This paper proposes a new mathematical method of
ionospheric delay estimation in single point positioning (SPP) using a
single-frequency receiver. The proposed approach focuses on the

Single point positioning (SPP) allows of the indication of an autonomous position of a receiver using code data from different Global Navigation Satellite Systems (GNSS). Code ranges are not ambiguous and do not require applying the precise method of ambiguity initialization (Bakuła, 2020; Cellmer et al., 2018; Kwaśniak et al., 2016; Nowel, et al., 2018). The principal problem of SPP stems from different types of errors degrading the Global Positioning System (GPS) signal between a rover and a specified satellite in a given epoch. Ionospheric delay contributes to the general GPS error budget by its volatility in the range of 40–60 m during daytime and 6–12 m at night (US Army Corps of Engineers, 2003).

The ionosphere consists of charged particles that appear because of the
ionization process (El-Rabbany, 2002; Awange, 2012). Problems with
ionosphere modelling come from difficulties between solar activity and the
geomagnetic field interactions (Xu and Xu, 2016). The basic concepts of the
GPS signals delay were briefly considered by Golubkov et al. (2018a, b) and Kuverova et
al. (2019).
To specify a suitable magnitude of delayed GPS signal along an appropriate
path between receiver and satellite, a proportional quantity such as total
electron content (TEC) has to be involved and defined as the linear integral
of the density of the particles alongside the ray path (Cooper et al.,
2019). The TEC unit is equal to 10

The authors propose the autonomous SPP approach with

On the contrary, empirical models do not significantly reduce the ionosphere influence in the GPS positioning as mathematical (deterministic) methods but can make real-time improvements by using the external data, e.g. coefficients transmitted in the navigation message to correct the signal pseudo-ranges. One of these is the Klobuchar algorithm (see Klobuchar, 1987), which compensates for 50 %–60 % of the ionospheric range error, utilizing a single-layer model of the ionosphere (Leick et al., 2015). In the current study, the authors wanted to treat the SPP method with the Klobuchar algorithm as a reference method because of its popularity and utility in GPS measurement. A significant improvement can be noted in the vertical component which is the most affected by the atmospheric delay. Setti et al. (2019) investigated the analysis of the Klobuchar model in the ionospheric delay reduction procedure utilizing code observation in point positioning. The algorithm works clearly when ionosphere activity is significant and improves vertical solutions by 67 %. For the horizontal components, the improvement using the Klobuchar algorithm is up to 9 % regarding the non-ionospheric model. It should be noted that GPS point positioning using the Klobuchar algorithm can degrade the position because of the constant value of the ionospheric delay (up to a 5 ns set) during nighttime.

High-quality representation of the ionosphere influence on positioning can
be obtained by the global ionospheric models (GIMs), used mostly in the
post-processing purposes as explained in Ciećko and Grunwald (2020). It
is worth noting that Abdelazeem et al. (2016) developed the regional
ionospheric model over the European area and implemented it in precise point
positioning (PPP), operating in real time using the real-time service (RTS)
products of the IGS. The results present
an improvement in the accuracy on the level of 40 % (under the
midlatitude region) in the 3-D position relating to the IGS-GIM. The
accuracy is higher primarily because of the better temporal and spatial
resolution of the model (15 min and 1

In sum, the motivation of this paper is to analyse a new mathematical method of ionospheric delay estimation to improve the SPP. The authors put forward the hypothesis to be independent of external data use in the meaning of the new method in the ionospheric delay calculation procedure.

In this section, the grounds of the commonly used SPP mathematical models using the Klobuchar algorithm and IGS TEC map will be introduced, along with the proposition of a new strategy of SPP determination by use of simple and autonomous method to estimate the ionospheric delay. This is followed by the appropriate algorithm presentations with suitable explanations. In addition, the accuracy analysis criteria will be described in view of the models' credibility procedure.

In this study, the Klobuchar model was adapted as a reference in the SPP
accuracy tests. Eight model coefficients transmitted via navigation message
are the primary components involved in the algorithm to reduce the
ionosphere effect in the SPP. The geodetic coordinates of the GPS antenna,
GPS observing time (in seconds) as well as azimuth and elevation of observed
satellites as viewed from the receiver are needed to be known. The formula
to calculate the vertical ionospheric delay based on the Klobuchar algorithm is
as follows (Hofmann-Wellenhof et al., 2008):

To obtain an ionospheric delay alongside the GPS signal travel path, the
mapping function should be employed. Thus, the concept of the ionospheric
point has to be expanded as a piercing point of the GPS wave path and the
ionospheric single layer on the specified altitude. Thus, the satellite
zenith angle at the piercing point

It should be also noted that the type of mapping function in the atmospheric effect calculation process contributes to the final solution accuracy as well. Allain and Mitchell (2009) examined the tomographic mapping function known as the Multi-Instrument Data Analysis System (MIDAS) with ionospheric effect determination for the single-frequency data. Research has shown that daily positioning errors are up to 50 % lower in comparison to positioning using the Klobuchar algorithm or IRI when the surrounding distribution of receivers are favourable. Regardless of the map type, dual-frequency observations allow for even greater precision of the ionospheric effect mitigation in the GPS pseudo-range measurement.

Therefore, the mapping function can be used as an inverse of the cosine
function (Hofmann-Wellenhof et al., 2008):

The second approach is SPP with ionospheric corrections computed based on
the IGS TEC map. This method is used to examine and verify the quality of
the new autonomous estimation method of the ionospheric effect in the SPP.
Consequently, ionospheric delay as the base formula in the zenith direction
can be introduced (Schüler, 2001):

Taking into account ionospheric delay as a proportional value to TEC and
proportional to the distance covered across the band, the relation of VTEC
and TEC can be defined (Leick et al., 2015):

Using Eqs. (5), (6) and (7), the ionospheric correction can be obtained in
the ray path direction between the rover and satellite:

The essence of the proposed modified SPP method lies in an estimation of
the

The last row is a pseudo-observation equation in
which VTEC

After many computational tests, it was assumed that the initial value of
VTEC

It is assumed in this method that the “observed” and approximate values
are equal:

The vector of unknowns receives an additional parameter in the adjustment
process:

The weight matrix has been prepared based on pseudo-range measurement error
which was assumed as 2.00 m and an appropriate satellite elevation angle. The
criterion of the minimal mask was implemented as 10

The basic statistical operator in the experiment is a distance of the
solution from the true position “

The covariance matrix of mean values computed from the whole observational
day is

Set of the results of the positioning models:

In this section, the explanation of the research concept will be done. Next, the appropriate numerical experiment in view of graphics and numeric settings will be presented. The parallel discussion about obtained results for appropriate interpretation will be made.

Set of the results of the positioning models:

The numerical experiment is based on real single-frequency code pseudo-range observations from GPS, namely C1C code data on the L1 carrier frequency (1575.42 MHz). Continuing, three different EUREF (Regional Reference Frame Sub-Commission for Europe) Permanent GNSS Network stations have been chosen in Scandinavia: two stations in Sweden – Visby (VIS) and Skellefteå (SKE) – and one in Norway – Vardø (VARS). The observational files and initial coordinates of receivers were gained from the BKG (Bundesamt für Kartographie und Geodäsie) GNSS Data Center. The parameters of precise satellite orbits (SP3 file), broadcast ephemeris data and atmospheric data were obtained by means of CDDIS (Crustal Dynamics Data Information System) – in fact, IONEX only in view of atmospheric data, as a source of IGS TEC map. The coordinates of points were treated as the true coordinates in the practical part of the experiment. The reference coordinates are presented in Table 1.

Actual coordinates of points.

In the models, the actual coordinates have been converted to the antenna phase centre to make a comparative analysis with the SPP results, where measurements were executed to the antenna phase centre.

Three different localizations allow checking how the modified SPP model works on different geodetic latitudes because of ionosphere activity changes, so its quality in the GPS code domain can be widely stated.

The research concept focuses on measurement on two different days in the cited locations. Therefore, three stations of the EUREF Permanent GNSS Network were employed for comparative analysis based on data from two parallel days.

Experiment concept.

To execute the numerical experiment of the research, the MATLAB environment from MathWorks was used. The “PostCalc” software developed by Dawid Kwaśniak was utilized as the base MATLAB programme. Next, the complementary changes were done by the authors of paper because of the numerical experiment requirement.

Figures 1–3 present the distribution of

Set of the results of the positioning models:

Following the experiment report, the next examined subject is SKE on
15 June 2019. Looking at Fig. 2a, the top part presents the

The last studied point is VARS00NOR. The first examined day is 15 June 2019.
The middle part of Fig. 3a demonstrates that the

Focusing on the mean errors of the final solution in the NEU system, we will
consider the average precision of the differences of the components

Average errors of the difference in the positions using the NEU system.

The error quantities of the difference in the positions were achieved for the
MSPPwithdVTEC and SPPwithITM on a close level. Separating the
horizontal and the vertical components of the position, the
MSPPwithdVTEC is characterized by comparable precision to
SPPwithKM in the north and east directions; therefore, the
additional estimated parameter in the code equation does not change the SPP
model enough to reduce its quality. The case is repeated in the context of
the vertical component

The main idea of this paper was to introduce the new method to estimate the ionospheric delay in the SPP without using the external data. Moreover, in the case of comparative analysis, two common approaches in SPP were employed: SPP with the Klobuchar algorithm and SPP with IGS TEC map. The first one was treated as a reference one. The SPP model with IGS TEC map was utilized to authenticate the proposed model in view of IGS TEC map use – defined as a high-quality product. The explanation of mathematical models and appropriate accuracy analysis criteria were done. Next, the numerical experiment using real code data from three different GNSS stations with discussion to interpret the obtained results was conducted. Referring to achieved solutions, the proposed approach can be defined as a simple and independent way to improve SPP. Moreover, the MSPPwithdVTEC can be employed in the procedure of determining the approximate position for the need of the single-epoch precise positioning.

Based on the mean distance of the solution from the true position, the MSPPwithdVTEC achieved improved GPS position in comparison to the basic SPPwithKM in each tested station. Moreover, the MSPPwithdVTEC acquires a similar level of error reduction to the SPPwithITM, which is the most satisfying in view of method authentication.

Finally, the results of the MSPPwithdVTEC confirm the potential use of the mathematical model in the SPP. The strategy should be developed in the future through the verification of model stability in the other stations since ionosphere changes are highly dependent on localization. Therefore, the proposed method of SPP can be recognized as a good forecast to become independent of external products delivering information about the ionospheric delay.

The data used in the numerical experiment are free and publicly accessible. The observation files were downloaded from the BKG GNSS Data Centre:

AF designed, prepared, and performed the research, processed the data, and wrote the manuscript. SC designed and controlled the experiment as well as results' analysis. KN constructed the research methodology and helped in the theoretical elaboration of the manuscript.

The authors declare that they have no conflict of interest.

This research has been supported by the Polish National Science Centre (grant no. 2018/31/B/ST10/00262).

This paper was edited by Lev Eppelbaum and reviewed by two anonymous referees.