# Estimates of Vertical Velocity Errors for IGS ITRF2014 Stations by Applying the Improved Singular Spectrum Analysis Method and Environmental Loading Models

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## Abstract

A reliable subtraction of seasonal signals from the Global Positioning System (GPS) position time series is beneficial for the accuracy of derived velocities. In this research, we propose a two-stage solution of the problem of a proper determination of seasonal changes. We employ environmental loading models (atmospheric, hydrological and ocean non-tidal) with a dominant annual signal of amplitudes in their superposition of up to 12 mm and study the seasonal signal (annual and semi-annual) estimates that change over time using improved singular spectrum analysis (ISSA). Then, this deterministic model is subtracted from GPS position time series. We studied data from 376 permanent International GNSS Service (IGS) stations, derived as the official contribution to International Terrestrial Reference Frame (ITRF2014) to measure the influence of applying environmental loading models on the estimated vertical velocity. Having removed the environmental loadings directly from the position time series, we noticed the evident change in the power spectrum for frequencies between 4 and 80 cpy. Therefore, we modelled the seasonal signal in environmental models using the ISSA approach and subtracted it from GPS vertical time series to leave the noise character of the time series intact. We estimated the velocity dilution of precision (DP) as a ratio between classical Weighted Least Squares and ISSA approach. For a total number of 298 out of the 376 stations analysed, the DP was lower than 1. This indicates that when the ISSA-derived curve was removed from the GPS data, the error of velocity becomes lower than it was before.

## Keywords

GPS seasonal signals singular spectrum analysis environmental loadings ITRF2014 dilution of precision## 1 Introduction

Most global navigation satellite system (GNSS) position time series show displacements with annual and semi-annual periodicities (e.g. Blewitt et al. 2001; Collilieux et al. 2007; van Dam et al. 2007). Those periodicities, or so-called seasonal changes, are produced by geophysical effects related to tides (solid Earth, ocean, atmosphere or pole), transportation of mass over the surface of the Earth (atmosphere- and hydrosphere-related non-tidal effects) or bedrock and antenna thermal expansion (Dong et al. 2002). However, the power at certain frequencies results from systematic errors rather than real geophysical effects. In part, these errors are due to the residual diurnal and semi-diurnal tidal signatures that are aliased into a discrete 24-h GPS solution (Dong et al. 2002; Penna and Stewart 2003). Moreover, unmodelled- or mismodelled oscillations can propagate into spurious features with periods that vary from a few weeks to 1 year (Penna et al. 2007). A draconitic year (Agnew and Larson 2007) with a period equal to 351.6 days (Amiri-Simkooei 2013) has also been found in a position time series from International GNSS Service (IGS) stations. It is an artefact of GPS solution as it has not been detected using very long baseline interferometry (VLBI) and satellite laser ranging (SLR) techniques, or in geophysical fluids (Ray et al. 2008). In addition, the quality of the GNSS position time series suffers from errors when considering the double differences and network adjustment including transfer from fiducial stations or inclusion of scale (Tregoning and van Dam 2005). Finally, each type of modelling (satellite orbits, the Earth Orientation Parameters, satellite Antenna Phase Center Variations, mapping functions, etc.) may introduce artificial oscillations into the time series. The comprehensive contributions of both geophysical sources and errors of models to the observed periodic (annual) changes in position were demonstrated by Dong et al. (2002).

Jiang et al. (2013) corrected the GPS height time series from three various environmental loading models (atmospheric pressure, continental water storage and non-tidal ocean loading) and tested their effectiveness for a set of 233 globally distributed GPS reference stations. They investigated the influence of the loading contributions on the IGS weekly reprocessed time series and found significant improvement in the values of weighted root mean square (WRMS) error. They also found that the RMSs of the times series after applying loadings show larger variations in the Northern Hemisphere than in the Southern Hemisphere, mainly due to differences in land–ocean distribution.

Previous research into environmental loading models and their contribution to ground displacements showed that some power of observed annual variation can be explained by hydrological (e.g. van Dam et al. 2012; Dill and Dobslaw 2013), atmospheric (e.g. van Dam and Wahr 1987) and non-tidal oceanic (e.g. Fratepietro et al. 2006) loadings. Non-tidal ocean loading (NTOL) can cause a 5-mm peak-to-peak variation in a vertical direction (van Dam et al. 1997) and can reduce the scatter in the GPS height changes of about 10% when it is subtracted (Zerbini et al. 2004). Van Dam et al. (2001) found that continental water storage (CWS) may explain more than half of the annual signal in vertical displacements derived from GPS. Loading deformation of Earth’s crust due to the redistribution of air masses can reach 20 mm (Petrov and Boy 2004). As much as 40% of the observed annual variation in vertical direction results from many cumulative factors, such as pole tide effects, ocean tide loading, atmospheric loading, non-tidal oceanic mass and groundwater loading (Dong et al. 2002). Apart from the mass loading due to the atmosphere, continental water storage and non-tidal ocean loading jointly contribute at a rate of about 53% to the annual amplitude of GPS vertical changes (Yan et al. 2009). The effects of non-tidal ocean and atmospheric loading on GPS displacement in vertical direction are comparable when considered separately and may reduce the GPS RMSs by 30% when they are combined (Williams and Penna 2011). When atmospheric tidal model (ATML) is improperly modeled, it may mitigate in the combined GPS solution introducing anomalous signal around draconitic, annual and semi-annual period (Tregoning and Watson 2009). Dach et al. (2011) proved that the repeatability of station coordinates improves at a rate of 20% when the atmospheric pressure loading (APL) is applied directly during the analysis of data and at a rate of 10% when the APL is employed as a post-processing correction comparing to a solution that does not take APL into account.

An accurate removal of the seasonal signals from the position time series is beneficial for the accuracy of the derived velocities. These velocities are used, e.g. to create the horizontal (e.g. Bogusz et al. 2014) and vertical velocity (e.g. Kontny and Bogusz 2012) field, to correct the mean-sea level (MSL) records for vertical land movements (VLM) (Teferle et al. 2002) or to create the kinematic reference frames, as the newest realization of the International Terrestrial Reference System (ITRS)—ITRF2014 (Altamimi et al. 2016). As far as their reliability is concerned, the velocities of GPS stations should be estimated simultaneously with the annual and semi-annual signals, as the seasonal changes will bias the value of velocity when they are not accounted for (Blewitt and Lavallée 2002; Bos et al. 2010). Moreover, all seasonal periods have to be modelled when noise analysis is performed afterwards, as they may significantly bias the noise parameters when neglected (Davis et al. 2012; Bogusz and Klos 2016). A seasonal model should also be considered during the reference frame definition, either terrestrial (Freymueller 2009; Collilieux et al. 2012) or celestial (Krásná et al. 2015) and residual annual oscillations are still significant when continental (Kenyeres and Bruyninx 2009) and regional networks (Bogusz and Figurski 2014) are processed. All of the aforementioned phenomena will transfer directly to the velocity and its error when they are un- or mismodelled. Recently, Santamaria-Gomez and Memin (2015) subtracted atmospheric, oceanic and continental water mass loading to access the effect on secular velocities estimated from the short time series. They found that at least 4 years of continuous data (in some regions this should be extended to 10 years) are needed to mitigate the effect of the inter-annual deformations on secular velocities.

Since we are not able to model these seasonal effects perfectly, we have to estimate seasonal curves for each station from the position time series to improve the accuracy of the derived velocities. The most common approach to modelling the seasonal signal on a station-by-station basis is to employ the weighted least-squares (WLS) to estimate the sine curves and then subtract them from the data. Due to the fact that some power of observed seasonal variation can be explained by environmental loadings, one can also subtract them from data to account for a signal that may vary with time and, therefore, cannot be entirely covered by a constant WLS-derived sine curve.

In this paper, we start by estimating the annual and semi-annual amplitudes of environmental (atmospheric, hydrological and ocean non-tidal) loading models (ELM) and of IGS ITRF2014 GPS position time series with the most common WLS approach. Furthermore, we demonstrate that assuming their constancy over time may lead to mismodelled seasonal phenomena. Therefore, we subtract the superposition of ELM from GPS position time series and prove that some power from the frequencies between 2 and 100 cpy is being artificially cut from the GPS data. To avoid this artificial loss in power, we mitigate the combined seasonal effect in the GPS position time series by an Improved Singular Spectrum Analysis (ISSA) approach, which is suitable to time varying effects. We employ the ISSA analysis to estimate the time-varying seasonal curve in environmental loading models and then subtract this curve from the GPS position time series, which results in no loss in power and allows us to model a seasonal curve that varies with time. We show that although the seasonal signal was removed, some residual oscillations still remain, but their time-variability is not as large as the one for loading models. Finally, we assess the estimates of vertical velocity errors for IGS ITRF2014 stations when the entire time-varying real seasonal signal is removed.

## 2 Data

### 2.1 GPS

First, a standard pre-processing for offsets, outliers and gaps was performed. The offsets were removed based on epochs defined by the IGS and also a few offsets that were unreported before were detected with the STARS algorithm (Rodionov and Overland 2005) and removed. A number of no more than 6 offsets were employed for one series. The interquartile range (IQR) approach was employed to remove outliers. Missing values accounted for a maximum of 8% of the entire data.

In this research we only focus on the vertical position time series from the IGS contribution to the ITRF2014, as the time-variability of height changes is much larger than the ones in the horizontal direction. However, the presented approach can also be successfully applied to the North and East time series.

Figure 1 presents detrended vertical displacements for IRKT (Russian Federation) and GUAT (Guatemala) stations. The IRKT station follows a long-period trend non-linear trend, which can be clearly noticed by changes in maxima around 2002 and between 2010 and 2012. IRKT station is situated within Lake Baykal region. Two large earthquakes have been already registered to affect the area of Irkutsk with magnitudes of M5.9 and M6.3. Those two happened in 1999 and 2008, which in its turn may mean that there is some stress in the rock mass with a very long period. We checked the long-term ISSA curve delivered during analysis and the trend is not linear. However, this non-linear trend did not filter into any other frequencies and did not affect the estimates of ISSA-curve we analysed in this research. In the following analysis, we only focus on seasonal estimates.

### 2.2 Environmental Loadings

In this research, we used the atmospheric (ATM), hydrological (HYDR) and non-tidal oceanic (NTOL) loading effects on surface displacements using recent global reanalysis models, respectively, ERA (ECMWF Re-Analysis) interim (Dee et al. 2011), MERRA (Modern Era-Retrospective Analysis) land (Reichle et al. 2011) and ECCO2 (Estimation of the Circulation and Climate of the Ocean version 2) (Menemenlis et al. 2008). The induced surface displacements were computed using Green’s function formalism (Farrell 1972) based on the Preliminary Reference Earth Model (Dziewonski and Anderson 1981). All loading computations were performed in the center-of-figure (CF) reference frame. For the atmospheric loading computation, we assumed that the ocean responds to pressure variations as an inverted barometer (IB) process. This approximation is assumed to be valid for periods exceeding few weeks. The mass conservation of the ocean is enforced by adding/removing a uniform ocean layer to compensate for the non-zero mean atmospheric pressure over the oceans (for details see Petrov and Boy 2004). We used a land sea mask at higher resolution (0.10°) than the resolution of the atmospheric model (about 0.70°). We also enforced the total mass conservation of the hydrological model by adding/removing a uniform ocean layer to compensate for any lack/excess of water on the land surface. The ECCO2 model conserves its volume, but not its mass, so we used a similar approach to conserve the ocean mass. For these, we used the ocean model at 0.25° resolution, i.e. higher than the hydrological model (0.66° in longitude and 0.5° in latitude). For more details, see http://loading.u-strasbg.fr/.

In further part of this research, for consistency, we refer to environmental loadings as ERAIN to atmospheric loading, MERRA to hydrological loading and ECCO2 to non-tidal oceanic loading. We also estimate the joint contribution of the above-mentioned environmental models, referred to as a superposition (SUP).

## 3 Methodology

*x*

_{0}is the initial value,

*A*is the amplitude of seasonal sine with phase shift

*ϕ*and residuals formed when a model was subtracted are denoted as

*ε*

_{ x }(this is also a so-called noise or stochastic part). GPS position time series contain different seasonal signals that may be significant for up to 9 harmonics of a tropical year (Bogusz and Klos 2016). The amplitude of vertical displacements may be as high as between tenths of a millimetre to 8 mm. Beyond the deterministic model, we consider here residuals

*ε*

_{ x }which are well approximated by white plus power-law process (e.g. Williams et al. 2004). These reveal the amplitudes of few millimetre and, therefore, may have a strong impact on the linear parameters that are estimated from the GPS position time series (e.g. Klos et al. 2016).

*a*is the set of principal components (PC) of the time series,

*E*is the set of eigenvectors retrieved by covariance matrices,

*M*is the length of the window that is sliding through the time series and dividing in this way the time series into a number of

*N*sets. Two approaches to create the covariance matrix can be distinguished: the BK algorithm (Broomhead and King 1986) which creates the covariance matrix by sliding the

*M*-point window through the entire data, and the VG algorithm (Vautard and Ghil 1989) for which the lagged-covariance matrix with a maximum lag of

*M*has the Toeplitz structure.

The SSA approach has been already successfully applied to extract modulated seasonal signals from weekly GPS time series (Chen et al. 2013) or to investigate the inter-annual temporal and spatial variability of atmospheric pressure, terrestrial water storage and surface mass anomalies observed with the GRACE gravity mission in Europe (Zerbini et al. 2013). Most recently, Xu and Yue (2015) investigated the effects of known geophysical sources (atmospheric and hydrological) and stated that they are insufficient to explain the annual variations in GPS observations. They also applied the SSA in combination with the Monte Carlo test to extract time-variable seasonal signals, however, it was concluded that this method was problematic, since by using this approach the SSA-filtered annual signal may contain a signal driven by colored noise. Therefore, the MCSSA signal may artificially remove some power and in consequence, reduce the amplitudes of flicker noise.

The key problem in the SSA approach is the choice of the size of the *M*-point lag-window. This size depends on the length of time series and frequencies of seasonal signal we intend to model. In this research, based on the tests we performed with the Akaike Information Criterion (AIC, Akaike 1974), we adopted a 2- and 3-year window adaptively, depending on the time series. This will resolve for periods between *M*/5 and *M* and will result in the determination of the annual and semi-annual oscillations which amplitude changes over time (Gruszczynska et al. 2016).

The SSA approach requires a continuous time series without gaps (Schoellhamer 2001). Shen et al. (2015) proposed an Improved Singular Spectrum Analysis (ISSA) to be appropriate to employ to incomplete time series with gaps. In this research, we analyse the GPS position time series and environmental loading models with the ISSA approach and employ the BK algorithm to create the covariance matrix. A set of frequencies that corresponds to each *k*-th EOF is estimated in this research with Welch’s periodogram (Welch 1967). Considering the length of the tropical year (365.25 days) and the latest determination of the length of draconitic year (351.6 ± 0.2 days according to Amiri-Simkooei, 2013), we would need 25.6 years of data to resolve them with spectral analysis. In this way, when the annual cusp is estimated, we mitigate the combined seasonal effect of a tropical plus draconitic year which in fact contribute the most to total data variance (Blewitt and Lavallée 2002; Bogusz and Figurski 2014). Whenever the annual cusp is being referred to in this research, we mean a combined effect of tropical and draconitic year which we are not able to differentiate due to a limited data span, since the longest time series available were 21.1 years long (YELL—Canada).

By using the ISSA approach, we estimated the annual and semi-annual seasonal signals that have major variability over time. The changeability of higher harmonics of a tropical year is not significant. Therefore, they can be accurately modelled with the WLS approach. Apart from time-changeable curves, we estimated a percentage of the total variance of the time series which is explained by particular ISSA-curve. In this way, we used the ISSA approach to model the seasonal signal (annual + semi-annual) in environmental loadings to estimate the combined effect of atmosphere and hydrosphere (ERAIN + MERRA + ECCO2). Afterwards, we subtracted this ISSA-curve from vertical GPS position time series. In consequence, we should be able to remove all time-changeable curves that arise from an impact of the environment. However, we realize that the environment is not the only source of seasonal signals in GPS position time series. Therefore, residual oscillations of a year and half-a-year were modelled with WLS.

Finally, the maximum likelihood estimation (MLE) approach was employed to derive the parameters of residuals when the ISSA-curve was subtracted. We used Hector software (Bos et al. 2013) to estimate both the spectral index and amplitude of power-law noise that GPS time series are likely to follow.

## 4 Results

In this section, we present the results of annual and semi-annual amplitudes estimates. We applied the WLS approach for GPS position time series and a superposition of environmental loadings (SUP). Then, we subtract the SUP directly from GPS vertical data and show that this approach can change the stochastic part leading in this way to the artificial bias of vertical velocities. Therefore, we apply the ISSA approach to retrieve the seasonal signal from the SUP and subtract it from the GPS position time series. Afterwards, we estimate the vertical velocity errors for selected IGS ITRF2014 stations with the seasonal signal being efficiently removed.

### 4.1 Seasonal Signal Constant Over Time

Environmental loadings are characterized by an annual sine curve with a median amplitude of 1.3, 2.0 and 0.3 mm for ERAIN, MERRA and ECCO2, respectively. Asian stations are characterized by the largest annual amplitudes for the ERAIN model. These amplitudes fall between 3 and 5 mm with a maximum of 5.7 mm for NVSK (Russia), while amplitudes for other stations range between 0 and 2 mm. The annual amplitudes of the MERRA model range from 0 to 8 mm. Considering the Southern Hemisphere, almost all annual sinusoids have an amplitude close to zero and only a few stations differ from this rule. These are BELE, IMPZ, BRAZ, CHPI (all situated in Brazil) and DARW (Australia) with amplitudes of 4.8, 7.8, 5.7, 3.3 and 3.2 mm, respectively. The hydrological model for the Northern Hemisphere is much more varied in amplitude than for the Southern Hemisphere with a maximum for ZWE2 and MDVJ (Russia) with amplitudes around 6 mm. European stations are characterized by amplitudes of the MERRA model between 0 and 6 mm. The amplitudes of the annual curve for the oceanic model range between 0.0 and 1.5 mm with a median of 0.3 mm. The maxima of the annual sinusoid of ECCO2 were found for HELG, VIS0 and TERS (all situated in Europe) with values of 1.5, 1.4 and 1.5 mm, respectively. Beyond the annual signal, environmental models are also characterized by semi-annual curves. These are equal to 0.3 mm for ERAIN, 0.2 mm for MERRA and 0.1 mm for ECCO2. When a combined effect SUP is considered (Fig. 2), the mean annual amplitude is equal to 2.8 mm with a maximum of 8.6 mm for NVSK (Russia). The overall picture of annual amplitudes is homogenous for all stations. Apart from NVSK which has a maximal amplitude, only a few of the 376 stations analysed stand out from others. These are BELE, IMPZ, BRAZ (Brazil), ZWE2, MDVJ, ARTU, NRIL and YAKT (Russia) with amplitudes of 5.3, 8.3, 5.6, 7.1, 7.0 and 7.4 mm, respectively. Apart from the annual curve, environmental loadings are also characterized by a semi-annual signal (not-shown here). Its amplitude ranges between 0.0 and 1.4 mm for ERAIN with a median value of 0.3 mm, between 0.0 and 1.4 mm for MERRA with a median value of 0.2 mm and between 0.0 and 0.4 mm with a median of 0.1 mm for ECCO2. The median of semi-annual signal for SUP is equal to 0.4 mm with a maximum of 1.8 mm for the SCH2 station (Canada).

A power-law behaviour with a spectral index close to flicker noise (Fig. 4) is a good approximation for GPS position time series (e.g. Zhang et al. 2014; Williams et al. 2004 or Klos et al. 2016). In cases where a pure white noise is wrongly assumed instead of a power-law noise, the velocity uncertainty is underestimated from 5 to 11 times (Mao et al. 1999). Conversely, if any seasonal signal remains unmodelled, it leads to an additional correlation to the GPS position time series leading in this way to an overestimation of velocity uncertainty which can be then inappropriately interpreted. In this instance too, if any obvious and clear time-variability remains unmodelled, it will additionally bring artificial correlation into GPS residuals leading to overestimation of velocity error. As stated by Santamaria-Gomez and Memin (2015), having subtracted the loading models directly from the GPS position time series, the data variance is reduced which is only a variance of white noise. However, when it is subtracted directly, loading models might influence the stochastic part of GPS data and in this way, change the noise parameters from flicker noise into less correlated white noise or into a non-stationary random-walk noise. If this is not taken into account in the noise modelling, the errors of velocities are going to be under- or over-estimated, respectively, and might be misinterpreted. Having subtracted the SUP model, we examined the noise parameters of residuals with MLE. All stations from a number of the 376 selected in this study, showed a change in spectral index when SUP was removed. A number of 104 from 376 data sets changed the spectral index of a value lower than 0.1 when SUP was removed. For a total of 210 out of 376 stations, the character of the stochastic part was moved towards a random walk with a maximal change of spectral index equal to 0.5 for station NICA (France). For the rest of the 376 stations, the noise parameters changed from flicker noise into less correlated white noise. For this group, the maximal change of spectral index was 0.5 for station SKE0 (Sweden).

### 4.2 Seasonal Signals Changing Over Time

The seasonal signal observed in the GPS position time series may vary slightly from year to year, since the geophysical causes that absorb the GPS data are not constant over time (Chen et al. 2013; Davis et al. 2012). Whenever the WLS approach is applied, it only estimates sinusoidal curves with amplitudes constant over time. In this section, using the Improved SSA (ISSA), we introduce the seasonal signals that change over time for the SUP loading model. Then, we subtract this curve from the vertical changes for GPS stations. We focus on annual and semi-annual curves, as their time-variability is the highest. Other harmonics of a tropical year may be modelled with the WLS approach afterwards.

To produce an overall view of how much the annual signal contributes to the variance of time series, we estimated the percentage of the total variance explained by the annual curves for ERAIN, MERRA, ECCO2, SUP and GPS, respectively. The annual signal explains on average 23% of the total variance of GPS vertical displacements. Considering IMPZ, BRAZ (Brazil), IRKT (Russia) and LHAZ (China), the annual oscillations, respectively, explain 64, 62, 54 and 53% of total variance. Stations situated in Asia are characterized by the highest contribution of the annual signal to GPS data. The annual signal estimated with ISSA explains on average 23% of the total variance of ERAIN. The greatest percentage of explained variance was found for DARW (82%), KARR (80%) (Australia) and for WUHN (78%) and SHAO (74%) (China). The percentage of total variance explained by the annual signal for ERAIN represents approximately 10% for stations in Europe. Annual oscillations estimated for the hydrological loading (MERRA) account on average for 68% of the total variance. Such a high contribution of the annual signal to the total variance of the loadings demonstrates that the continental hydrosphere may have a significant impact on the observed signals. For the majority of the stations in Europe, the percentage of total variance explained by the annual signal ranges between 83% and a maximum of 91%. The annual signal derived from non-tidal ocean loading models (ECCO2) explains on average 19% of the total variance for 376 selected stations. In Europe, the annual signal accounts for approximately 29% of the total variance of this loading.

Then, the ISSA-curve estimated for SUP was compared to the WLS-curve estimated for the vertical displacements of GPS data. Comparing to the ISSA-curve derived for loading models, the seasonal signal estimated with WLS for GPS position time series is underestimated by a maximum of 4.8 mm for KIT3 station (Uzbekistan).

To date, the GPS position time series were corrected for loading models by direct subtraction of the model from GPS data. Earlier in this chapter, we showed that this might absorb some power in the GPS data, leading in this way to incorrect estimates of velocity errors. In this research, we show a completely new approach to modelling the seasonal curve that comes from the SUP loading model. Time changeable curves are provided by the ISSA approach and then, subtracted from the GPS position time series. In this way, the stochastic properties of the GPS signal remain intact, and seasonal changes that arise from environmental loading are removed. Having removed this curve, one should still model the annual and semi-annual residual oscillations with the WLS approach, as these result not only from loading models, but also arise from GPS artefacts.

### 4.3 Dilution of Precision

*Γ*is the gamma function,

*N*is the number of data in the time series and Δ

*T*is the sampling interval.

## 5 Discussion and Conclusions

We studied 376 IGS ITRF2014 stations to measure the influence of applying atmospheric, hydrological and oceanic non-tidal loading models on the estimated vertical velocity. These loadings show a dominant annual signal and their superposition shows amplitudes of up to 12 mm. The highest amplitudes were noted for Asian stations but also three South American stations are characterized by amplitudes close to 11 mm, which is in agreement with the findings of Bogusz et al. (2015). We summed the hydrological, atmospheric and non-tidal oceanic model into one superposition (SUP) to examine the impact all models may have on the GPS position time series. For the SUP model, we found the annual amplitude larger than 7 mm for 5 stations in Asia and one station in Brazil.

Looking at each contribution separately, we found that the annual amplitude of the atmospheric loading, using the ERAIN model, can be as large as 6 mm, especially in Asia. For hydrological loading this value is 8 mm using the MERRA model. The highest amplitudes circa 4–8 mm were delivered for central European, north Asian and the northern group of North American stations. Also, BRAZ and IMPZ (Brazil) are characterized by an annual signal greater than 5 mm. Finally,using the oceanic ECCO2 model, we estimated that 13 out of 376 stations have an annual loading amplitude higher than 1 mm with three maximal amplitudes of 1.5 mm for three stations in Europe: VIS0 (Sweden), TERS (Netherlands) and HELG (Germany). The two latter stations were also noted by Williams and Penna (2011).

After subtracting the SUP directly from the GPS vertical position time series we obtained a reduction of more than 25% in RMS value for the majority of the 376 stations. We found an average reduction in RMS of 17%. The RMS reduction we estimated in this research is very similar to what was shown by Tregoning and van Dam (2005).

Next, we examined the noise properties of these residuals with MLE. Assuming a power-law behaviour, we found a maximum change in the spectral index of 0.5, which would cause an underestimation in velocity error, if this error was to be estimated from residuals after the SUP was removed. This change was mostly caused by the ERAIN model, which is characterized by autoregressive-like behaviour. If ERAIN contributes to SUP more than MERRA, it will cause a change in the noise parameters. On the other hand, if MERRA contributes more than ERAIN, the change in the noise parameters will not be so dramatic, due to its power-law-like character. Beyond a change in the stochastic part, we also obtained a decrease in the annual peak of the GPS position time series.

The loading models do not remove the seasonal signal complete from the GNSS time series and, therefore, we fitted another seasonal signal to the residuals using the ISSA approach and subtracted it from the GPS position time series. The ISSA approach was chosen, as seasonal signals from real geophysical sources are not constant over time. Naturally enough, we still modelled the residual oscillations of a year and half-a-year with the WLS approach, as the seasonal signal in GPS does not only result from environmental loading. We discovered that for environmental loading models, the annual signal explains up to 66% of a total variance of SUP for Europe, between 30 and 100% for Asia, up to 30% for North America and up to 100% for South America. The annual curve estimated for Antarctica explains 10% of the SUP variance. Having removed the ISSA curve from the GPS data, we examined the properties of residuals with MLE. We found a maximal difference in the spectral index between the GPS position time series and the GPS data without ISSA-seasonal of only 0.1. This means that we are able to reduce the seasonal peaks from environmental impact with no significant influence on the stochastic part of the GPS data.

The reliability of the GPS-based velocities determined from the time series are constantly improving due to the ongoing efforts in the refinement of equipment, processing strategies and the models of geophysical phenomena as well. With regard to this, we need to estimate the errors of velocity as reliably as possible. In this research, we estimated the dilution of Precision (DP) of the vertical velocities of the ITRF2014 series. For a total of 298 out of the 376 stations analysed, the DP was lower than 1. This indicates that when the SUP-ISSA-curve was removed from the GPS data, the error of velocity becomes lower than it was before. We were able to reduce the velocity uncertainty with no change in the stochastic part of data. It means that the change in velocity uncertainty arises only from a proper modelling of time-varying seasonal signal.

Concluding, we propose a two-stage solution of the problem of reliable subtraction of seasonal curves. It is important to implement the environmental loading and model a seasonal curve with an approach that allows for considering their changeability over time. Afterwards, the analytical methods still need to be implemented to remove the remaining seasonal oscillations. These two steps may provide an estimation of the vertical velocities that are used for kinematic reference frames and geodynamical interpretations as well as their reliable uncertainty assessment.

## Notes

### Acknowledgements

Anna Klos, Marta Gruszczynska and Janusz Bogusz are financed by the Polish National Science Centre, Grant No. UMO-2014/15/B/ST10/03850. Machiel Simon Bos is financially supported by Portuguese funds through FCT in the scope of the Project IDL-FCT-UID/GEO/50019/2013 and Grant Number SFRH/BPD/89923/2012. Jean-Paul Boy is partly funded by CNES (Centre National d’Etudes Spatiales), through its TOSCA program. Loading time series used here are available at EOST/IPGS loading service (http://loading.u-strasbg.fr). Maps and charts were plotted in the Generic Mapping Tool (Wessel et al. 2013). IGS time series were accessed from ftp://igs-rf.ensg.eu/pub/repro2.

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