Simulating VLBI observations to BeiDou and Galileo satellites in L-band for frame ties

2.1 Global antenna network

The navigation signals of GNSS satellites emitted in L-band are outside the nominal frequency range of the legacy S/X or the next-generation VLBI Global Observing System (VGOS). In order for antennas to be able to observe these signals, they require a dedicated receiver. The frequency catalog provided by the SCHED program^[2] gives a selection of frequency setups for active VLBI stations. Based on this catalog, we identified 16 VLBI antennas that are able to observe BeiDou’s B1 and B3 as well as Galileo’s E1 and E6 navigation signals. The distribution of the participating antennas is illustrated in Figure 1. The antennas Br (BR-VLBA), Hn (HN-VLBA), La (LA-VLBA), Mk (MK-VLBA), Nl (NL-VLBA), Ov (OV-VLBA), and Pt (PIETOWN) belong to the very long baseline array (VLBA). Ir (IRBENE), Jb (JODRELL2), O8 (ONSALA85), T6 (TIANMA65), Tr (TORUN), and Wb (WSTRBORK) belong to the European VLBI Network (EVN). Finally, Cd (Ceduna), Ho (HOBART26), and Ww (WARK12M) are part of the long baseline array (LBA).

Figure 1

Global network of 16 VLBI antennas utilized in the simulation study. The station names are given as 2-letter code.

While nominally all of these stations can observe the target frequencies and a few station have actually demonstrated to be capable of tracking and recording GNSS signals in VLBI-type experiments (Plank et al. 2017), each station would need to conduct preliminary tests. Compared to the faint quasar sources, GNSS satellite signals are considerably stronger. Although oversaturation could occur in some of the stations used, in general reducing sensitivity is generally easier than increasing it (Petrov et al. 2023).

2.2 Satellites

The target navigation satellites comprise the currently available constellations of BeiDou and Galileo satellites. At the time of this writing, there are a total of 46 operational BeiDou and 23 operational Galileo satellites. BeiDou was initially a regional navigation system covering China and neighboring regions by having satellites on geostationary (GEO) and inclined geosynchronous orbits (IGSO). By now, BeiDou provides a global coverage as satellites also move on mean Earth orbits (MEO) with an inclination of 5 5 ∘ . The Galileo satellites orbit in three MEO planes with an inclination of 5 6 ∘ . Figure 2 shows the number of visible BeiDou and Galileo satellites over stations in Oceania (Ho), Asia (T6), Europe (O8), and North America (La) over a period of 24-h and considering an elevation mask of 5 ∘ . The comparable large amount of GEO and IGSO satellites in the BeiDou constellation leads to a significantly higher number of visible satellites at Ho and T6. While the number of visible satellites is lower, O8 appears to still benefit from the high orbital altitudes of GEO and IGSO satellites in the BeiDou constellation. La shows the fewest visible satellites because it can only observe the MEO satellites of both constellations.

Figure 2

Number of visible BeiDou and Galileo satellites over stations in Oceania (Ho), Asia (T6), Europe (O8), and North America (La) over a period of 24-h considering an elevation mask of 5 ∘ .

Figure 3 depicts the navigation frequency bands of BeiDou and Galileo. The limited frequency ranges of the VLBI antennas restrict the number of frequency bands employed in this study, primarily due to the inability of EVN antennas to observe GPS’s L2 band or lower frequencies. Of the BeiDou frequency bands, we assume observations of the B1 signals at a center frequency of 1561.098 MHz for the second-generation BeiDou satellites and at a center frequency of 1575.42 MHz for the third-generation BeiDou satellites. For all BeiDou satellites, observations are additionally assumed of the B3 signals at a center frequency of 1268.52 MHz. These signals overlap with the employed Galileo frequency bands E1 at a center frequency of 1575.42 MHz and E6 at a center frequency of 1278.75 MHz. For reference, the remaining BeiDou and Galileo signals along with the L1, L2, and L5 signals of GPS are shown in Figure 3. We assume that the ionospheric delay can be removed by building a dual-frequency ionospheric-free combination as shown for BeiDou’s B1/B3 in Yang et al. (2018) and for Galileo’s E1/E6 in Duan et al. (2023).

Figure 3

Galileo, BeiDou, and GPS navigation frequency bands.

2.3 Quasar sources

In addition to satellites, we also add quasar sources to the simulation study. When performing VLBI observations to quasar sources, we fix the position of both the VLBI antennas and quasar sources as they are both in the celestial reference frame. In doing so, observations to quasars only contribute to the estimation of the troposphere and the station clock model without influencing the estimation of the antenna position. Only satellite observations determine the antenna position estimates.

We incorporated all available quasar sources from the SKED good geodetic sources catalog list.^[3] The catalog is a list of 366 sources that are commonly used for geodetic VLBI experiments. A subset is selected when imposing the constraint of utilizing only sources in the scheduling that exceed 0.5 Jansky. This is to ensure high signal-to-noise ratio (SNR) values with short integration times. As the source structure is desired to be small (e.g., Anderson & Xu 2018), a distinguishing characteristic of this source catalog is its compact source structure in the S- and X-bands.

In this study, we propose observing quasars in L-band as was carried out in Plank et al. (2017) and Petrov et al. (2023). We are aware that the source structure as well as flux density is frequency dependent and that the utilized catalog does not include L-band information. At this stage, we assume that a sufficient number of stable and strong sources in L-band can be found for these experiments. Furthermore, as we do not use quasars for antenna position estimates, the requirement for source position precision is lessened.

2.4 Scheduling

In this study, the scheduling is carried out using the VLBI scheduling software VieSched++ (Schartner & Böhm 2019). Observations to satellites and quasars are scheduled in an automated fashion, which implies that VieSched++ selects scans and sources based on weighted optimization criteria. The length of the experiment is chosen to be 24-h, which is the typical session length for a geodetic VLBI experiment. To establish an accurate troposphere model, it is required that the antenna spends a considerable amount of time pointing at many different directions in the sky (e.g., Schartner & Böhm 2020). Consequently, instead of dedicating lengthy observation periods to a single satellite as it traverses the sky, a satellite is observed with a duration as short as possible before switching to the next source. As the satellite observations can be made at extremely high SNR values in short integration times, satellite scans are scheduled with a duration of 10 s. The comparatively less bright quasars are scheduled with a fixed 60 s scan duration. One parameter to optimize is the ratio of satellite to quasar scans. Upon evaluating various ratios, we determined that about 80–90% satellite to 10–20% quasar scans yield the most favorable outcomes for station position estimates.

2.5 Simulated observations

The generated schedules are simulated in the Vienna VLBI and Satellite Software (VieVS; Böhm et al. 2018). As usual for simulations of geodetic VLBI observations, three main error sources are considered: tropospheric delay, station clock inaccuracies, and measurement noise. Additionally, we implement satellite orbital errors. The tropospheric turbulence is modeled by using the simulator’s standard parameters (Nilsson et al. 2007). We assume a uniform tropospheric refractive index structure constant C n of 1.8 × 1 0 − 7 m 1 ∕ 3 across all stations. The scale height is set to 2,000 m, and the wind speed is set to 8 m/s in an eastward direction. Clock inaccuracies are modeled with an Allan standard deviation (ASD) of 1 × 1 0 − 14 @50-min, consistent with the performance of current Hydrogen-masers. Measurement errors are modeled as white noise with a standard deviation of 50 ps ( ≈ 15 mm) for observations of both satellites and quasars. For satellite observations, we incorporate satellite orbit errors at the level of 2.5 cm, reflecting the final type orbit accuracy achieved by the International GNSS Service (IGS). These errors are modeled as an equal combination of systematic piecewise-linear offsets, and a random noise component. Initially, normally distributed offsets of the satellite from the true position are determined in 30-min intervals for the x -, y -, and z -directions. At the time of a scan, the offsets are interpolated, and noise is added. The offsets across all scans are scaled with a factor so that their standard deviation equals 2.5 cm. Finally, for each pair of stations, the geometric delays with respect to the true and offset satellite positions are determined, and the difference is added to the observable. We conduct 500 Monte Carlo simulations, enabling us to carry out a subsequent statistical evaluation based on mean values and variance.

2.6 Analysis

In a least-squares adjustment, the normal equations of the seven 24-h sessions are stacked to derive a 7-day solution. A troposphere and clock model are estimated for each individual session. The antenna coordinates are estimated from the combined solution of all sessions. Earth orientation parameters (EOPs) as well as satellite and quasar positions are held fixed.

To model the tropospheric delay, zenith wet delays are estimated at 15-min intervals for each station as PWLOs with a relative constraint of 1.5 cm. Troposphere gradients to account for azimuthal asymmetry were estimated at 30-min intervals with relative constraints of 0.05 cm and absolute constraints of 0.1 cm. Station clocks are estimated as PWLO with 30-min intervals, including a rate and quadratic term per clock. Clock estimates are constrained to 1.3 cm. A set of antenna coordinates for each station is estimated using all seven 24-h sessions. Only satellite observations are used for estimating the antenna coordinates. Observations of quasars are solely used for estimating the troposphere and clock models. Unlike conventional geodetic VLBI analysis, datum constraints are not required as the inclusion of satellites as sources and the fixation of EOPs eliminate the rank deficiency. Consequently, NNT, NNR, and no-net-scale (NNS) conditions are not applied. Since only satellite sources are used to determine the estimated VLBI antenna coordinates, these coordinates exist in the satellite reference frame.

In this study, the VLBI frame is defined as the set of antenna positions obtained purely from VLBI observations to quasars. We assume that VLBI experiments conducted outside the scope of this study provide station positions with high accuracy. When estimating the VLBI station positions from observations to signal emitting spacecraft while no constraints are imposed on the network, the estimated VLBI station positions are expressed in the spacecraft’s reference frame. For GNSS satellites, the estimated positions are in what we call the GNSS frame.

The integration of real observations from the actual antennas into the realization of a TRF could be achieved by including them at the level of normal equations, similar to the inclusion of local tie measurements. However, in this study, we adopt the analysis strategy presented in previous works, e.g., Plank et al. (2014) or Anderson et al. (2018). The accuracy of the inter-technique frame tie is evaluated at the level of VLBI antenna coordinates in the satellite reference frame. We assess the repeatability of antenna coordinates in the form of a standard deviation by means of 500 estimates of antenna coordinates. To enhance the geometrical interpretation of the repeatabilities, the antenna coordinates are transformed from a global XYZ-frame to a local East-North-Up (ENU) reference frame for each individual station.

3.1 Schedule and sky coverage

As an optimized scheduling is critical for geodetic VLBI experiments, we created a range of schedules to evaluate the optimal ratio of satellite scans to quasar scans. Within VieSched++, we experimented with the weights on satellite scans as well as different scan sequences. Figure 4 shows the performance of about 60 schedules as a percentage with respect to the percentage number of satellite scans. The performance is measured based on the repeatability of antenna positions for a single 24-h experiment. It is visualized relative to the best performing schedule (), which is utilized in the remainder of this study. This schedule has a percentage number of satellite scans of about 85%.

Figure 4

Performance with respect to the number of satellite scans in percentage of about 60 different schedules relative to the utilized schedule.

By increasing the number of satellite scans from about 15 to 90%, the performance increases significantly. Since quasars are used to estimate the station clock and troposphere model but not the antenna position, schedules with a small number of satellite scans include too few sources that contribute to a better estimation of antenna coordinates. Schedules with a number of satellite scans between about 80 and 90% show the highest performance.

While an excessive number of quasar scans lead to a decrease in performance, an insufficient number is also disadvantageous. One schedule with 100% satellite scans, meaning no quasars were scheduled, shows a performance of approximately 65%. Incorporating a certain number of quasar sources into the schedule enhances the results due to two primary factors: improved sky coverage and an increase in long-distance baseline observations.

In geodetic VLBI, it is important for each antenna to perform scans at a wide range of sky positions in order to improve the estimation of the troposphere, thereby reducing the correlation between tropospheric delay and station position estimates (Klopotek et al. 2020). Figure 5 shows the skyplots for Ho, T6, O8, and La for a single 24-h experiment. The top row differentiates between satellite and quasar sources. The bottom row illustrates the number of observations per source scan.

Figure 5

Skyplots showing the distribution of observed sources for a single 24-h experiment on local skies for Ho (HOBART26), T6 (TIANMA65), O8 (ONSALA85), and La (LA-VLBA). The top row represents a distinction between satellite and quasar sources. The bottom row shows the source positions colorcoded by the number of observations per source.

The number and distribution of sources and the number of observations per source vary considerably across the four stations. Due to the restricted inclination of satellites ( 5 5 ∘ for BeiDou and 5 6 ∘ for Galileo), each station exhibits gaps devoid of satellite sources in the polar regions. The scheduler, VieSched++, seems to compensate the absence of satellite sources in the polar regions with quasar scans, thereby improving the sky coverage. The same applies for regions with sparse numbers of satellite scans, such as in the southeast of O8. Besides improving the sky coverage, quasar scans also have the benefit of providing long-distance baseline observations connecting the troposphere and clock estimation of distant antennas. This is especially prominent for Ho and La. A majority of quasar scans for Ho are concentrated in the northeast, while quasar scans for La are concentrated in the southwest, establishing crucial links across vast distances. Furthermore, the sky coverage does not account for the number of observations generated by each source. For the illustrated antennas in Figure 5, the observations of quasars often yield a high number of observations. Consequently, each quasar source, on average, contributes numerous baseline observations and links regional networks across the globe. For instance, the average number of observations per satellite scan is approximately 3.0 for Ho and 6.3 for La. The average number of observations per scan for quasar sources is about 7.0 for Ho and 8.1 for La. Despite requiring six times the scan duration of satellite scans and only achieving a seemingly minor improvement in sky coverage, quasar scans significantly impact performance, as shown in Figure 4.

Figure 6 illustrates the baselines for each observation in the simulation. The regional networks in America and Europe share the highest number of baselines. The regional network in Oceania is not as interconnected with Europe and America as T6 in Asia. Favored by the short distance, the baseline with the highest number of observations is between La and Pt, sharing 1,008 observations.

Figure 6

Baselines between antennas in the global network colorcoded by the number of shared observations for a single 24-h session including both satellite and quasar scans.

3.2 Antenna position repeatability

The 24-h schedule is simulated seven times. The normal equations are combined to obtain a 7-day solution. Figure 7 illustrates the resulting antenna position repeatabilities for the participating antennas. Additionally, the number of scans and observations are given. The median 3D antenna position repeatability is 7.3 mm. Compared to America and Oceania, the repeatabilities for antennas in Europe and Asia are larger. However, the differences are less than 1 mm. Considering how much the number of scans and observations varies across the antennas in the network, the variation of the antenna position repeatabilities is small. While not addressed by the authors, this effect can also be seen in Wolf & Böhm (2023). It is mainly caused by not applying NNT or NNR conditions on the network. Other factors are also the inhomogeneous distribution of antennas and sources as well as the unequal slew speeds of the antennas.

Figure 7

Antenna position repeatabilities derived from the 7-day solution given in 3D as well as in the individual east, north, and up components in the local antenna coordinate frames. The antennas are grouped into antennas from America, Europe, Asia and Oceania. Referring to a single 24-h session, the blue and green data points represent the number of scans and observations to satellites, respectively.

Generally, the number of scans increases with higher antenna slew speeds. This is exemplified by Ir and Ww, which exhibit the fastest slew speeds among all antennas in the global network and have around 950 scans. Conversely, O8, Tr, and Wb exhibit slow slew speeds and a significantly lower number of scans, with around 400. Another factor influencing the number of scans is whether a station belongs to a regional network with other fast-slewing antennas. The stations in America have a higher slew speed than the average slew speed of all participating stations but are still significantly slower than Ww or Ir. However, their number of scans is similar because their regional network includes antennas with faster slew speeds. The number of observations reflects the number of antennas with which baseline observations can be established. The regional network in America is denser than the regional networks in other regions, resulting in more observations

For a subset of stations, the repeatability in the north component exceeds that in the east or height. This is attributed to the sensitivity of VLBI, which is given exclusively in the plane defined by the two stations and the source. Referred to as the network effect, Plank et al. (2014) found that a systematic distribution of scans causes biases in the determination of horizontal coordinates. For example, the north component for O8 is determined with the least accuracy. For O8, the skyplot in Figure 5 reveals a limited number of scans toward the north and south. These areas also show a lack of satellite scan. However, only satellite scans contribute to the determination of the antenna position. Since the majority of satellite scans are conducted toward the west, a limited number of north-south baselines are formed, resulting in a lack of sensitivity in this direction.

3.3 Limitations

In this section, we aim to identify the factors that may be limiting the precision of the results. The simulation results are based on the selected accuracies of the error sources. Figure 8 illustrates the averaged antenna position repeatability across all antennas in the network with respect to varying error values. The tropospheric structure constant ( C n ) represents a change in tropospheric turbulence. Similarly, variations in station clock precision, measurement noise, and satellite orbit errors are represented by the ASD, white noise, and inaccuracies in the satellite orbital position, respectively. For each error factor, we vary its value while keeping the other error factors at their default values, as described in Section 2.6. Each data point represents a 7-day solution.

Figure 8

Antenna position repeatability with respect to variations in the four error sources. The variations are with respect to the tropospheric structure constant for tropospheric turbulence, the ASD for clock inaccuracies, the white noise for measurement noise, and the orbit error for inaccuracies of the satellite orbit. The diamond shape data point ✦ at the center of each individual plot represents the default value used in the rest of the study.

The results demonstrate a strong dependence on the tropospheric turbulence. Reducing the tropospheric structure constant C n to 0.9 or eliminating tropospheric delay entirely improves the antenna position repeatability to 6.4 and 6.2 mm, respectively. Correspondingly, the antenna position repeatability deteriorates with a steep slope to 8.4 and 10.1 mm upon increasing C n to 2.7 and 3.6, respectively. Upon examining the clock inaccuracies, characterized by the ASD within a 50-min time interval, reducing the clock performance from 1 × 1 0 − 14 @ 50 min to 1 × 1 0 − 13.5 @ 50 min, the antenna position repeatability deteriorates from 7.3 to 8.1 mm. By hypothetically reducing the clock performance by a factor of 10 to 1 × 1 0 − 13 @ 50 min, the antenna position repeatability deteriorates to 16.3 mm (exceeding the figure’s y -axis limits). The performance shows only marginal improvement for clock systems with performance exceeding 1 × 1 0 − 14 @ 50 min. As noted by Petrachenko et al. (2009), the performance measured currently for H-masers and their associated clock distribution systems is typically better than that. For the previously discussed simulations, the measurement noise has been set to a value of 50 ps ( ≈ 15 mm). Reducing the measurement noise to 25 ps ( ≈ 7.5 mm) or eliminating it altogether results in only marginal improvements. However, increasing the white noise to 75 ps ( ≈ 22.5 mm) and 100 ps ( ≈ 30 mm) degrades the antenna position repeatability from 7.3 to 8.0 and 9.0 mm, respectively. Variations in the satellite orbit error show the strongest dependence of the antenna position repeatability on this error source. The antenna repeatability decreases to 5.2 mm in the absence of simulated orbit errors. Conversely, it increases to 11.0 mm when the orbit error is set to 5 cm.

A further potential limitation is the number and distribution of antennas in the global network. Figure 1 highlights a noticeable absence of additional antennas in the southern hemisphere. We hypothetically assume the addition of antennas equipped with dedicated L-band receivers at the Argentinian-German Geodetic Observatory (AGGO) and the Hartebeesthoek Radio Astronomy Observatory (HartRAO). Under the assumption of a similar slew speed to the stations in the VLBA, the two additional antennas in South America and Africa enhance the average antenna position repeatability from 7.3 to 4.8 mm.