Global Positioning System Coordinate Time Series Research Articles

Slow Slip Events Associated with Seismic Activity in the Hikurangi Subduction Zone, New Zealand, from 2019 to 2022

Slow slip events (SSEs) are geophysical phenomena primarily occurring in subduction zones. These events are often associated with seismic activity and can be detected by Global Positioning System (GPS). However, the relationship between SSEs and seismic activity remains unclear. To further investigate SSEs associated with seismic activity, we conducted SSE detection and inversion for the period from 2019 to 2022 on New Zealand’s North Island, where both SSEs and seismic activity frequently occur. By modeling daily GPS coordinate time series from 40 GPS stations and applying the Network Inversion Filter (NIF) method, we obtain surface displacements, cumulative slips, and slip rates for eight shallow SSEs. Subsequently, we conduct a statistical analysis of seismic activity concerning its spatial distribution and frequency before, during, and after SSE occurrences. The results indicate that SSE1 and SSE7 exhibited larger cumulative slips, at 14.35 and 7.20 cm, and surface displacements, at 4.97 and 2.53 cm, respectively. During their occurrences, the seismic frequency noticeably increased to 6.5 and 5.6 events per day in the Eastern Coastal Region (ECR) of New Zealand’s North Island. However, the other six SSEs, characterized by cumulative slips of less than 6 cm and maximum surface displacements of less than 2 cm, did not lead to a noticeable increase in seismic frequency during their occurrences in the ECR. In the Main Slip Regions (MSR) of these eight SSEs, a significant upward trend in seismic frequency was observed during their occurrences. Therefore, it can be inferred that in the ECR of New Zealand’s North Island, all SSEs result in an increased seismic frequency within their respective MSRs, but only significant SSEs impact the seismic frequency of the ECR. Monitoring shallow SSEs may contribute to the identification and recording of seismic activity.

Insights into spatio-temporal slow slip events offshore the Boso Peninsula in central Japan during 2011–2019 using GPS data

Using Global Positioning System (GPS) coordinate time series, we detect three transient slow slip events (SSEs) offshore the Boso Peninsula in central Japan during 2011–2019. To extract the tiny SSE signals obscured by the significant post-seismic deformation after the 2011 MW9.0 Tohoku earthquake, we develop a new GPS coordinate time series processing software to obtain these SSE-induced deformations from high-noise GPS data. In addition, we apply the principal component analysis-based inversion method (PCAIM) to get the spatio-temporal slip distribution of the three SSEs. The spatio-temporal evolutions of these slips reveal that the nucleation styles are different. Compared to the 2011 and 2018 SSEs, the 2013–2014 SSE displays faster slip spatio-temporal variation, deeper slip, shorter slip duration, minor seismic moment, and lower maximum slip rate. The 2018 SSE exhibits the most significant seismic moment, the maximum slip, and the maximum slip rate of these three SSEs. The spatio-temporal variations of the 2011 SSE are the most complex, containing two acceleration and deceleration phases. The slip zone expanded along the eastern side of the Boso Peninsula in the acceleration phase and shrank back in the deceleration phase. Furthermore, the recurrence interval of SSEs spans from 2.2 to 4 years during 2011–2019, suggesting that the recurrence interval might become shorter and non-periodic due to the enormous earthquake. After the 2013–2014 SSE, the recurrence interval of the SSE gradually returns to normal. Thus, we can infer that the SSE may occur every 4–7 years after the 2018 SSE if there is no large earthquake.

Fast approximation algorithm to noise components estimation in long-term GPS coordinate time series

Understanding the noise content of the Global Positioning System (GPS) coordinate time series is a prerequisite for a realistic assessment and uncertainty of unknown parameters. Variance component estimation methods [e.g., restricted maximum likelihood estimator (REML)] are used to assess the noise content of GPS coordinate time series. For large-scale data, namely over a wide range of spatial and temporal scales, the previous methods’ efficiency could significantly improve. Meanwhile, the estimation method, including repeated inversion of large matrices, has led to intensive computations and large storage requirements. By quantifying the REML estimator by decorrelation property of discrete wavelet transformation, the current research has offered FREML (fast REML) for accurate and fast approximation of noise content. For evaluating the method’s efficiency, 360 synthetic daily time series with different lengths $$N=2048$$ , 4096, and 8192 observation epochs were used. The time series composed of linear trends, periodic signals, offsets, transient displacements, gaps (up to 10%), and a combination of white, flicker, and random walk noises. The FREML algorithm’s outcomes were compared with existing software that uses a maximum likelihood approach to quantify the uncertainties (e.g., Hector). The results indicated that both methods provided equivalent results for noise components, unknown parameters (rate, offset, and transient displacement), and their uncertainties. Moreover, the FREML method reduced the computation time by a factor of 2–14 compared to Hector software, depending on the amount of data and missing epochs. For more assessment of the method, the FREML method was applied to the 36 real time series with noise models as (i) white plus flicker noise and (ii) combination of white, flicker, and random walk noises. The results demonstrated that the two methods’ outcomes were close, and the FREML method speeded up the estimation of noise and unknown parameters.

Vertical Displacements of the Amazon Basin From GRACE and GPS

AbstractThe extent to which Gravity Recovery and Climate Experiment (GRACE)‐recovered gravity anomalies can improve our understanding of Global Positioning System (GPS)‐measured vertical displacements is currently uncertain. To address this issue, we compared vertical displacements measured by 23 GPS stations in the Amazon basin with displacements estimated from GRACE geopotential fields. We show that despite poor correlation ( ) between rate estimates in GPS and GRACE‐derived displacement time series, further analyses reveal low bias between annual amplitude estimates and a scaling near 1. There is higher correlation ( ) between annual periodic motions, with near 1 to 1 agreement, but there is poor correlation ( ) and little agreement between semiannual amplitude estimates. Subtracting GRACE displacements from the GPS time series flattens the GPS power spectra, reducing the spectral index magnitude, on average, from (“fractional Brownian motion”) to (“fractional Gaussian noise”), suggesting that some fraction of the apparent GPS error correlation derives from mass loading signals that are not completely characterized by secular trends or seasonal periodic motions. From March 2011 to November 2016, we find a GPS and GRACE‐derived displacement combined average uplift of the Amazon basin of mm/yr and combined average annual periodic motion of mm. Deviations from a standard trajectory model for site motion are apparent in both data sets and appear to coincide with various flooding and drought events between 2011 and 2016, which suggests that the GPS coordinate time series record displacements driven by large‐scale climate oscillations.

A Sub-Regional Extraction Method of Common Mode Components from IGS and CMONOC Stations in China

There is always a need to extract more accurate regional common mode component (CMC) series from coordinate time series of Global Positioning System (GPS) stations, which would be of great benefit to describe the deformation features of the Earth’s surface with more reliability. For this purpose, this paper combines all 11 International Global Navigation Satellite System (GNSS) Service (IGS) stations in China with over 70 stations selected from the Crustal Movement Observation Network of China (CMONOC) to compute CMC series of IGS stations by using a principal component analysis (PCA) method under cases of one whole region and eight sub-regions. The comparison results show that the percentage of first-order principal component (PC1) in North, East and Up components increase by 10.8%, 16.1% and 25.1%, respectively, after dividing the whole China region into eight sub-regions. Meanwhile, Root Mean Square (RMS) reduction rates of residual series that have removed CMC also improve obviously after partitioning. In addition, we compute displacements of these IGS stations caused by environmental loadings (including atmospheric pressure loading, non-tidal oceanic loading and hydrological loading) to analyze their contributions to the non-linear variation in GPS coordinate time series. The comparison result shows that the method we raise, PCA filtering in sub-regions, performs better than the environmental loading corrections (ELCs) in improving the signal-to-noise ratio (SNR) of GPS coordinate time series. This paper raises new criteria for selecting appropriate CMONOC stations around IGS stations when computing sub-regional CMC, involving three criteria of interstation distance, geology and self-condition of stations themselves. According to experiments, these criteria are implemental and effective in selecting suitable stations, by which to extract sub-regional CMC with higher accuracy.

The Magnitude of Diurnal/Semidiurnal Atmospheric Tides (S1/S2) and Their Impacts on the Continuous GPS Coordinate Time Series

The site displacement due to diurnal and semidiurnal atmospheric tides (S1/S2) is often neglected in the routine precise GPS data processing. We recall the S1/S2 modeling method and show the magnitude of the S1/S2 tides under the Center of Mass (CM) frame. The results show that the annual amplitudes caused by both S1 and S2 tides exceed 1 mm for stations near the equator. The impact of S1/S2 on the 24-h Global Positioning System (GPS) solution is typically at the sub-mm level, and the scatter of the S1/S2 caused displacement difference increases with the decreasing latitude for northern hemisphere stations, among which the maximum Standard Deviation (STD) reaches up to 1.5 mm, 1 mm and 0.7 mm for the Up, East and North components, respectively, at low-latitude stations. We also find that 65% of the stations’ vertical velocity change caused by S1/S2 is larger than 1%, among which the maximum velocity variation rate reaches more than 40% for some coastal/island stations, while stations with the weighted root mean square reduced account for 65%, 39%, and 38% for the up, east, and north components respectively, in particular for most coastal/island stations. Furthermore, the S1/S2 correction could partly reduce the annual and the semi-annual signals of the global stacked vertical component together with the semi-annual amplitude of the east component. The power of some anomalous harmonics of 1.04 cycle per year also decreased a lot. These results further prove the benefits of S1/S2 correction in the precise GPS data processing.

Theory of second order stationary random processes applied to GPS coordinate time-series

The analysis of Global Positioning System (GPS) coordinates time series is a valuable tool in quantifying crustal deformations. The longer continuous GPS time series allow estimation of nonlinear signatures. As a matter of fact, besides the linear and periodic behaviors, other relevant signals are present in such time series as the so-called transient deformations. They can be related to, e.g., slow slip events, which play a crucial role in studying fault mechanisms. To give reliable estimates of these signals, an appropriate and rigorous approach for defining the deterministic and the stochastic models of the data is needed. We prove that the theory of the second order stationary random process (SOSRP) can be used to describe the stochastic behavior of the daily GPS time series. In particular, the second order stationarity condition has to be verified for the daily GPS coordinate time series to be described as a SOSRP. This method has been already used for modeling the gravity field of the earth and in predicting/filtering problems, and this work shows that it can also be useful for characterizing the colored noise in the GPS time series.

Effects of unmodelled tidal displacements in GPS and GLONASS coordinate time-series

This study demonstrates the effect of unmodelled (sub-)daily tidal displacement on GLONASS (GLObalnaya NAvigatsionnaya Sputnikovaya Sistema) coordinate time series in comparison with GPS (Global Positioning System). More than two propagated periodic signals appear in GPS and GLONASS coordinate time series in the presence of an unmodelled M2 /O1 tidal displacements. The propagation mechanism of Stewart et al. (2005) explains well most of the periodic signals. At a fortnightly period, an unmodeled M2 tidal displacement propagates into two long-period signals at 13.6x (x is a positive integer) and 14.76 days for GPS, while only a significant propagated periodic signal at 14.76-day is discernible for GLONASS. Similarly, significant propagated periodic signals at 13.6x and 14.19 days for GPS and only at 14.19-day for GLONASS are evident as a result of an unmodeled O1 tidal displacement. However, an unmodeled Mf (long-period) results in a strong power of similar magnitude at 13.6x-day (∼13.66-day) for both GPS and GLONASS solutions. The appearance of different periodic signals but as a result of the same unmodelled tidal displacement is attributed to the different ground repeat periods of the constellations, which is one of the factors of the propagation mechanism from Stewart et al. (2005). The latter explains the reason why the 13.6x-day fortnightly is present only for GPS solutions. Comparing the powers of the M2 aliased periodic signals at 13.6x-day and 14.76-day from a stacked spectra over all stations, the amplitude of the former is larger than the latter by an order of magnitude. The results of this study may infer that the 13.6x-day periodic signal in GPS/GNSS (Global Navigation Satellite System) derived products of the IGS (International GNSS Service) is a joint contribution of the propagation of unmodelled (sub-)daily tidal displacements and errors at longer periods with the former appearing to contribute more. We also demonstrated that significant reduction of the propagated periodic signals can be achieved from combined-system solutions.

Common mode error in Antarctic GPS coordinate time-series on its effect on bedrock-uplift estimates

Spatially correlated common mode error (CME) always exists in regional, or-larger, global positioning system (GPS) networks. We applied independent component analysis (ICA) to GPS vertical coordinate time-series in Antarctica from 2010 to 2014 and made a comparison with the principal component analysis (PCA). Using PCA/ICA, the time-series can be decomposed into a set of temporal components and their spatial responses. We assume the components with common spatial responses are CME. An average reduction of ∼40 per cent about the root-mean-square (RMS) values was achieved in both PCA and ICA filtering. However, the common mode components obtained from the two approaches have different spatial and temporal features. ICA time-series present interesting correlations with modelled atmospheric and non-tidal ocean loading displacements. A white noise plus power law (PL) noise model was adopted in the GPS velocity estimation using maximum likelihood estimation analysis, with ∼55 per cent reduction of the velocity uncertainties after filtering using ICA. Meanwhile, spatiotemporal filtering reduces the amplitude of PL and periodic terms in the GPS time-series. Finally, we compare the GPS uplift velocities, after correction for elastic effects, with recent models of glacial isostatic adjustment (GIA). The agreements of the GPS observed velocities and four GIA models are generally improved after the spatiotemporal filtering, with a mean reduction of ∼0.9 mm yr−1 of the weighted RMS values, possibly allowing for more confident separation of various GIA model predictions.

INTER-ANNUAL HEIGHT VARIATIONS OBSERVED BY GPS MEASUREMENTS IN TURKEY

Abstract. Turkey is located in Africa-Eurasia-Saudi Arabian plate converged areas with surrounding by the Black Sea and the Mediterranean Sea. Apart from its tectonic conditions, the climate is complexly varying according to regions. As an example while in the northern regions floods often endanger human life, and in the middle regions, drought is a serious situation. To understand the Earth system and its temporal changes, a reliable measurement of various geophysical processes and mass redistribution is needed, which are mostly related to the regional water cycle and coupling processes associated with a mass exchange between the oceans, the lands and the atmosphere at seasonal and inter-annual timescales. Nowadays, dense and continuous GPS (Global Positioning System) observations provide direct measurements to capture such signals. In this paper, continuous GPS coordinate time series and inter-annual height variations are obtained and investigated from Turkish CORS network called TUSAGA-Active with more than 140 stations from 01 Jan. 2010 to 01 Jan. 2016 processed by GAMIT/GLOBK software. Results show significant inter-annual variations of GPS height time series with a period of about 2.8 years at most GPS stations. Furthermore, some relationship between inter-annual height and rainfall as well as ENSO index will be further investigated, indicating that inter-annual height variations are mostly related to climate changes, such as drought. These results will contribute to understand continuous GPS measurement signals in Turkey as well as applications in near-real-time geohazards estimations.

Differences in GPS coordinate time series resulting from the use of individual instead of type-mean antenna phase center calibration model

It is well-known that the phase center of a Global Navigation Satellite System (GNSS) antenna is not a stable point coinciding with a mechanical reference. The phase center position depends on the direction of the received signal, and is antenna-and signaldependent. Phase center corrections (PCC) models of GNSS antennas have been available for several years. The first method to create antenna PCC models was the relative field calibration procedure. Currently only absolute calibration models are generally recommended for use. In this study we investigate the differences between position estimates obtained using individual and type-mean absolute antenna calibrations in order to better understand how receiver antenna calibration models contribute to the Global Positioning System (GPS) positioning error budget. The station positions were estimated with two absolute calibration models: the igs08.atx model, which contains typemean calibration results, and individual antenna calibration models. Continuous GPS observations from selected Polish European Permanent Network (EPN) stations were used for these studies. The position time series were derived from the precise point positioning (PPP) technique using the NAPEOS scientific GNSS software package. The results show that the differences in the calibrations models propagate directly into the position domain, affecting daily as well sub-daily results. In daily solutions, the position offsets, resulting from the use of individual calibrations instead of type-mean igs08.atx calibrations, can reach up to 5 mm in the Up component, while in the horizontal one they generally stay below 1 mm. It was found that increasing the frequency of sub-daily coordinate solutions amplifies the effects of type-mean vs individual PCC-dependent differences, and also gives visible periodic variations in time series of GPS position differences.

Study of the effects on GPS coordinate time series caused by higher-order ionospheric corrections calculated using the DIPOLE model

As one of the main error sources in high-precision Global Positioning System (GPS) data processing, higher-order ionospheric (HOI) delays cause significant effects on coordinate time series that cannot be ignored in analyses of long time series. Typically two geomagnetic models, DIPOLE model and International Geomagnetic Reference Field (IGRF) model, are used for calculating HOI corrections. This paper investigates the effects of HOI correction caused by the DIPOLE model on coordinate time series. GPS data from 104 globally distributed International GNSS Service (IGS) stations spanning from January, 1999 to December, 2003 were reprocessed following up-to-date processing strategies utilizing GAMIT and GLOBK software. Two coordinate time series solutions before and after applying HOI corrections using the DIPOLE model were derived for studying the effects in terms of seasonal variations and noise amplitudes. The results show that after applying the HOI corrections calculated with DIPOLE, the noise amplitudes of the coordinate time series increased, especially in the north and east directions, and the increased amplitudes of the flicker noise were larger than those of the white noise. Furthermore, spurious periodic signals that were probably introduced by the HOI corrections from the DIPOLE model were also found. Moreover, an apparent increase was confirmed for the power spectra of most of the stations, especially in the north direction, and the amplitudes of both the annual and semi-annual signals also increased in the north and east directions. It can be inferred that the quality of the external data sources such as the geomagnetic model might be the key factors that lead to the above results. The results also suggest that we should be very careful when the DIPOLE model is used for HOI corrections.

Reconstruction of gappy GPS coordinate time series using empirical orthogonal functions

AbstractData gaps are an unfortunate aspect of Global Positioning System (GPS) time series and that can (a) slow down the maximum likelihood estimation for stochastic noise modeling and (b) hamper signal processing algorithms that require evenly spaced data. Simply filling the data gaps by linear interpolation does not significantly bias the velocity estimates in the least squares analysis, whereas it may be less suitable for periodic term estimates for vertical GPS observations especially those that are sampled with wide gaps. We use iterative empirical orthogonal function‐based (EOF‐based) method to recover the gappy GPS coordinate time series and discuss their effects on periodic term estimates. Ordinary least squares estimation‐based (LSE‐based) interpolation (including trend, annual, and semiannual terms) is performed for comparison. We demonstrate our method with synthetic examples, as well as real daily coordinate time series of 10 GPS sites with more than 14 years of data. The optimal window size and number of dominant EOFs are determined using crossvalidation. We find that EOF‐based interpolation may bias amplitude estimates in the up component owing to the artificial signals introduced in the EOFs reconstruction. After skipping the “noisy” EOFs and including only EOFs that we are particularly interested, the bias becomes smaller. Moreover, EOF‐based interpolation outperforms LSE interpolation in the wavelet analysis of GPS time series particularly for situations with wide gaps. Offsets are also found to be essentially nuisance parameters that must be removed prior to applying interpolation.

Effect of the span of Australian GPS coordinate time series in establishing an optimal noise model

The span of coordinate time series affects the determination of an optimal noise model. We analyzed position data recorded for 10 continuous Global Positioning System (GPS) sites from 1998.0 to mid-2009 on the Australian Plate to estimate the best noise model and thereafter obtain the true uncertainties of the velocity, employing the maximum likelihood estimation (MLE) method. MLE was employed to analyze the data in four ways. In the first two analyses, the noise was assumed to be a combination of flicker noise and white noise for the raw time series and spatially filtered time series. In the final two analyses, the spectral indices and amplitudes were simultaneously estimated for a power law noise plus white noise model for the raw time series and spatially filtered time series. We conclude that the noise model of GPS time series in Australia can be best described as the combination of flicker noise and white noise. Velocity uncertainties fall below ∼0.2 mm/yr when the time span exceeds ∼9.5 years. A comparison of noise amplitudes and maximum likelihood estimation values between the raw and spatially filtered time series suggests that traditional spatial filtering to remove common-mode errors might not be applicable to the raw time series of this region.

Estimates of Minor Ocean Tide Loading Displacement and Its Impact on Continuous GPS Coordinate Time Series

The site displacement due to ocean tidal loading is regarded as one of the largest uncertainties in precise geodetic positioning measurements, among which the effect of minor ocean tides (MOT), except for the 11 main tidal constituents, are sometimes neglected in routine precise global positioning system (GPS) data processing. We find that MOT can cause large vertical loading displacements with peak-to-peak variations reaching more than 8 mm at coastal/island stations. The impact of MOT on the 24-hour GPS solution is slightly larger than the magnitude of MOT loading itself, with peak-to-peak displacement variation at about 10 mm for the horizontal and 30 mm for the vertical components. We also find that the vertical velocity of all the selected stations in the Southwest Pacific was reduced by more than 10% after considering the MOT effect, while stations with weighted root mean square reduced data account for 62%, 59%, and 36% for the up, east, and north components respectively, in particular for most coastal/island stations. Furthermore, MOT correction could significantly reduce the annual signal of the global stacked east component, the near fortnightly and the long-term periodic signals in the up component. The power of some anomalous harmonics of 1.04 cycle per year is also decreased to some extent. These results further proved the benefits of MOT correction in precise GPS data processing.

Effects on noise properties of GPS time series caused by higher-order ionospheric corrections

Higher-order ionospheric (HOI) effects are one of the principal technique-specific error sources in precise global positioning system (GPS) analysis. These effects also influence the non-linear characteristics of GPS coordinate time series. In this paper, we investigate these effects on coordinate time series in terms of seasonal variations and noise amplitudes. Both power spectral techniques and maximum likelihood estimators (MLE) are used to evaluate these effects quantitatively and qualitatively. Our results show an overall improvement for the analysis of global sites if HOI effects are considered. We note that the noise spectral index that is used for the determination of the optimal noise models in our analysis ranged between −1 and 0 both with and without HOI corrections, implying that the coloured noise cannot be removed by these corrections. However, the corrections were found to have improved noise properties for global sites. After the corrections were applied, the noise amplitudes at most sites decreased, among which the white noise amplitudes decreased remarkably. The white noise amplitudes of up to 81.8% of the selected sites decreased in the up component, and the flicker noise of 67.5% of the sites decreased in the north component. Stacked periodogram results show that, no matter whether the HOI effects are considered or not, a common fundamental period of 1.04 cycles per year (cpy), together with the expected annual and semi-annual signals, can explain all peaks of the north and up components well. For the east component, however, reasonable results can be obtained only based on HOI corrections. HOI corrections are useful for better detecting the periodic signals in GPS coordinate time series. Moreover, the corrections contributed partly to the seasonal variations of the selected sites, especially for the up component. Statistically, HOI corrections reduced more than 50% and more than 65% of the annual and semi-annual amplitudes respectively at the selected sites.

Noise analysis of continuous GPS coordinate time series for CMONOC

The Crustal Movement Observation Network of China (CMONOC) is one of the major scientific infrastructures, mainly using Global Positioning System (GPS) measurements, to monitor crustal deformation in the Chinese mainland. In this paper, decade-long coordinate time series of 26 continuous GPS sites of CMONOC are analyzed for their noise content using maximum likelihood estimation (MLE). We study the noise properties of continuous GPS time series of CMONOC for the unfiltered, filtered solutions and also the common mode signals in terms of power law plus white noise model. In the spatial filtering, we remove for every time series a common mode error that was estimated from a modified stacking of position residuals from other sites within ∼1000km of the selected site. We find that the common mode signal in our network has a combination of spatially correlated flicker noise and a common white noise with large spatial extent. We demonstrate that for the unfiltered solutions of CMONOC continuous GPS sites the main colored noise is a flicker process, with a mean spectral index of ∼1. For the filtered solutions, the main colored noise is a general power law process, indicating that a major number of the filtered regional solutions have a combination of noise sources or local effects. The velocity uncertainties from CMONOC continuous GPS coordinate time series may be underestimated by factors of 8–16 if a pure white noise model is assumed. In addition, using a white plus flicker noise model, the median values of velocity errors for the unfiltered solutions are 0.16 (north), 0.17 (east) and 0.58 (vertical)mm/yr, and the median values for the filtered solutions are 0.09 (north), 0.10 (east) and 0.40 (vertical)mm/yr.

Long GPS coordinate time series: Multipath and geometry effects

Within analyses of Global Positioning System (GPS) observations, unmodeled subdaily signals propagate into long‐period signals via a number of different mechanisms. In this paper, we investigate the effects of time‐variable satellite geometry and the propagation of a time‐constant unmodeled multipath signal. Multipath reflectors at H = 0.1 m, 0.2 m, and 1.5 m below the antenna are modeled, and their effects on GPS coordinate time series are examined. Simulated time series at 20 global IGS sites for 2000.0–2008.0 were derived using the satellite geometry as defined by daily broadcast orbits. We observe the introduction of time‐variable biases in the time series of up to several millimeters. The frequency and magnitude of the signal is dependent on site location and multipath source. When adopting realistic GPS observation geometries obtained from real data (e.g., including the influence of local obstructions and hardware specific tracking), we observe generally larger levels of coordinate variation. In these cases, we observe spurious signals across the frequency domain, including very high frequency abrupt changes (offsets) in addition to secular trends. Velocity biases of more than 0.5 mm/yr are evident at some sites. The propagated signal has noise characteristics that fall between flicker and random walk and shows spectral peaks at harmonics of the draconitic year for a GPS satellite (∼351 days). When a perfectly repeating synthetic constellation is used, the simulations show near‐negligible time correlated noise highlighting that subtle variations in the GPS constellation can propagate multipath signals differently over time, producing significant temporal variations in time series.

Assessment of noise in GPS coordinate time series: Methodology and results

We propose a methodology to assess the noise characteristics in time series of position estimates for permanent Global Positioning System (GPS) stations. Least squares variance component estimation (LS‐VCE) is adopted to cope with any type of noise in the data. LS‐VCE inherently provides the precision of (co)variance estimators. One can also apply statistical hypothesis testing in conjunction with LS‐VCE. Using the w‐test statistic, a combination of white noise and flicker noise turns out in general to best characterize the noise in all three position components. An interpretation for the colored noise of the series is given. Unmodelled periodic effects in the data will be captured by a set of harmonic functions for which we rely on the least squares harmonic estimation (LS‐HE) method and parameter significance testing developed in the same framework as LS‐VCE. Having included harmonic functions into the model, practically only white noise can be shown to remain in the data. Remaining time correlation, present only at very high frequencies (spanning a few days only), is expressed as a first‐order autoregressive noise process. It can be caused by common and well‐known sources of errors like atmospheric effects as well as satellite orbit errors. The autoregressive noise should be included in the stochastic model to avoid the overestimation (upward bias) of power law noise. The results confirm the presence of annual and semiannual signals in the series. We observed also significant periodic patterns with periods of 350 days and its fractions 350/n, n = 2, …, 8 that resemble the repeat time of the GPS constellation. Neglecting these harmonic signals in the functional model can seriously overestimate the rate uncertainty.

Noise in GPS coordinate time series

We assess the noise characteristics in time series of daily position estimates for 23 globally distributed Global Positioning System (GPS) stations with 3 years of data, using spectral analysis and Maximum Likelihood Estimation. A combination of white noise and flicker noise appears to be the best model for the noise characteristics of all three position components. Both white and flicker noise amplitudes are smallest in the north component and largest in the vertical component. The white noise part of the vertical component is higher for tropical stations (±23° latitude) compared to midlatitude stations. Velocity error in a GPS coordinate time series may be underestimated by factors of 5–11 if a pure white noise model is assumed.