Height Variation Depending on the Source of Antenna Phase Centre Corrections: LEIAR25.R3 Case Study

In this study, we compared two sets of antenna phase center corrections for groups of the same type of antenna mounted at the continuously operating global navigation satellite system (GNSS) reference stations. The first set involved type mean models provided by the International GNSS Service (release igs08), while the second set involved individual models developed by Geo++. Our goal was to check which set gave better results in the case of height estimation. The paper presents the differences between models and their impact on resulting height. Analyses showed that, in terms of the stability of the determined height, as well as its variability caused by increasing the facade mask, both models gave very similar results. Finally, we present a method for how to estimate the impact of differences in phase center corrections on height changes.


Introduction
Global navigation satellite systems are applicable in many areas of life and are probably the most popular method for positioning. Their continuous development introduced new methods and improved algorithms [1][2][3], allowing the determination of the accuracy of estimated position [4] and testing the possibilities of using three frequencies [1,5]. This development increased the accuracy of the positioning with shorter sessions. GNSSs are used for measurements wherever a stable and precise reference frame is required, such as landslides [6], and subsidence or secular tectonic movements [7]. They also have wide applications in Earth sciences, including in atmosphere research [8,9] or determining the position of the Earth's mass center [10]. All these works require the most accurate determination of parameters, which is particularly important for the integration of GNSS measurements with other remote sensors [11] or direct measurements [12]. To be able to provide millimeter accuracy, it is necessary to take into account the impact of the receiving system, especially the antenna.
The purpose of GNSS antennas is to convert the energy of the electromagnetic wave to electric current, which is then processed by the receiver. The currently constructed antennas are adapted to receive the wave at individual frequencies corresponding to the frequencies of individual GNSS systems. Recent systems enable reception waves on all available frequencies, e.g., L1/L2/L5 for GPS, G1/G2/G3 for GLONASS, E1/E5a/E5b/E5/E6 for Galileo, and B1/B3/B1C/B2a/B2b for BDS [13,14]. However, the most popular ones are those receiving the signal from two frequencies, and, because of their price and ensuring sufficient accuracy for many years, they will dominate the market. In theory, the best antenna is the one with isotropic distribution of radiation intensity; however, for technical reasons, it checked which of them gave the most stable results. We also made an attempt to show how differences in PCCs translate to differences in height.

Methodology
For our analysis, GPS observations collected at 24 selected stations belonging to EPN were used in this study. The analysis was performed on full-year 2015 data and focused on one type of antenna, the most popular of the ground antennas mounted at that time in EPN, the GNSS Choke Ring Antenna Type AR25 from Leica (IGS code: LEIAR25.R3) with Radome LEIT [31].
All GPS data were processed in the Gamit/Globk v. 10.60 software [32]. The processing strategy followed EPN Guidelines and corresponded to the contribution by the Military University of Technology to the EPN Repro2 project [26]. To correct the phase observations, two ground antenna phase center correction sets were used. The first one, igs08.atx, contains corrections for types of GNSS antennas recognized by IGS, where the values are means from several calibrations [22]. The second one, epnc.atx [23], contains corrections for individual antennas derived from individual calibrations made by Geo++ GmbH and collected by EPN Central Bureau (Belgium). The network alignment to the IGb08 frame was made using 14 additional stations ( Figure 1) using no-net-translation minimum constraint conditions. The processing strategy was the same for the reference stations and was manipulated for the remaining stations. For each calibration table, nine solutions with different elevation masks were prepared (Table 1).

Methodology
For our analysis, GPS observations collected at 24 selected stations belonging to EPN were used in this study. The analysis was performed on full-year 2015 data and focused on one type of antenna, the most popular of the ground antennas mounted at that time in EPN, the GNSS Choke Ring Antenna Type AR25 from Leica (IGS code: LEIAR25.R3) with Radome LEIT [31].
All GPS data were processed in the Gamit/Globk v.10.60 software [32]. The processing strategy followed EPN Guidelines and corresponded to the contribution by the Military University of Technology to the EPN Repro2 project [26]. To correct the phase observations, two ground antenna phase center correction sets were used. The first one, igs08.atx, contains corrections for types of GNSS antennas recognized by IGS, where the values are means from several calibrations [22]. The second one, epnc.atx [23], contains corrections for individual antennas derived from individual calibrations made by Geo++ GmbH and collected by EPN Central Bureau (Belgium). The network alignment to the IGb08 frame was made using 14 additional stations ( Figure 1) using no-net-translation minimum constraint conditions. The processing strategy was the same for the reference stations and was manipulated for the remaining stations. For each calibration table, nine solutions with different elevation masks were prepared (Table 1).   Before the investigation of height estimates, the PCC values for individual antennas were compared. The comparison procedure is explained in detail in References [19,33]. Firstly, the PCOs from epnc.atx were transferred to a common PCO with igs_08.atx according to Reference [19]. Then, independently for both GPS frequencies, PCVs were subtracted as EPN − IGS. Differences for L1 and L2 obtained in this way were used to calculate the final difference in PCCs (dPCCs) for "ionosphere-free" linear combination (LC). For selected stations, resulting dPCCs are presented in Figure 2. Before the investigation of height estimates, the PCC values for individual antennas were compared. The comparison procedure is explained in detail in References [19,33]. Firstly, the PCOs from epnc.atx were transferred to a common PCO with igs_08.atx according to Reference [19]. Then, independently for both GPS frequencies, PCVs were subtracted as EPN − IGS. Differences for L1 and L2 obtained in this way were used to calculate the final difference in PCCs (dPCCs) for "ionospherefree" linear combination (LC). For selected stations, resulting dPCCs are presented in Figure 2. As it was expected, the greatest differences between compared correction sets occurred at low elevations, where they were close to 20 mm ( Figure 2). This came from the original patterns (e.g., igs_08.atx), which adopted a , 90° 0 mm condition and, therefore, showed higher values at lowest elevations. Corrections at the lowest elevation also had the highest standard deviations [18]. The greatest differences between compared PCCs were for station VAE6, where the total RMS for LC dPCC was 4.9 mm. As already shown in Reference [27] for this station, the change of PCC from IGS type mean to individual calibration may affect the estimated height by over 15 mm. However, this was an exception. The mean RMS for all LC dPCCs was close to 2.0 mm ( . This common feature of analyzed dPCCs explains partly why the change between type mean and individual calibrations mostly impacts estimated heights. Analyzing the elevational discrepancies in dPCCs, one can distinguish the following two groups of patterns: • The first one, where the differences are very small and can be considered as insignificant (e.g., stations BORJ, NOR7). • The second one, where differences are positive (e.g., LEK6) or negative (e.g., OST6, VAE6) at low elevations and are gradually closer to 0 (e.g., OST6, VAE6). Such patterns have a direct impact on the results, as described in the next section.

Results
The main products of analysis were 18 coordinate time series per each station, two for each elevation mask. In this study, we focused only on the heights. Firstly, we checked the repeatability of As it was expected, the greatest differences between compared correction sets occurred at low elevations, where they were close to 20 mm ( Figure 2). This came from the original patterns (e.g., igs_08.atx), which adopted a PCV(φ, 90 • ) = 0 mm condition and, therefore, showed higher values at lowest elevations. Corrections at the lowest elevation also had the highest standard deviations [18]. The greatest differences between compared PCCs were for station VAE6, where the total RMS for LC dPCC was 4.9 mm. As already shown in Reference [27] for this station, the change of PCC from IGS type mean to individual calibration may affect the estimated height by over 15 mm. However, this was an exception. The mean RMS for all LC dPCCs was close to 2.0 mm (Table A1, Appendix A). The highest values occurred for the antennas at new Swedish stations. In general, dPCCs show an azimuthal symmetry with a lower consistency occurring at lower elevations (Table A2, Appendix A). Among all investigated stations/antennas, the mean azimuthal RMS, calculated as mean RMS from all RMSs for each elevation, was 1.12 mm (minimum = 0.02; maximum = 3.08 mm), whereas the elevational RMS, calculated as mean RMS from all RMSs for each azimuth, was 1.41 mm (minimum = 0.19 mm; maximum = 5.70 mm). This common feature of analyzed dPCCs explains partly why the change between type mean and individual calibrations mostly impacts estimated heights. Analyzing the elevational discrepancies in dPCCs, one can distinguish the following two groups of patterns: • The first one, where the differences are very small and can be considered as insignificant (e.g., stations BORJ, NOR7).
Such patterns have a direct impact on the results, as described in the next section.

Results
The main products of analysis were 18 coordinate time series per each station, two for each elevation mask. In this study, we focused only on the heights. Firstly, we checked the repeatability of the estimated heights in relation to the used elevation mask (Tables A3 and A4, Appendix B). Then, we excluded from further analysis all values that exceeded more than three times the initial values, which came from the solution where no mask was applied. The results obtained for the tested PCC tables were very similar. For solutions where no mask was applied, the individual calibrations gave better results for 15 stations, worse results for three stations, and the same results for seven stations (Figure 3). This outcome changed for higher masks (5 • and 10 • ), where we received better repeatability for type mean calibrations. For even higher masks, individual calibrations showed their advantage again. In general, the individual calibrations improved solutions for more cases. However, the improvements were very small at no more than 0.2 mm. the estimated heights in relation to the used elevation mask (Tables A3 and A4, Appendix B). Then, we excluded from further analysis all values that exceeded more than three times the initial values, which came from the solution where no mask was applied. The results obtained for the tested PCC tables were very similar. For solutions where no mask was applied, the individual calibrations gave better results for 15 stations, worse results for three stations, and the same results for seven stations ( Figure 3). This outcome changed for higher masks (5° and 10°), where we received better repeatability for type mean calibrations. For even higher masks, individual calibrations showed their advantage again. In general, the individual calibrations improved solutions for more cases. However, the improvements were very small at no more than 0.2 mm. The dependence between minimum observation angle and the estimated height uncertainty was quite obvious and came from the satellite geometry. Assuming that other factors like tropospheric delay are modeled the same way, we can interpret height differences between the solutions as the consequence of using PCCs. Therefore, we checked for which source of PCCs heights were more stable upon increasing the elevation mask (Table A5, Appendix C). We distinguished three groups of stations ( Figure 4). The first one (e.g., BORJ, VIL6) showed quite good consistency between solutions regardless of the mask. The second group contained stations (e.g., SOFI) which showed a systematic bias between solutions independently of the used elevation mask. The last group contained stations (LEK7, LOV6) for which a higher mask also caused higher differences between corresponding solutions.
It should be mentioned here that the height of individual stations varied in different ways. This was a combined effect of many factors, such as the satellite geometry, environmental effects, and the used antenna. There were stations for which height decreased when a higher mask was adopted, as well as the opposite where height increased. There were also more complicated cases, where the estimated height firstly decreased and then increased. Changes generally did not exceed a single centimeter, even for high elevation angles. The dependence between minimum observation angle and the estimated height uncertainty was quite obvious and came from the satellite geometry. Assuming that other factors like tropospheric delay are modeled the same way, we can interpret height differences between the solutions as the consequence of using PCCs. Therefore, we checked for which source of PCCs heights were more stable upon increasing the elevation mask (Table A5, Appendix C). We distinguished three groups of stations ( Figure 4). The first one (e.g., BORJ, VIL6) showed quite good consistency between solutions regardless of the mask. The second group contained stations (e.g., SOFI) which showed a systematic bias between solutions independently of the used elevation mask. The last group contained stations (LEK7, LOV6) for which a higher mask also caused higher differences between corresponding solutions.
It should be mentioned here that the height of individual stations varied in different ways. This was a combined effect of many factors, such as the satellite geometry, environmental effects, and the used antenna. There were stations for which height decreased when a higher mask was adopted, as well as the opposite where height increased. There were also more complicated cases, where the estimated height firstly decreased and then increased. Changes generally did not exceed a single centimeter, even for high elevation angles.

Discussion
The obtained results showed that there were some differences for various PCC tables. There was a clear correlation between dPCC patterns and estimated differences in height. For each antenna where the dPCC exhibited a higher discrepancy at lower elevations, the height was more affected. When the dPCC exhibited a negative value at lower elevation, the height differences were also negative (e.g., OST6, SOFI). Of course, for smaller dPCC values and higher elevation masks, this relationship was not so clear. Figure 5 presents the dPCCs for all analyzed antennas, together with height differences for one pair of solutions.
Based on the analyzed dPCCs, a linear relationship could be shown between the mean dPCC values and the resulting differences in height. For various sections of dPCC, the Pearson's correlation coefficient value was close to 0.9 ( Figure 6). If the mean dPCC was negative, the height change was also negative. In particular, for antennas where dPCCs at lower elevation were significant (>2 mm), the mean value corresponded to the expected height difference. We compared the mean value of dPCCs with the height differences. The mean value of dPCCs for the lower section (below 10°) was close to height differences. The mean difference was 0.8 mm with an RMS of 1.4 mm. On this basis, one can try to estimate the expected difference in height resulting from the use of different models of phase centers. However, any conclusions should be drawn very carefully, because, for higher elevations, dPCCs are generally smaller and the dependence is not so clear.

Discussion
The obtained results showed that there were some differences for various PCC tables. There was a clear correlation between dPCC patterns and estimated differences in height. For each antenna where the dPCC exhibited a higher discrepancy at lower elevations, the height was more affected. When the dPCC exhibited a negative value at lower elevation, the height differences were also negative (e.g., OST6, SOFI). Of course, for smaller dPCC values and higher elevation masks, this relationship was not so clear. Figure 5 presents the dPCCs for all analyzed antennas, together with height differences for one pair of solutions.
Based on the analyzed dPCCs, a linear relationship could be shown between the mean dPCC values and the resulting differences in height. For various sections of dPCC, the Pearson's correlation coefficient value was close to 0.9 ( Figure 6). If the mean dPCC was negative, the height change was also negative. In particular, for antennas where dPCCs at lower elevation were significant (>2 mm), the mean value corresponded to the expected height difference. We compared the mean value of dPCCs with the height differences. The mean value of dPCCs for the lower section (below 10 • ) was close to height differences. The mean difference was 0.8 mm with an RMS of 1.4 mm. On this basis, one can try to estimate the expected difference in height resulting from the use of different models of phase centers. However, any conclusions should be drawn very carefully, because, for higher elevations, dPCCs are generally smaller and the dependence is not so clear.
During the analysis, we checked which PCCs gave a more stable result. Two criteria were adopted for comparison. The first one referred to the height repeatability for particular variants of the applied mask. In our analysis, we showed that the repeatability does not depend on the used PCC source. For 17 stations, individual calibration gave better results, but the improvement should be considered as rather insignificant (see values in Appendix B). The second criterion was the height stability when the elevation mask was increased. In general, both sources of PCCs gave comparable results. For solutions from 0 • to 25 • , results were almost the same. The mean value of stability differences was 0.1 and varied from −2.7 mm to 2.8 mm. Therefore, one should conclude that the source of PCC has no effect on this. Including less accurate solutions (from 30 • to 40 • ), the differences were bigger (up to 9 mm), but the number of more stable stations was still similar for individual (11) and type mean (12)   During the analysis, we checked which PCCs gave a more stable result. Two criteria were adopted for comparison. The first one referred to the height repeatability for particular variants of the applied mask. In our analysis, we showed that the repeatability does not depend on the used PCC source. For 17 stations, individual calibration gave better results, but the improvement should be considered as rather insignificant (see values in Appendix B). The second criterion was the height stability when the elevation mask was increased. In general, both sources of PCCs gave comparable results. For solutions from 0° to 25°, results were almost the same. The mean value of stability differences was 0.1 and varied from −2.7 mm to 2.8 mm. Therefore, one should conclude that the source of PCC has no effect on this. Including less accurate solutions (from 30° to 40°), the differences were bigger (up to 9 mm), but the number of more stable stations was still similar for individual (11) and type mean (12) calibrations.
Most of the stations used in the analyses had available individual calibrations that were not  Most of the stations used in the analyses had available individual calibrations that were not included in the IGS08 models, especially the new stations in Sweden. For the selected antenna type, the IGS model was based only on 10 calibrations of five antennas. This is not a large sample, which may explain the received differences. For the current release, IGS14 [34], a type mean model for LEIAR25.R3 LEIT includes 54 calibrations of 28 antennas in total, as well as those installed at Swedish stations. Including more calibration makes IGS models more reliable. In our case, this did not change much. For antennas where negative bias was noted (Figure 5), the new release of the type mean model looked more consistent with individual calibration (Figure 7). However, for stations with a positive bias, we had the opposite effect. In general, all biases in height only changed by about +4 mm, which was already explained in IGSMAIL-7399 "Upcoming switch to IGS14/igs14.atx". Therefore, based on the received biases (Table A5, Appendix C, and Figure 5) we can now conclude the impact of changing from type mean to individual models (or contrariwise). For OST6 we saw an improvement (Figure 7) and the height difference was about 8 mm (instead of 12 mm for igs08). However, for JON6 (Sweden), it was close to 10 mm. stations. Including more calibration makes IGS models more reliable. In our case, this did not change much. For antennas where negative bias was noted ( Figure 5), the new release of the type mean model looked more consistent with individual calibration (Figure 7). However, for stations with a positive bias, we had the opposite effect. In general, all biases in height only changed by about +4 mm, which was already explained in IGSMAIL-7399 "Upcoming switch to IGS14/igs14.atx". Therefore, based on the received biases (Table A5, Appendix C, and Figure 5) we can now conclude the impact of changing from type mean to individual models (or contrariwise). For OST6 we saw an improvement ( Figure 7) and the height difference was about 8 mm (instead of 12 mm for igs08). However, for JON6 (Sweden), it was close to 10 mm.

Conclusions
This paper presents the impact of modelling the antenna phase centre on the estimated height. The comparison concerned two sets of phase center corrections used by the EUREF community: individual and type mean. During the analysis, we shown that for the most frequently used elevation masks (up to 10°), the consistency of dPCCs and the obtained height differences is high enough that the described method can be used for a first estimation of the impact of PCCs on the estimated heights.
Analyzes have shown that depending on the PCC source, the results may differ over 1 cm. However, this does not translate into repeatability of coordinates. It cannot be indicated which PCCs give significantly better results. The introduction of the new release of IGS type mean model did not change much. Only systematic differences have changed, which result from the differences between IGS08 and IGS14.

Conflicts of Interest:
The authors declare no conflicts of interest.
Appendix A Table A1. Total RMSs of the differences between compared phase center corrections for both GPS frequencies (L1 and L2) and "ionosphere-free" linear combination (LC). Values are in millimeters.

Conclusions
This paper presents the impact of modelling the antenna phase centre on the estimated height. The comparison concerned two sets of phase center corrections used by the EUREF community: individual and type mean. During the analysis, we shown that for the most frequently used elevation masks (up to 10 • ), the consistency of dPCCs and the obtained height differences is high enough that the described method can be used for a first estimation of the impact of PCCs on the estimated heights.
Analyzes have shown that depending on the PCC source, the results may differ over 1 cm. However, this does not translate into repeatability of coordinates. It cannot be indicated which PCCs give significantly better results. The introduction of the new release of IGS type mean model did not change much. Only systematic differences have changed, which result from the differences between IGS08 and IGS14.

Conflicts of Interest:
The authors declare no conflicts of interest.
Appendix A Table A1. Total RMSs of the differences between compared phase center corrections for both GPS frequencies (L1 and L2) and "ionosphere-free" linear combination (LC). Values are in millimeters.