1 Introduction

Global Terrestrial Reference Frames (TRF) are an imperative prerequisite to measure parameters such as time, positions, and velocities on the Earth since every geometric measurement requires a basis for the parameter estimation. Potential uncertainties in the TRF directly propagate into the parameters based on the TRF and hence restrict the comprehension of processes associated with the Earth. In worst case erroneous conclusions for relevant global phenomena such as the global sea level rise could be drawn.

The underlying TRF should be at least an order of magnitude more accurate than the derived parameters. The global sea level rise as the most prominent example, with the recently published value of 2.74 ± 0.58 mm/year (Rietbroek et al. 2016), should be based on a TRF which is highly accurate (1 mm) and stable in long-term (0.1 mm/year) as required by the Global Geodetic Observing System (GGOS) (Gross et al. 2009).

Since these requirements have not been met yet, simulation studies based on known stochastic processes as input data can contribute to understanding better the limiting factors in TRF determination (Schuh et al. 2016). Latest TRFs such as the ITRF2014 (Altamimi et al. 2016) with its predecessor ITRF2008 (Altamimi et al. 2011), the DTRF2014 (Seitz et al. 2016, predecessor: DTRF2008 Seitz et al. 2012), and the JTRF2014 (Wu et al. 2015; Abbondanza et al. 2016) were determined from a combination of the four space geodetic techniques: Doppler Orbitography and Radiopositioning Integrated by Satellite (DORIS), Global Navigation Satellite System (GNSS), Satellite Laser Ranging (SLR), and Very Long Baseline Interferometry (VLBI).

There are different strategies concerning the combination of the techniques. All latest TRFs were determined using local measurements at co-located sites on the ground (“local ties”—LT). However, it is known that there are discrepancies between the local measurements and the estimates from the space geodetic techniques (Altamimi et al. 2011; Thaller et al. 2011; Glaser et al. 2015a). One reason is the inhomogeneous data base of the LT (Ray and Altamimi 2005; Seitz et al. 2012) with sometimes limited stochastic modeling leading to a degradation of the combination results. Alternatively the techniques can be combined via parameters which can be observed by all techniques such as the Earth Rotation Parameters (ERP) known as “global ties” (GT, Seitz et al. 2012; Glaser et al. 2015b).

In view of GGOS, many investigations on TRF improvements based on simulations can be carried out. Besides simulations of future networks and expected improvements in the observation accuracies and their impact on the TRF, it is also important to get more detailed insights in the combination of the techniques within the TRF determination process. The geodetic datum of the network has to be realized in order to get a solution. Within the combination using either LT or GT, the datum is transferred from one technique to the other(s). For instance, the origin of the ITRF is realized solely by SLR which means that the translation information was transferred from SLR to the other techniques by using LT for the combination. In case of GT when combining ERP, orientation information can be transferred. Simulated observations are free of unknown systematic effects and allow to exactly tell how well the datum can be transferred.

In this paper, we focus on the combination of normal equation systems generated from simulated observations of the current ground network of VLBI and SLR spanning the period 2008–2014. Based on single-technique solutions (Sect. 2), we test different combination strategies applying LT (Sect. 3.1) with e. g. various stochastic models and GT (Sect. 3.2) with regard to the accuracy of the datum realization within the combination in view of GGOS requirements. Furthermore, the current ground network of VLBI is extended by adding artificial stations (Sect. 4.1) and additional observations (Sect. 4.2) and the impact on the TRF-defining parameters and their accuracies are investigated. Finally, we conclude our study with the most important findings in Sect. 5.

2 Single-technique solutions

2.1 VLBI-only

We commence our investigations from simulating VLBI observations that have actually taken place. Initially, we carry out simulations considering the most important random error sources in VLBI (wet tropospheric delay, station clock and measurement noise, Pany et al. 2011) to obtain interferometric group delays for the standard weekly IVS-R1 and IVS-R4 (International VLBI Service for Geodesy and Astrometry; Schuh and Behrend 2012) sessions spanning the period from 2008 to 2014. The simulated zenith wet delays are based on the turbulence model of Nilsson and Haas (2010); the station clocks are simulated as random walk plus an integrated random walk process with an uncertainty which is characteristic for hydrogen maser systems at current VLBI sites. The measurement noise is based on the formal errors of the real observations in order to ensure affinity of the simulated and the real observations, see Fig. 2. In total, observations for 28 globally distributed stations (Fig. 1) were simulated. The GFZ version of the Vienna VLBI software (Böhm et al. 2012), VieVS@GFZ (Nilsson et al. 2015), is used for the simulation (module: VIE_SIM) and the data analysis (module: VIE_LSM). The single-session normal equation systems (NEQs) generated with VIE_LSM are stacked to one NEQ within the VIE_GLOB module. Station positions, velocities, and ERP (pole coordinates and UT1-UTC) were estimated in a long-term solution to determine a VLBI-only TRF. The geodetic datum was realized by imposing no-net translation (NNT) and no-net rotation (NNR) conditions to the positions and velocities of the VLBI core stations, see Fig. 1. The core stations were selected considering their actual performance regarding the number of observations within 2008-2014 and their location aiming at a good global distribution. Five stations (AIRA, KASHIM11, KASHIM34, TIANMA65, PARKES) were reduced in the long-term solution since they have rather short observation time spans (only few months within 2008–2014, see Fig. 2) which does not allow to reliably estimate station velocities (Glaser et al. 2016). Important to note is that the simulation and the analysis of the VLBI observations was done with standards that are also consistent with SLR. They follow the IERS Conventions 2010 (Petit and Luzum 2010) including that atmospheric loading was not taken into account. ITRF2008 and IERS C04 08 EOP series (Bizouard and Gambis 2011) were used as a priori values.

Fig. 1
figure 1

Network of 23 VLBI (blue diamonds) and 38 SLR (red circles) stations with 10 local ties (yellow circles). Two co-located VLBI telescopes at Hobart, Tasmania and at Hartebeesthoek, South Africa, and the SLR stations Katzively and Simeiz, Ukraine, are overlapping. Filled symbols indicate core stations (VLBI: 10 and SLR: 9). The 5 reduced VLBI (see Sect. 2.1) and 3 reduced SLR (see Sect. 2.2) stations are not shown

Full size image
Fig. 2
figure 2

Formal error of delay (mm) of simulated VLBI data per station within the time span 2008–2014 according to real observations

Full size image

The parameters of a 14-parameter Helmert transformation between the VLBI-only TRF and ITRF2008 (Table 1) reveal that the VLBI-only solution is consistent with the ITRF2008 on the mm-level. All stations of the networks instead of only the core stations were considered for the transformation since all stations represent the network. Outliers have been rejected within the transformation applying a \(3\sigma \)-criterion.

2.2 SLR-only

The SLR simulations are based on the analysis of real SLR data to the satellites LAGEOS-1 and LAGEOS-2 over the period from 2008 to 2014. The real analysis provides the periods where SLR observations are taken in the network, and it gives the accuracy levels of each station in the network in each arc. The accuracy level is given by the orbital fit of the observations per station and per arc in terms of root mean squared (RMS) values. In the simulation these RMS values were used to generate white noise deviations to the theoretical observations. This means that the uncertainties of the simulated SLR observations are at the level of the uncertainties of real data. The observations are simulated only if the station was active in that arc in reality. The number of observations per station and per arc in the simulation is steered according to the relative percentages in reality by utilizing the EPOS software package (Zhu et al. 2004). We passed on following different occupations of some single stations, so that leaps of the coordinates and changes in velocities are prevented. We ended with 365 7-day arcs and simulated nearly 1,000,000 observations for the 41 stations depicted in Fig. 1. Figure 3 compiles arc-wise the RMS values of an orbital analysis with the simulated observations instead of the real ones over the study period. Concluding, we designed our simulations very close to the actual observations.

Table 1 14-parameter Helmert transformation parameters (translation \(T_x\), \(T_y\), \(T_z\), rotation \(R_x\), \(R_y\), \(R_z\), scale D and their temporal changes with their corresponding standard deviations) at reference epoch 2005.0 of VLBI- (top) and SLR-only (bottom) solution with standard deviation \(s_0\) of the unit weight of the transformation w. r. t. ITRF2008 considering all network stations
Full size table
Fig. 3
figure 3

Orbital fit of simulated SLR ranges (mm) to LAGEOS-1 after acceptance in precise orbit determination per station within the time span 2008–2014

Full size image

The simulated observations were analyzed and an SLR-only TRF was determined. Station positions and velocities as well as ERP (pole coordinates, UT1-UTC) were estimated. All other parameters such as the orbital elements were reduced. “Reducing or pre-eliminating of a parameter” means that this parameter was not explicitly estimated but is implicitly included in the normal equation system (Brockmann 1997). The geodetic datum was realized with NNR conditions imposed on the coordinates and the velocities of the SLR core stations, see Fig. 1. As with the VLBI-only solution (Sect. 2.1), stations with short observation time spans within 2008-2014 were reduced: Kunming, China (7820) with 4 months; Brasilia, Brazil (7407) with 6 months; Daedeok, Republic of Korea (7359) with 1 year of observations.

It should be emphasized that the same a priori values (ITRF2008, IERS C04 08) and reduction models were applied as for VLBI in order to ensure consistency. The parameters from a 14-parameter Helmert transformation (Table 1) indicate that the SLR-only TRF from simulated observations is consistent on the mm-level with ITRF2008. Compared to the Helmert parameters of the positions of the VLBI-only solution w. r. t. ITRF2008, the Helmert parameters of the SLR-only solution are smaller probably due to the fact that only white noise was simulated in case of SLR.

In addition, it should be noted that our simulations are based on single runs per observation in case of SLR and VLBI and are not obtained as repeatabilities from several simulations runs. The a posteriori information like standard deviations of the estimated parameters are taken from the covariance matrix. The standard deviations of the estimated parameters tell then how well the parameters could be estimated which is the same for the real observations. This also allows to compare with real TRFs such as the ITRF2008.

3 Combination results

The datum-free NEQs of VLBI and SLR are used as input for the combination applying LT (Sect. 3.1) and GT (Sect. 3.2) with appropriate datum constraints by using the EPOS software package.

Fig. 4
figure 4

7-parameter Helmert transformation parameters (translation \(T_x, T_y, T_z\), rotation \(R_x, R_y, R_z\), scale \(S_c\) and standard deviation \(s_0\) of the transformation) of the combined solution w.r.t. VLBI-only by using VLBI core stations in case of different a priori local tie standard deviations: \(s_{\text {LT}} = [0.01; 0.1; 1; 10; 100]\) mm (left) and different numbers of local ties: all 10 local ties, 6 northern (NH) and 4 southern (SH) hemisphere LT stations, 1 in Wettzell (WETZ) and 2 in Hartebeesthoek (HART) (right)

Full size image

3.1 Local ties

The datum-free SLR- and VLBI-NEQs were combined by using LT. Coordinates and velocities of the SLR and VLBI stations were estimated, the ERP were reduced before stacking the individual NEQs to minimize computation time. Ten LT at nine co-locations between SLR and VLBI (Fig. 1) were introduced as pseudo-observations in the combination. There are two VLBI telescopes at station Hartebeesthoek, South Africa. Unlike in a real TRF solution, where the measured LT are applied, we assumed the LT to be known (a priori coordinate differences are set to zero within the deterministic model). The stochastic model is based on uncorrelated observations with the same accuracy for all LT in order to investigate the impact of different homogeneous accuracy levels on the TRF. In addition, co-motion constraints were imposed since the velocities at co-located sites should reflect the same geophysical behavior.

Applying LT and co-motion constraints, the geodetic datum was realized as follows: The origin and its temporal change were realized from SLR and the scale and its temporal change inherently from the VLBI and SLR observations. The orientation and its temporal change were realized by imposing NNR conditions to the SLR core stations, see Fig. 1. In order to investigate how well the datum can be transferred among the techniques within the combination, we imposed the NNR conditions to the SLR core stations only and compared the combined solution with the VLBI-only solution.

In Fig. 4 the parameters of a 7-parameter Helmert transformation of the combined TRF w. r. t. the VLBI-only TRF considering just the core stations are depicted in the left part in case of applying different standard deviations of the LT. There are maximum differences of 25 mm in the rotations and 21 mm in the translations. The scale is not affected appreciably only by 1 mm when changing the a priori LT standard deviations. The transformation parameters for \(s_{\text {LT}} = [0.01; 0.1; 1]\) mm are very similar, therefore, there is not a large impact on the TRF-defining parameters. However, LT with inferior precision (\(s_{\text {LT}} = 100\) mm) lead to poor datum transfer. We can conclude that in case of known LT the a priori standard deviation of the LT should be 1 mm or better in order to best transfer the datum from one to another technique within the combination.

Another interesting point is the impact of different numbers of LT on the TRF-defining parameters. We investigated 5 different cases: all 10 LT stations, 6 northern and 4 southern hemisphere LT stations, 1 in Wettzell, and 2 in Hartebeesthoek. In all cases the LT have an assumed standard deviation of \(s_{\mathrm {LT}} = 1\) mm. Referring to Fig. 4 (right part), the largest differences in the Helmert parameters occur, if one LT only in Wettzell was applied in the combination. The differences between all LT and only LT on the southern hemisphere are \({\sim }\)9 mm in \(T_x\) and \(R_y\) evidencing the lack of LT on the southern hemisphere.

Fig. 5
figure 5

Estimated Earth rotation parameters (pole coordinates \(x_p\), \(y_p\), and UT1-UTC) of VLBI-(blue), SLR-only (red) and combined (green) solution w. r. t. IERS 08 C04 with mean and standard deviation (in brackets). The first UT1-UTC value for SLR was fixed to a priori

Full size image
Table 2 14-parameter Helmert transformation parameters (translation \(T_x\), \(T_y\), \(T_z\), rotation \(R_x\), \(R_y\), \(R_z\), scale D and their temporal changes with their corresponding standard deviations) at reference epoch 2005.0 between the combined and the SLR-only solutions evaluated at the SLR core sites when combining all ERP (above) or only pole coordinates (below) as a global constraint
Full size table

3.2 Global ties

Another combination strategy applying GT instead of LT was investigated. The following parameters were estimated: station positions, velocities and ERP (pole coordinates \(x_p\), \(y_p\), and UT1-UTC) whereby the ERP are common for both SLR and VLBI as they have been combined. VLBI, as the only space geodetic technique, is capable to observe UT1-UTC in an absolute sense. However, SLR is able to observe LOD (length of day) which is its time derivative with the precise relationship \(\mathrm{d}(\text {UT1-TAI})/\mathrm{d}t = -\varDelta \text {LOD}/86400\) s with \(\varDelta \text {LOD} = \text {LOD} - 86400\) s, e.g., Bloßfeld et al. (2014). The VLBI solution is based on R1 (24 h session starting on Mondays) and R4 (24 h session starting on Thursdays) sessions. Three ERP have been set up per 24 h session: midnight before, within and after the session. A loose relative constraint of 0.1 mas was introduced to stabilize the ERP estimates within the solution. However, one day per week (usually Sunday) is without ERP estimates from VLBI. In case of SLR the ERP are set up at midnight every day from 7-day arcs yielding 8 values per week. At the interval boundaries between the 7-day arcs one ERP value is based on two values: end of arc i and start of arc \(i+1\). This continuous piece-wise linear parametrization is equivalent to the parametrization as offset (UT1-UTC) and drift (LOD). If at least the first value of the continuous piece-wise linear polygon has been fixed to a priori, UT1-UTC can be estimated from satellite techniques, see also Thaller et al. (2007) for the estimation and combination of ERP from VLBI and GPS. The following conditions were imposed to realize the geodetic datum: The scale and its temporal change were realized inherently from the VLBI and SLR observations. The orientation and its temporal change were realized by NNR condition to the VLBI core stations, see Fig. 1. Since VLBI is not sensitive to the origin and GT only allow to transfer orientation information, NNT conditions to the VLBI core stations needed to be imposed to realize the origin of the combined network.

The ERP from the single-technique and the combined solution are depicted in Fig. 5. The scatter of the pole coordinates w. r. t. IERS 08 C04 of the VLBI-only solution is much larger than for SLR. The estimates of the pole coordinates from the SLR simulations are probably too optimistic as the only error source in case of SLR is white noise based on the level of orbital fit, see Fig. 3. Within the SLR-only solution, UT1-UTC was fixed to the first a priori value (IERS 08 C04). The time series of UT1-UTC of SLR is drifting away due to uncertainties in the real orbit determination which is the basis for the simulation, see also Bizouard and Gambis (2011) and Bloßfeld et al. (2014). Within the combination of ERP from SLR and VLBI, we can investigate if and how well datum information can be transferred from one to another technique within the combination applying GT.

The 14-parameter Helmert transformation parameters between the SLR-only and the combined solution are tabulated in Table 2. The combination of \(x_p\), \(y_p\), UT1-UTC (upper part of Table 2) leads to a significant rotation around the Z axis (\(\sim \)27 cm). This means that the orientation around Z cannot be transferred from VLBI to SLR within the combination of UT1-UTC. It should be noted that, when UT1-UTC of SLR and VLBI was combined, UT1-UTC of both VLBI and SLR series does not contain any a priori constraints, otherwise the combined solution will be biased. Since the combination of UT1-UTC did not lead to a sufficient datum transfer of \(R_z\), UT1-UTC was fixed to a priori for both SLR and VLBI. The combination of \(x_p\), \(y_p\) and an additional NNR condition around the Z axis to SLR core stations (lower part of Table 2) results in a much smaller rotation around Z of about −0.6 mm. Whereas including UT1-UTC to the GT distorts the combined solution (\(R_z = 269\) mm), the GT with only the pole coordinates does not significantly distort the combined solution since the rotations \(R_x\) and \(R_y\) are only 1 mm. The other transformation parameters in Table 2 are close to zero indicating that the origin and the scale of the combined network solution were not distorted by the combination applying GT.

Considering the co-located sites, the differences between the coordinates estimated from the single-technique and the combined solution can also tell how well the combination of the pole coordinates of VLBI and SLR was carried out, see Fig. 6. These differences at the 10 LT sites show that the datum can be transferred with 4.1 mm in north and sub-mm in the up component on average. The east component depends on the NNR condition around the Z axis and therefore cannot be evaluated. It should be noted that these differences are minimized if the station is a core station (indicated with “v” for VLBI and “s” for SLR in Fig. 6). For that reason, Hart (HART15M) shows larger differences in north and up compared to HartRao.

Fig. 6
figure 6

Coordinate differences in north, east and up component for the 10 local ties (Fig. 1) of combined w. r. t. VLBI-, SLR-only solution, respectively, with mean and [min max]. Core stations are indicated with v (VLBI) and s (SLR). East component depends on datum definition (NNR around Z)

Full size image

4 Extension of VLBI simulations

Simulation studies also allow to add artificial components to the existing data. In case of VLBI we extended the current state by additional stations (Sect. 4.1) and additional observations (Sect. 4.2) and investigated their impact on the TRF toward a TRF meeting the GGOS requirements.

4.1 Spatial extension

We focus here on the spatial extension of a 14 station current VLBI network (Fig. 7) and examine the impact on the (VLBI-)TRF and its accuracy when one artificial station was added to the 14 station network. In this network the station AGGO (Argentinean German Geodetic Observatory) in La Plata, Argentina, is included since it will be operational in the next months. Three different scenarios with additional stations were studied:

  • Tahiti, French Polynesia (co-located with GNSS, SLR, DORIS, VLBI planned, Biancale 2016a)

  • CGGN, close to Toro in Nigeria (co-located with GNSS, VLBI planned)

  • Medicina, Italy (existing VLBI station for cross checking)

Fig. 7
figure 7

VLBI station network with core stations in red and additional stations highlighted

Full size image

In order to simulate observations for new VLBI stations it is necessary to first set up schedules reflecting when which stations observe which quasar (called “scan”). This was done with the module VIE_SCHED within VieVS@GFZ utilizing the source-based strategy (Sun et al. 2014) observing four sources simultaneously. The properties, such as antenna sensitivity (system equivalent flux density—SEFD) and slewing speeds, of the current stations were set according to their individual current performance. The performance of the new stations Tahiti and CGGN was assumed according to next-generation antenna specifications with slewing speeds of 12 \(^\circ \)/s in azimuth and of 6 \(^\circ \)/s in elevation. The schedules serve as the basis for the simulation. In contrast to the simulation of the current ground network (Sect. 2.1), the number of observations and their formal errors could not be taken from actual observations. We assumed continuous observations with a formal error of 30 ps (\(\sim \)9 mm) over a time span of 7 days. The simulated observations were analyzed and finally, station positions and ERP were estimated. This was done four times for the 14 station reference solution (RS), RS plus Tahiti, RS plus CGGN, RS plus Medicina. The number of scans where the additional stations participated is similar to each other and compared to the other stations they are rather higher performing stations such as Fortaleza and Tsukuba. In total, the difference of the number of all scans compared to the reference solution is rather small (Tahiti +1%, CGGN +1%, Medicina −1%) ignoring the scans of the additional stations.

Table 3 Standard deviations s of the estimated parameters (station coordinates x, y, z, pole coordinates xp, yp and UT1-UTC) and the difference to the reference solution (w/o) in percent as well as the condition number \(\mathrm {cond}(\varvec{\varSigma }_{XX})\) of the variance-covariance matrix \(\varvec{\varSigma }_{XX}\) in case of the solutions with different additional stations from a 7-day global solution with the standard deviation of the unit weight \(s_0\)
Full size table

The standard deviations of the station positions and the ERP of the different 7-day global solutions are presented in Table 3. As far as the precision of the station positions is concerned, an improvement of about 13% in case of Tahiti, about 12% (CGGN) and about 11% (Medicina) can be noticed. In addition, the ERP improve by \(\sim \)17% (Tahiti), \(\sim \)11% (CGGN), and \(\sim \)7% (Medicina). The improvements in the station positions for the three cases are similar. In contrast, adding Tahiti to the network improves mainly the ERP estimates, especially \(y_p\). The smallest improvement occurs if Medicina was added to the network.

In order to draw a conclusion on the reliability of the network solution, the condition number of the variance-covariance matrix of the estimates including only station positions can be consulted. The condition number is defined as the ratio of the maximum and minimum eigenvalue: \(\mathrm {cond}(\varvec{\varSigma }_{XX}) = \lambda _{\max }/\lambda _{\min }\) taken from a singular value decomposition \(\varvec{\varSigma }_{XX} = \mathbf X \varvec{\varLambda } \mathbf X ^{\mathrm {T}}\) with \(\varvec{\varLambda }\) matrix of eigenvalues \(\lambda \) ordered by their value and eigenvectors \(\mathbf X \). The closer the condition number to 1 the better the numerical stability of the matrix and the larger the reliability of the solution. Comparing with last line in Table 3, the network solution with Tahiti is most reliable in a global sense compared to the reference solution. The network solution including Medicina is even characterized by a deterioration of the reliability of the network solution.

In addition, the quality of the datum realization of the reference frame in case of the networks with additional stations was assessed. We employed the method as presented in Sillard and Boucher (2001) which is based on the assumption that the variance-covariance matrix is decomposed in two parts. One part is associated with the datum realization of the solution (“reference system effect”), the other is independent of the datum realization. The datum part is expressed by the standard deviations of 7 parameters of a Helmert transformation as listed in Table 4. These standard deviations show how well the solution is sensitive to the respective parameter. Small values indicate a good sensitivity, large values the opposite. Relative to the reference solution, the standard deviations of the translations in x, y, z get smaller the most for Tahiti (\({\sim }-\)19%) followed by the solution with CGGN (\({\sim }-\)12%) and Medicina (\({\sim }-\)8%). The same applies to the rotations: −19% (Tahiti), −13% (CGGN) and −9% (Medicina). This shows that the network solution with Tahiti is most sensitive for translations and rotations compared to the reference solution. An improvement especially in \(R_y\) can be noticed which is in concert with the improvement in \(x_p\), see Table 3. There is not a large change in the standard deviations of the scale. In general, the standard deviation of the translations and rotations are up to 8 times worse than the one of the scale since VLBI is able to realize the scale inherently from the observations, whereas translations and rotations need to be realized by NNT and NNR conditions.

Table 4 Reference system effect expressed with standard deviations s of 7 parameters of Helmert transformations (translations \(T_x\), \(T_y\), \(T_z\), rotations \(R_x\), \(R_y\), \(R_z\), scale D) according to Eq. (18) in Sillard and Boucher (2001) in case of the solutions with different additional stations w. r. t. the reference solution (w/o)
Full size table

4.2 Temporal extension

The VLBI-only solution (Sect. 2.1) is based on standard observations twice a week (R1 and R4 24-h sessions) without including IVS Intensive sessions. Moreover, 24/7 operations are one of the envisaged goals of the VLBI Global Geodetic Observing System (VGOS, Petrachenko et al. 2014). To investigate the impact of a continuous temporal extension of data on the VLBI-TRF, daily 24h sessions featuring the current ground network and spanning the period 2008-2014 were scheduled, simulated and analyzed. The following parameters were estimated: station coordinates x, y, z and velocities vx, vy, vz, pole coordinates \(x_p,y_p\) and UT1-UTC. The standard deviations of the estimated parameters in case of the standard R1, R4 and the “every day” solution can be found in Table 5. The improvements in the coordinates and velocities by 47% can be mainly explained by the larger number of observations. In case of the every-day solution there are about 4 times more observations compared to the R1, R4 solution. Therefore, the standard deviation should get smaller by 50% (\(1/\sqrt{4}\)) which is approximately the case for the positions and velocities. The ERP shows a slightly larger improvement of about 65% on average. The largest improvement by 67% can be noticed for UT1-UTC. The reason is that the ERP were determined from a long-term solution consistently together with the station positions and velocities. If the station positions and velocities improve, the ERP improve as well. Furthermore, the ERP estimates also improve due to the stacking of the every day observations.

Table 5 Standard deviations s of the estimated parameters: station coordinates x, y, z and velocities vx, vy, vz, pole coordinates xp, yp and UT1-UTC and degree of freedom (DOF) of the standard R1, R4 and the “every day” solution within the time span 2008–2014
Full size table

The standard deviations \(s_0\) of about 1.4. in Tables 3 and 5 are in the investigated cases almost the same. This means for example that adding another station to the solution does not change the quality of the adjustment. It ensures the comparability of the different solutions. Moreover, the value of 1.4 slightly deviates from 1.0 which could indicate that the stochastic model can be improved. However, 1.4 is close to 1.0 and a common value for VLBI analysis of real data.

5 Summary and conclusions

The impact of different combination strategies on the datum realization of TRFs was investigated by simulating VLBI and SLR observations of the current ground networks in the time span 2008-2014. The observations were simulated to behave as close as possible as real measurements. Datum-free NEQs from consistent modeling on a session- and arc-wise basis containing station positions and velocities as well as the pole coordinates and UT1-UTC were combined to determine single- and multi-technique TRFs. On the basis of 14-parameter Helmert transformations, the VLBI-only and the SLR-only TRFs are consistent on the mm-level with ITRF2008.

Different combination strategies were applied involving LT and GT together with proper datum realizations. By utilizing known LT with varying precisions, we examined that the geodetic datum can be best transferred from one to the other technique if the LT standard deviation is at least 1 mm. This confirms the desired LT precision level of 1 mm (e.g., Boucher et al. 2015). Testing different numbers of LT demonstrates the lack of co-located sites between SLR and VLBI on the southern hemisphere by \(\sim \)9 mm differences in \(T_x\) and \(R_y\) w. r. t. the combination utilizing all available LT.

The combination applying the pole coordinates and UT1-UTC as GT instead of LT results in a significant rotation around the Z axis between the SLR-only and the combined solution. The reason for this is probably the missing sensitivity of SLR to UT1-UTC since UT1-UTC is correlated with the right ascension of the ascending node of the orbit and the zero meridian of the station network. Therefore, it is not sufficient to get UT1-UTC from VLBI in the combined solution. A reason for this can also be that the ERP estimates from VLBI are not available every day as the data are based on R1 and R4 sessions. Thus, UT1-UTC was fixed to a priori and an additional NNR condition around the Z axis was imposed and solely the pole coordinates act as GT. By doing so, the datum can be transferred with an accuracy on the mm-level in north and sub-mm-level in up component consulting co-located sites.

Another possibility to transfer datum information within the combination of different techniques is to consult space ties. In order to co-locate all four space geodetic techniques in space, satellite missions like GRASP (Bar-Sever et al. 2009, 2015) and E-GRASP/Eratosthenes (Biancale 2016b) are envisaged striving the anticipated goal to determine a TRF with unprecedented accuracy and stability meeting the GGOS requirements. Here, future simulation studies will also contribute considerably to investigate the accuracy of the datum realization.

Simulations further enable to extend the current infrastructure by e. g. adding artificial stations and observations in order to examine the impact on the estimated parameters and their accuracies and reliabilities. Adding a VLBI station to the current network improves the station positions by \(\sim \)13% in case of Tahiti, by \(\sim \)12% in case of CGGN, and by \(\sim \)11% in case of Medicina. The ERP estimates also improve by \(\sim \)17% (Tahiti), \(\sim \)11% (CGGN), and \(\sim \)7% (Medicina). The network solution with the additional stations in Tahiti and Nigeria is more reliable in a global sense since the variance-covariance matrix has a higher numerical stability. In contrast, adding Medicina results in a degradation of the reliability of the network solution. Furthermore, the network solution with Tahiti which is in a key position is most sensitive to the realization of the translations and rotations utilizing the method of Sillard and Boucher (2001). Here, the standard deviation of the translation of 2.0 mm and the rotation of 3.0 mm head for the GGOS requirements; however, more VGOS stations will be needed to achieve the 1 mm goal. VLBI observations every day improve the ERP estimates by \(\sim \)65% on average compared to standard twice-weekly (R1, R4) sessions (keeping in mind the 4 times larger DOF). The largest improvement in the standard deviations by 67% of UT1-UTC is worth to mention since VLBI is the only space geodetic technique which can determine UT1-UTC in an absolute sense. In order to get daily UT1-UTC values, the two sessions per week (R1, R4) have to be bridged with Intensive VLBI sessions including usually a single baseline. Therefore, every day VLBI observations of a global network will also lead to better results in terms of consistency. Until VGOS will be completely realized, a reduced duty cycle instead of 24/7 observations is planned (Petrachenko et al. 2014).

As shown by VLBI simulations in this study, it is of major significance to improve the network configuration of the TRF with regard to the GGOS goals which was also demonstrated by simulation studies of MacMillan et al. (2016) and Otsubo et al. (2016) and by the iterative most remote point method (Hase and Pedreros 2014). In order to coordinate activities dealing with these important issue, a Standing Committee on Performance Simulations and Architectural TradeOffs (PLATO) was established by GGOS belonging to the International Association of Geodesy.

Further studies will include the combination with simulated GNSS and DORIS observations of the current ground network as well as an extension of the networks with additional stations in order to make reliable conclusions on all techniques contributing to a global TRF for GGOS.