# Impact of the GPS orbital dynamics on spurious interannual Earth deformation

H. Ait-Lakbir, A. Santamaría-Gómez, F. Perosanz

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H. Ait-Lakbir, A. Santamaría-Gómez, F. Perosanz. Impact of the GPS orbital dynamics on spurious interannual Earth deformation. Geophysical Journal International, 2023, 235 (1), pp.796-802. 10.1093/gji/ggad268. insu-04831618

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# Impact of the GPS orbital dynamics on spurious interannual Earth deformation

H. Ait-Lakbir<sup>1</sup>, A. Santamaría-Gómez<sup>1</sup> and F. Perosanz<sup>2</sup>

Accepted 2023 June 29. Received 2023 June 23; in original form 2023 April 12

## SUMMARY

Global Positioning System (GPS) daily position time-series have a standard precision of a few millimetres. However, GPS position series contain large temporal correlations that impede the observation of subtle interannual Earth deformation. We show that the specific configuration of the GPS constellation, compared to other Global Navigation Satellite Systems (GNSS), contributes to the temporal correlation. Based on the analysis of observed and simulated GPS, Galileo, GLONASS and BeiDou orbits, we determine that the GPS orbital dynamics are more prone to interannual drifts caused by their higher sensitivity to the lunisolar gravitational resonance. This leads to substantial changes in the observation geometry over time, which, combined with mismodelled station-dependent systematic errors, results in a larger temporal correlation for GPS position time-series. Improving the weighting of the GPS observations may mitigate the effect of geometry, which is absent in other GNSS constellations.

**Key words:** Satellite geodesy; Time-series analysis.

## 1 INTRODUCTION

In the past decades, Global Navigation Satellite Systems (GNSS), mostly Global Positioning System (GPS), have significantly contributed to geosciences by providing long series of precise station coordinates that are used to examine crustal deformations. The station coordinates are modelled with a deterministic function accounting for both trend and seasonal signals, which mostly reflect underlying geophysical and measurement processes. However, it has been demonstrated that GPS positions exhibit temporally correlated errors that must be considered to obtain meaningful uncertainties of the deterministic parameters. For instance, assuming uncorrelated noise can lead to underestimated uncertainties in GNSS velocities by a factor of 5 (Mao et al. 1999; Williams 2003; Santamaría-Gómez et al. 2011; Langbein 2012; Klos et al. 2017).

Mao et al. (1999) and Williams et al. (2004) determined that the noise in GPS station positions comprises white and coloured, mostly flicker noise and possibly some amount of random walk noise. White noise predominates at high frequencies, while flicker noise predominates at low frequencies. The noise in GPS position time-series has been related to long-term and unaccounted-for displacements, such as site-specific random motion due to ground (Wyatt 1982, 1989) or monument instability (Williams et al. 2004; Beavan 2005; King & Williams 2009). Non-tidal surface loading can potentially contribute to GPS positioning noise, although the current models primarily explain noise at intraseasonal periods rather than long periods, where the variance of the state-of-the-art load models is

still far from the observed GPS variance (Rebischung et al. 2017; Memin et al. 2020; Gobron et al. 2021; Michel et al. 2021).

Despite being equally impacted by the same geophysical processes, the station positions determined from other space geodetic techniques, such as Doppler Orbitography by Radiopositioning Integrated on Satellite (DORIS), Satellite Laser Ranging (SLR) and Very Long Baseline Interferometry (VLBI) have demonstrated that a power-law process with a spectral index closer to white noise is a better stochastic model (Le Bail 2006; Feissel-Vernier et al. 2007; Ray et al. 2008; Klos et al. 2018). The reason why coloured noise seems specific to GPS observations remains a subject of ongoing research. Although the positioning noise variance is lower for GPS, it is worth noting that DORIS and SLR use different types of satellite orbits, which are absent in VLBI. This could potentially explain the type of positioning noise. Indeed, King & Watson (2010) used simulated GPS data and showed that a time-varying satellite geometry, associated with multipath errors may contribute to coloured noise in GPS positions. Yet, their work did not elaborate on the mechanisms behind the variations of the satellite geometry, and how they propagate positioning errors into long-period correlation.

Previous studies characterizing the coloured noise in GNSS position series have focused on GPS data. In this study, we use the four available GNSS constellations and demonstrate that part of the coloured noise is due to the design of the GPS constellation geometry. The geometry of the other GNSS constellations is significantly less prone to introduce coloured noise.

<footnote n="1"><sup>1</sup>GET, Université Paul Sabatier, CNES, CNRS, IRD, 31400 Toulouse, France. E-mail: hanane.ait-lakbir@get.omp.eu</footnote>

<footnote n="2"><sup>2</sup>Centre National d'Etudes Spatiales (CNES), 31400 Toulouse, France</footnote>
## 2 IMPACT OF THE GNSS CONSTELLATION GEOMETRY

To assess the sensitivity of coloured noise to the geometries of the different GNSS constellations, we run precise point positioning (PPP) simulations using precise GPS, Galileo and GLONASS RE-PRO3 orbits provided by the analysis centers of the International GNSS Service (IGS) from 2018 January through 2021 January to benefit from up-to-date orbital modelling. The BeiDou constellation is therefore omitted for this first test. Simulated GNSS observations above a cut-off elevation of 10° are computed every 900 s from the geometric distance between the real orbits and a network of 118 globally distributed stations. The PPP processing applies the usual linearization of the observation equations and is based on the solution of the corresponding normal equations by weighted least squares. A set of station coordinates is estimated every 24 hr, while we determine receiver clock biases per observation epoch.

To isolate the impact of the GNSS constellation geometry, observations are simulated by neglecting satellite clocks, phase ambiguities and tropospheric delays shared by all constellations. Furthermore, based on our tests with and without estimated troposphere parameters, neglecting them decreases the variance of the height noise, but not its type. We add an elevation-dependent random noise with zero mean and a systematic error following a non-zero mean depending on the elevations that may reflect antenna phase centre, atmospheric delay and multipath errors. This systematic error was simulated using the multipath model from King & Watson (2010) under the assumption that all stations are identical in terms of antenna characteristics (height and antenna gain) and environment (ground roughness and refractive indices).

Fig. 1 shows the stacked power spectra of the estimated position series for the up coordinates (see Figs S1 and S2, Supporting Information, for the horizontal coordinates). The power spectra are unnormalized, making them comparable in terms of variance. The spectral power of GPS-derived coordinates increases for long periods, following a power law with a steep slope. Conversely, with Galileo and GLONASS orbits, the position spectra slightly increase with low frequencies and approach more white noise. The same test performed with a zero-mean elevation-dependent random measurement noise only (Fig. S3, Supporting Information) does not result in coloured noise. These findings suggest that at least some amount of the coloured noise observed in GPS position series could be related to the GPS orbits themselves, in the presence of systematic errors and could disappear in position series from the other GNSS constellations.

Elevation-dependent weighting functions are commonly used to mitigate unmodelled measurement errors in GNSS data processing. We assess the impact of three weighting functions on the coloured noise: an elevation-independent function, an inverse sine function estimated from post-fit observation residuals and an optimal elevation-dependent function derived from a priori knowledge of the systematic errors. Fig. 1 shows that a post-fit estimated weighting function reduces the noise variance in positions by a factor of 10 compared to a deficient weighting. Yet, the post-fit weighting function does not significantly impact the power-law noise type. Long-period coloured noise is only mitigated by optimal weights that correspond to the actual statistical characteristics of the systematic errors. Therefore, reducing the coloured noise content in GPS positions may be achieved by either limiting site-specific systematic errors, such as multipath, antenna phase centres or tropospheric

<figure><img src="image_3.png" /><figcaption>**Figure 1.** Stacked power spectra of simulated station heights obtained from real GPS (top), Galileo (middle) and GLONASS (bottom) precise orbits. Spectra are given for different weighting strategies: equal, inverse sine and optimal.</figcaption></figure>

propagation or by using an optimized empirical weighting function not based only on post-fit residuals.

Galileo- or GLONASS-derived positions are impacted by systematic periodic signals up to 47 d caused by the repeat period of the constellation and the presence of the systematic error. Their magnitudes remain unchanged by the post-fit weighting function, but only with optimal weighting. Additional GPS peaks, already observed in King & Watson (2010), occur at approximately 10, 5, 3.5 and 2.3 d, which vary among the orbits produced by different analysis centres and do not directly relate to GPS orbital characteristics. The orbits calculated by the ESA (European Space Agency) analysis centre for the third IGS reprocessing campaign, including GPS, Galileo and GLONASS, were used to obtain the spectra in Fig. 1; however, these signals are also observed when using the IGS combined GPS orbits (Fig. S4, Supporting Information). Multiple factors affecting the orbits could explain these signals, such as the alignment to the terrestrial frame, which is mostly counterbalanced by the satellite clock and earth orientation parameters products, or error transfers from the other constellations. While these signals deserve further investigation, they do not affect the conclusions regarding the coloured noise over long periods.

Real positioning confirms the stronger GPS-specific coloured noise. For the same network of 118 stations, daily PPP solutions were computed between 2019 January and 2021 November using GRGS orbit and clock REPRO3 products (Katsigianni et al. 2019) and the GINS software provided by CNES. We are limited to almost 3 yr of data because of the availability of homogeneously reprocessed orbit and clock products for the different GNSS constellations, especially
Galileo. Position series were generated for GPS-only and Galileo- only integer-ambiguity solutions and GLONASS-only ambiguity- floating solutions. BeiDou data are not fully supported by GINS. For each PPP series, we analyse the noise content while fitting a kinematic model that includes a linear trend, position offsets and constellation-specific periodic signals: 4 draconitic harmonics with periods of 351.2 d for GPS, 355.6 d for Galileo and 353.2 d for GLONASS, and 5 harmonics of the constellation repeat periods, with periods of 10 sidereal days for Galileo and 8 sidereal days for GLONASS. As the GPS constellation repeats every 24 hr, no GPS harmonics were fitted.

Concerning the choice of the noise model, we analyse the goodness-of-fit of the flicker and white noise models by using a Bayesian Information Criterion (BIC), which gives the maximum log-likelihood corrected by the degrees of freedom of the functional and noise models and the number of observations used for the fitting. Fig. 2 shows the difference between the BIC for the flicker noise and the white noise models. A positive value indicates that the flicker noise model better describes the position residuals, which implies the presence of coloured noise.

According to this test, most of the GPS-only series contain flicker noise, while it is more difficult to distinguish it from white noise for Galileo- and GLONASS-only series. As the series have similar variance for GPS and Galileo, the BIC differences presented in Fig. 2 indicate that the background noise in GPS coordinates is indeed closer to flicker noise than to white noise. Ambiguity-fixed Galileo East positions, which also exhibit whiter noise, better indicate than GLONASS the nature of coloured noise in this direction. The ambiguity-float GLONASS East coordinates are noisier and coloured noise, if any, would be hidden by larger white noise. Despite the short period available for these tests, both real and simulated GNSS observations confirm that GPS-only series are likely to contain more coloured noise than Galileo-only or GLONASS-only series.

## 3 TIME-VARIABLE GPS CONSTELLATION GEOMETRY CAUSED BY RESONANT DYNAMICS

The distinct contribution of the geometry of different GNSS constellations on the coloured noise raises questions about the orbital factors explaining these results. We focus on the long-term orbital dynamics of the four available constellations, including BeiDou, and evaluate their impact on the coloured noise of position series. Fig. 3 shows the evolution of the eccentricity and inclination of the GPS, Galileo and GLONASS orbits over three years (2018–2020). These values are extracted from the IGS REPRO3 precise orbits.

GPS satellites show a wide dispersion with eccentricities rising up to 0.025 and inclinations ranging within a ± 2° interval around the nominal value of 55°. The GPS satellites are initially launched in their nominal orbits, but they drift in eccentricity and inclination within the operational interval (dashed area) published in the GPS Standard Positioning Service (SPS) Performance Standard (2020), except for the GPS satellite SVN45, which falls in the tolerance zone (dotted area). For the same period, the Galileo and GLONASS orbits show less variability within the constellation and remain closer to the published nominal values (Teunissen & Montenbruck 2017). No public information about the operational ranges of these two constellations could be found.

Secular orbital behaviour is primarily caused by gravitational perturbations. GPS orbital period exactly commensurate with Earth's

rotation produces a deep resonance, which increases orbit sensitivity to geopotential terms and generates long-periodic signals on the semi-major axis, which is corrected by maneuvers (Hugentobler et al. 2003; Ineichen et al. 2003). Resonances due to the gravitation of the Moon and the Sun also perturb the inclination and eccentricity. Characterized by a commensurability between the orbital elements of the satellites, and the positions of the Moon and the Sun (Hughes 1981), lunisolar resonances cause the eccentricity to drift over time for the Medium Earth Orbits (Deleflie et al. 2005, 2011; Rosengren et al. 2015). By conservation of energy, the eccentricity drift perturbs the other satellite's orbital elements such as the inclination (Deleflie et al. 2011; Hughes 1981). GPS maneuvers do not correct for the satellites' drifts in eccentricity and inclination. Therefore, we assume that the lunisolar resonance may account for most of the observed long-term orbital behaviour for GPS shown in Fig. 3.

To assess the impact of the lunisolar resonance on the coloured noise, we compare three simulation scenarios with the GPS, Galileo, GLONASS and BeiDou constellations. In the first scenario 'nom', we set the eccentricities and inclinations of the orbits at their nominal values. We keep them constant by propagating the Kepler equations of motion, but consider the precession of the orbital planes due to the Earth's oblateness (J2) which gives the draconitic periodicity of the GNSS constellations. In the second scenario 'geom', the eccentricities and inclinations remain constant but are replaced by the actual values for each satellite at the epoch 2018 January 1 for GPS and GLONASS, 2019 January 1 for Galileo and 2021 January 1 for BeiDou. This configuration reflects a realistic constellation geometry, at a specific epoch, affected by the gravitational and other non-gravitational accelerations since each satellite was launched. The choice of the products does not matter for this analysis as the orbit differences between analysis centers barely affect these parameters. The orbits are also propagated with the Kepler and J2 dynamics, that is, the 'nom' and 'geom' scenarios differ only in their constant constellation geometry because both are propagated without resonant dynamics. In the third scenario 'res', we use the same initial satellite positions as the 'geom' scenario. However, the orbit propagation uses a semi-analytical model for the secular time-derivative orbital elements considering the lunisolar gravitation (Chao & Gick 2004; Chao 2005).

We use the same setting as stated in the previous section to simulate the GNSS observations but with the orbits simulated in the three scenarios. To lessen the computational burden and because we focus on the long-term behaviour, the observations and daily positions are computed only every 7 d. We fit a power-law noise model to the simulated series where the spectral index and variance are estimated.

Fig. 4 shows the distribution of spectral indices estimated from the positions determined with the nominal, reference and resonant orbits. While we focus on the vertical coordinate component, the conclusions drawn also apply to the horizontal coordinates (see Fig. S5, Supporting Information). To investigate deviations of the constellation geometry from the nominal values, we compare the spectral indices of the estimated noise models using the nominal GPS satellite positions ('nom') and the real, but constant, GPS satellite positions ('geom'). The spectral indices with 'nom' orbits, centred at 0, indicate white or uncorrelated noise. In contrast, the mean of the spectral indices slightly shifts towards flicker when using a more realistic constellation geometry, as given by the 'geom' scenario. These results suggest that even non-evolving, deformed constellation geometries play a role in the emergence of coloured noise in station positions.
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Distribution of the BIC differences between flicker noise and white noise models for GPS-only (blue), Galileo-only (red) and GLONASS-only (green) for the East (left), North (middle) and Up (right) coordinate components. A positive value means flicker noise is statistically preferred over white noise.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 3.** Estimated inclination (horizontal axis) and eccentricity (vertical axis) from precise GPS, GLONASS and Galileo orbits. The shade represents the time from 2018 January 1 (light) to 2020 December 31 (dark). The operational (dashed line) and tolerance (dotted line) ranges for GPS eccentricities and inclinations are shown.</figcaption></figure>

Lunisolar resonant dynamics deform the geometry of the constellation compared to the nominal values and, more importantly, make the geometry gradually change with time, as shown in Fig. 3. Because the variations of the orbital elements due to lunisolar resonances depend on the satellite's eccentricity and inclination (Hughes 1980, 1981), an even more irregular evolution of the GPS constellation geometry occurs due to the dispersion of the eccentricity and inclination among the GPS satellites (Fig. 3). In Fig. 4, GPS-only position time-series show that a resonant dynamic in the 'res' scenario has a net negative impact on coloured noise. Conversely, the noise in Galileo, GLONASS or BeiDou positions remains uncorrelated with power-law spectral indices always centred near-zero for any scenario. The impact of the lunisolar resonance on the GPS constellation still arises when combining multi-GNSS observations (Fig. S6, Supporting Information).

## 4 DISCUSSIONS AND CONCLUSIONS

To infer geophysical displacements such as the tectonic rate, loading or slow-slip events, considering the coloured noise in GPS station positions is crucial. Currently, the standard practice is to describe the noise in decade-long GPS position series by a combination of white noise and coloured noise processes. However, considering coloured noise does not allow us to better observe subtle long-period nonlinear Earth deformation hidden by the noise. The only way forward is to understand the mechanism behind the coloured noise, including its origin and what fraction of it is generated by the complexity of the GNSS precise positioning generates remains unclear. The recent availability of precise positioning from new GNSS constellations allowed us to identify, for the first time, the contribution of the different geometry of GNSS orbits to coloured noise.

We found that GPS series differs from Galileo, GLONASS and BeiDou using real and simulated orbits. Through simulated orbits accounting for secular dynamics, we examined the long-term variations in satellite orbits. We show that the GPS constellation reacts strongly to resonances caused by lunisolar gravitational accelerations. In contrast, Galileo, GLONASS and BeiDou constellation geometries exhibit greater stability over time, remaining closer to their nominal configurations. GPS exhibits greater constellation geometry variations and introduces significant coloured noise in its positions compared to Galileo and GLONASS. We confirmed this conclusion by simulating GNSS observations between a network of ground stations and real GNSS orbits, which considered only the geometry of the satellites' positions. We identified three necessary, but not sufficient, factors that combined will generate a significant level of coloured noise in the station positions. First, the presence of systematic station-dependent errors whose non-zero mean depends on satellite elevation, such as antenna phase centre, multipath or trophospheric delays; second, a weighting function inadequate for the actual systematic errors, such as those derived from post-fit residuals; and third, a constellation geometry that changes gradually, such as the GPS constellation, which is consistent with the findings of King & Watson (2010). Our results demonstrate that if any of these
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Power-law spectral index distributions. Based on vertical coordinates with simulated orbits and observations under three scenarios: ‘nom’ (top), ‘geom’ (middle) and ‘res’ (bottom). Since Galileo, GLONASS and BeiDou orbits are close to nominal, the ‘nom’ case is ignored. For comparison, GPS histograms are displayed in light grey.</figcaption></figure>

three factors is missing, the fraction of coloured noise will signifi-
cantly reduce. Improving the weighting strategy would allow us to
reduce the coloured noise in position series from historical GPS
data.

The contribution of the geometry of the GNSS orbits alone is
likely not enough to fully explain the coloured noise content in
long GPS series. Longer Galileo series are needed to separate and
quantify the level of noise generated by the orbits from the noise
generated by additional processes common to all GNSS observa-
tions. This analysis will also benefit from the comparison with
BeiDou positioning. Our results complete the previous error budget
examining unmodelled site displacements during data processing as
sources of noise in GNSS positions. These include random ground
or monumentation motion (Williams *et al.* 2004; Beavan 2005; King
& Williams 2009) or non-tidal surface loading (Rebischung *et al.*
2017; Memin *et al.* 2020; Gobron *et al.* 2021). Our findings are a
step toward demonstrating that the constellation geometry may also
play a significant role.

Previous work has suggested the contribution of the GNSS or-
bits to the draconitic signals (Ray et al. 2008; Amiri-Simkooei
2013; Allahverdi-zadeh et al. 2016). In our study, we do not
observe these draconitic signals. We only accessed the secular
orbital contribution to positioning errors with a semi-analytical
model that neglects the orbital, lunar and solar periods. Further-
more, non-conservative accelerations such as solar radiation pres-
sure also affect the satellite orbital evolution (Valk et al. 2009;
McMahon 2011; Lemaître 2019). Addressing the orbital contri-
bution to these short-period signals observed in position series
would require deriving analytical dynamic equations that do not

average out over these periods; which is out of the scope of this
paper.

The differences in noise types among the constellations raise the question of how the coloured noise would evolve over time as new GNSS systems are integrated to determine the station positions. Notably, the amount of undetectable random walk noise, which is typically omitted in the standard noise models, limits the assessment of the uncertainty of linear rates estimated from GPS series. Long Galileo series with reduced flicker noise may provide constraints for the random walk noise hidden in the longer GPS series and therefore improve the estimated uncertainty of the tectonic rate at the same stations. Additionally, the standard noise model involves a flicker or power-law noise constant throughout the entire period. The integration of Galileo and BeiDou could result in a progressive evolution of the coloured noise content over at least a decade. Consequently, more complex noise modelling will be required for multi-GNSS position time-series.

A reduced flicker noise in GNSS positions will facilitate the
detection of interannual nonlinear geophysical processes, such as
years-long slow-slip events (Peng & Gomberg 2010; Kobayashi
2017; Ducellier et al. 2022), or interannual deformations resulting
from changes in groundwater storage (White et al. 2022). While
Galileo looks promising for better observing interannual Earth de-
formation, GPS series still contain reduced systematic signals at
higher frequencies and may be more reliable for geophysical pro-
cesses occurring at periods of a few days. Regarding the linear rates,
GPS observations will still provide better estimates due to the longer
time-series already available and only longer Galileo series will al-
low us to answer this question. Nevertheless, it is worth mentioning
that the uncertainty of the linear rate is strongly impacted by the estimation of position discontinuities as they obscure the coloured noise at long periods (Williams 2003; Griffiths & Ray 2016; Santamaría-Gómez & Ray 2021; Gobron et al. 2022) and these are common to all GNSS constellations.

## SUPPORTING INFORMATION

Supplementary data are available at GJI online.

### suppl.data

Please note: Oxford University Press is not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the paper.

## ACKNOWLEDGMENTS

We thank the two reviewers Prof Matt King and Dr Anna Klos and the handling editor Prof Jörg Renner for their comments to improve the quality and clarity of this manuscript. We also thank Dr Florent Deleflie from the Observatoire de Paris for his explanation of orbital resonance, Dr Paul Rebischung from IGN for providing the pytrf software that we used for the noise analysis, and Dr Jim Ray for his constructive comments on an earlier version of the manuscript. This work was supported by Region Occitanie and the Centre National d'Etudes Spatiales (CNES).

## DATA AVAILABILITY

The GNSS RINEX observation files, the REPRO3 precise orbit and clock products for GPS, Galileo and GLONASS and the broadcast files for BeiDou were obtained through the online archives of the IGS hosted by the Crustal Dynamics Data Information System (CDDIS) (Noll 2010). GNSS RINEX observation files can be found at https://cddis.nasa.gov/archive/gnss/data/daily/ the REPRO3 precise orbit and clock products for GPS, Galileo and GLONASS at https://cddis.nasa.gov/archive/gnss/products/repro3/ and the broadcast files for the BeiDou constellation at https://cddis.nasa.gov/archive/gnss/data/campaign/mgex/daily/rinex3/. The readers will find access instructions to CDDIS archives at https://cddis.nasa.gov/Data_and_Derived_Products/CDDIS_Archive_Access.html. The time-series of simulated and real PPP positions are available upon request to the authors.

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