Keywords

1 Introduction

Mass re-distribution in atmosphere, oceans, and the terrestrial branch of the global water cycle causes a deformation of the solid Earth and an associated change in the Earth’s gravity field, its orientation, and – most important for this study – the geometry of the crust. Surface loading is relevant to reach the accuracy goals of the Global Geodetic Observing System (GGOS) that aim at 1 mm accuracy and 0.1 mm/a stability. In line with the International Earth Rotation and Reference Systems Service (IERS) Conventions 2010 (Petit and Luzum 2010) non-tidal loading was not corrected in the recent GNSS reprocessing campaign of the International GNSS Service (repro3) and the current reference frame realization ITRF2020 (Altamimi et al. 2023). This study aims (1) to compare the ITRF2020 seasonal displacement signals against the loading-predicted surface deformation and (2) to assess the potential impact of associated corrections on the reprocessed GNSS solutions. The main focus of this contribution is, therefore, on the comparison between solutions with non-tidal loading corrections applied at the solution level (abbreviated as SOL) and at the observation level (OBS).

Focusing on large station networks Martens et al. (2020), Mémin et al. (2020), Gobron et al. (2021), Klos et al. (2021) and others investigated the impact of non-tidal loading corrections applied to previously determined coordinate time series. Overall, the various studies revealed significant RMS reduction and decreased amplitudes on different frequencies especially for the station height coordinates. Incorporating loading corrections directly in the observation modeling was investigated for example by Tregoning and van Dam (2005), Dach et al. (2011), Männel et al. (2019), and Glomsda et al. (2020). The overall advantage of this approach is that corrections are consistently applied to all estimated parameters including Earth rotation parameters and satellite orbits which is of course crucial for reprocessing efforts. A comparison of different loading corrections focusing on reference frames was performed by Glomsda et al. (2022). Furthermore, loading corrections were applied in the DTRF2014 and DTRF2020 realization (Seitz et al. 2022).

The contribution is structured in the following way. After briefly describing the GNSS processing strategy and introducing the ESMGFZ loading models (Sect. 2), we discuss the correction at the solution level (Sect. 3). Section 4 contains the results when applying the corrections at the observation level. Finally, the paper closes with a summary and some conclusions in Sect. 5.

2 GNSS Processing and Loading Corrections

The GNSS processing for this investigation relies on IGS’ third reprocessing campaign. As the involved Analysis Centers decided not to correct for non-tidal loading, the derived GFZ solution is a reference (abbreviated as REF) in this study (Männel et al. 2020, 2021). Table 1 summarizes the applied processing strategy. Compared to previous reprocessings and the operational products, the GPS phase center offsets and the reference frame were adjusted to the published Galileo offsets.Footnote 1 While this leads to an independent GNSS-based scale, this does not impact our non-tidal loading investigation. Overall, the GFZ repro3 solution covered 322 stations (on average 185 per day) and 132 satellites, including GPS, GLONASS (from 2012 onwards), and Galileo (from 2014 onwards). According to Rebischung (2021), the daily median formal errors for the station coordinates are 1.0, 1.0, 3.5 mm in North, East, and Up directions. To assess the correction at the observation level, we repeated the repro3, keeping all models but adding the ESMGFZ non-tidal loading corrections (OBS solution). While repro3 was initially performed from 1994 to 2020 (with an extension for 2021–2022) the repeated OBS solution is – related to computational efforts – limited to 2012.0–2016.0. Subsequently, REF and SOL (corrected the derived coordinates) solutions only contain the original repro3 results for these years. Consequently, the investigations in Sects. 3 and 4 are limited to 2012–2016. As the apriori troposphere delays were derived using GPT2 in GFZ’s repro3 solution we kept this processing option. For the implications of using a Global Mapping Function when investigating loading effects, we refer to Steigenberger et al. (2009).

Table 1 Summary of estimation and processing strategy (repro3)
Full size table

The Earth System Modelling group of Deutsches GeoForschungsZentrum (ESMGFZ) in Potsdam (http://isdc.gfz-potsdm.de/esmdata/loading) provides surface loading corrections based on models of the atmosphere, oceans, the terrestrial hydrosphere (Dill and Dobslaw 2013). The fourth model component ensures global mass balance by distributing the excess water mass from atmosphere and terrestrial water storage over the ocean considering loading and self-attraction via the sea level equation. The calculations are performed based on mass distributions provided by the deterministic numerical weather prediction model of the European Centre for Medium-range Weather Forecasts (ECMWF), the Max-Planck-Institute for Meteorology Ocean Model (MPIOM, Jungclaus et al. 2013), and the Land Surface Discharge Model (LSDM, Dill 2008). Corresponding surface deformations in North, East, and Up are provided with a spatial resolution of 0.5\(^\circ \) and a temporal sampling of three hours for the atmosphere and ocean and 24 h for the continental hydrosphere and the barystatic sea-level variations. The surface deformations are provided in the center of the Earth’s figure (CF) and the center of Earth’s mass (CM). As of today, the ESMGFZ non-tidal loading corrections are provided without uncertainties.

An implicit comparison between repro3 results (computed without applying non-tidal loading corrections) and loading models was performed by assessing annual and semi-annual signals derived from the aggregated ESMGFZ loading models (CF frame) against the periodic signals provided along with the ITRF2020 (ITRF2020 2022). For around 80% of the 1344 stations, ITRF2020 reports larger annual amplitudes than predicted by the loading models (Fig. 1). While this is not surprising as the ITRF seasonal coefficients also contain seasonal variations of the observation geometry, systematics from near-field, and thermo-elastic signals, the amplitudes of 52% of the stations agree within 40%. A similar value was reported by Männel et al. (2019) based on a stand-alone PPP solution. Besides the overall agreement, a few stations showed discrepancies. For example, for MAPA (Santana, Brazil) located near the Amazon river, the ESMGFZ overpredicts the annual amplitude by \(-\)4 mm in the North and \(-\)10 mm in the Up direction (computed as ITRF2020-ESMGFZ). In this case, overprediction may occur as the station is too close to the loading source. The stations in Wuhan, China (WUH2, WUHN) also show larger amplitudes in the up direction for the loading time series (\(-\)4.3 and \(-\)4.5 mm, respectively), potentially for the same reason. A large horizontal discrepancy of 7.2 mm in the North occurs for UTQI (Barrow, USA), potentially related to the short time series – UTQI was installed in 2017 – and monument-related periodic variations. A more detailed study on the coordinate variability was presented by Boy et al. (2022).

Fig. 1
figure 1

Comparison of annual amplitudes computed from ESMGFZ loading models and amplitudes provided in the periodic component of ITRF2020

Full size image

3 Corrections at the Solution Level

As non-tidal loading was not considered in repro3, users might correct the corresponding deformations using the available products, such as ESMGFZ. As shown by Glomsda et al. (2020, 2021) this is possible without introducing inconsistencies at the normal equation level, which is accessible via the provided SINEX files. However, given the required expert knowledge and software capabilities, most users might prefer to simply subtract loading corrections from the extracted coordinate time series, which we call correction at the solution level.

In the following, the seasonal signals and the coordinate variability (RMS of the time series) are investigated to assess the impact of applying loading corrections. While the first required the estimation of corresponding periodic signals – annual and semi-annual – in the time series assessment the RMS – computed as \(\sigma _{x} = \sqrt {\frac {1}{n-1}\sum _{i=1}^n (x_i-\bar {x})^2}\) – was investigated considerung a pure linear station behavior (i.e., using a linear trajectory model). In any case, significant coordinate discontinuities (larger than 2 cm) are considered. With the given dataset, four discontinuities related to antenna replacements (stations: GRAC, JOG2, POLV, SUTM) and one related to earthquakes (SANT, Illapel earthquake 2015) are applied.

Figure 2 shows the difference in annual amplitudes for all stations with more than 600 daily coordinate solutions between 2012.0–2016.0. For the horizontal amplitudes, the effect is small (overall below 0.5 mm), which is not surprising as non-tidal loading creates primary vertical displacements, and horizontal amplitudes are small. The reduction is, on average, from 1.5 to 1.1 mm and from 1.1 to 1.0 mm for North and East, respectively. Nevertheless, amplitudes in North and East are reduced for 217 (82%) and 166 (63%) out of 264 stations. For the vertical amplitudes, an overall reduction of \(-\)0.6 mm from 3.2 to 2.6 mm can be observed; 159 out of 264 stations show smaller amplitudes (Fig. 3). Semi-annual amplitudes, overall smaller in size, are reduced similarly. The mean coordinate variability derived by applying a linear trajectory model to the original repro3 (2012–2016) solution is 2.1, 2.1, and 5.1 mm in North, East, and Up direction. Subtracting non-tidal loading corrections before the time series adjustment leads to slightly reduced RMS values of 1.9, 2.0, and 4.8 mm (Fig. 4). Overall, the variability in North, East, and Up is reduced for 85, 70, and 68% of the stations. Few stations show larger amplitudes and RMS values after applying the corrections at the solution level. For the stations YELL and YEL2 located in Yellowknife, Canada the RMS increased by 1.8 and 2.7 mm. For YELL a similar behavior was reported already by Männel et al. (2019). Comparing the uncorrected coordinate time series and the loading models indicate significant shifts between the model and GNSS for Yellowknife and Wuhan, China (see also Fig. 6). For station YAKT (Yakutsk, Russia), ESMGFZ models and GNSS agrees but the results – RMS increased by 0.3 mm – are biased by unconsidered coordinate variations related to snow and ice on the antenna.

Fig. 2
figure 2

Correction at the solution level: difference in annual amplitude in (a) North, (b) East, and (c) Up direction. Differences are computed by subtracting the original repro3 solution from a solution where corrections are applied directly to the coordinates

Full size image
Fig. 3
figure 3

Correction at the observation level: difference in annual amplitude in (a) North, (b) East, and (c) Up direction. Differences are computed by subtracting the original repro3 solution from the solution where corrections are applied directly at observation level

Full size image
Fig. 4
figure 4

Correction at the solution level: difference in station coordinate RMS using a linear trajectory model. Differences are computed by subtracting the original repro3 solution from a solution where corrections are applied directly to the coordinates

Full size image

4 Corrections at the Observation Level

Figure 3 shows the difference in annual amplitudes for solutions OBS and REF. For the horizontal amplitudes, larger reductions are visible, from 1.5 to 0.8 and from 1.1 to 0.8 mm for North and East, respectively. Annual amplitudes in North and East are reduced for 215 (81%) and 191 (72%) out of 264 stations. An overall reduction of \(-\)1.3 mm from 3.2 to 1.9 mm can be observed for the vertical amplitudes. The geographic distribution reveals decreased annual amplitudes, especially for stations in South America (large signals in terrestrial water storage) and central Asia (significant atmospheric loading). A small fraction of the considered stations (5%) show larger amplitudes if correcting for surface loading. Similar to the SOL results, semi-annual amplitudes are reduced but are overall small in size. The mean coordinate variability derived by applying a linear trajectory model to the original repro3 (2012–2016) solution is 2.1, 2.1, and 5.1 mm in North, East, and Up direction. Correcting for non-tidal loading corrections at the observation level leads to slightly reduced RMS values of 1.8, 2.0, and 4.5 mm (Fig. 5). Overall, the variability in North, East, and Up is reduced for 90, 80, and 84% of the stations. Around 3% of the stations show RMS values increased by more than 0.5 mm; among them stations at Yellowknife (Canada). Compared with the SOL results, the stations in Wuhan show significant improvement.

Fig. 5
figure 5

Correction at the observation level: difference in station coordinate RMS using a linear trajectory model. Differences are computed by subtracting the original repro3 solution from the solution where corrections are applied directly at observation level

Full size image
Fig. 6
figure 6

Comparison of the model-based displacements (blue) and the derived GNSS time series (black) for stations BRAZ (Brasilia, Brazil), JFNG (Jiufeng, China), WUH2 (Wuhan, China), and YELL (Yellowknife, Canada) in figures (a, c, e, g); co-located stations WUHN and YEL2 are added to figure (e) and (g) (grey). Comparison of time series, original (grey), loading corrected at solution level (orange) and at observation level (purple) in figures (b, d, f, h). Results based on linear trajectory model, moving average for GNSS time series applied

Full size image

Figure 6 shows an individual comparison between GNSS and loading models and between uncorrected and corrected height time series for selected stations discussed already above. BRAZ (Brasila, Brazil) is strongly affected by hydrological loading with the models predicting more than 20 mm peak-to-peak variations. The GNSS time series represent that displacement quite well; a correlation factor of 0.81 is derived between the model and GNSS time series (smoothed with a monthly moving averageFootnote 2). Consequently, the corrected time series shows a significant reduction in annual amplitude (from 10.9 mm to 3.2 mm) and RMS (9.7 mm to 6.7 mm). Applying the correction at the observation level has no additional impact on the amplitude but reduces the RMS by additional 0.4 mm. For the stations in Jiufeng (JFNG) and Wuhan (WUH2) located within a distance of around 14 km the models also predict peak-to-peak variations of 20 mm driven by the hydrological loading. This strong signal is dominated by Yangtze and Han rivers and by the several hundred lakes within the Jianghan Plain. Despite data gaps in the GNSS, time series for JFNG, WUH2, and the co-located station WUHN agree, especially in 2015. However, the GNSS-based motion patterns differ significantly in phase; a shift of around 100 days (loading is ahead) is determined for JFNG and WUH2 with a correlation of 0.7 and 0.5 respectively. If applying the correction at the solution level, annual amplitudes increase from around 3–4 mm to 8–10 mm. Correcting at the observation level performs better for these stations leading to reduced amplitudes in Wuhan and only slight increases in Jiufeng. The stations YELL and YEL2 located close to Canada’s Great Slave Lake, also show large differences between GNSS results and model-based displacement series (GNSS is ahead by around 90 days). Applying loading correction leads to significantly increased amplitudes independent of the level at which the correction is applied.

5 Summary and Conclusions

Time-dependent mass variations in the atmosphere, oceans, and hydrosphere lead to significant deformations of the Earth’s surface. Associated displacement corrections are provided in dedicated non-tidal loading models. An initial comparison between seasonal signals determined from the ESMGFZ models and ITRF2020’s periodic functions revealed a good agreement with amplitude differences below 10 mm. Overall, the ITRF2020 amplitudes are larger than the loading amplitudes for around 80% of the stations while the amplitudes agree within 40% for half of the 1344 stations.

While the beneficial impact of applying loading corrections on GNSS-based time series has been shown in different studies, our focus is on the differences between correcting at the solution and the observation level. For the vertical amplitudes a significant reduction by 18% was found at the solution level, while a decrease by 42% was achieved at the observation level. Figure 7 compares the resulting RMS differences for both solutions with respect to the original repro3 solution. While the negative values indicate the overall improvement, the additional improvement related to the observation level is visible as the difference between the grey (SOL results) and red (OBS results) bars. Overall, applying non-tidal loading corrections at the solution level decreased the height RMS by 5% while corrections at the observation level lead to a 10% reduction.

Fig. 7
figure 7

Histogram of the RMS differences between original GNSS time series and solution derived with applying the loading corrections at the solution level (solid, grey), and at the observation level (red outline). The equivalent normal distribution was added

Full size image

Finally, we recommend to apply the corrections at the observation level whenever possible. This is also beneficial for the consistency between the simultaneously estimated station coordinates, satellite orbits, and Earth rotation parameters.