2.1 Single-frequency code-based positioning

For a Galileo satellite ($k$), the equation of code pseudo-range observation on the first frequency (E1) can be rewritten using Equation (2.1) as

(2.3)\begin{equation}P_{1,r}^{E,k} = \rho _r^{E,k} + c({\widetilde {dt}_r^E - {{\widetilde {dT}}^{E,k}}} )+ T_r^{E,k} + I_1^{E,k} + \varepsilon ({P_{1,r}^{E,k}} )\end{equation}

where $\widetilde {dt}_r^E$ and ${\widetilde {dT}^{E,k}}$ demonstrate the reformed receiver and satellite clock offsets which are

(2.4)\begin{equation}\widetilde {dt}_r^E = dt_r^E + b_{1,r}^E\quad \textrm{and}\quad {\widetilde {dT}^{E,k}} = d{T^{E,k}} + b_1^{E,k}\end{equation}

Equation (2.4) reveals that the reformed receiver clock offset contains both the actual receiver clock offset and receiver code hardware bias. Typically, they are estimated together as the reformed receiver clock offset in the estimation process due to their high correlation. In addition, an external clock source, such as precise ephemeris or broadcast ephemeris, is employed to correct the reformed satellite clock offset that contains satellite code hardware bias as well as the actual satellite clock offset. However, a particular signal or signal combination is utilised in the generation of satellite clock corrections provided in broadcast ephemeris and IGS precise ephemeris. For Galileo satellites, the reference signal combination is the dual-frequency ionosphere-free (IF) combination of code observations on E1 and E5a frequencies, while the similar IF combination on L1 and L2 frequencies is utilised to generate the satellite corrections for GPS satellites (Steigenberger et al., Reference Steigenberger, Hugentobler, Loyer, Perosanz, Prange, Dach, Uhlemann, Gendt and Montenbruck2015). Therefore, it is not possible to employ satellite clock corrections that embrace additional code hardware biases directly for single-frequency positioning. Accordingly, the satellite clock correction that is produced based on the dual-frequency IF combination on two distinct frequencies ($i = 1,\; 2$) is given as

(2.5)\begin{equation}dT_{IF}^{E,k} = d{T^{E,k}} + \frac{{f_1^2}}{{f_1^2 - f_2^2}}b_1^{E,k} - \frac{{f_2^2}}{{f_1^2 - f_2^2}}b_2^{E,k}\end{equation}

Using Equation (2.4) and Equation (2.5), satellite clock correction can be rearranged for single-frequency positioning as

(2.6)\begin{equation}{\widetilde {dT}^{E,k}} = dT_{IF}^{E,k} - \frac{{f_2^2}}{{f_1^2 - f_2^2}}({b_1^{E,k} - b_2^{E,k}} )= dT_{IF}^{E,k} - {T_{GD}}\end{equation}

where ${T_{GD}}$ indicates the timing (or total) group delay and it is also represented as proportional to $({b_1^{E,j} - b_2^{E,j}} )$, which indicates the differential code biases (DCBs) between the related frequencies (Bahadur and Nohutcu, Reference Bahadur and Nohutcu2021b). In the single-frequency positioning, the timing group delay is must be corrected for aligning the satellite clock correction. For this purpose, the broadcast ephemeris includes the timing-group delay corrections for single-frequency users. Furthermore, some IGS analysis centres generate and disseminate the DCB parameters routinely and they can be employed to align the satellite clock corrections with corresponding clock reference. After correcting the reformed satellite clock offset considering the timing-group delay, for single-frequency code-based positioning, Equation (2.3) can be rewritten as

(2.7)\begin{equation}P_{1,r}^{E,k} = \rho _r^{E,k} + c\widetilde {dt}_r^E + T_r^{E,k} + I_1^{E,k} + \varepsilon ({P_{1,r}^{E,k}} )\end{equation}

For the single-frequency code-based positioning, Equation (2.7) represents the functional model containing four parameters to be estimated, for example three position components and one receiver clock offset. On the other hand, the tropospheric delay is generally split into two parts, namely the hydrostatic (dry) and nonhydrostatic (wet) components. While the dry component can be corrected using the empirical models depending on the station position and atmospheric parameters, modelling the wet component is quite troublesome because of the rapid variations in water vapor content (Davis et al., Reference Davis, Herring, Shapiro, Rogers and Elgered1985). Therefore, the typical procedure is to estimate the wet tropospheric component as an additional unknown parameter. Also, it is necessary to correct the ionospheric delay in the single-frequency code-based positioning model since the IF combinations are not applicable. Finally, Equation (2.7) is given for the single-system single-frequency positioning. If the multiple constellations are integrated, it is required to introduce the inter-system bias (ISB) parameters for each additional system with respect to a selected reference constellation (Cai and Gao, Reference Cai and Gao2013; Li et al., Reference Li, Ge, Dai, Ren, Fritsche, Wickert and Schuh2015).

2.2 Single-frequency code-phase combination

Once phase observations are available together with code pseudo-ranges, a single-frequency code-phase combination, namely the single-frequency IF combination ($\mathrm{\Phi }$), can be constructed for the first frequency of Galileo as (Yunck, Reference Yunck1993)

(2.8)\begin{equation}\mathrm{\Phi }_{1,r}^{E,k} = 0.5\; ({P_{1,r}^{E,k} + L_{1,r}^{E,k}} )\end{equation}

From Equation (2.1) and Equation (2.2), the observation equation of single-frequency code-phase combination is written using the reformed receiver and satellite clock offset as

(2.9)\begin{equation}\mathrm{\Phi }_{1,r}^{E,k} = \rho _r^{E,k} + c({\widetilde {dt}_r^E - {{\widetilde {dT}}^{E,k}}} )+ T_r^{E,k} + \tilde{N}_1^{E,k} + \varepsilon ({\mathrm{\Phi }_{1,r}^{E,k}} )\end{equation}

where $\tilde{N}_1^{E,j}$ denotes the reformed ambiguity parameter which contains the integer ambiguity parameter together with code and phase hardware biases, and it is expressed by

(2.10)\begin{equation}\tilde{N}_1^{E,k} = 0.5\; \lambda _1^EN_1^{E,k} + 0.5\; c({B_{1,r}^E - B_1^{E,k}} )- 0.5\; c({b_{1,r}^E - b_1^{E,k}} )\end{equation}

Here, it should be mentioned that the satellite-dependent part of code and phase hardware biases, such as temporally stable part, is absorbed by the reformed ambiguity parameter, while the satellite-independent part, which is temporally variable, is absorbed by the receiver clock offset in this combination (Sterle et al., Reference Sterle, Stopar and Pavlovčič Prešeren2015; Pan et al., Reference Pan, Zhang, Liu, Li and Li2017b; Ge et al., Reference Ge, Zhou, Dai, Qin, Wang and Yang2019). After the correction of the reformed satellite clock offset with the timing group delay, the single-frequency code-phase combination is given by

(2.11)\begin{equation}\mathrm{\Phi }_{1,r}^{E,k} = \rho _r^{E,k} + c\widetilde {dt}_r^E + T_r^{E,k} + \tilde{N}_1^{E,k} + \varepsilon ({\mathrm{\Phi }_{1,r}^{E,k}} )\end{equation}

For the single-frequency code-phase combination, Equation (2.11) constitutes the functional model, which includes a float ambiguity parameter for each observed satellite as the additional estimated parameter to the single-frequency code-based positioning model. The single-frequency code-phase combination is presumed to be ionosphere-free because the ionosphere delays in phase and code observations have equal magnitudes in reverse directions. Furthermore, the noise level of code pseudo-range observations is considerably reduced with the single-frequency code-phase combination thanks to phase observations which have a lower noise level. Still, in the combination, phase ambiguities are no longer integer numbers because of the accumulation of hardware biases in the phase ambiguity parameters. Hence, the combination requires a relatively long initial period for converging the float phase ambiguity parameters, and therefore achieving the desired positioning accuracy.

3.1 Data description and satellite visibility

The daily observation dataset of 15 IGS stations over the two-week period of 3–16 January 2021 were employed in the experimental tests. The stations were randomly selected to represent the globe evenly, as much as possible. Figure 1 illustrates the geographical locations of the selected stations. All the stations are among the IGS MGEX network and are equipped with GNSS receivers that can record multi-GNSS observations, including Galileo and GPS satellites. In addition, the observation sampling interval of the observation dataset is 30 s. On the other hand, the real-time corrections disseminated by the Centre National d'Etudes Spatiales (CNES) through the IGS–RTS stream, namely the ‘SSRA00CNE0’ message, was used for the single-frequency positioning. CNES product contains real-time corrections for GPS and Galileo satellites as well as GLONASS and BDS.

Figure 1. Geographical locations of the selected IGS stations

Satellite visibility is one of the critical factors influencing the real-time single-frequency positioning performance. The visibility of Galileo satellites at the selected stations is investigated in comparison with the GPS constellation. Figure 2 depicts the station-based maximum, average and minimum visible satellite numbers per epoch for the GPS and Galileo constellation over the observation period. Considering each epoch over the two-week observation period, that is 40,320 individual epochs; minimum and maximum numbers indicate the fewest and greatest satellite numbers that appear per epoch during this period, while the average number is calculated as the mean of satellite numbers in all epochs. As can be observed from the figure, the station-based minimum satellite numbers range from 3 to 7 for the GPS constellation, whereas the minimum numbers of Galileo satellites are between 2 and 6. The maximum satellite numbers at the selected stations range from 11 to 14 for the GPS constellation, while the corresponding numbers alter between 8 and 11 for Galileo satellites. Also, the average numbers of visible GPS satellites are computed between 7⋅60 and 9⋅87. The station-based average numbers change between 4⋅47 and 7⋅47 for the Galileo constellation. Considering all stations, the average numbers of visible GPS and Galileo satellites are calculated as 8⋅93 and 6⋅61. Additionally, when the results of all stations are analysed, it can be said that the GPS constellation provides at least four visible satellites at nearly all epochs, excluding a few specific epochs. As regards Galileo, there similarly exist at least four visible satellites at almost all epochs for the selected stations, except for the ISTA and LHAZ stations. In these two stations, the percentages of epochs having at least four visible satellites within the whole epochs are considerably lower than those of the other stations. Especially, at the LHAZ station, the 16% of total epochs over the two-week observation period have three or fewer visible Galileo satellites. In addition, for the Galileo constellation, the average satellite numbers are calculated as 5⋅36 and 4⋅47 at the ISTA and LHAZ stations, which are the lowest averages. The result is not surprising because these two stations are located between the northern latitudes of 25 and 45 degrees, which is one of the weakest regions in terms of Galileo satellite visibility, depending on the longitude (Zhang et al., Reference Zhang, Tu, Liu, Hong, Fan, Zhang and Lu2019). In addition, the stations near the equatorial and polar regions, the NKLG, OHI3 and REYK stations, have average satellite numbers higher than 7 for the Galileo constellation. Consequently, although its satellite visibility changes depending on the station's location, the Galileo constellation has a considerable number of usable satellites at most stations utilised in the experiment.

Figure 2. Station-based maximum, average and minimum visible satellite numbers for the GPS and Galileo constellation during the observation period

3.2 Processing method

As a part of this study, an enhanced version of PPPH (Bahadur and Nohutcu, Reference Bahadur and Nohutcu2018), which enables real-time positioning processes, was employed to perform the Galileo real-time single-frequency positioning solutions. To compare the Galileo positioning results and to understand the contribution of the Galileo constellation, the observation dataset was also processed under two additional processing scenarios, which are GPS-only and GPS/Galileo combination. Since single-frequency receivers mostly require kinematic or dynamic positioning conditions, the kinematic mode, with a spectral density of ${10^2}\; {m^2}{s^{ - 1}},\;$was adopted in the processing of all positioning solutions. To access the real-time satellite products and observations, a Bundesamt für Kartographie und Geodasie (BKG) NTRIP client program (BNC) was utilised in this study (BKG, 2021). Table 1 presents the detailed processing strategies applied for the real-time single-frequency positioning processes.

Table 1. Processing details adopted in PPPH for real-time single-frequency positioning

3.3 Performance analysis of single-frequency code-based positioning

The results were analysed as regards positioning accuracy for the real-time single-frequency code-based positioning with Galileo satellites. For this purpose, the positioning error specified as the coordinate difference between the related positioning solution and ground truth was utilised to evaluate the positioning performance. Typically, IGS provides precise coordinates of its stations routinely and IGS weekly solutions were employed as the reference source for the precise station coordinates in this study. Accordingly, the positioning errors were computed in the local coordinate system for each separate epoch in which the positioning process was performed. As previously mentioned, to compare the Galileo positioning results, a similar procedure was employed for the GPS and GPS/Galileo positioning solutions. Figure 3 presents the horizontal (north and east directions) and vertical (up direction) positioning errors at the FFMJ station on 3 January 2021 for the GPS, Galileo, and GPS/Galileo single-frequency code-based positioning solutions. As shown in the figure, the GPS solution mostly has higher horizontal and vertical positioning errors in comparison with the Galileo solution. The positioning performance of single-system solutions is augmented with the GPS/Galileo integration, as expected. For the daily GPS, Galileo and GPS/Galileo solutions, root mean square (RMS) errors are calculated as 0⋅548, 0⋅399 and 0⋅375 m for the horizontal component. Similarly, for the vertical component, the RMS errors computed for the GPS, Galileo and GPS/Galileo solutions are 0⋅914, 0⋅881 and 0⋅783 m, respectively. The results denote that, compared with the GPS solution, the Galileo solution acquires better positioning accuracy.

Figure 3. Horizontal and vertical positioning errors for the GPS, Galileo and GPS/Galileo single-frequency code-based solutions at FFMJ station on 3 January 2021

The observation residuals computed after the estimation processes can also be used for the evaluation of positioning performance as they represent the harmony between the defined system model and measurements. Moreover, the observation residuals reveal the noise characteristics of observations, including orbit errors, multipath effect and unmodelled errors. Figure 4 illustrates the observation residuals computed for the GPS and Galileo single-frequency code-based positioning solutions at the FFMJ station on 3 January 2021. The figure also provides the RMS values calculated when taking all daily observation residuals into consideration. The RMS values of residuals are computed as 0⋅319 and 0⋅198 m for the GPS and Galileo observations. The observation residuals of Galileo are approximately 38% fewer than those of GPS, on average, which indicates that Galileo code observations have a considerably lower noise level in comparison to GPS observations.

Figure 4. Observation residuals computed for the GPS and Galileo single-frequency code-based solutions at FFMJ station on 3 January 2021

On the other hand, Figure 5 provides the distributions of positioning errors computed in the north, east and up directions for the GPS, Galileo and GPS/Galileo solutions. In the generation of distributions, epoch-wise positioning solutions of 15 stations during the two-week observation period were considered. In the figure, the error distributions are presented as the probability percentages which are based on the ratio of error frequencies to the total number of epochs, instead of actual error frequencies. Therefore, the positioning solutions at the whole epochs, 604,800 epochs in total (2880 epochs in a day over two weeks for 15 different stations) were used when generating the error distributions. Furthermore, the figure demonstrates the mean and RMS errors computed in each direction, respectively. As shown in the figure, the RMS errors are 0⋅707, 0⋅622 and 1⋅876 m in the north, east and up directions for the GPS solution, while the corresponding errors are computed as 0⋅515, 0⋅429 and 1⋅401 m for the Galileo solution. The results denote that the Galileo solution provides a better positioning performance by 27⋅1%, 31⋅0% and 25⋅3% in the north, east and up directions, respectively, in comparison with the GPS solution. The GPS/Galileo solution has the related RMS values of 0⋅477, 0⋅377 and 1⋅289 m, respectively, which demonstrates that it considerably augments the positioning accuracy of the single-system solutions in all directions.

Figure 5. Error distributions for the GPS, Galileo and GPS/Galileo single-frequency code-based solutions

As regards the GPS, Galileo and GPS/Galileo solutions, Table 2 demonstrates the station-based RMS values computed for horizontal, vertical and three-dimensional (3D) positioning errors considering the whole positioning processes over the two-week period. The table also demonstrate the average RMS values of horizontal, vertical and 3D positioning errors computed, taking all stations into consideration. Except for the AREG, NKLG and PTGG stations, the Galileo solution presents considerably better positioning accuracy in all positioning components than that of the GPS solution for all stations. The Galileo solution augments the positioning accuracy of the GPS solution by 25⋅8% on average when the 3D RMS errors are considered. Also, its improvement percentages for the horizontal and vertical components are 27⋅9% and 25⋅3%, respectively. The 3D improvements of the Galileo solution reach 63⋅5% and 69⋅6% for the OHI3 and REYK stations located near the polar regions where the visibility of Galileo is quite strong. When it comes to the GPS/Galileo solution, it has the best positioning performance in all the stations excluding the AREG station. The GPS/Galileo integration presents a better 3D positioning accuracy than the GPS and Galileo solutions by the ratios of 31⋅6% and 7⋅9%.

Table 2. Station-based RMS values of horizontal, vertical and 3D positioning errors for the GPS, Galileo and GPS/Galileo single-frequency code-based solutions (in:m)

3.4 Performance analysis of single-frequency code-phase combination

The observation dataset was also processed for the single-frequency code-phase combination with the GPS, Galileo and GPS/Galileo solutions. To converge the non-integer phase ambiguity parameters, an initial period is necessary in this combination, called the ‘convergence time.’ Therefore, the convergence time was also utilised in the assessment of positioning performance of the single-frequency code-phase combination together with the positioning accuracy. Figure 6 depicts the horizontal and vertical positioning errors at the FFMJ station on 3 January 2021 for the GPS, Galileo and GPS/Galileo single-frequency code-phase combination solutions. When the figure is analysed, it is apparent that the horizontal positioning accuracy of the GPS solution is mostly lower than that of the Galileo solution. The daily horizontal RMS errors are computed as 0⋅266, 0⋅184 and 0⋅173 m for the GPS, Galileo and GPS/Galileo solutions, respectively. In addition, the daily vertical RMS errors of the GPS, Galileo and GPS/Galileo solutions are 0⋅322, 0⋅285 and 0⋅206 m, respectively. The results indicate that the Galileo solution presents slightly better positioning accuracy than does the GPS solution horizontally and vertically, while the best positioning performance comes from the GPS/Galileo solution.

Figure 6. Horizontal and vertical positioning errors for the GPS, Galileo and GPS/Galileo single-frequency code-phase combination solutions at FFMJ station on 3 January 2021

Similarly, for the GPS and Galileo single-frequency code-phase combination solutions, Figure 7 illustrates the observation residuals computed at the FFMJ station on 3 January 2021. When compared with the single-frequency code-based positioning, the observation residuals are substantially smaller for the single-frequency code-phase combination, owing to the presence of phase observations. The RMS values computed from all daily observation residuals are 0⋅082 and 0⋅064 m for the GPS and Galileo solutions, which means that the observation residuals of Galileo are averagely 21⋅9% lower than those of GPS. The results confirm that the noise level of GPS single-frequency observations is substantially higher than Galileo observations.

Figure 7. Observation residuals computed for the GPS and Galileo single-frequency code-phase combination solutions at FFMJ station on 3 January 2021

On the other hand, Figure 8 provides the error distributions in local directions for the GPS, Galileo and GPS/Galileo single-frequency code-phase combination solutions. Here, it should be clarified that the positioning errors at the first-hour period of each positioning solution were excluded from the probability distributions to avoid the influence of unconverged phase ambiguity parameters. The RMS errors are 0⋅192, 0⋅262 and 0⋅403 m in the north, east and up directions for the GPS solution, respectively, while these errors are computed as 0⋅179, 0⋅260 and 0⋅347 m for the Galileo solution. The results reveal that the performance of Galileo solution is better than that of the GPS solution by 6⋅7%, 1⋅0% and 13⋅9% in the north, east and up directions, respectively. Likewise, the GPS/Galileo combination provides the best positioning performance with the RMS errors of 0⋅111, 0⋅141 and 0⋅246 m. Furthermore, the convergence time was assessed as an additional indicator of the performance of single-frequency code-phase combination. The time or epoch where the 3D positioning error falls under 1 m and does not pass over the 1 m threshold for the following 10 min was utilised as the convergence time criteria in this study (Bahadur and Nohutcu, Reference Bahadur and Nohutcu2021a). According to this definition, the average convergence times computed from all positioning results over the observation period are 66⋅68, 50⋅43 and 33⋅24 min for the GPS, Galileo and GPS/Galileo solutions, respectively. The results demonstrate that the Galileo solution has a 24⋅4% shorter convergence time on average in comparison with the GPS solution, while the GPS/Galileo solution presents the best convergence performance, thanks to the increasing number of visible satellites.

Figure 8. Error distributions for the GPS, Galileo and GPS/Galileo single-frequency code-phase combination solutions

Finally, Table 3 shows the station-based RMS values of horizontal, vertical and 3D positioning errors, considering all positioning processes during the two-week period for the GPS, Galileo and GPS/Galileo solutions. The Galileo solution, compared with the GPS solution, has a better positioning performance in all stations, except for the ABMF, DJIG, ISTA and LHAZ stations. Especially, the performance of the Galileo solution is considerably worse at the ISTA and LHAZ stations, which have the lowest average numbers for visible Galileo satellites over the two-week observation period. As the single-frequency code-phase combination is a rank-deficient model, which contains more unknown parameters than the number of measurements, the satellite number has a considerable influence on the positioning performance of this model. Therefore, it can be said that the poor positioning performance at these two stations mainly stems from the low visibility of Galileo satellites. Considering the 3D RMS errors of all stations for the single-frequency code-phase combination, the performance of the Galileo solution is better than that of the GPS solution by 9⋅4%. The positioning accuracy is enhanced with the integration of GPS and Galileo constellation significantly. The GPS/Galileo solution improves the 3D positioning accuracies obtained from the GPS and Galileo solutions by 41⋅2% and 35⋅1%, respectively.

Table 3. Station-based RMS values of horizontal, vertical and 3D positioning errors for the GPS, Galileo and GPS/Galileo single-frequency code-phase combination solutions (in:m)