Keywords

1 Combination of Space Geodetic Techniques

In the current realisation of the International Terrestrial Reference Frame (ITRF), Earth Orientation Parameters (EOPs) are heterogeneously determined. Polar motion (x-pole and y-pole) is estimated based on the combination of the four space geodetic techniques, whereas their rates are only based on two techniques, namely Global Navigation Satellite Systems (GNSS) and Very Long Baseline Interferometry (VLBI). Moreover, the Earth’s rotation angle (UT1-UTC) and Length of Day (LOD) are taken solely from the VLBI solution (Altamimi et al. 2016). In addition, the combination of troposphere parameters from VLBI, DORIS and GNSS through the use of tropospheric ties at fundamental sites is not implemented in ITRF’s combination strategy. Hence, a rigorous combination of all parameter types common to all techniques, with consistent EOPs and with appropriate inter-technique tropospheric ties, is still missing. A consistent estimation of the TRF, capable of exploiting the advantages of the dense GNSS network with continuous observations and excellent geometry, and the full set of EOP delivered by VLBI, is required to achieve higher precision levels following the requirements given in Rothacher et al. (2009), and it is a pre-requisite for the full exploitation of dedicated co-location satellite mission concepts, such as Delva et al. (2023). A complete definition of the standards, models and parametrisation required for the consistent processing of the different space geodetic techniques is presented by Rothacher et al. (2010), within the scope of the GGOS Germany initiative (GGOS-D). This work discusses the important aspects of a rigorous combination of space geodetic techniques, and emphasises the need for the computation of consistent time series of the parameters relevant to the different techniques, extending the parameter space to link geometry, Earth rotation, and gravity field. In their comprehensive work, Coulot et al. (2007) carried out an early attempt of combining GPS, VLBI, SLR and DORIS data on the observation level. With data covering one year (2002), their work strove to perform the combination by estimating parameters simultaneously, while making use of all their correlation information. Thaller (2008) performed a combination of VLBI, GPS, and SLR normal equations, during the CONT02 campaign, in order to estimate station coordinates, EOPs, and troposphere parameters. Her approach aimed at the homogenisation of the normal equations, through the use of identical a priori models in the estimation of the parameters common to the three techniques. Her work performed the combination at the normal equation level, with all common parameter types included, where the improvement of the combined solution w.r.t. the individual technique solution is evident. In particular, she accomplished a successful estimation of UT1-UTC and LOD, and the stabilisation of the determination of the height component of the coordinates thanks to the common estimation of troposphere zenith delays and gradients. More recently Diamantidis et al. (2021) performed a combination at the observation level of VLBI and GNSS data during the CONT17 campaign, using a unified piece of software based on a batch least-squares estimator. Their work reports an improvement in the coordinate repeatabilities, polar motion, and UT1-UTC of 25%, 20% and 30%, respectively, with respect to the single technique solutions. In a similar fashion, Wang et al. (2022) performs the integrated processing of VLBI and GNSS data, to achieve a combination at the observation level. The main characteristic of their approach was the use of the tropospheric ties among VLBI and GNSS co-located stations, where residual zenith wet delays (ZWD) and gradients for VLBI and GNSS were estimated. As their work used different tropospheric tie setups, the improvement of the coordinate repeatabilites range between 12% and 28%, while for EOPs it goes from 2% up to 18%.

2 Dataset and Processing Strategy

The test scenario to validate our strategy was the data of the Continuous VLBI Campaign 2017 (CONT17). CONT17 was a campaign of continuous VLBI sessions, carried out between November 28, 2017, and December 12, 2017. It was composed of three independent networks observed: two legacy S/X networks with 14 stations each, and one VGOS broadband network consisting of six stations (Behrend et al. 2020). For the scope of our work, we only used the two legacy networks. The geodetic VLBI data of this campaign were extracted from the corresponding NGS cards. Since we were only using the legacy networks, the processing of the data was performed using the S/X part of the source catalogue of the 3rd realization of the International Celestial Reference Frame (ICRF3) of Charlot et al. (2020). To complement the VLBI observations, we selected about 180 GNSS stations of the International GNSS Service (IGS) network (Dow et al. 2009) covering the same time interval, with several stations co-located with the VLBI telescopes (in most of the cases). The integrated processing of the different techniques is done at the observation level, which provides the most rigorous and consistent solution, especially, when all the possible ties are considered. To guarantee the consistency, it is best performed with a single piece of software capable of processing all the techniques with state-of-the-art models and identical parametrisation. To handle the processing of the VLBI and GNSS data at the observation level, we used a modified version of the Bernese GNSS Software v5.2 (Dach et al. 2015), capable of handling VLBI data. This so-called Bernese v5.2 – VLBI Version, inherits all the GNSS & SLR capabilities of the original version: Pre-processing, outlier detection, residual screening, time-series analysis, daily and session processing, generation of normal equations, and more. The main advantages of this development are: (1) the use of an identical parametrisation for all the techniques. (e.g. piece-wise linear estimates, offset-drift estimates, interpolation methods, …), (2) the use of identical models for all techniques, where applicable (e.g station motion, tropospheric refraction, loading, troposphere), with identical handling of parameter constraints, (3) appropriate datum definition such as No-Net-Rotation (NNR), No-Net-Translation (NNT), No-Net-Scale, fixed coordinates, and (4) the implementation of coordinate and tropospheric ties.

Table 1 shows a summary of the modelling and a-priori information used for the rigorous combination of VLBI and GNSS data. For the combination of the data, we estimated most of the common parameters with daily resolution: daily station coordinates using the NNR–NNT condition, daily EOPs: polar motion, UT1-UTC, LOD, and celestial pole offsets, and their corresponding rates of change. Zenith tropospheric delays were estimated with 1-h resolution and tropospheric gradients every 24 h. We estimated VLBI clock offsets piece-wise linearly with intervals of 3 h. Finally, for the 15-day rigorous combination, we additionally used the available terrestrial ties and our approach for tropospheric ties.

Table 1 Modelling and a-priori information used for the rigorous combination of VLBI and GNSS data
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3 Realisation of Tropospheric Ties

For the modelling of the troposphere, we used as a-priori values for the zenith hydrostatic delays and mapping function the data of the Vienna Mapping Function 1 (VMF1) (Bohm et al. 2006). The use of this type of modelling ensures that the zenith total delay (ZTD) difference between GNSS and VLBI at co-located stations, caused by the height difference, are modelled in advance. The residual wet delays were then estimated as one-hourly piece-wise-linear functions and the tropospheric gradients with daily resolution. In particular, for the baseline FD-VLBA–MDO1 the modelled mean \(\varDelta \)ZHD has a significantly large value of 91.7 mm, mostly due to the large height difference between the two stations (ca. 398 m). The statistics associated with the estimated \(\varDelta \)ZWD also have an inferior performance: 7.4 mm for the mean, 4.6 mm of standard deviation, and an RMS of 8.7 mm. This clearly shows that the height difference at the co-location site Fort Davis is too large to apply a tropospheric tie and the ZWD parameters for MDO1 and FD-VLBI cannot be stacked. At the remaining co-location sites, we observed that the \(\varDelta \)ZWDs were not correlated with the height difference, and that the ZWD mean values vary within \(\pm 5\) mm (excluding FD-VLBI–MDO1). These mean differences are shown in Fig. 1. We define the tropospheric tie as the difference in the tropospheric delay between the reference points of the VLBI and the GNSS antennas. Since the a-priori values of these delays are based on state-of-the-art global numerical weather prediction models, the difference between the delays at two stations caused by the height difference is modelled in advance (Wang et al. 2022) and only the delays caused by the residual troposphere should be considered. Moreover, the mean differences shown in Fig. 1 are not taken into account in the tropospheric ties, but are interpreted as resulting from the estimation uncertainty and from small systematic-effects that are not due to the troposphere or the troposphere delays modelling.

Fig. 1
figure 1

Summary of the mean zenith wet delay differences for co-located sites, w.r.t. the height difference in the baseline

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4 Optimal Weighting

An important aspect of the combination is the weighting of each technique, as the quality of the individual techniques varies considerably. The large contrast in the formal errors of each solution supports the need for an adequate inter-technique weighting. Our approach follows the idea of Thaller (2008), using coordinate repeatabilities as the base of the weights, since they are directly part of the terrestrial reference frame. First, the quadratic mean repeatability of the station coordinates for all co-located stations over the 15 days of the CONT17 campaign was calculated, as an indicator of the quality of the observations (and the solution):

$$\displaystyle \begin{aligned} r^2 = \dfrac{r^2_e + r^2_n + r^2_u}{3} \end{aligned} $$

With this, a relative weighting between techniques i and j was computed:

$$\displaystyle \begin{aligned}w_{rep_{ij}} = \dfrac{r^2_i}{r^2_j}\end{aligned} $$

Then, the sum of the main-diagonal elements of the normal equation matrix was calculated:

$$\displaystyle \begin{aligned}\overline{N} = \dfrac{1}{n_{crd}} \sum_{z=1}^{n_{crd}} N_{zz}\end{aligned} $$

where the parameter \(n_{crd}\) refers to the number of diagonal elements of the normal equation matrix. Since the weight is based on repeatabilities, only coordinate elements were considered. Moreover, only the coordinates of the co-location sites were used. Finally, the \(\overline {N}\) values of each technique were combined with the weight of the corresponding parameter, to obtain the weighting of technique j with respect to technique i:

$$\displaystyle \begin{aligned}w_{ij} = \dfrac{\overline{N}_i}{\overline{N}_j} \cdot w_{rep_{ij}}\end{aligned} $$

Table 2 shows the results of the calculation of the optimal inter-technique weight. For the data of the CONT17 campaign, an optimal weight for the VLBI NEQs of 0.276 was determined.

Table 2 Results of the calculation of the optimal inter-technique weight for the data of the CONT17 campaign. Repeatabilities are given for east, north and up components, respectively, in millimetres
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5 Validation of the Optimal Weighting

To test the adequacy of the weight determined in Sect. 4, we studied the performance of the repeatabilities of the combined solution, for typical cases of inter-technique weights, taking as reference the GNSS solution and using the parametrisation of Sect. 2. This is, in all the cases the GNSS solution had a weight of 1, while we vary the weight of the VLBI solution. A large number of cases was investigated, but four specific cases give the essence of the behaviour. These are: (1) \(100^{-2}\), meaning that the GNSS observations had a considerably larger contribution to the final solution. (2) 0.276, the “optimal weight” of Sect. 4. (3) 1, meaning that both techniques were equally weighted. (4) \(100^2\), meaning that the VLBI observations had a considerably larger contribution to the final solution. Figure 2 shows an example of the repeatabilities for two co-location sites, Pietown and Brewster (USA). For these two particular cases, the repeatabilities of the solution with the optimal weight shows a marginally better performance, especially when it comes to the height component when compared to the solution with equal weights. From these two examples, it is also noticeable that the solutions with larger weights for either VLBI or GNSS underperform when compared to the solution with optimal weights. Moreover, Fig. 3 displays the RMS of the repeatabilities for the combined solutions over the 15 days of the CONT17 campaign, when using different inter-technique weights, using exclusively the stations at co-location sites. While the repeatabilities of the horizontal components remain almost unchanged, there is an improvement in the height component when analysing all stations together (top plot of Fig. 3). The differences are more evident when looking at the stations separated by technique, especially for the GNSS case, where the height component of the solution with the optimal weight outperforms all the other solutions by more than 10%. Since the optimal inter-technique weight was based on the coordinate repeatabilites, it is fair to assume that its influence is not so evident in the remaining parameters.

Fig. 2
figure 2

Repeatabilities of the combined solution, for different inter-technique weights. The first three values (blue, red and yellow) are the repeatabilities for the GNSS station, while the remaining three (purple, green and cyan) correspond to the VLBI station

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Fig. 3
figure 3

RMS of the repeatabilities of the combined solution over the 15 days of the CONT17 campaign, for different inter-technique weights. The RMS value was calculated using all the co-located stations

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6 Realisation of the Coordinate Ties

A central element in the combination of space geodetic techniques is the use of coordinate ties (Sarti et al. 2013), and in particular, the quality with which they have been determined. To realise the coordinate ties, we used the ITRF2014 coordinates of the GNSS stations and add the coordinate ties to get VLBI coordinates. Then, we applied relative constraints to constrain the vector between the co-located VLBI and GNSS stations with a certain weight. This ensures that the coordinates were consistent with the coordinate tie values. To investigate their quality, we used as relative constraints the formal errors of the coordinates contained in the SINEX files of the coordinate ties of the ITRF2014 solution, from the IERS website.Footnote 1 These formal errors (\(\sigma _{snx}\)) were the starting point for the remaining test solutions. We calculated combined solutions with relative constraints of \(10^{1}\sigma _{snx}\), \(10^{-1}\sigma _{snx}\) and \(10^{-2}\sigma _{snx}\), and analysed the coordinate repeatabilities. Figure 4 shows an example of these repeatabilities for the sites Brewster (USA) and Fortaleza (Brasil). It is expected that a strong constraint on the coordinate tie causes the repeatabilities of the two co-located stations to converge to the same value. We observed that the quality of the coordinate ties varies among the co-location sites, and that different co-location sites have different responses to the relative constraint used. The two co-location sites shown in Fig. 4 represent this behaviour. For the baseline BR_VLBA–BREW, the original relative constraints (\(\sigma _{snx}\)) end up in different repeatabilities for the two co-located sites, especially for the up component. The same is true when using a softer relative constraint. However, when using stronger versions of \(\sigma _{snx}\), the repeatabilities of the two stations converge to the same (low) values. In contrast, the co-location baseline FORTLEZA–BRFT shows larger differences in the repeatabilities of the vertical component when using stronger values for \(\sigma _{snx}\) and does not converge to the same values for both co-location sites, indicating strong inconsistencies between the two techniques. In this case, a weaker constraint of the coordinate ties delivers the best results for this co-location site. Based on this analysis, we selected the optimal set of coordinate tie constraints, for each baseline at the co-location sites, so that it minimised the repeatabilities of the two stations, while trying to get them to converge to the same value. Finally, we calculated the RMS of the coordinate repeatabilities when using these appropriate constraints for the coordinate ties, and display them in Fig. 5. The benefits of the solution with optimal relative constraints are evident. The repeatabilities improve by 18%, 13%, and 14%, for the east, north, and height components, respectively (top plot of Fig. 5). When looking only at the GNSS stations, the improvements are 12% for the horizontal, and 11% for the vertical component. The largest improvement can be seen in the repeatabilities of the VLBI stations, with 21% 12%, and 17%, for the east, north and height component, respectively (bottom plots of Fig. 5).

Fig. 4
figure 4

Coordinate repeatabilities of the combined solution, regarding the type of constraint used for the coordinate tie. The first three values (blue, red and yellow) are the repeatabilities for the GNSS station, while the remaining three (purple, green and cyan) correspond to the VLBI station

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Fig. 5
figure 5

RMS of the repeatabilities of the combined solution over the 15 days of the CONT17 campaign using appropriate constraints for coordinate ties. All values in mm

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7 Differences of EOPs to IGS Solution

To assess the improvement in the EOPs we used the IGS final solutionsFootnote 2 as reference for the comparison, and the two 15-days rigorously combined solutions of Sects. 5 and 6 were analysed. The RMS of the differences between the daily EOP estimates and the IGS solution are displayed in Fig. 6. Both solutions agree with the IGS solution at approximately the same level for the LOD and polar motion rate parameters. However, there is a large improvement in both polar motion components: 36% and 42% for the X and Y components, respectively. It should be mentioned that it is difficult to find a solution that can be used as ground truth for a comparison, as the rigorously combined solution is expected to be better than any other solution.

Fig. 6
figure 6

RMS of differences of the daily estimated EOP, in the combined solutions, with respect to the IGS solution. Notice the different units (left-hand side of the plot) for each type of parameter

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8 Rigorous and Single Technique Solutions

The final step in the study of the rigorous combination is the comparison of the relevant parameters, to the single-technique solutions. Moreover, as an additional reference for comparison, we included a rigorous combination, where only coordinate ties were used. We start with the analysis of the RMS for the coordinate repeatabilities. While the combined results of both techniques may show a decrease in the performance of the rigorous solution with respect to the GNSS solution (top plot of Fig. 7) when separating the repeatabilities per technique, the benefits of the combined solution are more evident (bottom plot of Fig. 7). The improvement in the repeatabilities of the GNSS stations in the rigorous solution regarding the GNSS-only solution are 22%, 24%, and 19%, for east, north and height, respectively. Similarly, the improvement regarding the VLBI-only solution amounts to 2% and 14% for the north and height component, respectively. We also observe an improvement in the coordinate repeatabilities when comparing the rigorous solution with coordinate and tropospheric ties with the rigorous solution with only coordinate ties, as expected mainly in height, with the height component of the former improving the performance by 11% (only GNSS stations), 7% (only VLBI stations), and 6% (all stations included). Additionally, the RMS of the difference of the EOPs regarding the IGS final solution is investigated, and displayed in Fig. 8. Once again, the rigorous solution outperforms the single-technique solutions in the polar motion estimates, with an improvement of 35% and 9% regarding the GNSS-only solution, for the X and Y components, respectively, and 25% and 19% regarding the VLBI-only solution, for the X and Y components, respectively. The three solutions agree with the IGS solution at approximately the same level for the UT1-UTC, with the rigorous solution helping to improve the results in the LOD estimate: 48% and 10%, compared to the GNSS-only and VLBI-only solutions, respectively. Polar motion rates show a favourable tendency towards the rigorous solution: 20% and 2% for the rate of the X and Y component, respectively, compared to the GNSS-only solution, and 9% and 20% for the rate of the X and Y component, respectively, regarding the VLBI-only solution. The comparison of the rigorous solution with coordinate and tropospheric ties with the rigorous solution with only coordinate ties showed that both approaches yield similar results regarding the LOD estimation, with an improvement of the polar motion of 14% and 5%, for the X and Y components, respectively.

Fig. 7
figure 7

RMS of coordinate repeatabilities [mm] for the individual technique solutions, and the combined solutions

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Fig. 8
figure 8

RMS of daily EOP differences to IGS for the individual technique solutions, and the combined solutions. Notice the different units on the left side for each parameter

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9 Summary and Outlook

A rigorously combined solution for the estimation of geodetic parameters including GNSS and VLBI data has been achieved. This solution, based on the data of the CONT17 campaign plus GNSS/IGS data, profits from the use of coordinate ties with appropriate constraints, and troposphere ties at co-location sites with carefully chosen constraint levels, as well as from a tailored inter-technique weighting scheme based on the repeatabilities of the station coordinates. The combined solution was processed in a single state-of-the-art software, Bernese v5.2 – VLBI Version, where not only the a priori modelling and the parametrisation for both techniques was exactly the same, but also the full variance-covariance information of all the estimates, and the constraints for all the parameters were used throughout the estimation process. The combined solution with coordinate and troposphere ties generally improves the precision of all the estimated geodetic parameters. In particular, the repeatabilities of the station coordinates are improved by 22%, 24%, and 19%, for east, north, and height, respectively, compared to the GNSS-only solution, and by 2% and 14% for the north and height component, respectively, compared to the VLBI-only solution. Additionally, the EOPs estimates are also improved, with the rigorous solution outperforming the single-technique solutions in the polar motion estimates, by 35% and 9% compared to the GNSS-only solution, for the X and Y components, respectively, and 25% and 19% compared to the VLBI-only solution, for the X and Y components, respectively. The rigorous combination contributes to the stabilisation of the UT1-UTC, with the improvement of the LOD, showing a gain of 48% and 10%, compared to the GNSS-only and VLBI-only solutions, respectively. While there is an improvement when using tropospheric ties, further studies are required to improve the agreement among the VLBI and GNSS tropospheric estimates, which is currently at the level of 1–5 mm. Future activities will include an approach using variance component estimation for the weighting, the study of additional parameters in the combination, additional studies on the combination of intensive VLBI sessions with GNSS for the estimation of UT1-UTC, and the rigorous triple combination with SLR observations.