Keywords

1 Introduction

A global geodetic endeavor is to improve the space geodetic techniques contributing to the global Terrestrial Reference Frames (TRFs). The Global Geodetic Observing System (GGOS) sets requirements on TRFs of 1 mm accuracy and 0.1 mm/year stability and scale accuracy to 0.1 ppb and 0.01 ppb/yr long-term stability (Gross et al. 2009). However, these requirements are not fulfilled yet. To achieve this goal, there are proposed plans e.g., for next-generation GNSS (NextGNSS) constellations like the Kepler system proposed by the German Aerospace CenterFootnote 1 Footnote 2 (DLR). The Kepler constellation features in particular new optical sensors and precise inter-satellite observations via inter-satellite optical links allowing for perfect time synchronization (Giorgi et al. 2019; Glaser et al. 2020; Michalak et al. 2021). There are also plans to incorporate new observation types provided by Very Long Baseline Interferometry (VLBI) transmitters onboard the NextGNSS, which will be investigated in this work.

VLBI as one of the main space geodetic techniques observe extra-galactic radio sources (mostly quasars) and can determine the full set of Earth Orientation Parameters (EOP), i.e., Polar Motion (PM), UT1-UTC, and Celestial Pole Offsets. In contrast, GNSS cannot determine UT1-UTC in an absolute sense and can only determine the negative time-derivative of UT1-UTC, i.e., Length of Day (LOD). Placing a VLBI transmitter on a GNSS satellite can enable the VLBI stations to observe GNSS satellites besides the quasars possibly allowing to transfer UT1-UTC (Sert et al. 2022).

Studies have been performed to assess the benefits of placing a VLBI transmitter on the satellite (e.g., Plank 2013, Männel 2016, Anderson et al. 2018, Wolf et al. 2022). Plank et al. (2017) and Tornatore et al. (2014) focused on the direct observations of the GNSS signals with a VLBI network. Whereas, Hellerschmied et al. (2018) concentrated on observations to a dedicated VLBI transmitter on board a satellite. McCallum et al. (2016), Jaradat et al. (2021) discussed the technical aspects and challenges of signal generation on a Galileo satellite for VLBI observations. Mammadaliyev et al. (2021) simulated VLBI observations to a VLBI transmitter placed on a Low Earth Orbit (LEO) satellite in addition to the quasar observations. The future ESA mission GENESIS was approved recently, which aims to install a VLBI transmitter onboard a satellite for the co-location in space of all four space geodetic techniques (Delva et al. 2023).

In this work, we investigated the potential of a VLBI transmitter on one Galileo-like Medium Earth Orbit (MEO) satellite (semi-major axis: 29600 km) on geodetic products such as station coordinates, ERPs, and satellite orbits by examining different station networks. We will perform our analysis using dynamic Precise Orbit Determination (POD) with the EPOS-OC software (Zhu et al. 2004), which is also capable of simulating and processing all four main space geodetic techniques.

2 Scheduling

In VLBI, the observation plans (also referred to as schedules) are required as the initial step for conducting real and simulated observations, which are generated by dedicated software. Except for VieSched++ (Schartner and Böhm 2019), most of them were developed for classical geodetic VLBI purposes i.e. scheduling only the quasar observations and usually did not support scheduling of the VLBI to satellite observations. We adopted a strategy that was followed in Mammadaliyev et al. (2021) for scheduling observations to the MEO satellite and quasars. To generate a ‘quasar-only schedule’, the satellite observations were excluded from the ‘satellite + quasar’ schedule. This ensures that the number of quasar observations remains the same across different scenarios (Table 1).

Table 1 Scenario description and their parameterization
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We followed a source-based strategy for the source selection so as to obtain a uniform distribution in the sky. A total of 64 sources were selected by the scheduler, i.e., one source for every 64 sky segments of the equal area as done in Sun (2013). We created two sets of schedules based on two different station networks to assess the performance of station networks on the estimated parameters. The first network consists of 13 stations from the R1 VLBI sessions organized by International VLBI Service for Geodesy and Astrometry (Nothnagel et al. 2017), referred to as ‘Network A’. As network A has only three stations in the southern hemisphere, therefore for the second network, we added three stations in the southern hemisphere in addition to network A stations to improve the global distribution of the network geometry referred to as ‘Network B’. Figure 1 shows the participating stations of both networks. The priority is given to quasar observation over satellite observation as the primary objective of VLBI is to obtain UT1-UTC which can only be determined from quasar observations. Therefore, we scheduled in such a way that we have approximately 6500 quasar observations and approximately 1000 satellite observations in one day from network A (see Fig. 2). Similarly, there are approximately 9000 quasar observations and 1300 satellite observations from network B.

Fig. 1
figure 1

Station observation networks. The maroon triangle represents network A and the three additional stations are represented by a maroon inverted triangle. These additional stations together with network A comprise network B. The green dot represents the stations that are used for the datum realization

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Fig. 2
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Satellite and quasar observations as seen by the stations in one day (other days show a similar pattern)

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3 Estimation of Parameters

This study consists of three main scenarios and the acronyms assigned to the different scenarios as given in Table 1 will be used hereafter. In the first scenario, i.e. VLBI observations to quasars (VoQ), we imposed datum with No-Net Rotation (NNR) and No-Net Translation (NNT) conditions which are standard for VLBI to get a minimum constraint solution. The NNR and NNT conditions were imposed in the second scenario, i.e. VLBI observations to quasar + satellite (VoQS\(_{RT}\)). However, this results in an over-constrained solution as satellite observations are basically sensitive to the geocenter, and imposing NNT would not be necessary (e.g., Glaser et al. 2015). Therefore, we only imposed NNR in the VoQS\(_{R}\) scenario to achieve a minimum constraint solution.

3.1 Simulation

The simulation of VLBI observations uses the VLBI delay model recommended by IERS Conventions 2010 (Petit and Luzum 2010) for quasar observations, and the model for Earth satellites by Klioner (1991) for satellite observations. We assume perfect models in the simulations as the best-case scenario; this study did not include systematic errors in orbit modeling. The VLBI module of EPOS-OC software reads the observation epoch, baseline, and target source from the schedule as input to compute the group delays and keeps them for further processing. White noise for stations of 30 picoseconds was added to all group delays. We did not introduce tropospheric and clock modeling errors in this study, which is planned for future work. The simulated group delay observations (white noise added) were subsequently used to generate datum-free Normal Equations Systems (NEQs) for each day (one arc). This is followed for all 10 days, and further, these 10 generated datum-free NEQs were stacked together to obtain one stacked datum-free NEQ.

3.2 Recovery (Precise Orbit Determination)

To determine the satellite’s orbit with VLBI we use dynamic Precise Orbit Determination (POD) with EPOS-OC. This approach uses models of the forces, i.e., gravitational and non-gravitational, to calculate the sum of the acceleration forces acting on the satellite. There were several models used in our study, for example, EIGEN-6C (Shako et al. 2014) as Earth’s gravity potential, and the Solar Radiation Pressure (SRP) is taken care of by the empirical CODE (Center for Orbit Determination in Europe) orbit model (ECOM) in the reduced version (Beutler et al. 1994; Springer et al. 1999), to mention a few. We did not consider accelerations due to Earth’s albedo and atmospheric drag as their effects are quite small for MEO satellites. During simulation, the orbit of the satellite is first integrated. This integrated orbit represents the reference orbit to which observations are simulated which are subsequently used for the dynamic POD in the recovery run. The dynamic orbit determination process in EPOS-OC is done by least squares adjustment. In both VoQS scenarios, we also estimated satellite parameters such as the initial state vector of the daily arcs in the form of position and velocity, and reduced ECOM parameters.

3.3 TRF Solution

To obtain a TRF solution, we impose datum on the stacked NEQ (explained in Sect. 3.1) with NNR and/or NNT of 1 mm to the 13 common stations in both networks, i.e., all stations in network A and this stacked NEQ is inverted for estimating the final solution. The parameters estimated for the scenarios are illustrated in Table 1. We estimated one set of station coordinates from 10 days, daily polar motion (PX, and PY), and daily UT1-UTC for all the scenarios. We did not estimate source positions and kept them fixed to their a priori values.

4 Result and Discussion

4.1 Satellite’s Orbit

We computed the differences in orbital components for various scenarios (i.e., along-track, cross-track, and radial components) w.r.t. the reference orbit of the simulation (see Fig. 3) and computed the root mean square (RMS) for the 10 days (see Fig. 4). For network A the RMS values in the VoQS\(_{RT}\) case are 1 cm in radial, 0.8 cm in cross-track, and 1.5 cm in the along-track direction. The values in the VoQS\(_{R}\) case, whereby no NNT condition was applied, are remarkably larger with values of 3.5 cm, 2.7 cm, and 4.5 cm respectively. Significantly better values could be achieved with network B. The RMS values for the VoQS\(_{RT}\) scenario are 0.6 cm (radial), 0.5 cm (cross-track), and 0.8 cm (along-track). In the case without NNR, i.e. VoQS\(_{R}\), with network B we get again slightly larger deviations. With values of 1.9 cm, 1.5 cm, and 2 cm, however, significantly lower than for network A. We can deduce from the results that the addition of three stations in the southern hemisphere, i.e., network B, improved the orbital components in all three directions by up to 50 % across both VoQS scenarios. The network geometry plays a more vital role in NNR-only than in both NNR and NNT scenarios. For these VoQS\(_{R}\) scenarios (without NNT), we see improvements in radial, cross-track, and along-track directions of approximately 48 %, 42 %, and 54 % for network B compared to network A, respectively.

Fig. 3
figure 3

Time series of the differences in orbital components w.r.t. their reference orbit for both networks (for one day)

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Fig. 4
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RMS values of differences in orbital components w.r.t. their reference orbit for both networks (for 10 days)

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4.2 Station Coordinates

The formal errors of estimated station coordinate averaged over x, y, z for VoQ and VoQS\(_{RT}\) scenarios in 3D are around 2 mm for both networks (see Fig. 5). The results from these two scenarios have similar performances. Now we omit the NNT condition since this information should be given by the satellite observations. We obtain the results of the VoQS\(_{R}\) scenario, where the formal errors for network A increase strongly up to 8 mm. Whereas, for the same scenario for network B, the formal errors are up to only 2.5 mm. In VoQS\(_{R}\) scenario, the addition of three stations resulted in a reduction of formal errors in the 3D coordinate by 68 %.

Fig. 5
figure 5

Mean of formal errors in 3D station coordinates averaged over x, y, z for both networks and all scenarios

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The scenarios observed from network B have approximately 40 % more observations compared to network A due to three additional stations. To quantify the significance of the improvements in the estimated parameters from both networks, we computed the expected improvement due to different Degrees of Freedom (DoF). The anticipated improvement in the parameters from network B is approximate 15 % w.r.t. network A. So the improvements mentioned earlier can be considered significant as they are more than the expected improvement, i.e., 15 %.

4.3 Helmert Parameters

The Helmert transformation parameters between estimated station positions of VoQ, VoQS\(_{RT}\), and VoQS\(_{R}\) scenarios w.r.t. their a priori, and corresponding standard deviations (see Fig. 6) were computed. This is performed for the stations that participated in the datum definition (see Fig. 1). The Helmert transformation is used to see differences between the networks expressed by three translations, three rotations, and one scale factor. The translational parameters \(T_{x}\), \(T_{y}\), \(T_{z}\) and rotational parameter \(R_{x}\) for VoQS\(_{R}\) scenario from network A are approximately up to ±5 mm and from network B it reduces up to ±1 mm. Nevertheless, we can say that it is possible to realize datum on mm-level without imposing NNT condition. Furthermore, with better network geometry, the Helmert parameters do improve considerably w.r.t. a priori TRF.

Fig. 6
figure 6

Helmert transformation parameters (three translation \(T_{x}\), \(T_{y}\), \(T_{z}\), three rotations \(R_{x}\), \(R_{y}\), \(R_{z}\) and Scale S) of the estimated station positions (datum only) w.r.t. their a priori for both networks and all scenarios, along with their corresponding standard deviation (shown as error bars)

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4.4 Earth Rotation Parameters

The mean formal errors in PM and UT1-UTC for both networks and all scenarios are shown in Fig. 7. For VoQ and VoQS\(_{RT}\) scenarios from network A, the formal errors in ERPs are up to 40 \(\mu \)as, which corresponds to about 1.3 mm, and from network B, it is up to approximately 30 \(\mu \)as. In VoQS\(_{R}\) scenario, we observed relatively high formal errors in PM of up to 115 \(\mu \)as, specifically PY with network A. Whereas, for the same scenario for network B, the formal errors were up to 65 \(\mu \)as. By improving the global distribution of the station network, we observe a significant reduction w.r.t. expected improvement for the VoQS\(_{R}\) scenario in formal errors of PX, PY, and UT1-UTC by 38 %, 40 %, and 33 %, respectively.

Fig. 7
figure 7

Mean of formal errors in ERPs for both networks and all scenarios

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5 Conclusions and Outlook

We performed simulations of a VLBI transmitter on one next-generation GNSS satellite and by performing POD in addition to quasar observation for a period of 10 days. In this study, two different VLBI station networks were considered. The stations in network B consist of network A plus three stations in the southern hemisphere. In total, we formulated the study for three scenarios, i.e., VoQ scenario (quasar-only), VoQS\(_{RT}\) scenario (quasar + one satellite, NNR/NNT), and VoQS\(_{R}\) scenario (quasar + one satellite, NNR only). The expected improvement in formal errors of the estimated parameters due to more observations in network B w.r.t. network A is around 15 %. If the actual improvement is more than the expected value, it means that the parameters have improved due to the better network geometry and not because of more observations. We noticed that network geometry plays a vital role in estimating satellite orbits, station positions, and ERPs in this case.

By using simulated VLBI observations for POD, we recovered the orbit of the MEO satellite with cm-level accuracy. To quantify the effect of network geometry for the VoQS\(_{R}\) scenario on the results, a relatively poor network geometry as network A resulted in orbit recovery up to 5 cm (RMS value). The addition of three stations in the southern hemisphere, i.e., network B, significantly improved orbit recovery up to 60 %. If the network geometry for the VoQS\(_{R}\) scenario is good, one gets near the results of the VoQS\(_{RT}\) without having to impose NNT anymore. These are promising findings for upcoming satellite missions, which may be equipped with a VLBI transmitter.

The expected improvement due to an increase in observations because of the addition of one MEO satellite in the quasar + satellite (VoQS) scenario w.r.t. quasar-only (VoQ) scenario is 6 %. The station positions and ERP estimated from VoQS scenario do not lead to a considerable difference compared to VoQ scenario, as it is less than the expected improvement of 6 %. Upon further examination of different stations, it appears that some stations have got better, which is around the expected improvement, while others have not shown any improvement at all. This could be due to differences in satellite observations among the stations. It can be stated that the inclusion of satellite observations does not appear to have any adverse impact on the estimated parameters.

We observed the additional benefits of the addition of a MEO satellite such as, when VLBI observations are extended to satellite observations, it allows us to realize the origin of the reference frame, thus imposing that the NNT condition is not necessary. Based on the estimated Helmert transformation parameters, we realized datum on mm-level by imposing NNR condition only. The VoQS\(_{R}\) scenario results in translations (\(T_{x}, T_{y}, T_{z}\)) and rotation (\(R_{x}\)) of up to ±5 mm for network A. However, better network geometry, i.e. network B leads to significant improvement resulting in Helmert parameters around 1 mm. The mean formal errors in station positions for VoQS\(_{R}\) scenario from network A were approximate up to 6 mm. Here again, the better network geometry, i.e. network B leads to a reduction by up to 75 %. The ERPs formal errors show an improvement up to 40 % estimated from a better network geometry i.e., network B w.r.t network A. By looking at the results for station positions and ERPs, we can say that in the VoQS\(_{R}\) scenario, the improvements due to three additional stations are more significant w.r.t. the expected improvement than the VoQS\(_{RT}\) scenario.

In the following steps, we plan to combine GNSS and VLBI into one common space-tie satellite to investigate how well the datum information can be transferred from the GNSS network to the VLBI network. In addition, we would like to extend our study to next-generation GNSS satellites with optical links and future observation types. Furthermore, the impact of the VGOS stations and a lower noise level could also be studied.