## Abstract

The datum problem is a fundamental issue in the network adjustment when connecting a local measurement network to an external reference frame. Datum elements in 3D networks are scale, translation, and orientation. We consider here the local tie network at geodetic core stations, where the external reference frame is the latest ITRF realization, ITRF2014, in the mean epoch of terrestrial observations.

Accurate distance measurements are used for the determination of the network scale. Thus the improvement of its accuracy and the inclusion of weather measurements to account for refraction errors are essential. For rotation and translation of the network, we need external information. Angle observations are related to the coordinate system of the instrument (e.g. a tachymeter) which is usually aligned to the plumb line. Instruments have different vertical orientation at every station point and the direction of the plumb line does not coincide with the normal vector of the reference ellipsoid. Horizontally the observed set of angles are oriented in arbitrary or approximately oriented directions.

External information which is needed for solving the absolute orientation are datum points, providing the link to the global coordinate system, and correction terms for the vertical orientation (deflection of the vertical), which can be derived from combined terrestrial/GNSS observations, from a gravity based geoid model, or from astronomical observations.

In this article, we present the solutions/options for the datum problem in the framework of the EMPIR GeoMetre project using the example of the ITRF core stations Metsähovi and Wettzell using transformation-free approaches. The inclusion of distant targets is promising, since in small networks even a millimeter change in the coordinates of a datum point can significantly affect a local tie vector. It is shown that at both stations the determination of the deflection of the vertical using different techniques yield the same results within the measurement error.

### Keywords

- Datum problem
- Local ties
- Network adjustment
- Orientation
- Scale

The 18SIB01 GeoMetre project has received funding from the EMPIR programme co-financed by the Participating States and from the European Union’s Horizon 2020 research and innovation programme.

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## 1 Introduction

The local ties are currently used for the combination of the different space geodetic techniques: Global Navigation Satellite Systems (GNSS), Satellite Laser Ranging (SLR), Very Long Baseline Interferometry (VLBI), and Doppler Orbitography and Radiopositioning Integrated by Satellite (DORIS) system, to realize the International Terrestrial Reference System (ITRS). The local ties include the vectors connecting the reference points of space geodetic instruments. The tie vectors, in most cases with the full covariance matrices, in Solution Independent EXchange format (SINEX) are handled in combination as a fifth technique (Boucher et al. (2015)). The local ties allow in the combination the datum transfer among the techniques. That underlines the importance of accurate local ties. IAG working group WG 1.2.1 “Methodology for surveying geodetic instrument reference points” launched the Local Survey metadata project. The SINEX files of local ties are now collected with metadata about the site surveys (IERS 2021; ITRF 2021).

The reference points of VLBI and SLR telescopes are determined indirectly by monitoring measurements. Points rotating around the axes of the telescopes in several angle positions are monitored and included in the network. The reference points are then estimated in the reference frame of the local survey network using mathematical models (e.g. Harvey (1991), Sarti et al. (2004), Dawson et al. (2007), Lösler (2009), Kallio and Poutanen (2012), Lösler et al. (2021)). GNSS observations refer to the ARP (Antenna Reference Point) at least theoretically with certain uncertainty when antenna calibration tables with Phase Center Offset and Phase Center Variation are used. In tachymeter measurements, a common reference point can be achieved almost directly with the seamless network set up where special accurate adapters connect GNSS antennas and prisms. The small vertical offset can accurately be determined by leveling. The older permanent mast points like METS at Metsähovi on the top of the 20m mast are exceptions. As there is no seamless adapter, the ARP must be measured using the geometry of the antenna: vertically the bottom of the antenna and horizontally the left and right side of the antenna – as proposed in “IERS Technical Note No39” (Poyard 2017).

While the vectors between reference points form the local tie network, the local survey network, including pillar points and permanent GNSS points, connects the monitoring measurements to the same local frame (see Fig. 1). The more instruments the local tie network has, the more monitoring networks must be connected. Metsähovi is an example of a rather compact solution of a local survey network (Fig. 1). In Wettzell there are more instruments and the local survey network is more complicated (Fig. 2).

Because the local ties should be given in ITRF, the local survey network should be transformed to ITRF. Thus the datum definition of the local survey network plays a key role. For horizontal orientation, there is no other relevant option than the local GNSS/GPS network. For vertical orientation, the known Deflection of Vertical (DoV) as external information can be used. The geo-referencing methods at different co-location sites varies. There is a data field for referencing method in the above-mentioned metadata CSV-file (ITRF 2021). The short look inside the available files shows that without reading the more detailed reports it is almost impossible to say anything about the quality of the orientation or scale: how the GNSS points were situated in the area, were they permanent or temporary GNSS points, what was the session lengths of GNSS observations, or what is the quality of DoVs. The way how the coordinates of GNSS are constrained may influence the results too, and that depends on the software.

There are no limit values available for uncertainties of orientation angles or scale for local ties. The required accuracy of GGOS local ties is set to 1mm and the measurement accuracy 0.1mm (Gross et al. 2009). With these values, the local ties at co-location sites could be equated to the space geodetic techniques. If the distance between space geodetic instruments is 100m the 0.1mm measurement accuracy means 1 μrad (0.2″) for angles. The requirement of scale accuracy, 1 ppb (Gross et al. 2009) for GGOS, is hardly applicable to short distances in a local tie network.

The reference point estimation gives the uncertainty of the monitoring part but not the uncertainty of the datum definition part of the local survey network. It has been proved that reference point coordinates can be determined within 0.01mm precision using close-range photogrammetry (Lösler et al. 2021). Thus a small monitoring network, spanning a couple of meters, can be internally extremely accurate. However, the datum noise of the external reference frame, the uncertainty of realization of the local frame in ITRF, is not included in these measures. The influence of orientation and the scale uncertainty of the network must be included in the total uncertainty of local tie vectors.

One task of our work in the GeoMetre project is to mitigate the uncertainties of the orientation and scale of the local frame. The realization of the local frame in ITRF needs external information. Although the local survey network is accurate internally, the quality and handling of the external information influence the final alignment in ITRF. The details of the solutions at Wettzell and Metsähovi are presented separately: at first the Sects. 2 and 3 introduce the co-location sites. In Sects. 2.1 and 3.1 the local survey networks at the co-location sites are introduced and the subsequent sections describe solutions for the alignment problem datum element by datum element. Although the focus is on the datum problem, the results, the end product of the work, and the local tie vectors are presented at the end of the sections.

## 2 Wettzell

### 2.1 Local Network

The local survey network of the Geodetic Observatory Wettzell (GOW) consists of 30 deeply founded, shielded concrete pillars spread over the entire observatory. They tie the reference points of the space geodetic techniques together to provide local tie vectors for the mutual control and the combination of the different techniques (Fig. 2). Reference points of 3 VLBI antennas, 2 laser-ranging telescopes, 5 GNSS antennas, and 1 DORIS antenna must be measured and tied together. Fixed antennas as used for GNSS and DORIS can be surveyed more or less directly, either by temporarily removing the antenna or using permanent prism mounts for a seamless connection as proposed above. The permanent prism mounts are preferable, because it is not recommended to remove the antennas. The reference points of moving telescopes, however, have to be constructed as described above. This subset of points including the referencing pillars (monitoring setup) of all telescopes must be included in the network to obtain tie vectors between all space technique reference points. Only the 20-m radio telescope (RTW) has a physical representation of the axis intersection (i.e. the reference point), which can be surveyed directly from outside.

The survey of the most recent solution, which was submitted to the ITRF2020, was done in 2018. In addition to the local network, 3 pillars outside the observatory at distances between 300 and 1200 m were observed in terms of terrestrial survey and GNSS measurements. This setup was established to reduce the problem of orientation uncertainties on short GNSS baselines.

### 2.2 Transformation-Free Approach

Another step to overcome the datum problem is the transformation-free approach, which was optimized in the framework of the GeoMetre project. Usually selected pillars being part of the local survey network are temporally observed by GNSS measurements. The geocentric GNSS coordinates at these so-called pass points are used to perform a Helmert transformation of the local network to convert the local coordinates into the geocentric frame. In the transformation-free approach, no local coordinate system is required. The GNSS coordinates are introduced as datum points in the adjustment procedure, and all measured angles which refer to the local plumb line are used after correction of the local deflection of the vertical (DoV). The distances, which define the network scale as described above, were measured in a classical way, i.e. using a tachymeter and locally measured meteorological quantities to account for refraction.

### 2.3 Deflection of the Vertical

The vertical deflection describes the angular deviation between the direction of the gravity vector and the normal of the ellipsoid surface at a point on the Earth’s surface. It can be represented by its North-South component *ξ* (positive northward) and East-West component *η* (positive eastward). There are different techniques to determine the DoV: the astrogeodetic technique, being also known as Helmert DoV, the GNSS-leveling technique, and the gravimetric technique yielding the Molodensky DoV.

In the astrogeodetic technique, the physical (or astronomical) latitude (*Φ*) and longitude (*Λ*) are directly determined by the positions of stars relative to the local plumb line. On the other hand, the geometric (or geodetic) latitude (*φ*) and longitude (*λ*) are taken from GNSS measurements. The astronomic DoV components are simply calculated by the following relationships (e.g. Heiskanen and Moritz (1984)):

The astrogeodetic observations were done in August 2018 at GOW using the QDaedalus system mounted on two different tachymeters (Albayrak et al. 2021). On four different pillars on the observatory, seven sessions were performed at each pillar using 250–300 stars per session. The standard deviations are about 1 μrad (0.2″) for the TCRM1101 and 0.5 μrad (0.1″) for the TDA5005 instrument. Except for pillar WT03, the maximum differences between all sessions at one pillar are below 1.45 μrad (0.3″). Although there seems to be a slight gradient between the results at the four locations, an average value of − 10.57 μrad (−2.18″) for *ξ* and − 28.36 μrad (−5.85″) for *η* was used for the entire observatory. This is equivalent to an inclination of 30.25 μrad (6.24″) towards WSW (azimuth 249.6^{∘}) (Fig. 3).

The GNSS-leveling technique uses the differences between orthometric heights as determined by leveling and ellipsoidal heights as determined by GNSS. In small networks as considered here, these height differences form a plane that defines the orientation of the equipotential surface at ground level and the ellipsoid surface. It can either be calculated along GNSS baselines (Vittuari et al. 2016) or in a planar constellation by a least-square fit of a plane. Since the uncertainties of the GNSS analyses of a few millimeters in height are large as compared to the baseline length (300m maximum), this technique was not applied in our case.

The gravimetric technique is usually used to determine the the geoid or – in our case – the quasigeoid and thus represents the DoV at the Earth’s surface. The regional quasigeoid like BKG’s GCG2016 model usually reflects large-scale gravimetric structures. To account for small-scale structures, a local fine structure quasigeoid of 10 km radius around the GOW was calculated based on data from a densified gravity network, an improved topographic model, and regional GNSS/leveling co-location points as pass points (Schwabe 2019). First, a pure gravimetric model was computed. In a second step, the solution was fitted to regional pass points by introducing a plane to obtain a model being consistent with the national reference frame. This step is justified since only the deviations from the CGC2016 are considered in this context.

As compared to the GCG2016, the local model exhibits fine structures up to 11 mm over a 4 km distance (Fig. 4). In terms of short-wavelength geoid undulations, these errors account for trends up to 3 mm/km. However, these differences do not apply to the GOW. Hence, the quasigeoid surface at the GOW is well represented by the regional quasigeoid, the differences do not exceed 0.2 mm. Thus the dip of the quasigeoid surface at the observatory site is about 30 mm/km (30 μrad, 6.2″) towards an azimuth of 250^{∘} (Fig. 4), which is equivalent to −10.26 μrad (2.12″) and −28.19 μrad (−5.81″) or *ξ* and *η*, respectively. Within the measurement uncertainties, the astrogeodetic and the gravimetric technique yield the same results.

The astrogeodetic DoV components *ξ* and *η* were directly included in the used adjustment software PANDA. GNSS coordinates from selected points indicated in (Fig. 2) were used as datum point coordinates. The results, given as differences with respect to WTZR, are shown in (Table 1).

## 3 Metsähovi

The Metsähovi Geodetic Research Station has been upgraded during last years. It will be one of the northernmost Global Geodetic Observing System (GGOS) Core stations when the renewal process is completed. At Metsähovi there are two VLBI telescopes, the older telescope of Aalto University, and the new VLBI Global Observing System (VGOS) telescope owned by National Land Survey of Finland (NLS), a new SLR telescope, and two permanent GNSS antennas – all of them are part of the local tie network. In addition, there is a DORIS beacon, owned by the French government space agency, the Centre National d’Études Spatiales (CNES), about three kilometers away from Metsähovi (NLS 2022).

### 3.1 Local Network

The local survey network at Metsähovi consists of pillar points, mast points, temporary station points, and one ground marker on the floor under the radome of the SLR telescope, including a permanent GNSS point and a permanent Global Positioning System (GPS) point. Besides permanent stations, there were seven semi-permanent GPS points equipped with choke ring antennas on seamless network adapters. Most of the semi-permanent points were occupied for three months (8.–10.2020) during the network measurements. The terrestrial observations, local network, and monitoring network together (Fig. 5), consists of 2624 point-to-point measurements, most of them include two angles and slope distance, and 37 levelled height differences.

### 3.2 Scale

The network data include the scale information from tachymeter distance measurements. Terrestrial observations lack only orientation and translation.

Also, GPS vectors include a scale. However the scale of GPS vectors is not directly traceable to the definition of the meter in the metrological sense but the distance measurements of calibrated tachymeters with accurate refraction index derived from reliable air pressure, temperature, and humidity measurements provide the traceability.

One option in the adjustment is to use GPS vectors as observations. In that case, the GPS scale interfere with the scale coming from tachymeter distances and traceability is compromised. The other option, which we use, is not to use GPS vectors as observations at all. The GPS network was processed separately and the daily solutions with covariance matrices are combined into one solution, which is used as a datum. The inner constraints are formed for rotations and translation, thus the scale results from tachymeter distance observations.

The scale of the tachymeter, traceable to the definition of meter, is transferred from Nummela Standard Baseline (Jokela 2014). The application of the new technology from the Physikalisch-Technische Bundesanstalt (PTB) and CNAM (Conservatoire National des Arts et Métiers) yielding more accurate distances is planned, in the framework of GeoMetre project, for some of the baselines in the network with simultaneous spectroscopic temperature measurements by Technical Research Centre of Finland (VTT). We plan to adjust the network with some more accurate baselines measured with the new technique. Tachymeter distances can then be weighted appropriately or the scale parameter can be added to observation equations of tachymeter distances. One option is to determine the scale difference between the new instrument and tachymeter and if appropriate, to correct the tachymeter distances.

### 3.3 Translation and Orientation, Seamless Network

The orientation of the local survey network is basically coming from GPS. Permanent and semi-permanent GPS stations provide the datum of the network. The GPS daily solutions were processed with Bernese 5.2 (Dach et al. 2015) and combined with CATREF software (Altamimi et al. 2018). The coordinates and covariance matrix of the combined solution are used as a datum in inner constraint equations. Observation equations for angles and distances in the global Cartesian coordinate system form the observation part of the 3D-network adjustment. With the datum information in inner constraint equations, no further transformations after adjustment are needed. The new seamless adapters (Fig. 6) enable tachymeter measurements simultaneously with GPS. Individually calibrated GPS antennas with the seamless connection of the prisms mitigate the centring errors to less than 0.2mm.

We used two methods for handling the DoV. In the first method the angle observations are corrected to refer to the normal of the ellipsoid using the geoid model and the combined solution of the GPS-network is used for horizontal orientation only. In the second method we did not correct the DoV but solve for the instrument 3D-orientation at each set up.

#### 3.3.1 Using Geoid Model for Vertical Orientation

In the first method the Metsähovi local network station points are oriented vertically using the new local geoid model, FIN_Geoid_Geo-Metre_MH2000. It is an updated version of the high-resolution Finnish geoid model, FIN_EIGEN-6C4 (Saari and Bilker-Koivula 2018), with the inclusion of Metsähovi gravity data measured by Hannu Ruotsalainen and Selen Dayioglu in 2006 using a Scintrex CG-5 relative gravimeter (Ruotsalainen and Dayioglu 2006). The gravity data contain 122 points covering an area of 1 km × 1 km, with an average point distance of around 100 m. To match the main gravity data, the Metsähovi data has been transformed from the original epoch 1963.0 to 2000.0 and the tide system changed from mean-tide to zero-tide. The whole gravity dataset used in FIN_EIGEN-6C4 covers Finland and surrounding countries between 56.8^{∘} and 72.2^{∘} latitude and 15^{∘} and 36^{∘} longitude. The DoV, *ξ* = −22.32 μrad (−4.60″) and *η* = 33.92 μrad (7.00″), from the earlier used national geoid model, FIN2005N00 (Bilker-Koivula and Ollikainen 2009), differ slightly from the new ones but the difference means only 0.2mm in 100m. The components *ξ* = −22.44 μrad (−4.63″) and *η* = 36.08 μrad (7.44″) of the new geoid agree well with the astronomical ones measured in 1976–1980 (Ollikainen 1987): *ξ* = −22.49 μrad (−4.64″) and *η* = 35.39 μrad (7.30″). The geodetic coordinates of those from the adjustment of Jorma Jokela (Jokela et al. 2016) were firstly defined in the ED87 frame and later transformed to ETRF89 by Matti Ollikainen.

The angle observations refer to the plumb line and are corrected to refer to the ellipsoidal normal at the corresponding point. When we correct the angle observations due to the deflection of vertical, we fix the vertical direction of the instrument in each station point. The observation equations of sets of horizontal angles include orientation unknowns – one for each setup. The inner constraint matrix then includes four rows, one for horizontal orientation of the network and three for translation. The conversion of the geoid based deflections to the level of the earth’s surface can be neglected, since at Metsähovi its vertical distance to the geoid surface is less than 100m.

#### 3.3.2 Estimating the 3D Orientation of Instrument

In the second method the 3D orientation angles of each tachymeter set up were estimated, as unknown parameters in a network adjustment, without correcting angle observations using the geoid model. This kind of approach is usually applied in cases where the instrument alignment is not collinear with the plumb line or can not be controlled with compensators which is the case in some industrial or aboard a ship measurements. However, in the local survey networks at co-location sites, we level the instruments carefully and use compensators for controlling the alignment with the plumb line. Thus, with the assumptions that the instrument is aligned with the plumb line and our datum point coordinates are aligned with ITRF, we can solve for the deflection of vertical for each station point by estimating 3D-rotations for each instrument setup. The datum defect of the normal matrix in the network adjustment, in this case, is six. This approach is quite promising at least in small networks like at Metsähovi. However even 1 mm errors in coordinates of the datum points may cause a significant tilt to the network. At least the individual calibrations for antennas are needed. The seamless setup mitigates the centring errors.

#### 3.3.3 Results of Adjustments

The preliminary network adjustments have been performed using the in-house software with two different methods. In both cases, the resulting coordinates of the moving points in the telescope structure were used to estimate the local tie vector from MET3 GNSS ARP to the reference point of the VLBI antenna. There was about 1.5mm difference in height. Horizontally there is no significant discrepancy. The difference in the results of the two cases is due to the orientation of the network. The deflections of vertical components from the geoid model differ from estimated station point deflections *ξ* and *η*, − 11.6 μrad (−2.39″) and 21.8 μrad (4.50″), respectively, which explain the difference in the vertical component. The reason for the disagreement of estimated DoVs and DoVs interpolated from the new geoid model is the uncertainty in the GPS coordinates in the vertical direction and the small size of the network.

The network measured in 2014–2015, and the new, measured in 2020–2021 were combined and adjusted. The reference points of METSA13 (the new VLBI telescope) and METSAHOV (the old VLBI telescope) were estimated using the adjusted coordinates of the points rotating around the axes of the telescopes. The local tie vectors from terrestrial measurements are presented in Table 2 relative to the point MET3. The standard deviations of VLBI reference points include the noise of the monitoring. The noise of the datum (orientation) must be included to the total uncertainty. At Metsähovi the reference point of the new SLR telescope will be measured in near future.

## 4 Discussion

In this work two similar strategies to improve the handling of the datum problem at geodetic co-location stations are presented. At the GOW the transformation-free approach has successfully been tested. The required angles of vertical deflection were obtained with an accuracy of about 1 μrad (0.2″) by using the astrogedetic technique and were confirmed by the dip of fine structure quasigeoid surface in the vicinity of the observatory. The network orientation is realized by independent solutions from GNSS observations, which were introduced in the adjustment as datum points. Distant targets improve the accuracy of the horizontal orientation of the network.

At Metsähovi the functional model has been extended to solve for the instrument orientation at each station point. Using the GPS and terrestrial data set measured in 2020 and 2021, it is obvious that the orientation is not satisfyingly solved due to the uncertainties of GPS coordinates in the vertical direction influencing a small tilt to the whole survey network. The horizontal orientation is less sensitive to the tilt.

The other approach where the adjustment is done directly in the global frame without further transformation is suitable. The angle observations were corrected according to the DoV of the new Finnish geoid model. In preliminary adjustments of the Metsähovi network, all the permanent and semi-permanent GPS points were included in the datum in the inner constraint equations.

At both sites the poorly constrained vertical orientation from GNSS observations was significantly improved by the introduction of DoV data in the adjustment of terrestrial observations. Typical GNSS uncertainties of 2 mm in height over a baseline of 100 m result in an orientation error of 20 μrad (5″), while the determination of the local DoV using either the astrogeodetic or gravimetric technique can be done with an accuracy of 1 μrad (0.2″).

This approach can easily be adopted by other geodetic observatories. If a sufficiently accurate geoid or quasigeoid model is available, the local DoV can be taken from the normal of the (quasi-)geoid surface. But also the use of the astrogeodetic technique is a viable path. Modern instruments like the QDaedalus system allow for quick and accurate DoV measurements. As the datum problem seems to be the limiting factor in local tie determinations, a consequent integration of DoV observations in local tie solutions could help to further improve and harmonize ITRF solutions, in particular at GGOS core sites. The standard deviations of RP coordinates are a consequence of variance propagation in RP-estimation as an influence of the geometry, the number of VLBI antenna positions, and precision of measured target coordinates. The uncertainties of local tie vector components should also include the uncertainties of orientation and scale of the local survey network.

While the solution of these surveys, which were submitted to the ITRF2020, is based on classical tachymeter measurements, the next ITRF will contain data of the newly developed refraction-compensated distance meters being developed in the framework of the GeoMetre project to further improve the scale of the local ties and the traceability to SI units.

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## Acknowledgements

The very productive support of Cornelia Eschelbach and Michael Lösler is gratefully acknowledged. The FGI team thanks Jyri Näränen, Joona Eskelinen and Arttu Raja-Halli for their help at Metsähovi station during preparations and measurements.

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Kallio, U. *et al.* (2022). Datum Problem Handling in Local Tie Surveys at Wettzell and Metsähovi.
In: International Association of Geodesy Symposia. Springer, Berlin, Heidelberg. https://doi.org/10.1007/1345_2022_155

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