In order to estimate the impact of colored noise alone, we construct synthetic series using a subset of Eq. (1): xt=vt+Drandk,t. We first analyze the effect of the three parameters – the duration of the series (T), the spectral index (k) and the noise dispersion (D) – independently of the others. Figure 4 shows the velocity biases as a function of these three parameters. The worst values of velocity bias due to noise alone can reach v95=0.7 mm yr-1 for the shortest series (T<5 yr). For series longer than 15 years, all v95 are smaller than 0.1 mm yr-1. A near-exponential decrease in v95 is observed as a function of the duration of the series with a sharp slowdown from 15 years of data. The dependence of the velocity biases on noise parameters (k and D) shows an expected bias increase with smaller spectral indices (closer to -1) and higher noise amplitudes, with a near-exponential increase with D. Overall, the probability of velocity biases equal to or smaller than 0.1 mm yr-1 is p01=86 %. The 14 % of series with biases larger than 0.1 mm yr-1 is associated with the shortest and noisiest series.

A joint analysis of the parameters using a regression tree indicates their relative importance, with the most important being the series duration T (56 %) followed by the spectral index k (35 %) and the noise dispersion D (9 %). Figure 5 shows the tree classification (Fig. 5a) and the whisker plots of the associated leaves (Fig. 5b). The branches and the associated leaves are ordered in order of importance and leaf size from left to right. The comparison signs (>  <) or (<  >) are relative to each tree separation, with the sign on the left corresponding to the left branch and the sign on the right corresponding to the right branch. Hereafter, we limit the tree classification to three node levels in order to only highlight the primary controlling elements.

The tree classification shows that v95=0.1 mm yr-1 is achieved for over two-thirds of the series (leaves 1 and 2), corresponding to all the long series (T > 11.0 yr, leaves 1 and 2) and those with average durations and large spectral indices (6.1 < T < 11.0 yr, k > -0.6, Leaf 1). The overall velocity bias increases for the other leaves. v95=0.2 mm yr-1 is still reached for combinations of average durations and small spectral indices (6.1 < T < 11.0 yr, k < -0.6, Leaf 3) or short durations, large spectral indices, and low-noise amplitude (T < 6.1 yr, k > -0.7, D < 2.6 mm, Leaf 4). The remaining cases (short duration, small spectral index, high noise) represent less than 10 % of the samples and result in large biases with v95=0.4 mm yr-1 (Leaf 5) and v95=0.7 mm yr-1 (Leaf 6).

Additionally, a significant piece of information emerging from the regression tree analysis is the relatively low coefficient of determination R2∼0.5, which indicates that the combinations of the three model parameters (T, k, D) only explain about 50 % of the dispersion in velocity biases. This points out the strong effect of the stochastic noise generation, which alone accounts for about half of the velocity variability. In other words, for a given set of parameters, the generated time series will show variable characteristics (noise structures) that randomly impact the velocity estimations. We illustrate this point by estimating the dispersion of velocity biases for a sample of 300 series with constant parameters T=10 yr, k=-0.7, D=3.0 mm (belonging to Leaf 3 of the tree). The estimated velocities show an rms dispersion of 0.2 mm yr-1, of the same order as dispersion observed in Leaf 3 (Fig. 5b). This effect is more important if the series is short.

As noted in the introduction to Sect. 3, seasonal signals have very little effect on the velocity estimations. This is also true for seasonal signals added to series with random noise, which yield similar results to those presented above for noise alone (e.g., p01=86 %), with the seasonal parameters (A1/2 combinations) ranking with negligible importance in the tree classification (less than 1 %).

In order to test and estimate the effect of position offsets on velocity estimations, we analyze synthetic time series that include offsets added to seasonal signal and random noise (Eq. 1). This choice is justified by the very low effect of offsets alone (cf. introduction of Sect. 3) and the fact that this combination is representative of real data, thus providing useful estimations of the expected precision of actual velocities. In the case of real data, dealing with offsets requires either fixing their dates (from equipment logs or earthquake catalogs) or detecting their potential occurrences. In Sect. 4, we will come back to how to consider the latter. In this section, we quantify the two end-member cases in which we either do not know and therefore do not solve any offset or we know and solve all of them.

Real GPS time series are associated with an indeterminate number of offsets, which are classically included as instantaneous changes in position in the series inversion (cf. Eq. 1). Offset dates ti can be based on equipment logs, catalogs of earthquakes, or routines that detect them in the position series (Gazeaux et al., 2013). Here we propose a slightly different approach that does not consist of seeking where there are offsets but rather of seeking where there are none.

This simple principle is implemented by defining artificial offset dates that are regularly spaced in the series every Δd days. The series is then inverted to estimate all offset amplitudes (Ci) and their associated standard errors (σci) jointly with the other model parameters (velocity, seasonal signal, etc.). The offset with the smallest amplitude (CS) is then identified and a simple significance test is performed: CS≥b⋅σcs. If the amplitude (CS) is larger than its scaled standard error (b⋅σcs), the offset is considered significant. Because the test is performed on the smallest offset and the offset standard errors are similar in the majority of cases, we then consider that all offsets are significant and we keep them in the model. In the opposite case, the smallest offset is rejected and the inversion is redone with the remaining offsets in order to test the new smallest offset, until a significant offset is found or none remains.

This very simple approach can be implemented in most time series analysis and only requires an empirical calibration of the two parameters Δd and b. After several tests, we set the former to Δd=20 days, which corresponds to the lower limit before the method breaks down (i.e., too many undifferentiated offsets). The latter is set to b=20, which allows a good compromise between the detection of real offsets defined in the synthetic series and the detection of false positives (cf. Sect. 4.2).

This empirical calibration is not possible on real data, but considering that our synthetic data are representative of real data with the previous cautions, we can use this parameterization. In using Δd=20 days, all possible epochs are not tested. The assumption is that a real offset at any given epoch will be caught by the forced artificial offset located less than 10 days directly before or after. As such, we do not find the exact date of the real offset but its approximate date (within 10 days). This method cannot resolve real offsets situated within a few (10–20) days of each other. They will be lumped into a single artificial offset, but we assume that its effect on the estimated velocity will be a good proxy of the combined effect of the real offsets. This method is developed as a simple and efficient way to test the impact of offsets and their resolution on the velocity estimations. Several things could be done in future studies to improve it, including a finer calibration of the parameters, taking into account consecutive offsets, and an exhaustive scan of all epochs. Details on the parameter calibration and the detection levels are available in Supplement Sect. S2.

By applying our method to series with only one offset, it is possible to determine the conditions of offset detections. Overall, 67 % of the offsets are detected. The detection capacity depends primarily on the duration of the time series T, combined with the series noise amplitude D and the offset amplitude C. For the shortest time series (T < 6 yr), we detect 21 % of offsets. They correspond to the series with the largest offsets (C > 3.0 mm) and the smallest noise amplitudes (D < 2.1 mm). There is no offset detection in the series with large noise amplitude (D > 2.1 mm). For the time series of 6 to 18 years, we can detect offsets of small amplitudes (C=1.0–3.0 mm) in series with low noise levels (D < 2.1 mm) and large offsets (C > 4.0 mm) in all series. For the longest time series of more than 18 years, one widens the range of detection still further. Offsets larger than 3.0 mm are systematically detected and those between 2.0 and 3.0 mm are detected at 49 %. The very small offsets (C < 2.0 mm) are detected only in the low-noise series (D < 2.1 mm).

By applying our method to series with several offsets, the detection ability is decreased due to offset and noise interactions. Overall, the performance level is characterized by ca. 52 % of true detections (and so 48 % of missed detections) of the theoretical total number of offsets and about 20 % of false positives (cf. Supplement Sect. S2 for detection calibrations). These statistics are similar or slightly better than those of the most efficient automatic and manual detection methods analyzed in Gazeaux et al. (2013). Although not perfect, our method allows us to obtain robust and quantitative results and is suitable for processing of very large datasets such as our synthetic series or regional and global massive processing efforts that become increasingly common (e.g., Kreemer et al., 2014) and that could not be analyzed “by hand”.

The application of the offset detection method to a full dataset with multiple offsets, variable noise, and seasonal signals provides a sample that can be considered as close as possible to actual GPS data. We use this analysis to provide constraints on the potential velocity precision in real data. Overall, nearly two-thirds of series are associated with velocity bias smaller than 0.1 mm yr-1 (p01=61 %). This is lower than in the cases of noise alone (p01=86 %) or fully resolved offsets (p01=71 %) but significantly better than in the case of unresolved offsets (p01=33 %). The difference between the results of the offset detection method and those of the fully resolved offsets (ca. 10 %) is mainly associated with undetected offsets in the former.

For the regression tree analysis, the integration of a parameter associated with offsets is complex. Although these parameters (numbers total of offsets, of true and false detections, positions in the series, amplitudes) are known in our synthetic data, this is not the case in real datasets. Tests on several offset parameters indicate that the total number of offsets in the series (Noff) is both the simplest and the one with the highest prediction capacity. This new regression tree (Fig. 7) confirms the major role of the series duration (T 55 %) and noise dispersion (D 16 %) in explaining the variability in the velocities, but the total number of offsets now take the second position (Noff 25 %), above the noise dispersion. It is particularly worth noting that the number of offsets is in fact a binary predicator (splitting value Noff=0.5) corresponding to either the absence (Noff=0) or the presence (Noff≥1) of offsets in the series.

To first order, the regression tree results can be divided into three categories:

The lowest velocity biases (v95∼0.2–0.3 mm yr-1) are associated with either long (T > 8.0 yr) and low-noise dispersion (D < 2.3 mm) series or with series of intermediate duration (4.5 < T < 8.0 yr) with no offset (leaves 1 and 3). These represent over 42 % of the dataset.

On these bases, a fairly simple set of rules can be derived from the regression tree analysis that may be applicable to actual GPS data used for high-precision (sub mm yr-1) studies, considering the fact that series duration is the key parameter:

Duration of 8.0 years or more ensures a low-velocity bias in both horizontal (v95=0.2 mm yr-1) and vertical (v95=0.5 mm yr-1) components.

Finally, it is significant that no tree node exists that distinguishes very long series. In other words, the effect of the series duration is limited to ca. 4.5 and 8.0 yr. This is consistent with the observation made in the noise-alone analysis that the decay of the noise effect as a function of time stagnates ca. 15 to 21 years (cf. Fig. 4 and Sect. 3.1). Our results may indicate an overall lower limit on the velocity bias of ca. 0.1 mm yr-1 due to the colored nature of the time series noise. In other words, longer series may not be able to significantly reduce the velocity bias without additional efforts to whiten the noise through better data processing or taking into account pluri-annual signals. However, this hypothesis is only valid under the simple noise model (linear spectra, Eq. 2) used in our synthetic data. Alternative noise models exist that suggest a flattening of the spectra at long periods (e.g., Gauss–Markov model, Langbein et al., 2004), which would strongly limit the pluri-annual effect and allow a much stronger impact of long series duration. The actual nature of GPS noise at periods longer than 5–10 years is poorly defined (Santamaria-Gomez et al., 2011; Hackl et al., 2011) and is thus a major unknown in analyses of velocity precision.

For each series, the velocity standard error is calculated using the Williams (2003) generic expression for colored noise with a non-integer spectral index. In order to estimate the spectral index and amplitude of the colored noise, we use a simplified least-square inversion in which we fit a linear model to the series power spectrum limited to periods between 1/12 and T/2 years (with T the length of the time series). In contrast with a more complex nonlinear method, such as maximum likelihood, this simple approach does not solve for the noise crossover frequency and thus only provides a first-order estimate of the noise parameters and velocity standard errors.

We can test the robustness of these standard errors in comparison with their associated velocity biases by computing the ratio of the velocity bias to its standard error for each individual time series. A ratio of 1 corresponds to a standard error equal to its velocity bias; a ratio smaller (greater) than 1 corresponds to a standard error greater (smaller) than its velocity bias. Owing to our stochastic approach and assuming Gaussian distributions of the velocities and standard errors, appropriate standard error calculations should result in ca. 68 % of the ratio population smaller than 1 (i.e., 68 % of the velocity biases are included in their standard errors) and ca. 95 % of the population smaller than 2 (i.e., 95 % of the velocity biases are included in twice their standard errors). In our dataset, only 54 % of the ratio are smaller than 1 and 75 % are smaller than 2 (Fig. 8). These percentages are low and suggest that, on average, our velocity standard errors are too small by a factor of ca. 1.6.

This result is primarily controlled by the series spectral index, while the series duration and dispersion have little effect (Fig. 8). Series with indices ca. -0.6 > k > -0.9 are associated with ratio percentages close to the 68 and 95 % marks. In contrast, series with high indices (k > -0.6) present ratios that are too low especially for very high indices (k > -0.4). These results suggest that the simplified (linear spectra) approach yields reasonable results for series with near-flicker (k < -0.6) noise characteristics but significantly underestimates the standard errors for series with near-white (k > -0.4) noise.

The RENAG network comprises 74 stations whose equipment modifications are fully documented (cf. http://webrenag.unice.fr, last access: 29 March 2018), thus providing a good test case for our offset detection method. On the 222 time series with durations between 2.0 and 18.4 years, the comparison of detected offsets with the station logs show that a change in receiver is very rarely associated with an offset (only 6 % of the 137 cases), whereas a change in antenna causes an offset almost systematically (75 % of the 8 cases) with average amplitudes of 2.0–3.0 mm in the horizontal and ca. 13.0 mm in the vertical components. However, these percentages are not robust due to the small sample sizes (especially the antenna changes). A more robust analysis would require a larger dataset, as well as the distinction between equipment changes within large data gaps or near the ends of the time series. Additionally, the offset detection method could be improved to integrate the probability that an offsets occurs on all three components of the same station rather than individually as it is currently done.

The time series data of the 74 RENAG stations come from a Precise Point Positioning solution, combined with noise reduction using a regional common-mode technique (Masson et al., 2018; Nguyen et al., 2016). The time series of each station position component (north, east, up) are treated independently. We consider that the number of detected offsets is similar to the total number of offsets (Noff parameter in the regression tree), assuming that undetected offsets have small amplitudes and a small impact on the velocity estimations. This hypothesis is problematic for short series where the detection capacity is low (cf. Sect. 4.2) and for which it is likely that offsets were not detected, leading to a misclassification of series in Leaf 6.

Figure 9 shows a map of the RENAG stations with the v95 value associated with each component according to the tree leaves. Roughly half (53 %) of the 74 stations are associated with the highest precisions in the horizontal (north and east, v95=0.2 mm yr-1) and vertical (v95=0.5 mm yr-1) components. In a few cases (12 %), the east component is degraded to a slightly larger precision v95=0.5 mm yr-1. About one-third (30 %) of the stations correspond to cases with no detected offsets and identical precision in all three components, either v95=0.3 mm yr-1 or v95=0.6 mm yr-1 depending on the duration of the time series.

Recent studies of GPS data in western Europe have shown tectonic signals at the limit of GPS resolution. The most significant signal corresponds to a systematic uplift of 1.0–2.0 mm yr-1 in the central and northern regions of the Western Alps (Nguyen et al., 2016; Nocquet et al., 2016). The pattern of uplift and its lateral variations can provide important information on the associated dynamic (e.g., postglacial rebound versus slab tear; Chéry et al., 2016; Nocquet et al., 2016). Our analysis suggests that the 95 % confidence level of the RENAG velocities in the Alps is ca. 0.5 mm yr-1, which may still be too large to provide strong constraints on the dynamic processes. In parallel with the vertical signal, horizontal deformation is starting to emerge in the GPS data analysis that show radial extension rates ca. 0.2–0.5 mm yr-1 in the Western Alps and Pyrenees (Nguyen et al., 2016; Rigo et al., 2015; Walpersdorf et al., 2018). Such rates are at the limit of the 95 % confidence level estimated for individual RENAG stations (Fig. 9). This is especially true of stations in the French Jura, which show a relatively low precision v95≈0.6 mm yr-1 due to their recent installation and short time series (T < 3.5 yr). These examples highlight the importance of network redundancy and high station density in order to strengthen the deformation analysis by relying on several nearby stations to reduce aleatory noise in individual GPS time series.