Keywords

1 Introduction

GNSS velocities estimated by different groups differ due to several factors including the different strategies to compute the station positions from the raw GNSS observations, the completeness of the time series, their level of noise, the removed position discontinuities, and the alignment to a terrestrial reference frame. Among these factors, the detection and handling of position discontinuities that populate the GNSS time series has probably the biggest impact on the velocity estimates (Williams 2003; Griffiths and Ray 2016). Even when using the same GNSS position time series, it is common for different analysts to provide different velocity estimates and uncertainties, mainly due to the different choices of handling position discontinuities (Gazeaux et al. 2013).

Position discontinuities can be classified in four groups. The first group includes apparent position discontinuities generated when the GNSS antennas are replaced. They are an indication that the phase center corrections obtained from antenna calibrations do not reflect in general the actual antenna radiation pattern on the field, creating an apparent position change when antennas are replaced (Wanninger 2009). The second group includes a wide range of situations, often poorly understood, related to changes of the antenna environment, the signal tracking or the settings of the receiver’s frontend. The third group includes those situations where the position discontinuity is generated when estimating the station positions from the raw GNSS observations, for instance due to an incorrect use of the station’s metadata: outdated antenna calibration, wrong antenna model, wrong antenna orientation or wrong antenna eccentricity. Discontinuities from the previous three groups can introduce changes in the estimated station position that do not reflect any position change of the station’s benchmark. The last fourth group of discontinuities are generated by earthquakes, which can offset the position of the station’s benchmark if estimated in a conventional coordinate frame that is not affected by the earthquake itself. Not all the antenna changes, nor all the earthquakes, will introduce a significant position discontinuity. This fact, and the potential occurrences of discontinuities from the second and third groups, implies that different analysts will remove different sets of position discontinuities, even if they use exactly the same series (Gazeaux et al. 2013).

The International Association of Geodesy’s Joint Working Group 3.2 “Global combined GNSS velocity field” (2019–2023) was created to support the scientific community using GNSS velocities in fields such as tectonics, sea-level change, land subsidence and global isostatic adjustment (GIA) modelling. The objective of this JWG is to combine and compare the available global and regional GNSS velocity fields obtained by different groups from both network and precise point positioning (PPP) solutions.

This contribution presents the combined global GNSS velocity field, the methodology followed to combine the input velocity fields and the results of their comparison.

2 Input GNSS Velocity Fields

Nineteen GNSS velocity fields have been gathered and used in the combination (Table 1). The geographical distribution of each velocity field is shown in Figure 1 of the supplemental material.

Table 1 List of the input velocity fields with the number of sites retained in the combination, the geographical extension and the file format
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The input velocity fields considered for the combination include:

The inclusion of large global velocity fields in the combination, like NGL and JPL (see Table 1), helps making more robust the alignment of sparse regional velocity fields, like SIRGAS, by maximizing the number of sites used in the alignment to the ITRF2020. The number of common sites between each input velocity field is given in Table 1 of the supplemental material. Velocity fields with a geographical extension smaller than the continental scale were not considered in the combination due to limitations in their accurate alignment to a global reference frame (see Table 3 of the supplemental material).

The numbers of sites given in Table 1 correspond to the sites that were considered from each input velocity field. The sites retained for the combination have at least 5 years of data and a constant velocity, i.e., sites with velocity discontinuities were excluded. The threshold of 5 years of data was considered as a tradeoff between including the maximum number of sites and minimizing the secular velocity errors due to interannual deformation (Santamaría-Gómez and Mémin 2015). Other criteria to exclude sites were null velocity uncertainty or velocity uncertainty larger than 1 mm/yr, which should reject sites with strong non-linearity, excessive amount of discontinuities or substantial missing data.

The sites retained in each input velocity field were verified for duplicates, i.e., sites with the same 4-character IDs located at different places. Comparison of the coordinates of all the sites resulted in 223 duplicated sites being renamed (see Table 2 of the supplemental material).

After the verification of duplicated sites, the coordinates of each site were still slightly different across the input velocity fields, mainly due to the chosen reference epoch, which was unknown in most of the input velocity fields. The coordinates of the sites in each velocity field were replaced by a set of common coordinates. This way, the weighting of each velocity field in the combination process will only include the contribution of the velocity residuals.

A final preprocessing of the input velocity fields involved their conversion into SINEX format when necessary. Most of the input velocity fields are provided as ASCII tables in topocentric coordinates (see Table 1). These velocity fields were transformed into Cartesian coordinates while keeping the velocity covariances between the coordinates. Still in the case of velocity fields provided as ASCII tables, the velocity covariances between sites are unknown and were set to zero. For the velocity fields already provided in SINEX format, the covariances between sites were also set to zero to avoid down weighting these velocity fields.

3 Combination Method

The velocity field combination was carried out using the CATREF software (Altamimi et al. 2016) and involved four steps.

In the first step we attributed an a priori weight to each input velocity field, i.e., before any comparison of the velocity estimates. The approach followed to compute the velocity uncertainties varies among the input velocity fields, and they do not necessarily reflect realistic velocity errors in an absolute sense. They can nevertheless be assumed to reflect the relative proportions of the velocity errors of different sites within a given velocity field. Therefore, variance factors were computed to harmonize the uncertainties of the input velocity fields. The a priori variance factors were estimated with the starting assumption that all input velocity fields have the same average velocity error. Assuming an average velocity error of 0.1 mm/yr and 0.3 mm/yr for horizontal and vertical velocities respectively, the average 3D velocity error was set to \( \sqrt{0.1^{{}^2}+{0.1}^{{}^2}+{0.3}^{{}^2}}=\sqrt{0.11} \) mm/yr. The a priori variance factor of each input velocity field i was thus computed as:

$$ {VF}_i=\frac{0.11}{\sigma_{3{D}_i}^2} $$
(1)

where \( {\sigma}_{3{D}_i}^2 \) represents the average squared 3D velocity uncertainty of the velocity field i. Due to the different numbers and geographic distributions of sites among the input velocity fields, the average 3D uncertainty was obtained from a subset of common sites between the input velocity fields.

In the second step, the input velocity fields were iteratively combined using the variance factors from the first step. Drifts in the origin, orientation and scale of each input velocity field were estimated with respect to the ITRF2020 velocity field. After each iteration, sites with velocity residuals larger than 0.7 or 2 mm/yr for the horizontal and vertical velocities, respectively, were considered as outliers and removed from the next iteration. These thresholds were chosen as being 7 times the a priori average velocity error in the first step, so that only extreme velocity errors were removed. Only one site per velocity field and only one velocity estimate per site present in several velocity fields was removed at each iteration.

Once there were no more outliers to eliminate, in a third step, a posteriori variance factors were estimated based on the variance of the residuals of each velocity field. The outliers iteratively removed in the second step ensure the gaussianity of the velocity residuals, from which the a posteriori variance factors were obtained.

In the fourth and last step, a final iterative combination was carried out using the a posteriori variance factors from the third step to weight the input velocity fields. Due to the change of the weights of the input velocity fields, all the previously removed outliers were included and assessed again. Velocity outliers were removed using similar criteria as in the second step. For each iteration, we removed one velocity outlier per site, but only for those sites for which the outlier velocity could be identified unambiguously, i.e., sites having more than two velocity estimates available and a velocity residual that is significantly larger than the rest. This way, the combined velocity field includes sites for which the estimated velocity among the input velocity fields is ambiguous, which is reflected by a larger repeatability and uncertainty of the combined velocity. In addition, since changes of the weight of each input velocity field barely affects their estimated alignment, the alignment parameters (origin, orientation and scale drifts) of the velocity fields were held fixed to the ones obtained in the second step. This way, multiple velocity outliers could be removed from each input velocity field in the same iteration, which drastically reduced the number of iterations needed with no loss of rigor in the results. In total, 223 velocity outliers were removed, ranging between 0 and 55 outliers among the input velocity fields (see Table 7 of the supplemental material). The final combined velocity field was obtained from the last iteration once all the velocity outliers were removed.

4 Results of the Combination

Table 2 shows the square-roots of the a posteriori variance factors used to weight each input velocity field. Applying these (square-root) variance factors to each input velocity field allows comparing them in terms of their average 3D velocity uncertainty (also in Table 2), which is inversely equivalent to their relative weight in the combination. Most of the input velocity fields have comparable a posteriori 3D median velocity uncertainties, mostly in the range 0.2–0.4 mm/yr. The two rightmost columns of Table 2 shows the WRMS of the 3D velocity differences between each input velocity field and the combined field, with and without correcting their alignment bias (see Table 3 of the supplemental material). Only stations present in several velocity fields were used to compute the WRMS. This WRMS can be interpreted as the distance between the combined and each input velocity field, also taking into account their misalignment.

Table 2 Square-roots of the a posteriori variance factors of each input velocity field, their median weighted 3D velocity uncertainty, in mm/yr, the 3D velocity WRMS with respect to the combined field, in mm/yr, and also including the alignment bias (raw WRMS)
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The alignment parameters of the input velocity fields with respect to the ITRF2020 are well determined, with uncertainties in general smaller than 0.05 mm/yr, rising up to the level of 0.1 mm/yr for some of the regional velocity fields with a small number of sites (see Table 3 of the supplemental material). We also noticed a larger translation and scale drift in some of the regional velocity fields, most likely due, in part, to the correlation between the seven transformation parameters for velocity fields of small geographical extent.

Figures 1 and 2 show the combined horizontal and vertical velocity fields, respectively. The velocity estimates and their uncertainties can be found in Table 4 of the supplemental material. The combined field includes velocity estimates at 13,395 sites, from which 2,877 were present in more than four input velocity fields. These sites were used to assess the quality of the combined velocity field via the repeatability of site velocity across the input fields. The velocity repeatability can be considered as an alternative assessment of the velocity precision compared to the formal velocity uncertainty obtained from variance propagation of the models fitted to the GNSS position time series. For instance, the velocity repeatability includes the effect of the different time series used by the different groups and the different models fitted to them. However, the velocity repeatability is only available at those sites for which several velocity estimates are available in the input velocity fields. The velocity estimates for 7,207 sites are provided by a single velocity field, most of them by the NGL (5,791) and the EPND (921) velocity fields.

Fig. 1
figure 1

Horizontal velocities of the global combined field

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Fig. 2
figure 2

Vertical velocities of the global combined field

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Figure 3 shows the histograms of the horizontal and vertical velocity repeatabilities (WRMS of the velocity residuals) per site. The median repeatability of the velocity estimates is 0.17 and 0.27 mm/yr for horizontal and vertical velocities, respectively; in close agreement with the values used as a priori average velocity errors. The estimated velocity repeatability per site can be found in Table 5 of the supplemental material.

Fig. 3
figure 3

Histograms of repeatability in mm/yr of the horizontal (in blue) and vertical (in red) velocities

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The combination also helped to identify 327 sites out of 2,877 (11 %) for which the velocity repeatability is either larger than 0.45 mm/yr for horizontal velocities or larger than 0.75 mm/yr for vertical velocities (3 times the median values). The velocity repeatability per site was obtained after removing velocity outliers with deviations of several mm/yr, exceeding 1 cm/yr in a few cases. The rate of removed velocity outliers is typically less than 1% of the estimated velocities, but rises up to 10% for some velocity fields (see Table 7 of the supplemental material). There are several reasons that may explain a poor velocity repeatability, like noisy time series, the removal of different discontinuities and/or non-linear deformation due to post-seismic, hydrological loading or glacier discharge, among other processes (Riva et al. 2017; Gobron et al. 2021; Young et al. 2023). These sites should be used with caution, especially if their estimated linear velocities are extrapolated for long periods of time, as is common practice in sea-level (Wöppelmann et al. 2014) and GIA studies (King et al. 2022). By simple extrapolation, around 11% of the sites of the combined velocity field, but also of each input velocity field, might have poorly estimated velocities that may or may not be reflected by their formal velocity uncertainty.

5 Conclusions and Outlook

The International Association of Geodesy’s Joint Working Group 3.2 derived a global combined GNSS velocity field aligned to the ITRF2020 that gathers nineteen global and regional GNSS velocity fields computed by different groups. The combined field provides velocity estimates for 13,395 sites with at least 5 years of observations, completing by 1,849 sites the largest individual velocity field available.

More importantly, the combination allowed comparing the velocity estimates from the different groups and assessing their repeatability at 2,877 sites for which more than four velocity estimates were available. A median velocity repeatability at the level of 0.15 and 0.25 mm/yr was obtained for the horizontal and vertical velocities, respectively. However, up to 11% of the sites show poor velocity repeatability exceeding 3 times the median values. The estimated velocities at these sites should be used with caution.

For almost half the sites in the combined velocity field, the quality of the velocity estimates is given exclusively by the formal velocity uncertainty from a single velocity field. In this regard, the combination also provides estimated variance factors, based on the velocity repeatability across the input fields, which can be used to adjust the formal uncertainties of each input velocity field.