Keywords

1 Introduction

The Finnish Geospatial Research Institute is operating the Metsähovi Geodetic Research Station (MGRS) which is a core site of the Global Geodetic Observing System (GGOS 2023). A superconducting gravimeter (SG) has operated at the station continuously since 1994. The first SG, GWR-T020, operated at the site from 1994 until 2016. A new dual sphere SG OSG-073 with two sensors, was installed in early 2014 to the same laboratory on a pier three meters apart from the SG-T020. Unfortunately, the OSG-073 operated only until May 2015 when it had to be sent back to the manufacturer for a total redesign. The solution was to separate the two sensors of the iOSG-073 into two separate gravimeters: iGrav-013, replacing the SG-T020 on the original pier, and iOSG-022 installed on the second pier. SGs have proven to be very good instruments to study a variety of geophysical phenomena and offer a great tool to observe small periodical effects like free oscillations of the Earth and solid Earth and ocean tides (for a review see e.g., Hinderer et al. 2015).

Tidal signal is the largest periodic signal in the gravity time series and needs to be removed from the gravity data to be able to study other geophysical phenomena, e.g., crustal loading effects affecting other geodetic measurements like satellite laser ranging (SLR) and Global Navigation Satellite Systems (GNSS). To achieve best results, a local tidal model is necessary. In previous analysis of the SG gravity data, we have used a local observation based tidal gravity model referred as ME18, produced from the tidal analysis of the SG-T020 gravimeter data (see most recent results in Virtanen and Raja-Halli 2018). The old model ME18 included 45 tidal wave groups between annual Sa and quarter diurnal M4 tides. In the model ME18, the ocean tide loading was not separately analysed, hence the ocean tides were intervened with the body tides. To establish a new local tidal model and study the contribution of ocean tides, we use in this study the 5.5 years of gravity data collected by the new SGs and the ETERNA-X-ET34-v80 (Wenzel 1996; Schüller 2015; Schüller 2020) Earth tide software to simultaneously compute the contributions of the body tides and five different ocean tide loading models. After removing the tidal signal, the largest remaining signal is due to environmental mass changes. In Metsähovi the residual environmental signal is mostly dominated by the non-tidal effects of the atmosphere and the Baltic Sea, and mass changes in the local hydrology. The effect of environmental mass changes on the gravity at Metsähovi have been previously studied in e.g., Virtanen (2001), Virtanen and Mäkinen (2003), Mäkinen et al. (2014) and Olsson et al. (2009). In this analysis we adopt a simpler approach and use local groundwater level and the Baltic Sea level height at the Helsinki tide gauge only as regression parameters in the tidal analysis. Further analysis of the hydrological gravity effects is out of scope of this study. Several tidal analyses have been made with using SG data from different gravimeters. However, this is the first tidal analysis from the data of the new SG’s at Metsähovi also providing information on the drift and overall performance of the instruments.

An earlier very extensive analysis of ocean tidal loading at Metsähovi was done with the data from SG-T020 together with several SG stations around the globe by Boy et al. (2003). It was discussed that poorly modelled Baltic Sea and Arctic Sea might be the cause to discrepancies between the ocean tidal loading models and observations. Metsähovi is 15 km from the coast of the Baltic Sea which is a shallow estuary where tidal amplitudes are negligible compared to non-tidal sea level changes.

More recently, tidal analysis of SG data has been studied in Meurers et al. (2016) in which the temporal variation of the tidal parameters was analysed by using the data of several central European SGs. Recent local tidal gravity studies have been also carried out by Crossley et al. (2023) for the SG-046 at the Apollo Lunar Laser Ranging facility in USA, Luan et al. (2022) in Kunming, China, and Hinderer et al. (2020) in Djougou, Benin. In Hinderer et al. (2022) a comprehensive analysis is presented for eight SGs operated at the J9 gravity observatory in Strasbourg, France.

We adopt a similar approach as authors mentioned above and present the first tidal analysis of the new data from the iGrav-013 and iOSG-022, with a separate analysis where local groundwater and Baltic Sea level at Helsinki tide gauge were used as regression parameters. We compute the tidal amplitude factors and phases for the body tide wave groups from Sa to M4, and the ocean tidal loading of the 11 main tidal waves (Ssa, Mm, Mf, Q1, O1, P1, K1, N2, M2, S2, K2) for the different ocean tide models.

2 Data Processing

2.1 Gravity Data

The two new SGs were installed to the MGRS in 2016 three meters apart, inside the same laboratory, and have been operating and producing data with 1 Hz sampling rate since. For the tidal analysis we use the SG data from first of January 2017 until 22nd of August 2022 in total of 2059 days. The year 2016 was omitted from the processing due to the large initial drift of the instruments and disturbances caused by the repeated adjustments and installation procedures done on the instruments.

The raw 1 Hz voltage signal recorded by the gravity sensors is first converted to gravity variations in nm/s 2 with a scale factor acquired from simultaneous absolute gravity (AG) observations. In this study we use the calibration values of −945.27 ± 1.49 nm/s2/V and −887.40 ± 1.40 nm/s2/V for iGrav and iOSG, respectively, where V is the voltage recorded by the gravimeter. These values are the weighted means of a least squares adjustment of the SG gravity signal to 1825 absolute gravity set values measured during six AG campaigns made in 2018 between February and November. Absolute gravity measurements were made with the FG5X-221 and each measurement campaign lasted between 4–9 days, see Virtanen et al. (2014) for details of the calibration process. In addition to the scale factor, the instrumental time delays were determined with the help of the gravimeter manufacturer GWR (Richard Warburton 2017, personal communications). We followed the step procedure described in Van Camp et al. (2000), by injecting 6 step signals of known voltage to the gravimeter feedback coil during fifth of December 2017. From the resulting observations of these steps, we got time delays at zero frequency for iOSG-022 τ = 6.75s ± 0.19s, and τ = 6.83s ± 0.09s for iGrav-013.

In the gravity pre-processing we follow the remove and restore workflow described in detail e.g., Virtanen (2006), Hinderer et al. (2015) and Virtanen and Raja-Halli (2018) to achieve a continuous and un-disturbed time series best suitable for tidal analysis. We have used the Tsoft-software package (Van Camp and Vauterin 2005) for the pre-processing of the data.

First, the empirical tidal gravity model ME18 is removed from the gravity signal and the air pressure related effects are reduced by subtracting the local barometric pressure changes with an admittance factor of −3.1 nm/s2/hPa. Next step is to correct the time series for distinct steps and spikes, and occasional gaps by linear interpolation from the residual (Fig. 1). However, as pointed out in Hinderer et al. (2002), these pre-processing steps may cause significant differences in the resulting time series depending on the chosen correction strategy, for instance, careless correction of a step in the data may cause significant change in the overall trend of the time series. Here, the benefit of two close-by SG’s is evident, as we can compare the time series to cross-validate the changes in the gravity time series and distinguish even very small signals caused by instrumental disturbances. We have done the removal of outliers as an iterative process to minimize errors in the corrections and to achieve as clear time series as possible for tidal analysis: first we have corrected for spikes larger than 10 nm/s2/min and offsets larger than 10 nm/s2/min. These spikes and offsets are caused mainly by instrument maintenance like removing accumulated ice from the SG dewar or due to large earthquakes. Second, to clean the signal even further, we subtracted one gravimeter time series from the other to reveal additional smaller instrumental disturbances. This method allowed us to distinguish and correct the data for spikes larger than ~3 nm/s2/min and steps of ~1 nm/s2/min. However, the iOSG-022 has time periods lasting several days with overall noise level above ~3 nm/s2/min caused mainly by mechanical vibrations of the cold head, in these cases more conservative corrections were made. Also, in some cases even after comparing the time series it was hard to judge whether the difference between the two gravimeters is due to instrumental effect or due to a real physical phenomenon, e.g., snow accumulation on the roof of the laboratory can cause 20 nm/s2 gravity effect which can be unevenly distributed (Virtanen 2001).

Fig. 1
figure 1

A 1 month sample of the gravity data of iGrav-013 and iOSG-022. On the left, the raw gravity signal with full tidal signal, and on the right the residual after removing the tidal gravity model ME18 and air pressure effects. Adjustments of the cold head in the iOSG has produced the clear spikes and steps visible in the iOSG data which were removed in the data pre-processing

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After the above-mentioned corrections, the removed tidal model and air pressure signal are restored to the gravity signal to produce a continuous gravity time series for the tidal analysis. The sampling rate is further reduced to 1 h with a least squares (LSQ) lowpass filter. First the 1 Hz data is decimated to 1 min sampling rate with a LSQ lowpass filter with 0.00833 Hz cutoff and 504 s window, and secondly to 1 h data with 12 cycles per day cutoff and 480 min window to avoid aliasing effects.

2.2 Environmental Data

To monitor the environmental mass changes in the vicinity of the gravity laboratory we have installed 11 boreholes for measuring the water table level in the sediments and in the crystalline bedrock (for details see Virtanen and Raja-Halli 2018). For this study we use only water table measurements from a borehole BH2 in the bedrock within few tens of meters of the gravimeter to give a proxy of the local hydrological changes. The measurements made at 0.1 Hz are resampled to 1 h after correcting for the atmospheric pressure and outliers. To account for the non-tidal gravity effects caused by the close-by Baltic Sea, we use hourly sea level data from the Helsinki tide gauge 30 km’s from the station (Finnish Meteorological Institute Open data). The groundwater level and tide gauge data are used as regression parameters in the ETERNA-X analysis. For a more comprehensive description of the loading effects caused by the Baltic Sea we refer to Virtanen and Mäkinen (2003) and Virtanen (2004).

3 Ocean Tide Loading Models

In our analysis we compare five different ocean tide models, three recent models, EOT20 (Hart-Davis et al. 2021), FES2014b (Lyard et al. 2021) and TPX09v5a (Egbert and Erofeeva 2002), and two older models DTU10 (Cheng and Andersen 2010) and FES2004 (Lyard et al. 2006) which is still routinely used in the absolute gravity analysis in Finland. We used the ocean tide loading constituents calculated by the Onsala Ocean Tide Loading Provider (Scherneck 2022; Bos and Scherneck 2013) which determines the 11 main tidal load vectors for Ssa, Mm, Mf, Q1, O1, P1, K1, N2, M2, S2, K2 wave groups by using a visco-elastic Earth model (Kustowski et al. 2008) with Green’s functions (Bos and Scherneck 2013). The ocean tide load vectors are directly implemented on the ETERNA-X analysis to get ocean tide contribution to the gravity signal.

4 Tidal Analysis

The tidal analysis was done with the comprehensive earth tide software package ETERNA-X-ET34-v80 (Schüller 2015; Schüller 2020) which allows the simultaneous analysis of body and ocean tides. In the analysis we have used the DEHANT-DEFRAIGNE-WAHR non-hydrostatic inelastic Earth model (DDW-NHi) (Dehant et al. 1999) and the HW95 tidal potential catalog (Hartmann and Wenzel 1995) with 12,361 constituents. In the analysis the gravity effect of the polar motion, i.e., pole tide, was removed by using the daily Earth Orientation Parameters (EOP) from the International Earth Rotation and Reference System Service (IERS 2022). The local air pressure was treated as a regression parameter in the analysis. For comparison we performed separate analysis with the local groundwater level and Helsinki tide gauge data included as regression parameters.

The ETERNA-X allows a wide range of possibilities on choosing the tidal wave groups to be analyzed depending on the length and quality of the time series. We have followed a wave grouping scheme proposed in the Ducarme and Schüller (2018) where a conservative wave grouping “Y04-R04-safe1” is proposed for time series with lengths from 4 to 9 years. This wave grouping includes 91 Earth body tide wave groups from Sa to MK4 wave groups. To assess the quality of the analysis of an individual wave group we have used the Correlation Root Mean Square Error Amplifier (CRA) values. ETERNA-X calculates CRAs for all analyzed wave groups. We inspected, that the CRA values for all wave groups were below 2 and hence acceptable under the criterions laid out in the detailed description in Ducarme and Schüller (2018) and ETERNA-X documentation (Schüller 2020).

5 Results

From the tidal analysis we get amplitude factors and phases for the body wave groups from the annual Sa-wave group, up to wave group M4 and a fit of the amplitudes and phases for the ocean tidal loading wave groups. The resulting tidal parameters for the main tidal constituents are shown in the Appendix 1, Tables 2 and 3. For the gravity time series processing the average of the amplitudes and phases of the five ocean tide models (shown as the mean in Fig. 3 and in Appendix 2) is used to overcome the possible shortcomings of individual models.

The residual gravity series are presented in Fig. 2. The instrumental drift of the gravimeters was linear and stable as the first year of operation was omitted from the analysis. The drift was determined with a second order Chebychev polynomial in the ETERNA-X analysis. The drift rate includes both the instrumental drift but also the linear gravity change of approximately −7 nm/s2/year caused by the post-glacial rebound (Bilker-Koivula et al. 2021). The resulting residual RMS, drift and regression parameters are presented in the Table 1.

Fig. 2
figure 2

The final residual gravity signals in nm/s2 after removing the body and ocean tides, pole tide, pressure effects and instrumental drift. Red: iGrav+100 nm/s2, black: iOSG-100 nm/s2. Lighter coloured and dotted lines are the residuals reduced with groundwater and Helsinki sea level. Blue: The difference between iGrav and iOSG

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Table 1 Gravity residual RMS, drift, regression coefficients and correlation between air pressure and tide gauge, for the two different analysis, where GW and TG represent the analysis where groundwater and Helsinki tide gauge were removed through regression
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A clear decrease in the RMS was achieved through reduction of groundwater and sea level, especially for the iGrav. However, there is a clear shortcoming in our analysis as we are using only the local groundwater as a proxy for all hydrological effects and omitting larger scale loading effects as well as local effects of snow and water in the soil layers above the bedrock. There also remains large differences between the gravity signal of the two instruments visible in the Fig. 2. Further investigation is required to understand whether these are due to e.g., local hydrology or some instrumental effects.

Metsähovi is 1,000 km from the ocean and the Baltic Sea is a shallow estuary with a very low tidal amplitudes, hence the amplitudes of the ocean tides at Metsähovi are mainly below 1 nm/s2 with the maximum for the M2 being ~4 nm/s2. In the Fig. 3 we present the results for the five ocean tide models following the presentation used in Hinderer et al. (2020) and Luan et al. (2022): we plot the percentage rate of the vector X which is the excess in the gravity residual compared to the amplitude of the ocean tide model in the period range of the given wave group after removing the modelled body and ocean tide. The large remaining residuals, i.e., large X-vector, and differences between the models in the P1 and K1 ocean wave groups are believed to be due to poorly modelled Baltic Sea, which has been also discussed in Boy et al. (2003) and Lyard et al. (2021). The results agree well with the earlier results of Boy et al. (2003). In Appendix 2, Tables 4 and 5 show the resulting mean of the ocean tidal load vectors from the analysis and the residual vectors together with the X-vector. Results show clear differences between iGrav and iOSG which might be due to the disturbances and instrumental noise in the iOSG data, but also a small error in the calibration of the instruments might be the cause.

Fig. 3
figure 3

The amplitude of the residual vector X for the iGrav-013 (left) and iOSG-022 (right) for the five ocean tide models. TOP: without GW and TG, BOTTOM: GW and TG as regression parameters. The X is the excess in the residual gravity signal as a percentage rate of the model amplitude after removing the modelled body and ocean tide factors

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6 Discussion and Conclusions

In this study we have carried out the first tidal analysis of the new SG’s iGrav-013 and iOSG-022 at the MGRS and a comparison with five ocean tide models with a regression with local groundwater and Helsinki tide gauge. The high noise level caused by the instrumental noise in the data of the iOSG disturbed the tidal analysis. The residual of the iOSG exhibited large deviations compared to the iGrav. A more careful analysis is required to distinguish whether these differences are due to data processing, instrumental or environmental origin. It is clear from the analysis that instrumental disturbances and mass changes in the close vicinity of the gravimeter can produce large discrepancies between the two gravimeters.

Differences between the gravimeters might also arise from errors in the scale factors which are now based on set values of AG measurements done in 2018. Further investigation on the accuracy and stability of the scale factors is required. Next step is to combine all absolute gravity measurements made during 2017–2022 for a more comprehensive scale factor and SG drift determination by using the individual drops in the AG measurement rather than sets.

We found large deviations between the ocean tide loading models and the observed signals especially in diurnal P1 and K1 wave groups, where remaining signal was in some cases more than double of the model tidal amplitude, Fig. 3 and Appendix 2. Boy et al. (2003) explain these discrepancies to be possibly due to poor modelling of the Baltic Sea and Arctic Ocean in the ocean tide models. In addition, the non-tidal effects in the Baltic Sea can reach up to 30 nm/s2 (Olsson et al. 2009) and can have periodical properties through seiche waves which have periods close to diurnal and semi-diurnal bands (Metzner et al. 2000), hence interfering with the tidal analysis.

Seasonal signal was clearly decreased especially in the iGrav residual after removing groundwater and tide gauge signals through regression. However, this is not physically realistic to reduce hydrological effect by using only these data but served more as a proxy for this analysis. For further improvements it is necessary to include in the analysis the global hydrological and atmospheric contribution through e.g., by using EOST loading service (EOST 2023), a more detailed local hydrological modelling and a more detailed analysis of the loading effects caused by the close-by Baltic Sea.