The ULR-repro3 processing considered the advances that occurred over the past 7 years, since the second IGS reanalysis campaign (Rebischung et al., 2016). It complies with the highest international standards, which were adopted by the IGS for the third reprocessing campaign (http://acc.igs.org/repro3/repro3.html, last access: 5 January 2023). The new modelling and corrections were implemented in the GAMIT/GLOBK software packages (Herring et al., 2015, 2018), in particular the International Earth Rotation and Reference Systems Service (IERS) linear pole model adopted in 2018 and the high-frequency (sub-daily) Earth Orientation Parameters (EOP) tide model from Desai and Sibois (2016). Table 1 provides a summary of the main modelling features and corrections applied in the ULR-repro3 reanalysis.

The remaining aspects of the ULR-repro3 data analysis strategy align with the approach used in Santamaria-Gomez et al. (2017); thus, they are only briefly outlined in the following in order to understand the analyst choices for geophysical application and interpretation. For each network of stations, double-differenced GPS phase observations were processed in the ionosphere-free linear combination of measurements on L1 and L2 frequencies. To minimise the impact of mismodelled low-elevation tropospheric delays, satellite observations below 10∘ were not considered. This cut-off angle aims to mitigate the limitation due to ground antennas without absolute calibration (13 % of the antennas in the ULR-repro3 network). These antennas have a relative calibration (with respect to an antenna with absolute calibration) converted to absolute considering only elevation-dependent phase centre variation (PCV) down to 10∘. For the other (calibrated) GNSS antennas, phase centre offsets with azimuth-dependent and elevation-dependent absolute PCV corrections were applied (igsR3_2135.atx; IGSMail by Arturo Villiger, 2020). Satellite-specific antenna phase centre offsets and block-specific nadir-angle-dependent absolute PCVs were applied for the transmitting antennas.

The first-order ionospheric delays were removed using the ionosphere-free linear combination observations, whereas the second and third orders were corrected using the International Geomagnetic Reference Field model (Alken et al., 2021) and total electron content maps from the IGS IONEX (ionosphere exchange) files. For the tropospheric delays, a priori hydrostatic zenith delays at the ellipsoidal surface were obtained for each station from the new gridded Vienna Mapping Function (VMF1) grids (Böhm et al., 2006). They were then reduced to the station heights using the GPT2 model (Lagler et al., 2013). The residual zenith tropospheric delays were adjusted at 1 h intervals (i.e. 25 parameters per day) for every station using a piecewise linear model, assuming that the unmodelled wet component dominates. Both the hydrostatic and wet zenith tropospheric delays were mapped to the observation elevations using the VMF1 functions. The azimuthal asymmetry in the tropospheric delay was accounted for by estimating a linear change in gradients (north–south and east–west) over each day and station using the mapping function from Chen and Herring (1997).

The phase observations were weighted by elevation angle in the first iteration and then by elevation angle and station-dependent scatter of the phase residuals obtained from the first iteration. The double-differenced phase ambiguities were adjusted to real values except when they could be confidently fixed to integer values (more than 85 % fixed). Within the same inversion, GNSS satellite orbital parameters were adjusted using 24 h arcs, IGS orbits as a priori values, and loose constraints consistent with the station position constraints (free-network approach). Non-gravitational constant and once-per-revolution accelerations on the satellites were adjusted too, using the ECOMC (Empirical CODE Orbit Model, where CODE stands for the Center for Orbit Determination in Europe) model. This model is a combination of the ECOM1 and ECOM2 models (Springer et al., 1999; Arnold et al., 2015) with specific parameters constrained in post-processing. Nominal satellite attitude corrections were applied, except during eclipse periods where yaw rates were modelled (Kouba, 2009). Phase rotations due to changes in the satellite antenna orientation away from the Earth-pointing direction were also applied (Wu et al., 1993). Regarding the Earth orientation parameters (pole position, rate, and length of day), these were estimated daily with a priori values from the IERS Bulletin A. Modelled diurnal and semi-diurnal terms were added to the a priori pole and UT1 values following the IERS Conventions (Petit and Luzum, 2010).

Note that neither loading displacements due to atmospheric tides nor non-tidal (atmospheric, oceanic, hydrologic) loading displacements were corrected during the first step, which aimed at estimating daily station positions from the GNSS measurements. By contrast, the displacements of the crust due to solid-Earth and pole tides (solid Earth and ocean) were corrected following the IERS Conventions (Petit and Luzum, 2010). Crustal motion due to the ocean tide loading was corrected too, using the tidal constituents computed by the EOST Loading Service at each station from the FES2014b model (Lyard et al., 2021).

Figure 2 shows the number of stations selected for ULR-repro3 with GNSS observations available each day over the time period considered (2000.0–2021.0, in decimal years), ranging from 110+ to nearly 500 stations. For computational efficiency, the stations were split into several (up to 10) regional subnetworks, each having between 29 and 70 stations processed independently. For the reader interested in this technicality, Fig. S1 shows the regional subnetworks distribution for the day of 1 January 2018. An additional subnetwork of globally distributed stations was considered to allow the daily combination of the regional subnetwork results in a unique daily global solution. This global subnetwork was made up of IGS reference frame stations, each of which also appeared in one – and only one – of the regional subnetworks. In turn, one regional subnetwork included one IGS reference frame station at least but could include more depending on the total number of subnetworks. Moreover, to strengthen the physical link between regional subnetworks, two stations from adjacent regional subnetworks were also included, i.e. one station from one nearby subnetwork and another from another nearby subnetwork, exclusive of the stations in the global subnetwork. All the subnetworks vary day by day depending on the station data actually available for the day considered. This network strategy has changed compared with past ULR reanalyses, benefitting from the experience of the Massachusetts Institute of Technology (MIT) IGS analysis centre.

The loosely constrained station positions and tropospheric delays for the common stations and the satellite orbital and Earth rotation parameters estimated from the subnetwork data analyses were combined using GLOBK (Herring et al., 2015) to obtain the daily global solutions, which include all stations available each day with their positions expressed in a common but yet undetermined terrestrial frame. These daily global solutions were then stacked into a long-term solution using the CATREF software package (Altamimi et al., 2018) with a time-dependent functional model that included translation, rotation, and scale transformation parameters between daily and long-term frames, estimated simultaneously with the mean station positions (at the reference epoch 2010.5), annual and biannual signals, and velocities. The scale parameters, which represent the mean height changes of all the sites, are available upon request, especially for users interested in global sea level rise.

Note that position offset discontinuities (mostly due to equipment changes and earthquakes) as well as station velocity changes and post-seismic displacements (PSDs) were added to the above modelling, where appropriate. As experienced analysts still tend to perform better than automatic methods (Gazeaux et al., 2013), the position offsets were identified and adopted via expert visual assessment using all positioning components (i.e. including the north and east components). To facilitate this task, the equipment changes reported in the GNSS station logs were considered along with the co-seismic displacements larger than 2 mm predicted with the earthquakes database and modelling by Métivier et al. (2014). When a position discontinuity was detected in a time series, the station position was estimated separately before and after the discontinuity along with the offset amplitude. The velocities before and after each position discontinuity were tightly constrained (0.01 mm yr-1), unless a velocity discontinuity was suspected. In the latter case (less than 2 % of the GNSS stations considered in ULR-repro3), no constraint was applied, and different velocities were estimated for each period of data around the discontinuity.

The above procedure also included manual editing to identify (and remove) outliers as well as additional non-documented position offset discontinuities. It was iterated until convergence (expert visual assessment). Overall, 1.2 offset discontinuities were detected per decade and per station, mostly caused by equipment changes (66.8 %) and earthquakes (19.6 %), whereas the remaining 13.6 % were flagged as unknown (Fig. 3) due to the lack of available metadata.

The long-term terrestrial frame, in which the estimated velocities are ultimately expressed, was finally aligned to the ITRF2014 (Altamimi et al., 2017) by applying minimal constraints to all the transformation parameters (translation, rotation, scale, and their rates) with respect to the positions and velocities of a stable subset of about 35 well-distributed reference frame stations. This step resulted in daily position time series expressed in the ITRF2014 frame for all (601) stations considered in ULR-repro3. From this set of position time series, only stations with more than 3 years between two consecutive position discontinuities and with data gaps not exceeding 30 % were retained for the next step as input (546 stations, among which 161 are reference stations and 457 are near a tide gauge).

The last step was the estimation of the parameters of interest (primarily station velocities) and their uncertainties, where both a functional and a stochastic model were adjusted to each of the position time series found using the procedure described in Sect. 2.2.1 on a station-by-station basis, as follows: 1xt=xref+vxt-tref+∑i=1N0aiH(t-ti)+∑j=13sjsin2πτjt+cjcos2πτjt+∑d=18sdsin2πτdt+cdcos2πτdt+∑f=13sfsin2πτft+cfcos2πτft+∑k=1NPSDPSDk(t), where xref is the position at the reference epoch tref, defined arbitrarily as the mid of the observation period considered (2000.0–2021.0); vx is the linear velocity; Ht-ti=0 if t<ti1 if t≥ti is the Heaviside function that multiplies the position offset ai; τj=1j is the period (in years) of the seasonal term j (annual, biannual and triannual); τd=PD365.25 is the period (in years) (PD in days) of the draconitic signals; and τf=PF365.25 is the period (in years) (PF in days) of the fortnightly signals. PSDkt=aklog⁡1+t-tkτk if PSD model is logak1-e-t-tkτk if PSD model is expa1klog⁡1+t-tkτ1k+a2k1-e-t-tkτ2k if PSD model islog⁡+exp⁡a1klog⁡1+t-tkτ1k+a2klog⁡1+t-tkτ2k if PSD model is log⁡+loga1k1-e-t-tkτ1k+a2k1-e-t-tkτ2k if PSD model is exp⁡+exp⁡ In this step, an additional and independent time series editing was considered to eliminate any possible remaining unreliable estimates from the previous step. The position estimates were compared to a running monthly median. Any epoch with a position showing a difference from the median exceeding 5 times the median absolute deviation in at least one component was discarded.

The stochastic model considered a linear combination of white noise (WN) and the power-law (PL) process (WN + PL), whose parameters (the stochastic process amplitudes and the spectral index of the power-law process) were estimated using the restricted maximum likelihood estimation method (Patterson and Thompson, 1971; Koch, 1986; Gobron et al., 2022). To obtain realistic stochastic parameter estimates, non-tidal atmospheric loading (NTAL) displacements were also subtracted from the position time series prior to this adjustment, following the recommendation of Gobron et al. (2021). These NTAL displacements were obtained from the Earth System Modelling team of the German Research Centre for Geosciences in Potsdam (Dill and Dobslaw, 2013).

The functional model included an intercept, a linear trend (velocity), the position offsets identified in the previous step, three seasonal terms (annual, biannual, and triannual), periodic terms at the first eight harmonics of the GPS draconitic year (351.4 d; Ray et al., 2008), and three fortnightly terms with respective periods of 13.62, 14.19, and 14.76 d (Penna and Stewart, 2003; Amiri-Simkooei, 2013). The parameters of this functional model and their uncertainties were estimated using the weighted least squares estimator with the inverse of the estimated WN + PL model covariance matrix as the weight matrix. During the observation time span, some stations (44) recorded significant co-seismic offsets and transient post-seismic signals, in which case the modelling was further extended to include velocity changes, and logarithmic or exponential decay functions according to the observed time evolution.