Combining GPS and VLBI for inter-continental frequency transfer

No technique is capable of providing absolute clock parameters at each site. Therefore it is necessary that one clock at one selected station in a VLBI network is kept fixed and all other clocks of the other stations in the network will be related to this reference clock. By doing so, clock differences can be obtained from geodetic post-processing, i.e. the adjustment of the unknown parameters. Since the clock difference parameters need to be estimated together with the other unknowns, one needs to pay attention that neither unmodeled effects bias the clock estimates, nor correlations among the unknowns absorb clock variations in other parameters. Moreover, the parametrization of the clock unknowns has to reflect the physical behaviour of the clock, i.e. providing a temporal sampling which can follow a random walk (in phase) noise process.

The software package 'c5++' [13] has been developed with the purpose of supporting the combination of space geodetic data from VLBI, GPS and SLR on the observation level and also to allow processing of single-technique solutions. Since c5++ uses consistent geodetic/geophysical models [14] for all space geodetic techniques and all analysis steps, it is guaranteed that differences of the estimated parameters that are derived from e.g. VLBI and GPS do not originate from different models or inconsistent corrections, but are directly related to the performance of each technique. Moreover, the possibility to combine data on the observation level allows to study the impact of combining techniques, with the purpose to improve the target parameters. The concept of combination on the observation level has been successfully demonstrated in [15] and will be applied here (see section 3.3).

In a first attempt, the complete CONT11 VLBI data set was analysed with the Calc/Solve software [16] following a standard network analysis approach, as described in [17]. Even the previously mentioned 'problematic' stations (section 3) were included in the analysis. The hydrogen maser at Wettzell (Germany) was used as reference clock for the network and relative clock estimates w.r.t. this reference were estimated for all stations as daily second order polynomials with additional continuous spline corrections with a temporal resolution of 20 min. Since the original data correlation and analysis was done in daily batches, the resulting clock estimates were not continuous over day boundaries. To generate continuous series of clock estimates, the daily solutions were fitted together using overlapping estimates, thus removing potential discontinuities at day boundaries.

In parallel, GPS data recorded with the co-located GPS receivers at the CONT11 stations, all connected to the same frequency standard as the corresponding co-located VLBI-system, were analysed with the NRCAN-PPP software [18]. The analysis procedure is described in [17] and gave GPS clock estimates with 60 s temporal resolution for all analysed stations for the whole CONT11 campaign. These time series were differenced with respect to the solution for the GPS receiver at Wettzell (WTZZ) in order to produce time series of relative clock estimates on the network baselines corresponding to the VLBI solution.

The individual time series of clock differences derived from VLBI and GPS were then analysed and modified Allan deviation (MDEV, [19]) values were derived for each baseline in order to estimate the frequency link stabilities. The results show that VLBI and GPS perform similarly with best case one day instabilities on the order of 10⁻¹⁵ [17]. Figure 3 depicts as an example the results for a subset of 5 baselines connecting to Wettzell. GPS and VLBI derived MDEV values are shown in black dots and orange squares, respectively. It is evident that technical problems at Tsukuba affect the stability of the solution. The TSKB receiver lost lock on day 3 of CONT14, thus the GPS-derived MDEV values involving Tsukuba are deteriorated since the technical problems were not accounted for in this first study.

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In order to assess both short- and long-term frequency stabilities, the data were processed with c5++ in non-overlapping batch solutions with three consecutive days, i.e. covering a period of 72 h each. Thus, the whole CONT11 campaign could be analysed in five 72 h batches. This was possible since the whole campaign had been re-correlated with a consistent delay model that prevents discontinuities at the day boundaries [20]. As mentioned before, VLBI analysis requires that one station clock in the network is fixed to an arbitrary constant value, which is chosen normally to be zero. Doing so, all estimated clocks in the network represent the clock difference between that station's clock and the clock of the reference site. The clock reference should be assigned to a station which is known to show no clock break during an entire session. Again, as in the previous analysis described in section 3.1, the clock at Wettzell (Germany) was selected as reference. Another characteristic of VLBI analysis relating to clock estimates is the irregular temporal spacing of the original observational data. Depending on antenna and receiver characteristics and the flux density of the observed radio sources, optimized observation schedules require different scan-length and varying idle/slewing times. Moreover, hardware problems or operational problems can lead to larger temporal gaps during which a single station is not observing together with the other network sites. In order to accommodate for these difficulties, which could lead to singularities of the design matrices used in the data analysis, it is common to apply constraints on certain parameters.

For clock estimates, which are parametrized as piece-wise linear offsets (PLO) CL (t_i) here, a weak constraint on the rate was used, i.e.

$$\begin{eqnarray}&&\begin{array}{c} CL\left({{t}_{i-1}}\right)-CL\left({{t}_{i}}\right) =0.0\pm {{\sigma}_{\text{rate}}}. \\ \end{array}\end{eqnarray} \tag{ 1 }$$

The constraint's weight, i.e. the inverse of the formal error σ_rate, was chosen so that the clock estimates were smoother than the physical variation of the atomic clock at the VLBI site. If VLBI shall be used for frequency transfer, the proper choice of σ_rate is very important, in order to guarantee that the estimated frequency instabilities really represent the clock performance and are not dominated by the applied constraints [9]. The constraints themselves and the selection of the temporal resolution of the PLO clock model are other issues that need to be considered properly. A too-high temporal resolution of the PLO model bears the consequence that most of the estimated values are strongly dominated by the choice of the constraint rather than by the observational data.

In addition, zenith wet delays (ZWDs), troposphere gradients and station coordinates need to be estimated in order to obtain clock estimates that are not degraded by the correlations among these parameters. The c5++ software allows to process un-differenced GPS observations with the precise-point positioning (PPP) approach [21, 22], using the same geophysical models as used for VLBI analysis. PPP does not require GPS satellites in common-view and becomes independent of the length of the baselines that are defined by the VLBI network geometry. Thus, even inter-continental frequency links can be established by this analysis method. However, satellite orbits and clocks need to be fixed to IGS final products [23] since such parameters can not be estimated for this study, due to the small number of GPS receivers within the CONT11 network.

Since GPS observations are available with a regular and high sampling rate, clock parameters can be estimated without constraints. Moreover, PPP allows to directly access the behaviour of the station clock and frequency transfer between two stations can be performed by differencing the PPP clock solutions of the two sites. Other parameters like zenith troposphere delays or station coordinates have to be estimated together with the clock model, subject to the same correlations as VLBI.

Combination on the observation level reflects the concept of using all available geodetic data at a site in order to overcome potential deficits of a single technique. Since co-located instruments often share the same infrastructure (e.g. frequency standards) and are expected to experience the same environmental conditions, in particular the atmospheric conditions, one can estimate parameters related to these influences in a common model which only considers biases between the different techniques. It was demonstrated in [15] that the combination of GPS and VLBI on the observation level leads to a small but significant improvement of the geodetic target parameters and in particular improves the stability of the site coordinate time series. Since troposphere and clock parameters were estimated in [15] by a station-wise model it was obvious to investigate how the combination of these two techniques on the observation level can be utilized for the benefit of frequency transfer. As described in [15], troposphere parameters can be estimated with a common model that considers an offset between GPS and VLBI, caused by the height difference between the VLBI and GPS reference points, resulting in a hydrostatic delay difference which remains very stable over time.

On the other hand, a common clock model can be estimated by combining VLBI and GPS only if one considers that instrumental delays are not stable over time but are influenced by temperature-induced changes in reflections (cable multipath) and electrical lengths; for VLBI these can also depend upon the orientation of the antenna. It was suggested in [15] to estimate these inter-technique delay changes by a PLO model with a temporal resolution that is sufficient for representing at least a diurnal signal. A low temporal resolution, i.e. with a PLO step-width of more than 12 h, bears the risk of not modelling all inter-technique delay changes properly. However, a high temporal resolution reduces the benefit of combining GPS and VLBI since a large fraction of the information provided by the second technique gets absorbed into this parameter. Therefore, the proper choice of the temporal resolution for this crucial parameter is explored in section 4.3.

Another issue that arises from the combination of different observation types is related to the question on how to weight the data of each technique. In order to handle this problem with a proper stochastic model one needs to select an estimation strategy as described in the following.

3.3.1. Variance component estimation.

As the parameter estimation in c5++ relies on a Gauss–Markov model [24] it is possible to implement methods to determine the variance components of different groups of observations. The idea of variance component estimation (VCE) goes back to the beginning of the last century [25] and has seen many improvements and refinements based on different algorithms. Following this concept, it is possible in c5++ to estimate the variance component for VLBI observations, station-based GPS variance components for both GPS code- and carrier phase observations, as well as a component that gives proper weight to the local tie information (see section 3.3.2). In order to realize VCE for parameter adjustment problems with several thousand parameters, the simplified algorithm proposed in [26] has been implemented in c5++ . Thus, one can determine the variance factor $\sigma _{ii}^{2}$ of the i-th group of observations by

$$\begin{eqnarray}&&\sigma _{ii}^{2}=\frac{v_{i}^{T}{{P}_{i}}{{v}_{i}}}{{{r}_{i}}}\end{eqnarray} \tag{ 2 }$$

where v_i denotes the residual vector of the prior adjustment and P_i is the corresponding variance-covariance matrix. Together with the degree of freedom r_i of the i-th group, one can iteratively determine the variance components while solving the least-squares adjustment problem. Since correlations among the observations are not considered here one can update the variance component parameters after each iteration without the need of maintaining the large variance-covariance matrix as required for the original algorithm described in [25].

3.3.2. Local ties.

Figure 4 depicts the de-trended clock differences from the single-technique VLBI and GPS c5++ analysis for the third 3 d batch. Only baselines connecting to Wettzell are shown here in order to get an idea about the magnitude and behaviour of the clock differences in the network. A good agreement between both techniques is observed. The same small scale clock variations (see baseline WTZZ-ONSA) as well as large scale patterns (see baseline WTZZ-WES2) can be identified in both single technique results, pointing out the potential of combining both techniques for the purpose of frequency transfer. One can also notice that an observational data gap at the Tsukuba VLBI site causes the estimation to rely on constraints, resulting in a straight line during most of September 21, 2011. This means that in a combined analysis, GPS will bridge the VLBI data-gap and this implicitly support this station which would otherwise strongly rely on the choice of the constraints. Figure 5 depicts the frequency transfer instabilities as obtained from the VLBI and GPS c5++ analysis for all possible links in the reduced CONT11 network (6 stations). The recorded GPS data were decimated from 30 s to a regular sampling rate of 5 min, in order to exactly meet the temporal resolution of the estimated clock model, which has been also set to 5 min (see table 1). VLBI observations were available with a more irregular and sparse temporal sampling which enables us to estimate station clocks only with a temporal resolution of 30 min, given that constraints stabilize the solution in case no data is available over a longer period (see section 3.2). Thus, one needs to take into account that for shorter averaging periods, VLBI and GPS frequency transfer stabilities can not be compared directly. Despite this drawback, it becomes clear that on almost all baselines the individual VLBI and GPS solutions provide the same frequency transfer stability for averaging periods of 4 h or longer. This is in good agreement with the results presented in [17] where the complete CONT11 station network (more than 6 stations) was analysed with two individual analysis software packages applying slightly different processing strategies (see section 3.1). Based on the results from the single technique solutions one can conclude that VLBI reaches a frequency transfer stability similar to GPS for longer averaging periods, but does not allow a sufficiently dense temporal sampling of the clock estimates in order to provide any information for shorter averaging periods. Similarly to the concept presented in [27], where GPS and TWSTFT were combined, one could assume that a combination of GPS and VLBI will lead to a frequency transfer performance that benefits from the strength of both techniques.

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Combining VLBI and GPS on the observation level is the most natural way to estimate common parameters at geodetic co-location sites. With this approach, biases and technique-specific considerations can be taken into account before the observations are combined directly within the adjustment process. Two of such inter-technique biases are those related to troposphere and clock variability. The former one being mainly caused by different heights of the reference points of co-located instruments, which translate into a hydrostatic delay that can be either modeled or, as done in the present study, estimated as a constant offset during each 72 h batch analysis. The latter offset, hereafter denoted as Δ_clk(t), is expected to be caused by temperature-induced cable delay variations and can either be estimated as a constant offset or parametrized by a PLO model with sufficient temporal resolution to follow cable delay changes. It is obvious that the choice of this inter-technique delay potentially absorbs any benefit that could result from the estimation of a common clock between the two techniques. A too-short temporal resolution of Δ_clk(t) bears the risk that a large fraction of valuable information from the second technique is not being reflected in the estimated clock. On the other hand, an insufficient model representation of the mostly diurnal cable length changes potentially degrades the combined solution and the obtained clock parameters. Therefore, different choices for the temporal resolution of the inter-technique cable delay model Δ_clk(t) have been used and their impact on the frequency transfer stability has been evaluated.

In order to evaluate whether the combination of VLBI and GPS leads to an improvement of the frequency transfer stability w.r.t. the GPS-only solution, the ratio

$$\begin{eqnarray}&&\kappa \left(\tau \right)=\frac{\frac{1}{N}\underset{N}{\sum} \text{MDE}{{\text{V}}_{\text{GPS}}}\left(\tau \right)}{\frac{1}{N}\underset{N}{\sum} \text{MDE}{{\text{V}}_{\text{GPS}+\text{VLBI}}}\left(\tau \right)}=\frac{\underset{N}{\sum} \text{MDE}{{\text{V}}_{\text{GPS}}}\left(\tau \right)}{\underset{N}{\sum} \text{MDE}{{\text{V}}_{\text{GPS}+\text{VLBI}}}\left(\tau \right)}\end{eqnarray} \tag{ 3 }$$

is defined, where N is the number of baselines that are averaged. The ratio κ(τ) describes the average improvement/degradation when combining VLBI with GPS on the observation level for frequency transfer, compared to a GPS-only solution. Since the intervals for the PLO clock model were set to 5 min (see table 1) in both solutions one can compute κ(τ) in a straightforward way. Improvements in the overall frequency transfer performance will then be reflected as κ(τ) > 1, whereas degradations can be recognized when κ(τ) < 1. Since the data were processed in batches of 3 consecutive days one would expect to have N = 5·6·5/2 = 75 baselines when analysing the 15 d long CONT11 campaign with the reduced 6 station network depicted in figure 2. However, since the GPS receiver at TSKB lost lock on the third day of CONT11 and the GPS receiver at HRAO had a similar problem on the 5th day, these two stations were excluded from the first and second 3 d batch solutions, respectively. Thus, the total number of baselines over which κ(τ) can be computed reduces to N = 65. Figure 4 shows how well the clock differences match those obtained from the single-technique GPS PPP solution. VCE helps to determine technique specific weights that integrate VLBI observations in a way which does not degrade the frequency transfer performance, mainly determined by GPS. In order to study this in more detail one needs to evaluate κ(τ) for the different processing options. Figure 6 depicts κ(τ) for solutions with different choices for the temporal resolution of the inter-technique cable delay model Δ_clk(t). Overall, it can be seen that the combination of VLBI with GPS tends to improve the average frequency transfer stability w.r.t. the GPS-only solution. However, as anticipated in the previous section, the choice of the temporal resolution of Δ_clk(t) is crucial. The use of an interval length of 1 h for the PLO of Δ_clk(t) absorbs almost all benefit gained from adding VLBI. On the other hand, it is clearly visible that daily estimates or parametrization as a constant lead to a degradation of the short term stability while improving the long-term stability more than any of the other choices for the temporal resolution of Δ_clk(t). In general, one can see that VLBI improves the frequency transfer stability for averaging periods between 3000 and 20 000 seconds as well as for periods close to one day. The latter improvement can be explained by the fact that VLBI helps to smooth the jumps introduced by day boundary discontinuities of the used IGS orbit and clock products. However, one needs to consider also the lower significance (higher uncertainty) of the MDEV at the far end of the long averaging period domain. The improvement between 3000 and 30 000 seconds is thought to have its origin in the parametrization of tropospheric estimates, which become more robust against data artifacts, when combining VLBI and GPS. In addition, a temporal resolution of 12 h or longer for Δ_clk(t) leads to an improvement for averaging periods of 12 h, which might relate to the orbital period of GPS satellites.

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In order to assess which baselines benefit from combining the two techniques, κ(τ) was re-sampled to 28 logarithmically equally spaced data points, before the mean for each baseline was computed. This approach allows to derive the average improvement/degradation for a given baseline for which the frequency transfer performance gain is evaluated. Figure 7 depicts these results, displayed as a net chart. It becomes clear that for some baselines the gain of adding VLBI strongly depends on the choice of the temporal resolution of the inter-technique clock delay Δ_clk(t). It can be concluded that having a half-daily or daily resolution for the PLO model of these delays leads to an improvement on almost all baselines, bearing in mind that this comes mostly from averaging periods longer than a few hours and is accompanied by a degradation in the shorter time domain (see figure 6). It is also worth to mention that there is hardly any improvement on the shortest baseline in the network, i.e. WTZZ-ONSA, which can be explained by the fact that any error introduced by orbit or satellite mismodeling affects the PPP solutions at both sites approximately at the same level and thus cancels out to a large extent when forming the baseline differences and computing the frequency transfer stability over this link. On the other hand, on long intercontinental baselines it can be seen that κ(τ) is always larger than 1, which indicates that VLBI helps to improve these frequency links.

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We close this section by noting that use of the Allan Deviation statistic produced similar results and that the question of optimization might be a fruitful subject of further study.