Keywords

1 Introduction

The International Terrestrial Gravity Reference System (ITGRS) of IAG will be defined based on the instantaneous acceleration of free fall expressed in the International System of Units (SI) (Wziontek et al. 2021). The conventional quantity “acceleration of gravity” is then derived by a set of corrections. The International Terrestrial Gravity Reference Frame (ITGRF) should be realized by observations using absolute gravimeters (AG) and a set of conventional models for the correction of temporal changes (tides, atmosphere, polar motion). Accessibility to the users is ensured by a compatible infrastructure with reference stations as the main components. Reference stations provide a long-term stable absolute gravity reference function by monitoring the seasonal gravity variations using a superconducting gravimeter in combination with repeated AG observations or in future by continuously operated AQG, respectively.

Comparisons of absolute gravimeters are essential for the ITGRF to guarantee the traceability and compatibility of observations with AGs. The gravity reference is realized based on a set of precise absolute measurements during comparisons (e.g. Jiang et al. 2012).

The Regional Comparison of Absolute Gravimeters WET-CAG2021, organized as an Additional Comparison beyond the scope of the CIPM MRA (CCM 2015), was held in Germany at the Geodetic Observatory Wettzell (GOW) of the German Federal Agency for Cartography and Geodesy (BKG) in autumn of 2021. All measurements in the New Gravity Laboratory of 2010 were collected between September 7 and December 2, 2021 (Falk et al. 2022).

Due to the pandemic, the number of participants and the measurement schedule at GOW could not be fixed and optimized in advance. Finally, 9 absolute gravimeters of 4 different types were compared. Overall, two teams from National Metrology Institutes (NMIs) or Designated Institutes (DIs) and 7 teams from geodetic institutions participated in WET-CAG2021. The DI from Czech Republic (VÚGTK/RIGTC) participated here with FG5X-251H, but in past key comparisons with FG5-215H. A statistically significant and stable difference of (1.8 ± 0.3) μGal between these gravimeters has been estimated from almost 70 repeated absolute measurements at the station Pecný (Czech Republic) between 2017–2021. Therefore, the difference has to be taken into account for linking the WET-CAG2021 to key comparisons.

The comparison results were calculated independently by BKG and VÚGTK/RIGTC. Two processing strategies were followed as described in Pálinkáš et al. (2021). In one solution, labelled as ICN, all compatible gravimeters contribute to the reference values according to their harmonized uncertainties. In the other solution, labelled as KCN, the absolute level of the comparison reference values is determined only by gravimeters belonging to NMI/DIs which also provide the link to a key comparison while all other instruments stabilize the solution only with gravity differences. Although WET-CAG2021 is not a metrological key comparison and therefore a separation into groups of AGs is not mandatory, both approaches were followed to demonstrate the impact on the mean comparison level.

Six instruments have participated in both, the EURAMET. M.G-K3 (Falk et al. 2020) and WET-CAG2021. The comparison WET-CAG2021 was conducted to ensure the compatibility of the AGs, to check the long-term stability of the FG5/X gravimeters and for the first time to evaluate two commercial quantum gravimeters AQG (Ménoret et al. 2018). In contrast to the AQG-A laboratory device, the AQG-B has also been designed for field use. The characterization of the uncertainty budget for these instruments is still under investigation.

The deviation of a particular gravimeter from the reference values (RV) is usually expressed by the Degree of Equivalence (DoE). Since this additional comparison is not within the scope of CIPM MRA, the standard uncertainty, denoted by u, is given with a coverage factor of k = 1 as is usual in geodesy. In metrology k = 2 is common, which was applied in recent key comparisons of AGs. Sigma (σ) denotes the standard deviation or the error estimates, respectively obtained from error propagation for the parameters from the least-squares adjustment.

All measurement data and processing results can be found in the comparison report (Falk et al. 2022). Results are given in microGal (μGal) as unit of acceleration of gravity, 1 μGal is equal to 1 × 10 –8  m/s2.

2 Absolute Gravity Measurements

Each gravimeter measured at a minimum of two or generally, three different sites. Some gravimeters occupied the same site at multiple times to estimate the setup error. The absolute gravity measurement graw is the mean free-fall acceleration at the specific measurement height and was corrected for the Earth and ocean tides (zero-tide system), the effect of atmospheric mass variations using the local measured air pressure record and an admittance factor of −0.3 μGal/hPa, and the reference air pressure, polar motion, vertical gravity gradients above the measurement site, in accordance with the IGRS Conventions 2020 (Wziontek et al. 2021). The tidal parameters are the same as in previous comparisons (e.g. Falk et al. 2020) and were estimated from 10 years of continuous measurements of the superconducting gravimeter GWR SG-029 (2000–2010). This device is operated at the Old Gravity Laboratory, about 200 m away from the comparison site. Further, systematic instrumental corrections, e.g. for FG5/X the speed-of light correction, the self-attraction correction and the laser beam diffraction correction, were applied individually by each participant of the comparison together with processing of their observations. Corrections to the AQG measurements were applied following the latest instructions of the manufacturer.

In total, 43 final gravity values with associated uncertainties at the measurement height have been submitted. The transfer to the common comparison height of 1.250 m was based on vertical gravity gradients (VGGs) determined before and during EURAMET.M.G-K3 comparison in 2018 (Falk et al. 2020). The used comparison height of 1.25 m above the bench mark was defined as approximate mean of the effective instrumental heights of FG5 and FG5X gravimeters (Wziontek et al. 2021). As long as the AQG have distinct larger uncertainties as FG5/X gravimeters the comparison reference height should be close to 1.25 m to minimize the contribution of the transfer errors to the comparison reference values.

Residual temporal gravity variations not covered by models are accounted for by the continuous record of the superconducting gravimeter SG-030 as shown in Fig. 1. A correction for each AG observation epoch has been applied according to Fig. 1, together with including a contribution of 0.3 μGal to the uncertainty budget. This correction resolved the effect of temporal gravity variations reaching up to 2 μGal and allowed for a duration of the comparison over 12 weeks.

Fig. 1
figure 1

The residual gravity variations observed during the comparison with the superconducting gravimeter GWR SG-030 referred to October 20, 2021 12:00 UT as the comparison reference time. Red dots mark the mean values at the reference time of absolute gravity observations

Full size image

3 Data Elaboration

A least-squares adjustment was performed with the gravity values at the reference comparison height (g) and their associated uncertainties (u) as input. The direct observation equation for each gravimeter i with a bias δ i at the site j is

$$ {g}_{ij}={g}_j+{\delta}_i+{\varepsilon}_{ij} $$
(1)

The weights w ij for the stochastic model are derived from the respective uncertainties w ij = u o 2 /u ij 2 where u o is the unit weight. As the set of observation equations has no unique solution for δ, a constraint, which can be interpreted as definition of the CRV is required (Pálinkáš et al. 2021). Here, the weighted constraint

$$ \sum_{i=1}^n{w}_i\ {\delta}_i=d $$
(2)

was used, where the weights w i are normalized by the condition Σ w i = 1, and d is the linking converter (Jiang et al. 2012).

Similar to Pálinkáš et al. (2021), two different solutions (ICN/KCN) were processed. All gravimeters contribute to the definition of reference values of the ICN solution. This solution is independent on other comparisons, thus d = 0. For this solution, in case of Czech DI (VÚGTK/RIGTC) the reported data of FG5X-251H have been used without considering the bias to FG5-215H. The corresponding bias of (1.8 ± 0.3) μGal needs to be applied only when the link to previous key comparisons is accounted for, where FG5-215H and not FG5X-251H took part. The KCN solution is considering only the group of gravimeters belonging to NMI/DIs for the definition of comparison reference values in accordance with CCM (2015). The other gravimeters, practically, are only contributing with gravity differences and are thus neither included into the constraint nor in the determination of the linking converter d. Consequently, weights for non-NMI/DI gravimeters are all set to zero in Eq. (2). As proposed by Pálinkáš et al. (2021) declared uncertainties of those FG5/X gravimeters lower than 2.4 μGal were changed to this limit in the ICN solution. In case of the KCN solution, only the uncertainties of non-NMI/Dis were harmonized.

Following Pálinkáš et al. (2021) we used a correlation coefficient of 0.75 to account for correlations of repeated observations of the same instrument, reflecting the typical ratio between repeatability and uncertainty for all gravimeters included in this comparison. This includes the AQGs, as a similar ratio between uncertainty and reproducibility as for the FG5/X was found (Table 3). Note, that the parameter named repeatability in Pálinkáš et al. (2021), in this paper will be changed to a more correct term short-term reproducibility, because it also describes the variability of results due to setup error. The respective covariances for a particular AG i are then obtained from the harmonized u i j as cov = 0.75 u i j , min 2, where u i j , min is the minimum of all u ij of that AG. This approach can be understood as if the measurements of a particular gravimeter carried out within a few days are affected by the same systematic errors. So multiple measuring results of one instrument at one site after a new independent setup can be included with negligible influence on CRV, DoE and associated uncertainties, but providing more precise information about the instrument’s short-term reproducibility.

The linking converter d in Eq. (2) is conventionally taken to be zero in CCM key comparisons. However, regional comparisons have to be linked to a CCM comparison by at least two AGs (CCM 2015), therefore d was computed as weighted average of the biases of the respective gravimeters. Links were established to EURAMET.M.G-K3 held 2018 also at Wettzell (Falk et al. 2020) and to CCM.G-K2.2017 at Beijing, China (Wu et al. 2020) by two gravimeters (Table 1). Here, the link of FG5X-251H to FG5-215H was ensured by applying a bias of (1.8 ± 0.3) μGal, determined based on four-years of repeated measurements by both gravimeters at the station Pecný, Czech Republic.

Table 1 Linking converters as weighted mean of DoEs determined at the EURAMET.M.G-K3 (Falk et al. 2020) and CCM.G-K2.2017 (Wu et al. 2020) of joint NMI/DI participants of the three comparisons. Due to the bias of 1.8 ± 0.3 μGal applied to FG5X-251H original measurements, these results are in the level of FG5-215H
Full size table

4 The Geodetic Approach (ICN-Solution)

For the ICN solution (Table 2), all gravimeters were included in the weighted constraint (Eq. 2) with weights related to harmonized uncertainties. The weighting matrix of the observation equation (Eq. 1) introduced the correlation coefficient of 0.75 for all observations of a particular AG.

The consistency of measurements was checked based on the reported uncertainties using the compatibility index En which is defined as the ratio between the difference of the measured gravity value (gij) and the reference value RV (gj) at a site and its uncertainty

$$ {E}_{ij}=\frac{\left({g}_{ij}-{g}_j\right)}{u\left({d}_{ij}\right)} $$
(3)

Here, the uncertainty of deviations u(d ij ) was achieved from error propagation accounting for the correlations between observations (Pálinkáš et al. 2021).

As in previous comparisons, the expanded uncertainty was used here and, therefore, an absolute value of En larger than 2 indicates that a measurement is incompatible at a 95% confidence level, as the difference is not covered by the (expanded) uncertainties.

Table 2 ICN-solution comparison reference values (ICN-RV). The constant value 980,836,900.0 μGal was subtracted from the ICN-RV. u is the uncertainty at 68.3% confidence level computed as root mean square of standard deviations σ (from the least-squares adjustment). The reference height is 1.250 m. The ICN-RVs refer to October 20, 2021 12:00 UT as the mean time of the comparison
Full size table

With the exception of one measurement the harmonized compatibility indexes are all below 2, indicating consistency with the harmonized uncertainties. Only one observation of FG5-218 (DA) was found to be incompatible and consequently, the uncertainties of all measurements with this instrument were increased further by 50% to reduce the impact on the mean reference level. No measurements have been removed, as all showed a good short-term reproducibility (Eq. 5), and therefore not affecting the gravity differences. The Degree of Equivalence (DoE, Jiang et al. 2012)

$$ \vspace*{3pt}{D}_i=\left[\sum {w}_{ij}\left({g}_{ij}-{g}_j\right)\right]/\sum {w}_{ij}\vspace*{3pt} $$
(4)

is computed as the weighted average difference between the measurements of a gravimeter i and the RV at site j. Its uncertainty is computed by error propagation, again accounting for correlations (Pálinkáš et al. 2021). The DoEs are identical with the biases δ estimated from least squares adjustment when no measurement is excluded from the adjustment.

The short-term repeatability (Jiang et al. 2012) was computed for each AG from the differences between each observation with the respective CRV as the standard deviation

$$ {R}_i=\sqrt{\sum {\left({g}_{ij}-{g}_j\right)}^2/\left(\ \mathrm{n}-1\right)\ } $$
(5)

It allows to assess whether an incompatibility is caused by systematic deviation of an instrument. The measurements of FG5-218 show with reproducibility of 0.7 μGal a high precision, but with DoE of −4.6 μGal a lower accuracy. Since its DoE exceeds significantly the (harmonized) uncertainty, the FG5-218 was excluded from the constraint (Eq. 2). The final DoEs with an uncertainty at the 68.3% confidence level are presented in Table 3 and Fig. 2.

Table 3 Degrees of Equivalence (DoE) for ICN-solution of the gravimeters participating in the WET-CAG2021. The standard uncertainty U DoE is obtained from error propagation considering correlation of 0.75 between measurements of a particular gravimeters. The short-term reproducibility of each gravimeter during the comparison is also presented
Full size table
Fig. 2
figure 2

Joint presentation of Degrees of Equivalence (DoE) in μGal of the gravimeters participating in the WET-CA2021 comparison calculated from the difference between the gravimeter measurements and the CRV or KCN-RV for the corresponding pillar for both solutions. The error bars represent the standard uncertainties (UDoE) at 68.3% confidence. In the ICN-solution the incompatibility of FG5-218 with its DoE has been solved by down weighting it in the constraint. These KCN results are linked to EURAMET.M.G-K3. FG5X-251H* is given in the level of FG5-215H (FG5X-251H + 1.8 μGal)

Full size image

5 The Metrological Approach (KCN-Solution)

The KCN solution is obtained similar to the ICN solution. Nevertheless, only two gravimeters were included in the constraint Eq. (2) with the weights 0.54 for FG5X-251H and 0.46 for FG5-242. By setting the weights of the other gravimeters to zero in the constraint, they actually only contribute as relative gravimeters. As for the ICN solution underestimated uncertainties were harmonized. Declared uncertainties below 2.40 μGal were set to this value, but only for non-NMI/DI gravimeters. The final DoE with uncertainty at the 68.3% confidence level are presented in Fig. 2 and Table 5, applying linking converters to both, EURAMET.M.G-K3 and CCM.G-K2.2017.

6 Results and Conclusions

The primary objective of the Additional comparison WET-CAG2021 was to validate the long term stability of the FG5/X gravimeters relative to EURAMET.M.G-K3 and CCM.G-K2.2017. Also, for the first time, two commercial quantum gravimeters were included in such a comparison, which allows for an assessment of the compatibility of these two fundamentally different principles of measurement. However, as the characterization of systematic effects for both AQGs by the manufacturer represents a work-in-progress, we consider this a preliminary result.

Fig. 3
figure 3

Joint presentation of the Degrees of Equivalence (DoE) of EURAMET.M.G-K3 and of the KCN-solution (linked to EURAMET.M.G-K3) of WET-CAG2021. FG5X-251H* is given in the level of FG5-215H (FG5X-251H + 1.8 μGal), which participated at EURAMET.M.G-K3

Full size image

Three solutions including nine gravimeters are presented which mainly differ by the definition of the absolute level of the comparison reference values. The ICN solution is independent of previous comparisons and documents DoE of the seven FG5X gravimeters in the range of −4.6 μGal and +3.1 μGal, while for the quantum gravimeters the DoE ranges between −5.8 μGal and +6.8 μGal.

Both KCN-solutions differ by only 0.4 μGal, where the link to EURAMET.M.G-K3 results have higher RVs and lower DoEs. Using that link, the DoE of FG5/X gravimeters vary within −6.3 μGal and +1.3 μGal while the quantum gravimeters range between −7.5 μGal and +5.0 μGal.

Eight participating gravimeters are equivalent in all presented solutions taking into account their associated uncertainties. Although the measurements of FG5-218 are characterized by a high short-term reproducibility, the determined DoE is neither within twice of the declared nor the simple harmonized uncertainties and shows a significant bias of about −5 μGal. For both quantum gravimeters AQG-A02 and AQG-B02 the compatibility indexes are below 2, indicating consistency with the reference values, although the deviations are larger. This validates that the new technology based in atom interferometry corresponds with the declared uncertainties.

The short-term reproducibility for the FG5/X gravimeters varies between 0.3 to 1.3 μGal. The AQGs short-term reproducibility is with 2.7/3.9 μGal, resp. slightly lower than half of the declared uncertainties. These results confirming the assumption of a correlation between measurements of the same instrument of 75%.

For most of the FG5/X gravimeters a stable DoE (within 1 μGal) between both Wettzell comparisons (2018 and 2021) is demonstrated as shown in Fig. 3. This suggests that biases different for each gravimeter are reproducible and a stability within a few years could be achieved. This is important for the realization of the ITGRF and for studies, where the gravity rate of changes plays a crucial role (Van Camp et al. 2011; Olsson et al. 2019). However, biases may change when key components of an AG are replaced or maintained. For instance, the obvious changes of the DoE for FG5-101, FG5-301 and FG5X-220 between comparisons may be related to an exchange of the original collimator by another commercial product. The impact on the diffraction correction, e.g. according to Kren and Pálinkáš (2022), has not yet been considered. A rigorous quantification of such effects and an adequate correction of the measurements needs to be established to improve the stability of AGs over time. This also affects the accuracy of comparison reference values (Tables 2 and 4) that can only be enhanced to better than 1 μGal (k = 1) if instrumental corrections of all participating instruments are carefully investigated and consequently applied.

Table 4 KCN-solution comparison reference values linked to EURAMET.M.G-K3 using linking converter of (−1.15 ± 1.85) μGal and also to CCM.G-K2.2017 using linking converter of (−0.73 ± 1.88) μGal related to 2 NMI/DI gravimeters. The constant value 980,836,900.0 μGal was subtracted from the KCN-RV. The uncertainty u is at 68.3% confidence level computed as root mean square of the squared standard deviations σ (from the least-squares adjustment) and the squared uncertainty of the linking converter. The reference height is 1.250 m. The KCN-RVs refer to October 20, 2021 12:00 UT as the mean time of the comparison
Full size table

The ICN solution show a 1.8 μGal lower comparison reference values as the KCN solution, which seems not to be negligible. Nevertheless, this difference is within the uncertainties of CRVs. The reference values of the ICN solution are lower mainly due to the fact that the participating gravimeters are providing lower gravity values than those used for the link to key comparisons. Generally, the reference values always depend on the set of gravimeters participating at comparisons (Table 5).

Table 5 Degrees of Equivalence (DoE) for solution KCN of the gravimeters participating in WET-CAG2021. The standard uncertainty U DoE is obtained from error propagation considering a correlation of 0.75 between measurements of a particular gravimeter. FG5X-251H* is given in the level of FG5-215H. Results using different links (Table 1) are presented
Full size table

The WET-CAG2021 documents a high reproducibility for all FG5/X (0.8 μGal in average), even if a bias exceeds the uncertainty and that the biases of most FG5/X are stable over more than 3 years, compared to EURAMET.M.G-K3 (2018). For the first time it has been demonstrated that quantum gravimeters AQG are in equivalence with their declared uncertainties and reach short-term reproducibility of about 3 μGal (AQG-A02: 2.7 μGal, AQG-B02: 3.9 μGal).

WET-CAG2021 demonstrates the importance of additional (or non-metrological) comparisons to confirm the long-term stability of AGs and their traceability (by the KCN solution). Therefore, additional comparisons and comparison stations should become a key component of ITGRF in order to monitor AGs used for its realization. Finally, it is demonstrated again that the duration of a comparison can extend over several weeks by including gravity variations recorded by a SG.

7 Outlook

The research on AQG systematic effects is ongoing and the manufacturer is actively improving its procedure for the characterization of systematic biases and uncertainties. In spring of 2022, an improved characterization of both the AQG-A02 and AQG-B02 was preformed, resulting in an increased systematic uncertainty of 11 μGal for both instruments. Furthermore, with the new systematic bias corrections, the results of AQG-A02 and AQG-B02 are reduced by 5.6 μGal and 11.9 μGal, respectively. As these characterizations took place after the comparison data submission deadline, they were not considered in the analysis of the comparison. However, due to the larger measurement uncertainty of the AQGs compared to the FG5/X gravimeters, the impact on the RVs, and thereby DoE results for the other instruments, is negligible.