at the receivers.

In this paper we presented approaches to integrate a simulation model for satellite localisation into an existing simulation framework using ray optical pathloss predictions. As a result we are now able to simulate and do ray-optical analyses of GNSS-Satellites. In the future we extend the ray tracer further to enable multipath options as scattering. Additionally, setting up an own GNSS-Receiver based on Software-Defined-Radios (SDR) allows to extract intermediate states, like pseudoranges to improve and evaluate here proposed methods.

The data presented in this article are available from the authors upon request.

The computer simulations, measurements and results are carried out by BF. Scientific investigations and editorial hints were given by MS and ND. Mentorship including editorial hints were provided by TK.

The authors declare that they have no conflict of interest.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the special issue “Kleinheubacher Berichte 2020”.

This open-access publication was funded by Technische Universität Braunschweig.

This paper was edited by Madhu Chandra and reviewed by Pierre Reisdorf and Matthias Vodel.