GNSS scale determination using calibrated receiver and Galileo satellite antenna patterns

The reference frame of a global terrestrial network is defined by the origin, the orientation and the scale. The origin of the ITRF2014 is defined by the ILRS long-term solution, the orientation by no-net rotation conditions w.r.t. the previous reference frame (ITRF2008), and the scale by the mean values from global VLBI and SLR solution series (Altamimi et al. in J Geophys Res Solid Earth 121:6109–6131, 2016). With the release of the Galileo satellite antenna phase center offsets (PCO) w.r.t. the satellites center of mass (GSA in Galileo IOV and FOC satellite metadata, 2019) and the availability of new ground antenna calibrations for GNSS receivers, based on anechoic chamber measurements or on robot calibrations, GNSS global network solutions qualify to contribute to the scale determination of terrestrial networks, as well. Our analysis is based on global multi-GNSS solutions of the years 2017 and 2018 and may be seen as “proof of concept” for the contribution of GNSS data to the scale determination of the terrestrial reference frame. In a first step, the currently used Galileo PCO estimations (Steigenberger et al. in J Geod 90:773–785, 2016) are compared to the released PCO values, which show discrepancies on the decimeter-level. Eventually, the published Galileo PCOs are used in an experimental solution as known values. GNSS-specific PCOs are estimated, as well, for GPS and GLONASS, together with the “standard” parameters set up in global GNSS solutions. From the estimated network coordinates, a time series of daily scale parameters of the terrestrial network is extracted, which shows an offset of the order of 1 ppb (parts per billion, corresponding to a height difference of 6.4 mm on the Earth’s surface) w.r.t. to the ITRF2014 network and an annual variation with an amplitude of about 0.3 ppb.


Introduction
According to the International Earth Rotation and Reference Systems Service (IERS) Conventions (Petit et al. 2010), the realization of a terrestrial reference system (TRS) leading to the corresponding terrestrial reference frame (TRF) includes the definition of: The realization of the International TRF (ITRF) is based on four space geodetic techniques and is regularly updated by including the most recent data (Altamimi et al. 2016). The underlying techniques are: -Very Long Baseline Interferometry (VLBI).
One important aspect when realizing a reference frame is the realization of the scale. Up to now, the scale of the ITRF is defined by VLBI and SLR. A priori unknown satellite antenna phase center offsets (PCO) prevented the use of GNSS for the scale estimation. The dynamic GNSS satellite orbits are, w.r.t. to the center of mass (COM), well defined by the laws of celestial mechanics. The GNSS measurements refer, however, to the phase center of the transmitting antenna. They are linked to the COM of the satellites by the PCOs, describing the offset between the COM and the phase center of the antenna. Because the scale and the zcomponents of the receiver and the satellite antenna PCOs are strongly correlated (Rebischung 2014;Bruni 2016;Zhu et al. 2003), the scale can only be estimated if calibrations for both, the ground and space segment, are available. An alternative way to reduce the correlation between the two parameter types is to add measurements from Low Earth Orbiters (LEO) into global GNSS analyses (Haines et al. 2015). The European GNSS Agency (GSA) released as the first system provider the satellite antenna calibrations of the Galileo satellites including PCO and Phase Variations (PV) (GSA 2019). Therefore, the scientific community has no longer to rely on estimates. Up to now only, GPS and GLONASS data were used in the International GNSS Service (IGS, Johnston et al. 2017) contribution to the various ITRFs. With the availability of multi-GNSS calibrations for the ground antennas, by robot (Wübbena et al. 1997(Wübbena et al. , 2000 and anechoic chamber calibrations (Zeimetz and Kuhlmann 2008), and by including Galileo to the next IGS contribution for the ITRF, the problem of uncalibrated PCOs and PVs for GPS and GLONASS satellite antennas can be circumvented, offering the potential for GNSS observations to contribute to the scale determination of future ITRF releases.
This article presents the results of a reprocessing effort to assess the potential of GNSS for scale determination using calibrated antenna patterns on the ground and the space segment. When GSA first disclosed the patterns in 2017 for the Galileo In Orbit Validation (IOV) satellites, only chamber calibrations for receiver antennas were available, covering Galileo in addition to GPS and GLONASS. During 2019, in preparation of the IGS contribution to the next ITRF version 2020, Geo++ published a set of robot calibrations including Galileo. To study the impact of the two methods of antenna calibration, a two-year reprocessing was preformed twice, once using robot calibrations and once using chamber calibrations only, allowing to study the feasibility of the two methods for scale determination and to analyze the impact of the different calibration techniques.
Section 2 introduces the two antenna pattern calibration techniques. Satellite antenna patterns provided by GSA and chamber and robot multi-GNSS receiver antenna calibrations are introduced and the available data sets are presented. Section 3 is dedicated to the reprocessing effort which has been performed to assess the potential of the GNSS data to contribute to the scale determination. Section 4 addresses the inner consistency of the antenna calibrations using the so-called inter-system translation biases (ISTP), which introduces for each station an offset for x, y, and z between the coordinates from GPS and from all other GNSS. The main focus of Sect. 5 is on the scale and PCO estimation. Whenever using the term PCO, we only mean the z-component. Zhu et al. (2003) showed that the satellite antenna PCOs and scale are strongly correlated. Therefore, it is difficult to estimate both values simultaneously. An arbitrary scale for the ground coordinates can be partially absorbed by estimating the satellite PCOs. If the scale is changed by +7.4 parts per billion (ppb) (station height + 5 cm), then this would lead to a satellite PCO change of −1 m and the solution would be still consistent. One way to reduce the correlation between the two parameter types is to introduce additional GNSS measurements from LEO satellites (Haines et al. 2015) or to use calibrated satellite antenna PCOs.

Antenna calibrations
The main problem of previous reprocessing efforts organized by the IGS for ITRF updates was that neither the scale nor the satellite PCOs were known (Ray et al. 2013). The satellite PCOs were adjusted to the ITRF scale based on VLBI and SLR. The corresponding satellite PCOs were estimated and made publicly available (Schmid et al. 2007. Meanwhile, the situation has changed. New GNSS have been launched and are operational. GSA has released the satellite antenna patterns for all Galileo satellites as the first GNSS provider. The antenna patterns for the regional Quasi-Zenith Satellite System (QZSS) were released by the Cabinet Office, Government of Japan, in 2017 (CAO 2017). Because QZSS is a regional system, its contribution to a global solution is limited. It is, therefore, not used in this study. The next two subsections briefly introduce the satellite antenna calibrations and the available multi-GNSS receiver antenna calibrations (chamber and robot).

Satellite antennas
Before GSA released the Galileo satellite antenna offsets and patterns, the community relied on estimated PCO values ) similar to the case of GPS and GLONASS. The ground antenna patterns for Galileo were adopted from GPS L1 and L2 calibrations. Comparing the released PCOs and the estimated values referring to the ionosphere-free (IF) linear combination reveals notable discrepancies (see Table 1). Note that the estimates from Steigenberger et al. (2016) are rounded to cm. On average, the PCOs for the z-component from the chamber calibrations are about 16 cm smaller than the estimates derived from GNSS data. The main reason could be: 1. Missing antenna calibrations for Galileo for ground antennas. GPS L2 offsets and patterns have been used for the E5a signals due to the lack of available calibrations. If there is a systematic difference between L2 and E5a over all antennas, this discrepancy would influence the satellite PCO estimations. 2. Incompatible scale between GPS and GLONASS (based on the ITRF 2014 scale) on the one hand and Galileo on the other hand. Scale differences would also be partially absorbed by the satellite PCOs.
The Galileo constellation was initiated in 2011 by launching the first two IOV satellites (https://www.gsc-europa.eu/ system-service-status/orbital-and-technical-parameters, accessed November 21, 2019). Later on, in 2012, two more IOV satellites were launched. The constellation has been systematically augmented with the Full Operational Capability (FOC) satellites and reached its nominal constellation in 2018. The last satellites started to transmit in early 2019. Therefore, there are several years available with an almost full Galileo constellation prior to 2018 and since 2019 the full constellation is available. The next ITRF solution will be based on reprocessed data up to the end of the year 2020 or beyond.

Receiver antennas
As mentioned above, the availability of adequate calibrations is essential for scale determination. Menge et al. (1998) showed that the lack of absolute receiver antenna patterns  can result in networks scale differences of up to 1 cm per 1000 km. The antenna calibrations associated with the ITRF 2014 are mainly based on robot calibrations for GPS and GLONASS L1 and L2 observations ).
To take advantage of the satellite antenna calibrations for Galileo, calibrations are needed for the ground network, as well. Chamber-calibrated receiver antennas covering the full spectrum of frequencies provide a potential set of calibrations. In 2019, Geo++ released a first set of robot calibrations including Galileo while preparing the next IGS contribution to the ITRF 2020.

Chamber calibrations
Before multi-GNSS robot calibrations became available in 2019, chamber calibrations were the only available source for Galileo E5 antenna patterns. A data set of more than 250 individual patterns were collected and used to create mean antenna calibrations. In total, type-mean calibrations for 36 antenna / random combinations were created ( Table 2). The available receiver antenna calibrations cover about 50% of the IGS sites. For our scale study, this leads to the network shown in Fig. 1 with a total of 183 sites and a subset of 97 sites capable of tracking Galileo (as of January 1, 2017).

Robot calibrations
Our analysis is based on 37 robot calibrations (Table 2) including Galileo. Because the igs14.atx antenna model file is based on robot calibrations, the data set has been extended by adding the GPS/GLONASS only calibrations. For data processing, only observations with calibrations were used. If a receiver-tracked Galileo but no corresponding calibrations were available, the observations were skipped. This leads to the network shown in Fig. 2 with a total of 296 sites and a subset of 94 sites capable of tracking Galileo (as of January 1, 2017).  Table 3 lists the PCO up-component for the IF for all systems. For GPS L1/L2, for GLONASS L1/L2, and for Galileo E1/E5 are used. The two data sets are using different datum definitions for their patterns. Applying a zero-mean condition over all PV and removing removal of a constant term allows to compare the calibrations from the two techniques. From Table 3, we conclude that on average the PCOs are consistent within 1 mm. For individual antenna types, however, the differences may reach values up to 7 mm.

Case study covering 2017 and 2018
To study the potential of using calibrated antennas in space and on ground, data covering the years 2017 and 2018 were processed. The latest IERS and IGS standards were used. Table 4 summarizes the settings.
Our analysis consists of two solutions, one using chamber calibrations and the other one using robot calibrations only for the receiver antennas. The IGS14 antenna model, a collection of robot calibrated type-mean patterns for GPS and GLONASS L1 and L2, was modified replacing 37 GPS/GLONASS antenna patterns with their newly released

Processing strategy
Our study is based on the Center for Orbit Determination in Europe (CODE) GPS and GLONASS Final IGS operational processing (Dach et al. 2017). The processing scheme was extended to include Galileo based on CODE's MGEX solutions (Prange et al. 2020). The processing was performed using the Bernese GNSS software . Table 4 lists the key characteristics of our processing scheme. The generation of our solution can be divided into two parts. The first part performs the double-difference network solution over one day (24 h). In the second part, the daily solutions are combined into a 3-day solution by combining the underlying 1-day Normal Equations (NEQ). This step stabilizes the orbits and the Earth Orientation Parameters (EOP) (Lutz et al. 2016). Datum definition is achieved by applying a no-nettranslation and a no-net-rotation condition (three translation and three rotation constraints) to a well-defined list of ITRF positions (Altamimi et al. 2016), after having removed outliers, i.e., stations with more than 1 cm horizontal and more than 3 cm vertical differences, from the reference station list (IGS14-the IGS-specific realization of the ITRF2014 (Rebischung and Schmid 2016)). Figure 3 shows the total number of reference stations and reference stations including Galileo for both solutions. The datum of the solution using chamber calibrations is based on average on 100 sites, where 50 of them are tracking Galileo. The solution based on robot calibrations is based on around 150 fiducial sites where 50 track Galileo in analogy to the other solution.

Solutions
Two sets of solutions were produced to study the impact of satellite antenna calibrations. The first set, subsequently labeled as ROB, is based on robot calibrations for the receiver antennas. The second set using chamber calibrations for the ground antennas is labeled CHA. The number of Galileo satellites grew from 13 to 20, the number of GPS and GLONASS satellites remained stable at 32 and 24 satellites, respectively, for the 2-year test period (Fig. 4).

Receiver PCO verification
The scale determination depends on the antenna calibrations.
A simple way to test the consistency of the individual antenna patterns is to generate GPS-, GLONASS-and Galileo-only solutions and to compare the resulting coordinates. Ideally, the differences between all coordinate sets would be zero. We used a similar approach by analyzing the observations of all GNSS in the same parameter estimation procedure, and by setting up so-called inter-system translation parameters (ISTP) in the observation equations . The ISTPs are offsets (x, y, and z) pointing from the GPS to the GLONASS or Galileo phase center of the same site. Because we allow for such a vector for each station, the corresponding satellites of that system can be shifted by an arbitrary value.
To avoid such arbitrary shifts, a system-specific datum definition needs to be introduced, in analogy to the geodetic datum. System-dependent multipath effects, potential deficiencies in Experiment 1a (using the ITRF 2014 scale) introduces ISTPs using six zero-mean conditions, three translations and three rotations for datum definition. A non-zero average of the ISTPs vertical component indicates that the scales of the used satellite PCOs are not consistent among the three systems. A mean offset for Galileo has to be expected and the offsets for GLONASS should be close to zero as the GPS and GLONASS PCOs are aligned to the ITRF2014 scale, whereas the Galileo PCOs are based on chamber calibrations. The main purpose of this experiment is to assess the impact of the inconsistent scale between the ITRF2014 and the chamber-calibrated Galileo satellite antenna pattern on the station coordinates. As opposed to Experiment 1a, the Experiment 1b introduces adjusted PCO values based on a Galileo-induced scale (Sect. 5.2 below). The outcome of the experiment is expected to have no mean offsets in the up component of the ISTPs, as the satellite PCOs of the three systems are mutually consistent. The ISTP values reflect the consistency between the individual type-mean antenna calibrations w.r.t. the overall scale introduced with the Galileo scale. Experiment 2 introduces, in addition to the ISTPs, an elevation-and GNSS-specific Troposphere Bias Parameter (GTRP) using the WET-VMF model (Böhm et al. 2006), w.r.t. GPS. The GTRP accounts for deficiencies in the elevation-dependent PV antenna patterns as well as deficiencies in the troposphere models and system-dependent multipath effects. Table 5 lists the ITSP and GTRP values of the experiments. Experiment 1a compares the consistency of robot and chamber calibrations w.r.t. current IGS14 antenna models (no changes to GPS and GLONASS, extended by the Galileo PCOs from GSA). The ISTP results reveal, as expected, that the consistency between GPS and GLONASS is better for robot than for chamber calibrations. This seems to be obvious because the GPS and GLONASS PCOs for the satellites were estimated using robot calibrations for the ground antennas. In case of Galileo, we see a clear discrepancy of about ± 7 mm for both calibration types. This is a result of different scales between the GPS/ GLONASS and Galileo PCO values. The differences between the GLONASS and Galileo ISTPs from robot and chamber calibrations show an average discrepancy of 2 mm. Experiment 1b removes the discrepancy between GPS/ GLONASS and Galileo scale which is reflected in the ISTPs from Experiment 1a. By adjusting PCOs with a system-specific PCO offset for GPS and GLONASS (introducing a Galileo scale), the inconsistency between the different GNSS is removed leading to average offsets below 1.2 mm. Experiment 2 introduces GTRPs as additional parameters to absorb the nadir-dependent variations to test the PVs. From the latter test, we may conclude that the PVs and PCOs are consistent for all GNSS and for both calibration methods. With an average GTRP offset below 1 mm, we may assume that the calibrations are mutually consistent and that they may be used to transform the scale from one GNSS to another.

Results
The variation of the ISTP values is indicators for the consistency of different type-mean receiver antenna calibrations. The average ISTP values per antenna type for Experiments 1b and 2 as well as the GTRA in the case of the latter experiment are listed in Table 6. The values depend on the different antenna types and vary with the used calibration method. The ISTP and GTRP values for antenna ASH701945C_M NONE, for robot and chamber calibrations, are above 2 cm. The antenna was used by 4 stations and revealed a poor daily repeatability. The robot calibrations showed a good consistency below 5 mm except for the antennas JAVRINGANT_DM NONE and TRM115000.00 NONE with a discrepancy of −8 mm and 5 mm, respectively. The chamber calibrations show a similar behavior. With the exception of the antennas JAVRINGANT_DM NONE (discrepancy 9 mm), JAVRINGANT_G5T NONE (−9 mm), and TRM59800.00 SCIS (−5 mm), all antennas show an average value below 5 mm. Overall, we may state that we have a good agreement between GPS and Galileo (and GLONASS) among the antenna types as most of the offsets are in case of chamber calibrations below 4 mm and even below 3 mm for robot calibrations. The station-wise repeatability of the daily ISTP estimations of Experiments 1b and 2 are shown in Fig. 5. The average 1 − σ scatter (derived from the 0.16 and 0.84 quantiles to be insensitive to outliers) of the ISTPs is about 2.6 mm for chamber calibrations and 2.7 mm for robot calibrations. Adding the GTRP parameter in Experiment 2 leads to a roughly 1.7 times larger scatter.

Scale estimation
After generating the 3-day NEQs based on the two calibration techniques (robot calibrations (ROB) and chamber calibrations (CHA)) for 2 years, we estimated the corresponding scales. We used NEQs which include, among other parameters, coordinates and PCOs. The orbits and Earth Rotation Parameters (ERP) were pre-eliminated. We estimated a GPS, a Galileo (GAL) and a GLONASS (GLO) scale by constraining the corresponding PCOs in the NEQs to their a priori values. The coordinates were then compared to the ITRF 2014 coordinates using the reference sites only (outliers were removed during the datum definition) estimating three translations, three rotations and a scale. A positive scale means that the estimated coordinates are above the ITRF and a negative scale that the estimated coordinates are below. Altamimi et al. (2016) showed that daily estimated VLBI scales show an annual sinusoidal patterns. The same characteristic is observed in the GNSS-derived scale parameters. Therefore, the overall scale was estimated in combination with an annual signal (a sine and a cosine term). The results are summarized in Table 7. In addition to our estimates, the table also contains the scale and the annual signal of the ITRF 2014, which is based on all available space techniques (Altamimi et al. 2016). The amplitude of about 0.3 ppb emerging from the GNSS techniques is similar to the amplitude resulting from the International VLBI Service for geodesy and astrometry (IVS, Nothnagel et al. 2017) but the GNSS-derived annual signal has a phase shift of about 80 •  (Collilieux et al. 2010;Altamimi et al. 2016). The RMS of the estimated scale is about 0.1 ppb. Figure 6 shows the time series of the scale estimated by constraining the Galileo PCOs for both solutions, ROB and CHA.
For the scale comparison between the ITRF2014 and the individual solutions, potential long-term drifts have to be taken into account. For SLR and VLBI, the drifts w.r.t. ITRF2014 are provided by Altamimi et al. (2016). The intrinsic drift for GNSS is primarily given by the consideration of PCOs which are assumed to be constant (Collilieux and Schmid 2013;Rebischung and Schmid 2016). We assumed the scale drift to be close to the one between the igs14.atxbased GNSS solutions and the ITRF2014 with +0.026 ppb, leading to a total drift between 2010 and 2018 of 0.21 ppb (Rebischung and Schmid 2016).
The ROB solution shows the expected behavior. Constraining the PCOs to GPS or GLONASS leads to a small-scale difference. This is due to the fact that the introduced IGS14 GPS and GLONASS PCOs are aligned to the ITRF2014 using robot calibrations. The scale difference w.r.t. ITRF2014 is 0.25 ppb, which is close to the expected difference of +0.21 ppb when taking the drift from the year 2010 onward into account. The Galileo scale is 1.41 ppb and would, reducing it by the rate of +0.026 ppb/year, be about 1.20 ppb in 2010. The PCOs for GPS and GLONASS are derived from the ITRF 2014 solution and should be consistent with the current ITRF scale. Figure 7 shows the daily scale estimations for SLR, VLBI and DORIS for the ITRF 2014 (Altamimi et al. 2016) and our estimated GNSS scale (using the drift presented in Rebischung and Schmid (2016)).
The scale discrepancy between the GPS and the Galileo solution is 1.65 ppb for chamber calibrations. The ROB solution has a difference of 1.16 ppb. The RMS of the scale determination is the same in both solutions. Both solutions are internally fully consistent. Robot and chamber calibrations are not the same, however, and may have differences in the PCOs up to 7 mm. (Sect. 2.3). The IGS relies mostly on robot calibrations for its contribution to the ITRF as, in particular for older antenna types, only robot calibrations are available.

Phase center offset (PCO) estimation
The network scale is a function of the PCOs of the satellite and receiver antennas. If the receiver antennas are fully calibrated and if the PCOs for the satellites of one GNSS are available, we can transfer the scale of the GNSS with the known PCOs to the PCOs of the other constellations via the ground antennas. Introducing the PCOs of one GNSS defines the PCOs of all other GNSS, as well, via the scale of   Figure 8 shows the system-wise PCO offset estimates for 2017-2018 for the ROB and CHA solutions. The annual signal can be seen when inducing the scale trough ITRF coordinates. Constraining the GPS or Galileo PCOs instead removes the annual signal, because it is absorbed by the coordinates. This can be observed in the scale determination of the individual solutions (Table 7). The scatter of the PCOs is small (CHA: 2.6 mm, ROB: 1.5 mm). For the ROB solution, more than 300 stations were used, only up to about 200 sites for the CHA solutions, which might explain the difference in scatter between the two solutions.

Summary
Our analysis shows that a GNSS-based scale can be extracted with a daily repeatability of 0.12 ppb. The Galileo scale w.r.t. ITRF2014, evaluated at the mean epoch of our experiment (January 1, 2018), is between 1.03 (CHA) and 1.41 (ROB) ppb. The annual variation in the scale over the two years is similar in amplitude to the one observed by the IVS, and it has a significant phase shift. The difference between the CHA and ROB solutions is 0.38 ppb, which may be caused by the different networks (multi-GNSS calibrations for different antenna types) and potential systematic errors in the calibrations. For GNSS, we are in the comfortable situation to have two independent types of calibrations available allowing it to compare them. We conclude that the Galileo scale can be used for GNSS solutions using calibrated ground receiver antennas. The procedure has the potential to contribute to a TRF scale. The selection of the receiver antennas is an important part of our method. For the sake of consistency, we propose to rely mainly on robot calibrations, because such a TRF contribution covers GNSS back to at least 1994. The PCO estimation by fixing Galileo leads to a change of −15.0 and −14.3 cm for GPS and GLONASS PCOs when using robot calibrations, and −22.1 and −25.8 cm for chamber calibrations. The formal error of the estimates, when stacking the NEQs of the years 2017 and 2018, is about 1.4 mm. Introducing the PCO corrections for GPS and GLONASS, according to Table 8 leads to a consistent set of PCOs which are in agree-

Conclusions
The scale determination of the ground tracking network with GNSS became feasible with the disclosure of the Galileo satellite antenna patterns on April 25, 2019 and with the availability of multi-GNSS calibrations for receiver antennas. At present, two sources of receiver antenna calibrations are available. Robot and chamber calibrations support mean-while the Galileo system and can be used in future IGS contributions to new ITRF releases. Robot and chamber calibrations show overall small discrepancies in the derived PCOs and the network scale. Robot calibrations are closer to the current ITRF 2014 solution.
A GNSS TRF scale may be derived relying on either robot or chamber calibrations. The Galileo scale difference w.r.t. ITRF2014, evaluated at the mean epoch of our experiment (January 1, 2018), is 1.4 ppb in case of robot and 1.0 ppb in case of chamber calibrations. The VLBI scale difference for January 1, 2018, is taking the drift into account, 0.84 ppb. The extrapolated scale of the ILRS solution is −0.84 ppb.
ISTP may be used to test the compatibility of different GNSS antenna calibrations for individual ground antennas and to check the consistency of the individual GNSS scales induced by their PCOs. Using the chamber-calibrated Galileo satellite antenna pattern together with the IGS14 PCOs of GPS and GLONASS will introduce and average offset between GPS/GLONASS and Galileo of about 7 mm. After re-adjusting the PCOs to the Galileo scale, the ISTP are reduced, as expected, to roughly 1 mm. The Galileo ISTP values for the individual type mean antennas show a scatter of about 3 mm.
Introducing known PCOs from at least one GNSS, e.g., Galileo, is sufficient to estimate the network scale and to adjust the PCOs of the other GNSS accordingly. The question remains, however, whether the Galileo scale may be transferred backwards in time using GPS and GLONASS PCOs. This could be achieved by using the estimated PCOs and introduce the values from 2017-2018 to time periods with no or only limited Galileo constellation. It would be much better, however, if calibrated GPS and GLONASS PCOs would be disclosed in analogy to those of Galileo.