The GNSS measurements are strongly affected by ionospheric effects, due to the signal propagation through ionosphere; these effects could severely degrade the position; hence, a model to limit or remove the ionospheric error is necessary. The use of several techniques (DGPS, SBAS, and GBAS) reduces the ionospheric effect, but implies the use of expensive devices and/or complex architectures necessary to meet strong requirements in terms of accuracy and reliability for safety critical application. The cheapest and most widespread GNSS devices are single frequency stand-alone receivers able to partially correct this kind of error using suitable models. These algorithms compute the ionospheric delay starting from ionospheric model, which uses parameters broadcast within the navigation messages. NeQuick is a three-dimensional and time-dependent ionospheric model adopted by Galileo, the European GNSS, and developed by International Centre for Theoretical Physics (ICTP) together with Institute for Geophysics, Astrophysics, and Meteorology of the University of Graz. The aim of this paper is the performance assessment in single point positioning of the NeQuick Galileo version provided by ESA and the comparison with respect to the Klobuchar model used for GPS; the analysis is performed in position domain and the errors are examined in terms of RMS and maximum error for the horizontal and vertical components. A deep analysis is also provided for the application of the exanimated model in the first possible Galileo only position fix.

Global Navigation Satellite System (GNSS) provides, with global coverage and in all weather conditions, three-dimensional coordinates, velocity, and time synchronization for users equipped with a receiver/processor [

Statistical ranging error budget for GNSS single-frequency receiver [

Error source | 1 |
---|---|

Ephemeris data | 2.1 |

Satellite clock | 2.1 |

Ionosphere | 4.0 |

Troposphere | 0.7 |

Multipath | 1.4 |

Receiver noise | 0.5 |

| |

User equivalent range error | 5.3 |

The ionosphere is the main error source in GNSS measurements. At times, the range error of the troposphere and the ionosphere can be comparable, but the variability of Earth’s ionosphere is much larger and more difficult to model.

The range of the ionospheric error can vary from a few meters to 30 meters at the zenith, depending on observation epoch and latitude [

The ionosphere is the ionized region of the upper atmosphere, which extends from about 50 km to more than 1000 km where the density of free electrons and ions significantly influences the propagation of electromagnetic radio frequency waves [

Ionospheric profile and ionospheric regions.

Different methods can be adopted to minimize the ionospheric effect, such as the use of

dual-frequency technique,

augmentation system,

ionospheric model.

The ionosphere refractive index depends on the operating frequency so a dual-frequency GNSS user can take advantage of this feature and correct the measurements with an estimation of ionospheric delay obtained by a linear combination of dual frequency measures (this technique is referred to as iono-free). This method is the most effective but it cannot be used in a single frequency receiver [

Alternatively, GNSS receivers can obtain the ionospheric correction through augmentation systems (such as Differential GPS—DGPS, or Satellite-Based Augmentation System—SBAS), based on differential corrections, computed by a single station or by a network, and broadcasted to the receivers via terrestrial or satellite radio link.

Augmentation systems involve the use of complex architectures, while the algorithms using broadcast ionospheric parameters, within navigation messages, are cheaper and easy to be implemented in commercial single frequency receivers. Klobuchar developed the first algorithm for ionospheric correction in the mid-1970s for the GPS single frequency user, able to correct approximately 50% of the ionospheric range error [

NeQuick is an ionospheric model developed at the Aeronomy and Radiopropagation Laboratory of the Abdus Salam International Centre for Theoretical Physics (ICTP) - Trieste, Italy, and at the Institute for Geophysics, Astrophysics and Meteorology of the University of Graz, Austria. It has been considered as an evolution of the DGR profiler proposed by Di Giovanni and Radicella [

The remainder of this paper is organized as follows. The Position Velocity and Time (PVT) algorithm is described in Section

GNSS positioning is based on the one-way ranging technique: the time of travel of a signal, transmitted by a satellite, is measured and scaled by the speed of light to obtain the pseudorange (PR) as follows:

Using trilateration technique it is possible to obtain the navigation solution composed by the tridimensional receiver coordinates and the receiver clock offset relative to system time scale.

The measurement equations (

A least squares method is used to solve (

In this work GPS single point position is carried out and the PR are corrected for the satellite related error, tropospheric effect using Hopfield model, relativistic and Sagnac effects; the ionospheric delay is computed using Klobuchar and NeQuick G model, in order to compare their performance.

Single frequency users have to correct as much as possible the first-order ionospheric term, which accounts for more than 99.9% of the total ionospheric delays [

The dispersive effect of the ionosphere on GNSS observables causes a code delay obtained from
^{16} el/m^{2}), and

Below the official ICA for GPS (Klobuchar) and for Galileo (NeQuick G) are described.

Ionospheric Klobuchar model is an algorithm developed, around the mid-70s by Klobuchar, for single-frequency GPS receivers to correct approximately 50% of ionospheric delay [

In a SLM the STEC is calculated at the Ionospheric Point (IP) which is a geographic point, obtained by the intersection between the propagation direction (ray path, also called Line Of Sight—LOS) and the average height of the ionosphere. The projection on the surface of the ionospheric point is the Subionospheric Point (SIP) depicted in Figure

Klobuchar SLM.

Klobuchar model provides a different estimation for the daytime and nighttime ionospheric delay (in seconds) along the SIP vertical direction, starting from eight coefficients (within GPS navigation message), which describe the worldwide ionosphere behavior [

NeQuick is a semiempirical model that describes spatial and temporal variations of the ionospheric electron density. It is based on the DGR ionospheric profiler [

NeQuick model uses the peaks of the three main ionospheric regions (E, F1, and F2) as anchor points [

In the NeQuick model, the electron density above 100 km and up to the F2-layer peak (the bottom side region) is calculated by a modified DGR profile formulation which includes five semi-Epstein layers [

Taking advantage of the increasing amount of available data, the NeQuick model has been continuously updated with changes involving the formulation of some specific parameters although conceptual structure of the model remains unaltered [

NeQuick 1 constitutes the original version recommended by the ITU-R. At first, it has been improved regarding the bottom side [

NeQuick algorithm was originally developed to be used with monthly average solar flux index F10.7 (solar radio flux per unit frequency at a wavelength of 10.7 cm); so in order to use NeQuick model in real time application such as GNSS ionospheric correction computation, the monthly average F10.7 index has to be replaced by a daily input parameter in order to take in account both daily variation of the solar activity and the user’s local geomagnetic condition. This daily input parameter is the so-called effective ionization level (^{−22} (W)(m^{−2})(Hz^{−1})) [

Since the coefficients

Subsequently, on the basis of calculated

Effective ionization second-order optimization.

The method adopted to estimate

In order to carry out a performance comparison in position domain between NeQuick G and Klobuchar ionospheric corrections, the accuracy analysis of estimated coordinates obtained from a positioning using a single frequency receiver is performed. To evaluate the efficacy of the ionospheric models also the position without ionospheric correction (called “no-iono”) is considered. The static single point positioning approach has been conducted in postprocessing, due to the NeQuick coefficients unavailability; the analyzed data are related to the period from 9 to 11 May 2012, using three stations located in different geomagnetic latitudes. The period considered had a low geomagnetic activity; in fact the Average planetary (Ap) index—a daily planetary-scale measure of magnetic activity—achieves low values equal to 26, 10, and 10 for 9, 10, and 11 May 2012, respectively. Table

Test stations coordinates.

ID | City | Location | Latitude (deg, min, sec) | Longitude (deg, min, sec) | Height (m) | MODIP (°) |
---|---|---|---|---|---|---|

NAIN | Nain | Canada | 56° 32′ 13.07′′ N | 061° 41′19.38′′ W | 33.480 | 60.855 |

PANG | Naples | Italy | 40° 49′ 24.374′′ N | 014° 12′ 58.111′′ E | 122.659 | 48.579 |

AREQ | Arequipa | Peru | 16° 27′ 55.861′′ S | 71° 29′ 34.068′′ W | 2488.922 | −5.717 |

The utilized data for NAIN and AREQ stations are products available from IGS data centers [

The used software belongs to a Toolbox developed by PANG (PArthenope Navigation Group—

The work flow used for position processing is shown in Figure

Algorithm scheme.

Finally in order to analyze the impact of the ionospheric corrections estimated by the different models in position domain, the main statistical parameters (RMS, mean error, and maximum error) are calculated for both horizontal and vertical components.

In this section the positioning results for 3 days data sets, recorded with GNSS receivers, are summarized. Figures

Horizontal position error (AREQ station).

Horizontal position error (PANG station).

Horizontal position error (NAIN station).

The north and east errors of the AREQ station have a similar value, while the error position along the North-South direction of the PANG and NAIN stations is higher than the error on the East-West direction, due to the satellite geometry, confirmed by the analysis shown in Figure

North and east DOP of all three stations.

In Figures

Epoch-by-epoch Horizontal and Altitude Errors (AREQ Station).

Epoch-by-epoch Horizontal and Altitude Errors (PANG Station).

Epoch-by-epoch Horizontal and Altitude Errors (NAIN Station).

For the AREQ station the three configurations analyzed are characterized by similar performance in the horizontal component with a slight improvement for the NeQuick G model; the blue line is lower than the others, although the NeQuick G model has a higher maximum error. For the vertical component the effect of the correction provided by the two models is more evident, the red and blue lines are lower than the black one, and the differences between the models are of centimeter order.

For the PANG station the NeQuick G model guarantees improvements with respect to the Klobuchar model in terms of RMS in the horizontal component (passing from 2 to 1.5 meters) and maximum errors for the vertical (from 11.5 to 10 meters) and horizontal (6.7 to 5.9 meters) components. From Figure

For the NAIN station the NeQuick G and Klobuchar models have similar performance, the differences between the models are of submetric order, the use of the ionospheric models reduce the RMS error for both horizontal and vertical component with respect to the no-iono configuration.

From Figures

Finally, to summarize the performance of each ionospheric model adopted, in Figure

Summary results.

Ionospheric model | Station | Up RMS (m) | Hor RMS (m) | Up mean Error (m) | Hor mean Error (m) | Up max Error (m) | Hor max Error (m) |
---|---|---|---|---|---|---|---|

Without iono correction | NAIN | 4.317 | 1.934 | 3.812 | 1.438 | 11.234 | 9.417 |

PANG | 5.723 | 2.461 | 4.998 | 1.978 | 16.397 | 8.620 | |

AREQ | 8.3178 | 1.905 | 7.313 | 1.604 | 17.222 | 4.195 | |

Overall | 6.348 | 2.107 | 5.379 | 1.055 | 17.222 | 9.417 | |

| |||||||

Klobuchar | NAIN | 1.958 | 1.548 | 1.559 | 0.408 | 10.597 | 7.861 |

PANG | 2.436 | 2.053 | 1.900 | 1.634 | 11.542 | 6.731 | |

AREQ | 2.176 | 1.763 | 1.720 | 0.723 | 7.982 | 3.985 | |

Overall | 2.193 | 1.793 | 1.722 | 0.764 | 11.542 | 7.861 | |

| |||||||

NeQuick G | NAIN | 2.057 | 1.510 | 1.597 | 0.774 | 11.234 | 8.305 |

PANG | 2.834 | 1.477 | 2.358 | 0.271 | 10.081 | 5.926 | |

AREQ | 2.065 | 1.974 | 1.668 | 0.694 | 8.869 | 5.035 | |

Overall | 2.335 | 1.672 | 1.863 | 0.325 | 11.234 | 8.305 |

Overall statistical results (all stations).

Analyzing the final statistical results, showed in Table

there is a benefit in positioning if a ionospheric model is used, primarily for the vertical component and for equatorial stations;

NeQuick G correction guarantees for medium latitude, high latitude and overall stations a more accurate (horizontal RMS) positioning in the horizontal component;

for the up component NeQuick G correction obtains a better RMS performance only for equatorial station and the best result for overall maximum error.

With NeQuick G a vertical error slightly larger than Klobuchar case is observed for the PANG and NAIN stations, located at middle/high geomagnetic latitudes, while for AREQ station, placed near the geomagnetic equator, a slight vertical improvement relative to Klobuchar is noted. This result shows a worse performance of NeQuick model far away from geomagnetic equator but to definitely demonstrate this concept, analysis of more data relative to a much larger amount of stations placed at different latitudes is required.

The position of a receiver using only E1 Galileo measurements has been carried out on 12th of March 2013; hence the performance analysis of the aforementioned models was performed in the position domain using only Galileo observables. The data were stored at the Joint Research Centre (JRC) station placed in Ispra (Italy), whose coordinates are shown in Table

JRC station coordinates.

ID | City | Location | Latitude |
Longitude |
Height (m) | MODIP (°) |
---|---|---|---|---|---|---|

JRC | Ispra | Italy | 45° 48′ 37.30′′ N | 8°37′47.80′′ W | 279.00 | 52.072 |

Around 10 AM Central European Time (CET) for about two hours, all four Galileo satellites have been visible from Ispra and the position solution has been obtained with 10 meters of accuracy.

In Figure

Horizontal position error (JRC station).

The horizontal position is characterized by a bias toward North caused by not good geometry, due to the only four Galileo satellites available, and by larger values of NDOP with respect to EDOP (as in the GPS cases shown in Figures

The three models are characterized by similar performance summarized in Table

JRC Summary results.

Ionospheric model | Station | Up RMS (m) | Hor RMS (m) | Up mean Error (m) | Hor mean Error (m) | Up max Error (m) | Hor max Error (m) |
---|---|---|---|---|---|---|---|

No-Iono | JRC | 12.31 | 16.58 | 9.14 | 13.19 | 48.05 | 55.65 |

Klobuchar | JRC | 17.24 | 13.24 | 6.87 | 14.26 | 54.12 | 56.30 |

NeQuick G | JRC | 18.33 | 13.31 | 6.89 | 15.70 | 54.37 | 57.37 |

In Figure

Horizontal and altitude errors (JRC station).

In the position domain, using only the Galileo satellites, NeQuick G and Klobuchar models are characterized by a similar performance for all the considered parameters; both models guarantee a slight improvement in the horizontal component with respect to the no-iono configuration, while for the vertical component the models do not enhanced the positioning [

In this study the positioning performance of a single frequency GPS receiver using two ionospheric models, Klobuchar, and NeQuick G, and without ionospheric corrections are statistically analyzed.

In absence of the coefficients to forecast the effective ionization level

From the results obtained it emerges that NeQuick G and Klobuchar models get comparable performance in position domain. Specifically, although positioning through NeQuick G model achieves up RMS and mean error values slightly greater than Klobuchar ones, in general it offers a better behaviour than Klobuchar model in the horizontal positioning.

The position solution with 10 meters of accuracy is obtained using only Galileo pseudorange measurements as expected; in this case the model analysis is only a preliminary study due to the low number of samples.

The future studies of the authors will be focused on the use of ionization level

The authors wish to acknowledge the European Space Agency/ESA-ESTEC. The software used for part of the work presented in this paper have been provided by the European Space Agency. The views presented in the paper represent solely the opinion of the authors and should be considered as research results not strictly related to Galileo or EGNOS Project design. The authors wish to acknowledge Sandro M. Radicella Head Aeronomy and Radiopropagation ICTP Laboratory (Trieste, Italy) who gave a useful guide. The authors wish to acknowledge the IPSC institute of the JRC for providing Galileo data.