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MultiGNSS clock combination with consideration of inconsistent nonlinear variation and satellitespecific bias
Earth, Planets and Space volume 74, Article number: 142 (2022)
Abstract
As part of the International GNSS Service (IGS) multiGNSS Pilot Project (MGEX), precise orbit and clock products for multiGNSS constellations have been submitted by several analysis centers (ACs) since 2012. Based on the 30 s satellite clocks from 6 MGEX ACs, the multiGNSS clocks are combined and the consistency of the AC clocks is assessed in this study. Usually, a linear transformation between the combined and AC solution is used for clock combination, and the clock residuals of the AC solution w.r.t the combined solution are used to determine the weights of the AC. However, any inconsistent satellitespecific bias or nonlinear variations in the clocks induced by the AC’s processing strategy can contaminate the linear transformation as well as the determination of the weight. In this study, the analysis center and satellitespecific bias (ASB) of the MGEX AC clock solutions is first identified and estimated by using observations from globally distributed stations. Moreover, the clock solutions with nonlinear variations induced by the reference clock or nonconstant intersystem bias (ISB) are corrected by aligning the clocks to the selected reference solution before clock combination. With the correction of the ASB, the rootmeansquare of the clock residuals decreases significantly and reaches 14\(\)26, 37\(\)91, 33\(\)48 and 12\(\)44 ps for GPS, GLONASS, BDS2 and Galileo, respectively. In general, the consistency of AC solutions w.r.t the combination reaches 8\(\)16, 27\(\)58, 13\(\)27 and 9\(\)36 ps for GPS, GLONASS, BDS2 and Galileo in terms of the standard deviation. Finally, the individual AC and combined orbit as well as clock solutions of different constellations are assessed by precise point positioning, and the combined multiGNSS solutions show competitive performance with the best AC solution in terms of both the positioning accuracy and stability of the reference frame parameters. However, inconsistent scale parameters of both the AC and combined solutions are identified and require more investigation.
Graphical Abstract
Introduction
Precise Point Positioning (PPP) with a single receiver can provide positioning with centimetertomillimeter accuracy globally (Zumberge et al. 1997; Kouba and Héroux, 2001; Zhao et al. 2021). Precise orbit and clock solutions are the prerequisites of PPP and can be obtained from the International GNSS Service (IGS) (Dow et al. 2009) as well as its analysis centers (ACs). Currently, only the combined orbit and clock solutions of GPS as well as orbits of GLONASS are provided by IGS. Although the development of the European Galileo, China BeiDou Satellite Navigation System (BDS), and other constellations have noticeably increased the number of satellites in space and the performance of the positioning, navigation and timing (PNT) service, combined solutions, particularly for clocks, are lacking.
IGS has already made considerable effort to improve GNSS combined solutions in terms of precision, consistency and robustness. In early 1993, the GPS orbits and Earth Orientation Parameters (EOPs) from seven ACs were collected to validate the feasibility of orbit combination (Springer and Beutler 1993). Combined GPS orbits, as IGS legacy products, have been available since 1994, and orbit comparison has also pushed forward AC improvements not only in strategy, but also in dynamic and observational models (Beutler et al. 1994, 1995; Griffiths 2019). Motived by the competitiveness of GPS orbit combination, and to meet the satellite clock requirement for precise navigation, GPS clock products were also combined by IGS (Kouba et al. 1995). To meet the demand of highrate satellite clocks (e.g., orbit determination of lowearthorbit spacecraft), 30s GPS satellite clocks were submitted by ACs, and their combination is supported by IGS using a new program based on an improved strategy (Kouba 2000; Kouba and Springer 2001). The combination of AC station clocks is also allowed in the new clock combination program for the development of a stable time scale (i.e., IGS Time) together with satellite clocks (Ray 1999; Petit and Arias 2009).
In contrast to the orbits constrained by dynamic models, satellite clock estimates are prone to possible discontinuities and outliers caused by unmodeled observation errors and high correlations between clocks and phase ambiguities. Moreover, inconsistent attitude modes, especially for satellites during the eclipse season adopted by ACs, can result in inconsistency. This issue was investigated by Loyer et al. (2017), and ACs are encouraged to provide the quaternions of satellite attitudes used in precise orbit and clock determination (Loyer et al. 2021). In addition, the phase center corrections used by ACs can generate clock inconsistencies between ACs. Currently, the GPS clock combination conducted by the IGS analysis center coordinator (ACC) based on the strategy proposed by Kouba and Springer (2001) generally consists of four steps: (1) the alignment of individual AC clock solutions to the IGS reference frame with correction of the orbital errors as well as origin offset; (2) alignment of the individual ACs’ clocks to a common reference time frame for the rapid convergence of the clock combination; (3) combination using an iterative outlier detection and rejection scheme based on weighted averages, with the weights determined by the absolute values of AC deviations with respect to the average; (4) realignment of the combined clocks to the IGS time scale (Kouba and Springer 2001). When the clocks are aligned to the reference time scale in steps (2) and (3), it is assumed that the clock differences between the two solutions vary linearly. As the variations of the selected reference clock are absorbed by the clock estimates, the potential inconsistent nonlinear variation induced by the reference clock should be corrected before clock combination.
Following the success of GPS clock combination as well as the development of the multiGNSS system, clock combination for multiGNSS has attracted much interest from the GNSS community. The first proposal was the combination of GLONASS clocks. Unlike with GPS, the Frequency Division Multiple Access approach is used by GLONASS, and the interfrequency code bias (IFCB) hinders the consistent combination of GLONASS clocks. To overcome this, Song et al. (2014) introduced an additional constant parameter in clock combination to reduce significant differences in the analysis center and satellitespecific bias (ASB). The day boundary discontinuity was obviously reduced even when equal weights were used for ACs to generate combined solutions (Song et al. 2014). The ASB in GLONASS clock products was also analyzed by Chen et al. (2017), and a robust leastsquares estimation for clock combination was proposed. Beyond GLONASS, the ACs within the International GNSS Monitoring and Assessment System (iGMAS) have already provided global quadconstellation orbit and clock solutions, and the urgent need for combined solutions makes weighted averaging orbit and clock solutions possible (Chen et al. 2015). On the other hand, orbit and clock combination for multiGNSS is also being undertaken by IGS, and experimental multiGNSS combined orbits were first released at https://igs.org/acc/experimentalMultiGNSScombinations (Sakic et al. 2018; Masoumi and Moore. 2019; Sośnica et al. 2020). However, the orientation correction with respect to the operational IGS reference frame is not applied during multiGNSS orbit combination (https://igs.org/acc/experimentalMultiGNSScombinations). With the development of PPP with Integer Ambiguity Resolution (PPPAR), the correlation of clocks and phases as well as code biases has been achieved (Geng et al. 2019), and the PPPAR working group was established in 2018 by the IGS Workshop at Wuhan; it is in charge of the combination of clocks as well as phase biases. An experimental combination based on satellite clock and phase bias products from six ACs over a week was conducted by IGS (Banville et al. 2020), and the strategy used was summarized in detail by Pan (2021). In addition to the combination in postmode, to satisfy the needs of available and reliable realtime ephemeris corrections, two kinds of combined orbits and clocks are also provided by IGS based on the weighted average and Kalman filter (Mervart and Weber 2011).
For the experimental multiGNSS combined clocks, the IGS mainly focuses on the simultaneous combination of clocks and code/phase bias products to advance the PPP with ambiguity resolution (Banville et al. 2020). However, the clock differences between two solutions (e.g., the AC and combined clocks) are assumed to vary in a linear pattern during clock combination, which is not always the case for multiGNSS clocks. In this study, we focus on clock combination in postprocess mode for a multiGNSS constellation based on products from IGS MGEX ACs, with consideration of ASB and nonlinear variation in clock solutions. Following the description of IGS MGEX clock products from each AC, the algorithms for multiGNSS clock combination are presented. Subsequently, the consistency of an individual AC clock solution is assessed. Furthermore, the performance of combined orbit/clock solutions is evaluated by PPP with 96 selected globally distributed reference stations, and the accessible terrestrial reference frame parameters with respect to the IGS frame are also analyzed. Finally, the study is summarized.
Data and multiGNSS clock combination strategy
This section starts with the collection and overview of multiGNSS orbit/clock products of ACs. Then, the strategy for clock combination is presented with a focus on the a priori correction, inconsistent ASB and nonlinear variations. In addition, the weighting for individual constellations is designed.
MGEX orbit and clock products
IGS initiated the MGEX to track, collate and analyze all available GNSS signals (Montenbruck et al. 2017). Currently, 7 ACs are officially providing solutions for multiGNSS constellations, i.e., the Centre National d’Etudes Spatiales/Collecte Localisation Satellites (CNES/CLS), the Center for Orbit Determination in Europe (CODE), Deutsches GeoForschungsZentrum Potsdam (GFZ), the Information and Analysis Center (IAC), the Japan Aerospace Exploration Agency (JAXA), Shanghai Observatory (SHAO), and Wuhan University (WHU). In addition, the European Space Agency (ESA) releases multiGNSS solutions on its own server. In this study, the orbit and clock products from CNES/CLS, CODE, ESA, GFZ, JAXA, and WHU over the whole year of 2018 are collected for clock combination and positioning. As IAC began to submit solutions in the second half of 2020 and the sampling rate of clock solutions is 300 s for SHAO while it is 30 s for other ACs, these two ACs are excluded from clock combination.
Table 1 lists the status of the multiGNSS orbit and clock products used for clock combination. There are 4 ACs, i.e., CODE, ESA, GFZ and WHU, providing precise orbit and clock products for five constellations (i.e., GPS, GLONASS, BDS, Galileo and QZSS). However, the GEO satellites of BDS are not included in the CODE and ESA products. In addition to GPS and GLONASS, the satellite orbit and clock of Galileo and QZSS are supported by CNES/CLS and JAXA, respectively. As only a regional BDS constellation (BDS2) was available from those ACs in 2018, only BDS2 is analyzed in this study. Integer clocks are provided by CNES/CLS, whereas the other ACs’ clocks are similar to those of the IGS legacy clock.
MultiGNSS clock combination
For clock combination, the AC raw clock solution \({C}_{a0}^{s,i}\) can be expressed as
where \({C}^{s,i}\) is the sum of real physical clock offset and common hardware delays including the timeinvariant part of code/phase hardware delays and the timevarying phase hardware delays, which is common for all ACs clocks. The superscript \(s\) and \(i\) indicate satellite \(i\) of constellation \(s\). \({B}_{a}\) is the ACspecific time scale, which is usually taken from a selected station clock to avoid rank deficiency during clock determination. \({d}_{a,cor}^{s,i}\) is the a priori correction applied before clock combination to compensate for the inconsistency between AC and combined orbits. Generally, \({d}_{a,cor}^{s,i}\) includes corrections for inconsistent frames (i.e., \({d}_{a,frm}^{s,i}\)), attitude mode (\({d}_{a,att}^{s,i}\)) and phase center offset (\({d}_{a,pco}^{s,i}\)), which are subsequently substituted with \({d}_{a,cor}^{s,i}={d}_{a,frm}^{s,i}+{d}_{a,att}^{s,i}+{d}_{a,pco}^{s,i}\). \({ASB}_{a}^{s,i}\) is the AC and satellitespecific bias, which is not common within different AC, and possibly induced by the AC’s processing strategy (e.g., inconsistent observation networks with mixed receivers used for ACs).
A priori correction for AC clocks
To maintain the consistency between the combined orbit and clock, the following formula proposed by Ferland (1999) is adopted for the frame correction of individual AC clock:
where the superscripts \(s\) and \(i\) indicate the constellation and satellite, respectively; \({{\varvec{O}}}_{a}^{s,i}\) and \({{\varvec{O}}}_{cmb}^{s,i}\) represent AC \(a\) and the combined orbits, respectively; \({R}^{s,i}\) is the distance of the satellite in the EarthCenteredEarthFixed (ECEF) coordinate system; \(c\) is the velocity of light; and \({\varvec{D}}{{\varvec{X}}}_{a}\) is the geocenter offset obtained by comparing the AC station coordinates with respect to the combination. Considering that not all AC station coordinates are available during the combination experiment, no geocenter correction is applied for AC clock solutions.
To analyze the impacts of frame correction on clock combination, the clock differences between the two solutions are derived with the following formula:
Figure 1 shows the intersatellite and interAC clock differences of ESM and COM solutions without and with frame correction for the first day in 2018. It is demonstrated that the nonlinear variations in double differenced clock induced by different orbit errors are significantly reduced once frame correction is applied. Compared to the differences without frame correction, the mean standard derivation (STDev) of the clock residuals is reduced to 0.049, 0.082, 0.026 and 0.037 ns from 0.088, 0.111, 0.233 and 0.099 ns for GPS, GLONASS, BDS2 and Galileo, respectively. It also indicates that performing a quality assessment of clock products by using the approach of Eq. (3) regardless the frame correction is not rational, especially for newly established constellations (e.g., BDS and Galileo).
Corrections induced by inconsistent attitude modes and phase center offsets (PCOs) are obtained from Eqs. (4) and (5), respectively:
where \({\varnothing }_{a}^{s,i}\) and \({\varnothing }_{ref}^{s,i}\) are the yaw angles of the AC and the reference value, respectively. \({f}^{s,i}\) is the frequency of ionospherefree combination (Loyer et al. 2021). \(PC{O}_{a,Z}^{s,i}\) and \(PC{O}_{ref,Z}^{s,i}\) are the ZPCO values of the individual AC and the reference AC, respectively.
Because the PCO values are described in the satellite body fixed system, a minus sign is added in Eq. (5). Since the attitude products are not available for any ACs during the clock combination period of 2018, the correction of \({d}_{a,att}^{s,i}\) is not applied for AC clock solutions. For the GPS and GLONASS PCOs, conventional values from the IGS ATX file are adopted by all ACs, and there is no need to consider \({d}_{a,pco}^{s,i}\) for these two constellations (i.e., \({d}_{a,pco}^{s,i}=0\)). For the BDS and Galileo constellations, there are some discrepancies among ACs during the combination period. For example, ESA and WHU adopted the PCO values of BDS2 from the selfcalibration, while the PCOs from IGS ATX were used for other ACs. There is a small difference between the Galileo PCO values adopted by CODE and by IGS ATX. It is also noted that \({d}_{a,pco}^{s,i}\) is linearly related to \({ASB}_{a}^{s,i}\) and can be calibrated simultaneously.
Analysis center and satellitespecific bias correction
Obvious ASBs in the clock residuals after frame correction are noticed in Fig. 1 and would contaminate the combined results. Because different observation networks with mixed receivers are used for ACs, the ASBs induced by the pseudorange noises, hardware delays and signal distortions are inconsistent among ACs.
In this study, we calibrate the ASB as a daily constant using an identical software package and observations from a global network. Using the precise products from the AC, the ionospherefree observation accounting for the ASB parameter can be expressed by
where \({P}_{IF,r}^{s,j}\) and \({L}_{IF,r}^{s,j}\) are the ionospherefree pseudorange and phase measurements, respectively, and \({\varepsilon }_{{P}_{IF},r}^{s,i}\) and \({\varepsilon }_{{L}_{IF},r}^{s,i}\) are the observation noises. \({\rho }_{r}^{s,j}\) is the geometric distance between satellite \(i\) and station \(r\) with the correction of relativistic, tidal loading and phase center offset. \({d}_{r}\) is the receiver clock including the real physical clock offset and time scale \({B}_{a}\). Considering that the \({C}_{a0}^{s,i}\) is consistent with the orbit of AC \(a\), no frame correction is employed for \({C}_{a0}^{s,i}\). \(IS{B}_{r}^{s0\_s}\) is the daily constant intersystem bias (ISB) between systems \(s\) and \(s0\) (i.e., GPS). \({T}_{r}^{s,i}\) is the tropospheric delay. \({\lambda }_{IF}^{s}\) and \({N}_{r}^{s,i}\) are the wavelength and ambiguity of the ionospherefree phase, respectively.
For the estimation of ASB, 96 globally distributed stations with the capacity to track multiGNSS signals are selected, as shown in Fig. 2. The 300 s dualfrequency code and phase measurements of L1 and L2 for GPS, as well as QZSS, G1 and G2 for GLONASS, B1I and B2I for BDS2, and E1 and E5a for Galileo, are used to conduct float PPP in static mode. The satellite orbits and clocks are held fixed with AC products, and to separate the ASB, ISB and receiver clock, the sum of the ASB for all satellites in each constellation is constrained as zero. The IGS antenna file (i.e., igs_2136.atx) is adopted to correct the phase center variation, and any inconsistency in phase center values between AC and IGS antenna files is absorbed by the estimates of the ASB.
Figure 3 illustrates the ASB difference and daily bias of double differenced clock with respect to COM solutions for the first day in 2018, and the correlations are also presented. Due to the fact the number of satellites for ASB constraint are different among ACs, an average bias of all satellites is removed from the ASB difference of each constellation. As only three constellations are available for GRM and JAM, their results are not shown here. It is noticed that most of the ASB differences for GPS and Galileo satellites are within 1.0 ns, whereas those of GLONASS and BDS2 satellites are scattered across larger values, and the maximum can reach more than 3.0 ns for some satellites. A high correlation between the clock and ABS difference can be identified, and the correlation coefficients for each constellation are larger than 0.95 for all ACs. Hence, the ASBs estimated based on the proposed approach can be used to calibrate the clock solutions.
Inconsistent nonlinear variation of clocks
With a priori correction as well as ASB bias calibration, the corrected clock solution \({C}_{a1}^{s,i}\) for satellite \(i\) of AC \(a\) can be obtained as
where \({C}_{a0}^{s,i}\) is the raw clock solution; \({B}_{a}\) is the ACspecific time scale, which is common to all satellites and usually depends on the reference clock, e.g., a single receiver or satellite clock, used by the AC.
Considering that the \({B}_{a}\) of the clock solution may be arbitrary, it is necessary to keep \({B}_{a}\) compatible with different constellations. To analyze the time scale of individual ACs, the single difference between ACs of a and b can be obtained with the following formula:
where \(\nabla\) represents the single difference operator. Since \({C}^{s,i}\) is only related to satellites, \(\nabla {C}_{a,b}^{s,i}\) can be canceled. Assuming that \({B}_{a}\) is only an ACspecific parameter, and since stations equipped with stable atomic clocks are usually used as reference stations, the difference \(\nabla {B}_{a,b}\left(t\right)\) at epoch \(t\) can be fitted by a linear model:
where \(\nabla {O}_{B}\) and \(\nabla {D}_{B}\) are the intercept and drift of \(\nabla {B}_{a,b}\), respectively, and \(\Delta t\) is the elapsed time between \(t\) and the reference epoch.
Figure 4 presents the GPS and GLONASS singledifference clocks of individual ACs with regard to COM for the first day of 2018, and GPS satellites are used to obtain the coefficients of the time scale difference (i.e., \(\nabla {O}_{B}\) and \(\nabla {D}_{B}\)). The residuals of the GPS clocks are near zero, with an STDev less than 0.15 ns, except for the JAM solution. For the GPS and GLONASS clock differences of the JAM solution, significant nonlinear but consistent variations can be observed, which are most likely induced by a clock reference station equipped with an unstable clock. It is inferred from the GLONASS clock residuals that all ACs estimate the ISB as constant except GFZ, and a temporal variation in GLONASS with a range of 3.0 ns can be observed. Meanwhile, a clock jump is noticed for satellite R21 presented as yellow line and a few outliers are also found for the GRM solution, which indicate that outlier detection and transformation parameters of piecewise should be implemented for clock combination.
IGS usually uses a linear transformation between the AC and the combined solution in the clock combination to remove the differences in time scales among ACs (Kouba and Springer 2001). To obtain reliable weights from the clock residuals and meet the prerequisite of linear transformation for clock combination, it is essential to remove the inconsistent nonlinear variation induced by the time scale and ISB. In this study, one AC with a stable time scale and constant ISB is selected as a reference, and the inconsistent nonlinear variation in the other ACs (e.g., GFZ) is estimated with the following formula:
where \({C}_{ref}^{s,i}\) is the clock solution of the reference AC. \({dC}_{a,nl}^{s}\) is the nonlinear estimate for the AC clock solution of \(a\). \({O}_{a}^{s}\) and \({D}_{a}^{s}\) are the bias and trend of the time scale. The estimates of \({dC}_{a,nl}^{s}\) and \({O}_{a}^{s}\) in Eq. (11) are linearly correlated. To separate them, the following constraint is added at each epoch:
where \(n\) is the number of common satellites for constellation \(s\).
The clock difference of the solution GFM with regard to the COM is presented in Fig. 5 with the correction of constellationspecific nonlinear estimates using Eq. (11). It is clear that the nonlinear variations in the clock residuals disappear, and the mean STDev decreases to 0.048 and 0.076 ns from 0.121 and 0.711 ns for GPS and GLONASS, respectively. Equation (11) can also be used to align the nonlinear time scale of the JAM solution to the reference, and the STDevs of GPS and GLONASS are reduced from 1.149 to 0.048 ns and from 1.148 ns to 0.081 ns, respectively.
ACspecific weight determination for each constellation
Since the quality of the clock product depends on the processing strategies employed by individual ACs and different performances of AC clock solutions within constellations is noticed, the weight of an individual AC (i.e., \({w}_{a}^{s}\)) for each constellation is determined separately by
where \({\sigma }_{a}^{s}\) is the rootmeansquare (RMS) of the clock residuals for constellation \(s\) of AC \(a\). The clock residuals are obtained by aligning the individual AC to the combined solution as follows:
where \({C}_{cmb}^{s,i}\) is the combined clock and \({C}_{a2}^{s,i}\) are the AC solutions with the correction of \({d}_{a,frm}^{s,i}\), inconsistent ASB and nonlinear variation. \({O}_{a2}^{s}\) and \({D}_{a2}^{s}\) are the transformation parameters, i.e., intercept and drift of an individual constellation. \({v}_{a2}^{s,i}\) is the AC clock residuals compared to the combined solution.
The flowchart for multiGNSS clock combination proposed in this study is illustrated in Fig. 6. First, ASBs are estimated using global stations and AC orbit and clock products. Afterward, the frame correction is computed by Eq. (2) and applied to the individual AC clock solution. Second, the individual solution is aligned to a common time scale to speed up the convergence of the clock combination, as proposed by Kouba and Springer (2001). In this study, the final GPS combined clock provided by IGS is selected as the reference to keep the time scale of the combined solution consistent with IGS time scale. Moreover, the inconsistent nonlinear time scale applied in the AC solution is also removed by alignment to IGS. Third, one AC with constant ISB estimates is selected as a reference, and the constellationspecific nonlinear variation is calibrated using Eqs. (11) and (12). Fourth, an iterative combination is employed to adjust the weights of the individual AC solutions until no outliers are detected or the maximum iteration number (i.e., 3) is reached, and a final combination and comparison is performed.
Assessment of the combined clock solution
The clock solutions from the ACs are compared to the combined clocks in this section. Moreover, the observations from the global stations shown in Fig. 2, as well as individual ACs and combined orbits and clocks, are input to PANDA package for daily PPP of the whole year of 2018 (Liu and Ge 2003).
Consistency of multiGNSS clocks
Figure 7 shows the daily STDev with respect to the combined solutions for each AC. For clear illustration, the daily results are smoothed using a moving window with a 7day width. Because no more than 4 satellites are available for QZSS during the experiment, the results of QZSS are not presented. The yearly median of STDevs is also listed in the legend of Fig. 7 for each AC. For GPS, similar performances are obtained for different ACs, and the consistency between the ACs and the combined solution is in the range of 8\(\)16 ps. An obvious decrease can be observed after 22 June 2018 for COM, and it may be caused by the ambiguous integer solution for MGEX clocks (Dach et al. 2019).
During the selected period, the STDevs of GLONASS are approximately 4 times larger than those of GPS. The ESM shows the best consistency of 27 ps, followed by GFM, WUM and COM, where the GRM and JAM solutions are approximately 60 ps for GLONASS. For BDS2, the GEO satellites are excluded from the statistics, and the consistency of the other satellites can reach 20 ps. However, there are frequent and noticeable jumps for the BDS2 clocks, especially for the COM solution. This could be partly caused by an inaccurate attitude and solar radiation pressure model that was used until a dedicated model was activated at the beginning of July (Dach et al. 2019).
For Galileo, the COM, ESM and GFM solutions show good consistency, with a STDev of approximately 10 ps. An obvious improvement can be seen for COM when the fixedambiguity clock products were generated at the end of June. The GRM solution also benefited from fixed undifferenced phase observations after 7 October 2018 (Perosanz et al. 2018), and the STDev decreased from 40 to 15 ps. The median STDev reached 25 ps for the WUM solution.
To show the validity of the ASB calibration approach proposed in this study, we recombined the clock without ASB correction. Table 2 lists the RMS values of the clock residuals for different constellations with and without ASB correction. Compared to the solutions without ASB correction, the RMS of solutions with ASB correction noticeably decreases, especially for GLONASS and BDS2. For GPS, BDS2 and Galileo, RMS values smaller than 50 ps are achieved for AC solutions with ASB correction. The corresponding values of GLONASS are larger than the others, and the RMS is approximately 37\(\)91 ps. The RMS of GLONASS and BDS2 can reach a few nanoseconds for some solutions without ASB correction (i.e., COM and ESM for BDS2, GFM and GRM for GLONASS). Generally, using the ASB correction approach proposed in this study, the RMS of clock residuals can be reduced by approximately 85\(\)88%, 76\(\)98%, 90\(\)98%, and 69\(\)93% for GPS, GLONASS, BDS2 and Galileo, respectively.
The ASB residuals could also be used as an internal metric of validation of the ASB correction approach. First, the ASB biases are estimated for each AC with its orbit and clock products, and a combined ASB is generated by means of averaging based on different ACs. Second, the combined satellite clock solutions (i.e., W) are obtained based on the processing strategy illustrated in Fig. 6. Additionally, the combination without ASB correction applied to the AC solutions is generated (i.e., WO). Third, both the W and WO solutions are used to estimate the ASB again as in the first step, and the estimates are compared to the ASB combination. Figure 8 shows the daily RMS of the ASB residuals for each constellation. It is clear that the agreement between the ASB combination and the estimates from the W solution has a significant improvement compared to the WO solution, and the RMS decreases from 46, 100, 667, 85 ps to 19, 34, 86, 13 ps for GPS, GLONASS, BDS2 and Galileo, respectively. This confirms the validation of ASB correction for AC solutions before clock combination.
Precise point positioning with multiGNSS combined orbit and clock
In this section, static daily PPP is performed with the combined multiGNSS orbit and clock solutions of the whole year of 2018 for the global stations shown in Fig. 2. The PPP strategy is similar to that described for ASB estimation. It should be noted that the ASB is no longer estimated. The positions are compared to the IGS daily coordinates, and a 7parameter Helmert transformation is applied to remove the systematic errors.
Position accuracy
To assess the combined orbit and clock consistency, Table 3 lists the statistics of position accuracy for different constellations. Due to an insufficient number of satellites, BDS2 and QZSSonly PPP is not conducted. In general, GPS achieves the best position accuracy in the three singleconstellation tests, and the average RMS reaches 0.31, 0.17 and 0.45 cm in the east, north and up components, respectively. Compared to GPS positioning, the position accuracy of GLONASS and Galileo is slightly lower. The yearly average RMS is 0.49, 0.31 and 0.69 cm in the three components for GLONASS, whereas they are 0.52, 0.40 and 0.90 cm for Galileo. Compared to the GPSonly PPP, the multiGNSS solutions improve the accuracy by 16–26% in the east component (Table 3). Due to the high stability and accuracy of the combined orbits and clock solutions, the competitive PPP results can be obtained compared with the best AC solution.
Reference frame parameters
The consistency of the reference frame achieved by a precise orbit and clock can be evaluated by comparing the transformation parameters regarding the International Terrestrial Reference Frame (i.e., IGS14). Figure 9 illustrates the translation parameters derived from PPP with a combined orbit and clock. The daily translations of PPP fall into the range of \(\pm 3\mathrm{ mm}\) for each constellation, and the STDevs are less than 1.5 mm for all three components. Among the singlesystem solutions, the translations of all components have a negative bias, except the Ytranslation of Galileo. Additionally, translations derived from the combined GPS solution are more centered than those of GLONASS and Galileo. Due to the relatively high position noise of Galileo, the translations for Galileo are the most scattered. For PPP using the combined multiGNSS orbit/clock, the translations vary similarly to those of a single GPS constellation.
The correlation coefficients within translations of different PPP solutions are computed and the coefficients are listed in Table 4. Although the translations are contaminated by the noise and lower accuracy of the position results, especially for Galileo, a moderate correlation is still seen within the translations of different constellations. Due to the contribution of individual constellations in the multiGNSS integrated PPP, the correlation coefficients of translations between singleconstellation and multiGNSS solutions are expected to be larger than that between two constellations. The moderate correlation within translations also indicates the high consistency between the combined orbit and clock.
For the rotation parameters presented in Fig. 10, the Galileo solutions have the largest rotations, whereas those of GPS have the smallest. However, there is no obvious difference for rotations derived from PPP using the combined orbits and clocks of different constellations, and most of the daily rotations are in the range of −50 to 50 uas. Compared to the correlation coefficients of translations, a lower correlation is noticed for rotations (Table 4), which can be explained by the fact that the AC orbits are aligned to the IGS final orbit during orbit combination.
For the scale parameters demonstrated in Fig. 11, inconsistent scale biases are identified for different constellations, and the yearly averages are −0.06, 0.06 and −0.26 ppb for GPS, GLONASS and Galileo, respectively. Compared to the scales derived from GPS and GLONASS solutions, the scales for Galileo solutions are more scattered. Nevertheless, a larger correlation between Galileo and GPS solutions than that between GLONASS and GPS is obtained (i.e., 0.23 versus 0.09, Table 4).
Table 5 lists the statistics of transformation parameters of multiGNSS integrated PPP for comparison between the ACs and combined solution (WMC). First, an offset larger than 1.0 mm is observed for some ACs, i.e., −1.8 mm (DY) for COM, 1.2 mm (DX) and −1.5 mm (DZ) for GRM, and −3.8 mm (DZ) for WUM solutions. Since the multiGNSS combinations are dominated by ACs, it is expected that a consistent negative bias for the translations of PPP using combined orbits/clocks can be obtained. Additionally, offsets larger than 1.0 mm (i.e., 33 uas for an equatorial point with an Earth radius of 6371 km) are obtained for the RX of the GRM and the RY and RZ of the JAM. Considering that the orientation of the combined orbits is already aligned to the IGS terrestrial frame, no significant offsets are identified for the WMC solution. All the scales derived from PPP using the multiGNSS orbits/clocks of ACs show a negative bias, which may be related to the antenna phase correction, and further study is needed. Finally, the combined multiGNSS orbit and clock determine the robust PPP results, and the STDevs of the transformation parameters are competitive with those of the best AC.
Impact of the proposed strategy for clock combination on PPP results
To make a comparison between the combined clocks proposed in this contribution and the solutions with traditional strategy without consideration of the ASB and nonlinear variation, static and kinematic PPP with float ambiguity were conducted for the first 7 days of 2018. A least square filter was adopted to determine position of epoch by epoch with an interval of 30 s, and an initial constraint of 50 m was used for the position parameters of PPP in static and kinematic mode. Moreover, the process noise of 5 m was employed for position estimates for kinematic PPP. The estimates of ISB was taken as daily constant for multiGNSS integrated PPP. The station coordinates from the static daily PPP were used as true values, and the positioning errors of 24 h are computed. Due to the convergence, the positioning results of the first hour were excluded for statistics. Some epochs with positioning errors larger than 5sigma or number of satellites less than 5 were also removed.
Figures 12 and 13 demonstrate the positioning error of multiGNSS integrated PPP in static and kinematic mode, respectively, for station STK2 during the period of January 1, 2018. An obvious bias is noticed, especially in the up component, which should be induced by the uncalibrated ASB and nonlinear variation related to time scale for the combined clocks. Befitted from the proposed strategy of clock combination (i.e., WMC clocks), an improvement of 0.2, 0.1 and 0.6 cm are obtained for the east, north and up component, respectively.
For kinematic PPP of station STK2 with combined clocks without consideration of ASB and nonlinear variations (i.e., RAW clocks), an obvious fluctuation occurs at the time between the 6th and 10th hour in the up component (Fig. 13). Compared to the PPP with RAW clocks, a notable decrease of RMS is obtained in the up component for PPP with WMC clocks, and the RMS of position residuals are 0.8, 0.9 and 2.0 cm for the three components.
Table 6 summarizes the statistic results of PPP tests with single and integrated constellations. Due to insufficient satellites available to conduct Galileoonly and BDSonly PPP, only the singleconstellation PPP results of GPS and GLONASS are presented. Compared to the PPP results with RAW clocks, the average improvement of GPSonly positioning accuracy in the three components is less than 6% (i.e., 5.7% for static mode) for PPP with WMC clocks. However, the average improvement could reach 11.1% for GLONASSonly PPP in static mode, which can be explained by the optimal weights designed for ACs when ASB and nonlinear variations in RAW clocks are considered. Compared to the PPP results of individual constellation, the multiGNSS integrated PPP with WMC clocks in static mode achieves the smallest positioning errors, and the RMS values are 0.63, 0.36 and 1.77 cm in east, north and up components, respectively. The smallest positioning error is also obtained for PPP in kinematic mode, and the corresponding RMS are 1.63, 1.16 and 3.82 cm in the three components. Compared to the results of PPP with RAW clocks, a slight decrease of RMS is found for multiGNSS PPP with WMC clocks, and average improvements of 9.8 and 5.3% are achieved for PPP in static and kinematic mode, respectively. It could be explained by the fact that the inconsistent nonlinear variation within multiGNSS in RAW clocks could not be compensated by the daily constant ISB estimates.
Discussion and conclusions
MultiGNSS combined clocks are derived with solutions from IGS MGEX ACs and ESA for the whole year of 2018. The inconsistent ASB and nonlinear variation induced by the time scale and nonconstant ISB estimates are addressed in this study to achieve reliable clock combination.
Hauschild et al. (2019) pointed out that satellite clock products suffer from chip shape distortions in the ranging signals when the clocks are determined by a network with mixed receivers. As IGS stations with different receivers are used by ACs for precise orbit and clock determination, the analysis center and satellitespecific bias is prone to contaminating the satellite clock solution. Fortunately, this bias can be calibrated with the proposed approach, and a high consistency between the clock and ASB difference is found.
Two factors are identified in the inconsistent nonlinear variation in clock solutions in this study: one is the time scale, and the other one is the nonconstant ISB estimated in clock computation. Since an arbitrary time scale (e.g., a reference station clock) is usually selected for the satellite clock solution, inconsistent nonlinear variations from the reference clock are common for all satellites and can also be extracted from the average of the fitting residuals. However, the inconsistent nonlinear variations induced by the second factor are always constellation dependent and can only be eliminated by aligning the clocks to a reference solution. With respect to modeling the ISB as a constant, estimating the ISB with a random walk can achieve better positioning accuracy (Liu et al. 2019). Once the ISBs are estimated with a random walk for all ACs, there is no reference clock to be selected for the calibration of the nonlinear variations. Hence, a temporal ISB should be added in the current ASB calibration approach for the correction of inconsistent nonlinear variations.
With respect to the combined solution, the interAC clock agreement is 8–16, 27–58, 13–27 and 9–36 ps for GPS, GLONASS, BDS2 and Galileo, respectively. Outliers up to several nanoseconds sometimes occur for ACs, which demonstrates the necessity of clock combination. The analysis center and constellationspecific weight based on clock residuals is limited by a large ASB, which can be calibrated using the proposed approach. When the ASB is corrected, the RMS of the AC clock residuals obviously decreases, and consistencies of 14–26, 37–91, 33–48 and 12–44 ps for GPS, GLONASS, BDS2 and Galileo, respectively, are obtained. The average RMS of the ASB residuals also decreases from 46 to 19, from 100 to 34, from 667 to 86 and from 85 to 13 ps for GPS, GLONASS, BDS2 and Galileo, respectively.
Daily static PPP with float ambiguity was performed for different constellations to assess the combined multiGNSS orbit and clock while addressing the positioning accuracy and reference frame consistency with regard to the IGS terrestrial frame. For individual constellations, the best position accuracies of 0.31, 0.17 and 0.46 cm in the east, north and up directions, respectively, are achieved for GPS. The position accuracy of the east component is improved by approximately 1 mm for multiGNSS integrated PPP. The consistency of the combined orbit and clock is also validated by the correlation between translations. However, marginal correlations and offsets of 0.1–0.3 ppb are identified for the scale parameter within the PPP results of different constellations due to inconsistent observational and dynamic models. Generally, the combined multiGNSS orbit/clock shows competitive performance with that of the best AC in both positioning accuracy and reference frame parameters.
Further improvements are expected as an increasing number of ACs provide multiGNSS orbits, clocks and other related precise products (e.g., attitudes), and an improved orbit model and consistent correction could also benefit the combined solution. The history of multiGNSS combined orbit/clock solutions since 2018 will be generated in the near future.
Availability of data and materials
MultiGNSS precise orbits and global station observations are available by an anonymous user via ftp://igs.gnsswhu.cn. The precise products of ESA can be obtained from http://navigationoffice.esa.int/products. The combined orbit and clock solutions can also be freely accessed at ftp://igs.gnsswhu.cn/whu/MGEX.
Abbreviations
 AC:

Analysis center
 ACC:

Analysis center coordinator
 ASB:

Analysis center and satellitespecific bias
 BDS:

BeiDou Satellite Navigation System
 EOP:

Earth Orientation Parameter
 GNSS:

Global Navigation Satellite System
 IFCB:

Interfrequency code bias
 iGMAS:

International GNSS Monitoring and Assessment System
 IGS:

International GNSS Service
 IRC:

Integer recovery clocks
 ISB:

Intersystem bias
 MGEX:

MultiGNSS Pilot Project
 PPP:

Precise point positioning
 PPPAR:

PPP with Integer Ambiguity Resolution
 PNT:

Positioning, navigation and timing
 RMS:

Root mean square
 STDev:

Standard deviation
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Acknowledgements
The IGS MGEX and its analysis centers are greatly acknowledged for providing the multiGNSS products and observations.
Funding
This work was supported by National Natural Science Foundation of China (41974035, 42030109, 42204019), Young elite scientists sponsorship program by CAST (2018QNRC001) and Fundamental Research Funds for the Central Universities (2042021kf0064).
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Contributions
All the authors contributed to the design of this study. Guo Chen and Qile Zhao came up with the idea. Guo Chen processed the clock combination experiments, generated the results and wrote the draft. Jing Guo revised the manuscript, Na Wei participated in the analysis of transformation parameters, Min Li and Jun Tao participated in collection of MGEX orbit and clock products, and carried out the precise point positioning. All authors read and approved the final manuscript.
Authors' information
Guo Chen is a postdoctoral researcher at GNSS Research Center of Wuhan University. He received his PhD degree at Wuhan University in 2019. His current research mainly focuses on multiGNSS products combination and performance evaluation.
Jing Guo is an associate professor at GNSS Research Center of Wuhan University. He received his doctor degrees at Wuhan University in2014, and worked as research associate at Newcastle University in 2017. Currently, he works on GNSS data processing, particularly the precise orbit determination for GNSS satellites, and is also responsible for the routine data processing of IGS MGEX and iGMAS at Wuhan University.
Na Wei is an associate professor at GNSS Research Center of Wuhan University. She received her PhD degree at Wuhan University in 2011. Her current research mainly focuses on reference frame and surface loading.
Min Li received a Ph.D. degree from Wuhan University in 2011. He is currently a professor of GNSS Research Center of Wuhan University, and his main work focuses on GNSS satellite orbit determination and multiGNSS positioning.
Qile Zhao is a professor of GNSS Research Center of Wuhan University. He received his PhD degree at Wuhan University in 2004. His current research interests are precise orbit determination of GNSS and low Earth orbit satellites, and multiGNSS highprecision positioning.
Jun Tao received his master's degree from Wuhan University, Wuhan, China, in 2019, where is currently a Ph.D. candidate at the School of Geodesy and Geomatics. His main research interest is realtime GNSS positioning and its application.
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Chen, G., Guo, J., Wei, N. et al. MultiGNSS clock combination with consideration of inconsistent nonlinear variation and satellitespecific bias. Earth Planets Space 74, 142 (2022). https://doi.org/10.1186/s40623022017026
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DOI: https://doi.org/10.1186/s40623022017026