Each of the GPS-derived time series consists of the deterministic (functional) and stochastic part. We propose that the deterministic part includes all periodicities from 1st to 9th harmonics of residual Chandler, tropical and draconitic periods and compare it with commonly used calculations of the annual and semi-annual tropical curve. Then, we address the issues of whether all residual periodicities, as proposed here, need to be taken into consideration when performing noise analysis. We use the position time series from 180 International GNSS Service stations obtained at the Jet Propulsion Laboratory using the GIPSY-OASIS software in a Precise Point Positioning mode. The longest series has 22.1 years of GPS daily solutions. The spectral indices range from –0.12 to –0.92, while the median values of “global” spectral indices are equal to: –0.41 ± 0.15, –0.38 ± 0.12 and –0.33 ± 0.18 for North, East and Up components, respectively. All non-modelled geophysical processes or non-included artificial effects in time series lead to an underestimation of errors of velocities, but also to changes in the velocity values themselves. The proposed assumption of seasonals subtraction caused the Akaike information criterion values to show a decrease in the median value of 30 %, which in fact means that all the seasonals mentioned here must be taken into account when analyzing noises. Finally, we noticed that there are some of the GPS stations that improved their velocity uncertainty even of 56 %.

The velocity estimates and their uncertainties derived from position time series of Global Navigation Satellite System stations are affected by seasonal signals and their harmonics, and the statistical properties, i.e., the stochastic noise, contained in the series. If the deterministic model in the form of linear trend and periodic terms is not accurate enough to describe the time series, it will alter the stochastic model, and the resulting effect on the velocity uncertainties can be perceived as a result of a misfit of the deterministic model. The effects of insufficiently modeled seasonal signals will propagate into the stochastic model and falsify the results of the noise analysis, in addition to velocity estimates and their uncertainties. We provide the general dilution of precision (GDP) of velocity uncertainties as the ratio of uncertainties of velocities determined from to two different deterministic models while accounting for stochastic noise at the same time. In this newly defined GDP, the first deterministic model includes a linear trend, while the second one includes a linear trend and seasonal signals. These two are tested with the assumption of white noise only as well as the combinations of power-law and white noise in the data. The more seasonal terms are added to the series, the more biased the velocity uncertainties become. With increasing time span of observations, the assumption of seasonal signals becomes less important, and the power-law character of the residuals starts to play a crucial role in the determined velocity uncertainties. With reference frame and sea level applications in mind, we argue that 7 and 9 years of continuous observations is the threshold for white and flicker noise, respectively, while 17 years are required for random-walk to decrease GDP below 5% and to omit periodic oscillations in the GNSS-derived time series taking only the noise model into consideration.

The coordinate time series determined with the Global Positioning System (GPS) contain annual and semi-annual periods that are routinely modeled by two periodic signals with constant amplitude and phase-lag. However, the amplitude and phase-lag of the seasonal signals vary slightly over time. Various methods have been proposed to model these variations such as Wavelet Decomposition (WD), writing the amplitude of the seasonal signal as a Chebyshev polynomial that is a function of time (CP), Singular Spectrum Analysis (SSA), and using a Kalman Filter (KF). Using synthetic time series, we investigate the ability of each method to capture the time-varying seasonal signal in time series with different noise levels. We demonstrate that the precision by which the varying seasonal signal can be estimated depends on the ratio of the variations in the seasonal signal to the noise level. For most GPS time series, this ratio is between 0.05 and 0.1. Within this range, the WD and CP have the most trouble in separating the seasonal signal from the noise. The most precise estimates of the variations are given by the SSA and KF methods. For real GPS data, SSA and KF can model 49–84 and 77–90% of the variance of the true varying seasonal signal, respectively.

We reassess the absolute and relative sea level changes at 38 tide gauge stations in the earthquake-affected Western North Pacific for the 1993–2015 period, focusing on the vertical land motion (VLM) which is crucial for narrowing the gap between these estimates. In this area, simply discarding all earthquake-affected sites, one overestimates the average regional sea level rise by more than 0.5 mm/year. Disregarding VLM would lead to misestimating local sea level trends between 0.2 and 7.6 mm/year. If accounted for, but modeled as linear during the entire time span, VLM leads to regional absolute sea level rise errors of up to 0.4 mm/year. Therefore, we introduce a new methodology that better represents the Global Positioning System (GPS)-derived nonlinear VLM by accounting for co-seismic offsets, changes in the vertical velocities and post-seismic transient. Also, for the first time, a combination of white and power-law noises is added to this nonlinear model to derive proper uncertainties of VLM. We find a maximum difference of 15.3 mm/year between pre- and post-seismic vertical velocities. The GPS-sensed vertical co-seismic displacement approaches 36 mm. Assuming the changes in vertical velocities and displacement due to the tectonic movements is not accounted for, and then, estimating absolute sea level rise from tide gauges can result in an error of 10 mm/year. Introducing a new nonlinear VLM model improves absolute tide gauge sea level estimates by 20% on average. Finally, for the reconstructed Western North Pacific sea level, altimetry agrees best with tide gauge data corrected employing the new nonlinear VLM model.

We examine the sensitivity of the Global Positioning System (GPS) to non-tidal loading for a set of continental Eurasia permanent stations. We utilized daily vertical displacements available from the Nevada Geodetic Laboratory (NGL) at stations located at least 100 km away from the coast. Loading-induced predictions of displacements of earth’s crust are provided by the Earth-System-Modeling Group of the GFZ (ESMGFZ). We demonstrate that the hydrological loading, supported by barystatic sea-level changes to close the global mass budget (HYDL + SLEL), contributes to GPS displacements only in the seasonal band. Non-tidal atmospheric loading, supported by non-tidal oceanic loading (NTAL + NTOL), correlates positively with GPS displacements for almost all time resolutions, including non-seasonal changes from 2 days to 5 months, which are often considered as noise, intra-seasonal and seasonal changes with periods between 4 months and 1.4 years, and, also, inter-annual signals between 1.1 and 3.0 years. Correcting the GPS vertical displacements by NTAL leads to a reduction in the time series variances, evoking a whitening of the GPS stochastic character and a decrease in the standard deviation of noise. Both lead, on average, to an improvement in the uncertainty of the GPS vertical velocity by a factor of 2. To reduce its impact on the GPS displacement time series, we recommend that NTAL is applied at the observation level during the processing of GPS observations. HYDL might be corrected at the observation level or remain in the data and be applied at the stage of time series analysis.

Long series of Zenith Wet Delay (ZWD) obtained as part of a homogeneous re-processing of Global Positioning System solutions constitute a reliable set of data to be assimilated into climate models. The correct stochastic properties, i.e. the noise model of these data, have to be identified to assess the real value of ZWD trend uncertainties since assuming an inappropriate noise model may lead to over- or underestimated error bounds leading to statistically insignificant trends. We present the ZWD time series for 1995–2017 for 120 selected globally distributed stations. The deterministic model in the form of a trend and significant seasonal signals were removed prior to the noise analysis. We examined different stochastic models and compared them to widely assumed white noise (WN). A combination of the autoregressive process of first-order plus WN (AR(1) + WN) was proven to be the preferred stochastic representation of the ZWD time series over the generally assumed white-noise-only approach. We found that for 103 out of 120 considered stations, the AR(1) process contributed to the AR(1) + WN model in more than 50% with noise amplitudes between 9 and 68 mm. As soon as the AR(1) + WN model was employed, 43 trend estimates became statistically insignificant, compared to 5 insignificant trend estimates for a white-noise-only model. We also found that the ZWD trend uncertainty may be underestimated by 5–14 times with median value of 8 using the white-noise-only assumption. Therefore, we recommend that AR(1) + WN model is employed before tropospheric trends are to be determined with the greatest reliability.