The GNSS station information and daily position time series are obtained from global GNSS solutions produced by Nevada Geodetic Laboratory (NGL). These solutions are based on continuous observations from stations in the International GNSS Service (IGS) network together with many regional GNSS networks (Blewitt et al., 2018). For this study, we use stations with at least three years of valid daily coordinate solutions within 2000–2023. This threshold provides a basic requirement for estimating annual and semiannual components, while retaining a sufficiently large station sample for global analysis.

Because the present model requires land-based soil-temperature forcing and shallow subsurface properties, we exclude stations located in Antarctica, Greenland, permanent ice-covered regions, glacier-covered regions, and their surrounding buffer zones. These sites are not included in the released dataset or subsequent statistical analyses because their surface thermal boundary conditions and shallow subsurface properties cannot be reliably represented using the current global input datasets. In contrast, stations located in coastal or lake-side environments are retained as long as valid land-based temperature forcing and subsurface properties are available.

Stations affected by seasonal snow cover are not systematically removed, because doing so would exclude many otherwise useful GNSS stations in northern Europe, Canada, the northern United States, northern Japan, and mountainous regions. Instead, the influence of seasonal snow cover is treated as a source of uncertainty and discussed as a limitation of the current product. After this screening, the dataset contains approximately 14 000 stations distributed across the major ice-free land regions and a wide range of climate settings (Fig. 2). The same screening criteria are applied to all subsequent statistical analyses and figures.

The temperature-driven thermoelastic deformation is controlled by near-surface temperature perturbations that propagate downward through heat conduction, produce thermal strain, and are converted into vertical elastic displacement at the ground surface. The timing of the temperature forcing is therefore important, especially when daily TED corrections are derived from temperature fields that vary within the day. In this study, soil temperature from ECMWF ERA5 reanalysis is used as the thermal forcing (Hersbach et al., 2018). The shallowest ERA5 soil-temperature layer, representing 0–7 cm depth, is used to prescribe the near-surface upper thermal boundary forcing. Since ERA5 soil temperature is used rather than air temperature, part of the effects associated with surface insulation processes, including seasonal snow cover, is implicitly included in the thermal forcing. However, snow depth and snow thermal properties are not explicitly represented in the current model, which may introduce additional uncertainty in regions with strong seasonal snow cover. ERA5 soil-temperature fields are used on the 0.25° × 0.25° regular latitude–longitude grid. For stations close to coastlines or large lakes, ocean and lake grid cells are not used for the upper-boundary forcing. Only valid land grid cells with soil-temperature values are considered.

To account for within-day temperature variability and reduce the potential aliasing associated with a single globally fixed sampling epoch, the hourly ERA5 soil-temperature record is sampled at four equally spaced epochs per day. The selected 0–7 cm soil temperatures are then used as the upper-boundary forcing at 6 h intervals. This treatment captures the main diurnal-scale variability relevant to daily TED estimates while keeping the global computation tractable. In addition to the 0–7 cm layer, the ERA5 soil-temperature layers at 7–28, 28–100, and 100–289 cm are used as auxiliary information on the subsurface temperature profile and the deeper thermal background (Lu et al., 2026b).

Thermal diffusivity controls how temperature variations attenuate and lag with depth, while linear thermal expansion coefficient and elastic parameters control how the resulting thermal strain is converted into surface displacement. To describe the shallow subsurface conditions at each station, we use SoilGrids-derived depth-dependent soil properties (Hengl et al., 2017). SoilGrids provides soil-property predictions for six standard depth intervals: 0–5, 5–15, 15–30, 30–60, 60–100, and 100–200 cm. These layers allow the main vertical variations in shallow soil properties within the upper 2 m to be represented. In this study, we use SoilGrids 2.0, which provides global land soil-property maps at 250 m spatial resolution (Poggio et al., 2021). Ocean and lake areas are not assigned valid SoilGrids values; therefore, for stations close to coastlines or large lakes, only valid land soil-property values are used to construct the layered subsurface column.

For each GNSS station, the SoilGrids-derived depth profiles are extracted and used to estimate the thermophysical parameters required by the model, including thermal conductivity and volumetric heat capacity, from which thermal diffusivity is derived. Elastic and thermoelastic parameters, including Young's modulus, Poisson's ratio, and the linear thermal expansion coefficient, are assigned according to the corresponding material class and then mapped onto the layered structure. Below 2 m, globally consistent high-resolution soil-property information is not available from SoilGrids. The use of 2 m as the lower limit of the SoilGrids-derived layers therefore reflects the depth range of the available global soil dataset, rather than an assumption that thermoelastic effects vanish below this depth. In the model, the medium below 2 m is represented by a homogeneous bedrock background with prescribed thermal and elastic properties. This treatment preserves the shallow-layer contrasts represented by SoilGrids while using a simplified description of the deeper thermal and elastic background.

It should be noted that SoilGrids does not provide globally consistent and reliable shallow subsurface properties beneath ice sheets, glaciers, or ice shelves. Stations located in these regions and their surrounding buffer zones are therefore excluded from the released dataset and the subsequent statistical analyses. The present subsurface parameterization also does not explicitly include time-varying soil moisture. Because soil moisture can modify thermal conductivity, volumetric heat capacity, and effective thermal diffusivity, its influence is treated as a source of uncertainty and is partly reflected in the parameter-sensitivity tests.

We compute temperature-driven vertical TED using FEMFL. The detailed formulation and numerical implementation of FEMFL are described in Lu et al. (2026b); here we summarize the main settings used to generate the global TED dataset. For each GNSS station, the shallow subsurface is represented by a one-dimensional, vertically layered soil–bedrock column. The model is run independently for each station and does not use a horizontal finite-element mesh, so the spatial sampling of the dataset is determined by GNSS station locations. The layered column consists of SoilGrids-derived layers in the upper 2 m and a homogeneous bedrock background below. The thermophysical, elastic, and thermoelastic parameters of each layer are assigned using the SoilGrids-based parameterization described in Sect. 2.3 and are assumed to be uniform within each layer. The half-space is assigned prescribed thermal and elastic properties and represents the deeper thermal and elastic background where globally consistent station-scale stratigraphic information is not available.

The released TED product represents subsurface thermoelastic deformation only. It does not include thermal expansion of the above-ground structures, such as concrete pillars, steel masts, rooftop monuments, buildings, or antenna-support structures. These monument-related effects can be important at individual stations and have been reported in previous studies (e.g., Romagnoli et al., 2003; Yan et al., 2009, 2010; Lu et al., 2026b). They are not included in this release because globally complete information on monument height, material, geometry, anchoring condition, and local thermal environment is not available for most GNSS stations. The TED series should therefore be interpreted as a subsurface thermoelastic correction, rather than a complete correction for all temperature-related site effects.

The thermal part of the model is based on one-dimensional transient heat conduction. Temperature and heat flux are continuous across layer interfaces. The upper boundary temperature is prescribed using ERA5 soil-temperature forcing, while the lower boundary represents a stable deep thermal background and reduces boundary effects associated with the finite model depth. The model is driven at 6 h intervals, so within-day temperature variations can be considered while the final product remains a daily TED time series. As an order-of-magnitude estimate, the characteristic thermal penetration depth can be approximated by δ=κP/π, where κ is thermal diffusivity and P is the forcing period. For typical near-surface diffusivities of 0.3 × 10⁻⁶ to 1.0 × 10⁻⁶ m2s-1, diurnal temperature variations mainly affect the upper 0.1–0.2 m, whereas annual temperature variations can reach about 1.7–3.2 m. The SoilGrids-based layering in the upper 2 m therefore captures the main shallow subsurface contrasts, while deeper annual-scale thermal effects are represented in a simplified way by the homogeneous bedrock half-space. In the mechanical part of the model, temperature changes produce thermal strain within each layer, and the finite-element system is solved to obtain the corresponding vertical displacement at the ground surface. To reduce the influence of uncertain initial deep-temperature conditions, a spin-up calculation is applied before generating the final TED series for 2000–2023.

GNSS coordinate time series are typically provided at daily resolution, whereas near-surface temperature and the resulting thermoelastic response depend strongly on local solar time. If temperature forcing is sampled at only one fixed epoch, stations at different longitudes correspond to different local times, and part of the diurnal temperature signal may be aliased into longer-period components. To reduce this effect, FEMFL is driven by ERA5 soil-temperature forcing sampled at 00:00, 06:00, 12:00, and 18:00 UTC, producing four TED realizations for each station. Each realization is decomposed into annual, semiannual, and residual components. The annual and semiannual components are averaged using amplitude–phase vectors to treat the cyclic nature of phase consistently and to preserve the mean magnitude and timing of the seasonal signal. The residual component is averaged from the four 6 h samples within each day. The final daily TED series is obtained by adding vector-averaged harmonic component and the daily averaged residual component, producing a correction series compatible with daily GNSS height time series.

To consistently quantify the key amplitude characteristics of the TED product and its correction impact on GNSS vertical time series, we compute the annual-cycle amplitude and phase, the nonseasonal variability, and the WRMS improvement of GNSS residuals. For each TED series u(t), we fit an annual and semiannual harmonic model with a linear term: 1u(t)=a0+bt+∑k=12[ckcos⁡(ωkt)+sksin⁡(ωkt)]+ϵ(t) where a0 is the constant term, b is the linear term, ω1 and ω2 correspond to the annual and semiannual frequencies, respectively, and ϵ(t) denotes the residual. The annual amplitude and phase are defined as: 2aAann=c12+s122bϕann=atan2(s1,c1).

To quantify the remaining variability after removing linear term and the annual/semiannual harmonics, let u^(t) denote the fitted reconstruction. The residual is r(t)=u(t)-u^(t), and the nonseasonal variability is measured by the root-mean-square (RMS) of r(t): 3RMSnonsea=1N∑i=1Nr(ti)2 where N is the number of valid samples.

For the GNSS height time series x(t), we quantify scatter using the weighted root-mean-square (WRMS): 4WRMS=∑i=1Nωix(ti)-x‾ω2∑i=1Nωi5x‾ω=∑i=1Nωix(ti)∑i=1Nωi where the weights are ωi=1/σi2; when observation uncertainties are unavailable, we use equal weights (ωi=1). The WRMS reduction rate is defined as: 6ΔWRMS(%)=WRMSraw-WRMScorrWRMSraw×100% where WRMSraw and WRMScorr denote the WRMS of the original GNSS vertical time series and the WRMS after applying the TED correction alone, respectively.

In the parameter-sensitivity analysis, we apply a ± 10 % relative perturbation to a selected key parameter at each station and recompute TED. The difference between the perturbed and baseline solutions in the annual cycle is then quantified as: 7ΔAann=Aann(±10%)-Aann,Δϕann=ϕann(±10%)-ϕann where the perturbed and baseline solutions are compared in terms of their annual-cycle amplitude and phase. For summary statistics, we use the absolute amplitude and phase differences to represent the sensitivity magnitude.

The station-based characteristics of the FEMFL TED product are shown in Fig. 3. The annual amplitudes are mostly within about 0.2–1.2 mm, with locally larger values exceeding 1.5 mm in regions with strong seasonal temperature contrasts. Relatively large signals are mainly found in mid-latitude continental areas, including parts of North America, Europe, East Asia, and Australia, whereas humid maritime regions and low-latitude areas generally show weaker annual responses. The annual phase shows a clear hemispheric contrast, consistent with the opposite seasonal heating cycles in the Northern and Southern Hemispheres. In addition to the fitted annual component, we use the difference between the 95th and 5th percentiles of the full daily TED series to describe the robust peak-to-peak variability. This metric reduces the influence of occasional extreme values while retaining the main range of the modelled displacement. The robust peak-to-peak variability commonly reaches 2–4 mm in continental regions and locally approaches 5–6 mm. After removing linear term and the annual and semiannual harmonics, the remaining nonseasonal TED variability is mostly below 1 mm, but can locally reach millimetre level. These results indicate that the TED product is dominated by annual signal at most stations, while nonseasonal variability remains detectable in some regions.

The temporal behaviour of TED is further illustrated using four representative stations, AARS, SHAO, KSCO, and BRO1 (Fig. 4). These examples show that the modelled displacement is generally dominated by a clear seasonal cycle, with nonseasonal variations superimposed on the annual signal. The linear, annual, and semiannual terms describe the dominant seasonal structure of the TED series, while the residuals retain variability that is not represented by these fitted components. The peak-to-trough displacement reaches about 1–2 mm at AARS, SHAO, and BRO1, and is larger at KSCO. After removing the fitted components, the residual TED is generally below 1 mm, with some variations approaching 0.5–1.0 mm. These examples show that the released product contains both the dominant seasonal thermoelastic response and smaller nonseasonal components.

The representative time series in Fig. 4 also show low-frequency variations that appear as trend-like signals over 2000–2023. To examine the possible origin, Fig. 5 compares the long-term trends in ERA5 soil temperature with those in the TED series. The spatial pattern of TED trends is broadly consistent with that of the ERA5 soil-temperature trends, particularly over continental regions with pronounced temperature variations. Among the four ERA5 soil-temperature layers, the 7–28 cm layer shows the strongest station-scale relationship with TED trends (Pearson r = 0.719). This relationship becomes slightly stronger when using 5° × 5° grid-cell median values (r = 0.758). These results suggest that the low-frequency variations in TED mainly reflect the long-term variability of the temperature forcing and the associated subsurface thermal response, rather than being dominated by numerical artefacts.

Figure 6 compares the annual amplitude of TED with monument thermal expansion and non-tidal loading displacements. The median annual amplitude is about 0.50 mm for the traditional homogeneous FEM solution and about 0.71 mm for the FEMFL product, indicating that introducing depth-dependent subsurface properties generally increases the annual TED amplitude. The FEMFL TED amplitude is comparable to monument thermal expansion, whose median value is about 0.40 mm for stations with available monument information. Compared with non-tidal loading, TED is generally smaller than hydrological loading, but its annual amplitude overlaps with the lower-to-middle ranges of NTOL and NTAL. These comparisons suggest that TED is a small but measurable correction component for high-precision GNSS height analysis.

The boxplot comparison in Fig. 6 shows the overall amplitude difference between FEM and the FEMFL solutions. To further examine how these differences are distributed spatially, Fig. 7 maps the station-by-station differences in annual amplitude, robust peak-to-peak variability, and annual phase. Both solutions use the same temperature forcing and temporal sampling strategy, so their differences mainly reflect the effect of introducing depth-dependent thermophysical, elastic, and thermoelastic properties. The annual-amplitude difference, defined as FEMFL minus homogeneous, is generally within about ± 0.2 mm at most stations, but locally reaches 0.3–0.4 mm. The peak-to-peak displacement difference shows a similar spatial pattern, suggesting that subsurface layering affects not only the annual component but also the overall range of the TED series. The annual-phase difference is expressed as the minimum angular difference between two solutions and is mostly within several tens of degrees. These spatially coherent differences indicate that subsurface layering modifies both amplitude and the timing of TED, especially in regions where shallow soil–bedrock contrasts are more pronounced.

The TED dataset is evaluated here from three aspects: influence of the SoilGrids-based subsurface parameterization, sensitivity to key thermal and elastic parameters, and spectral behaviour of the modelled time series. These tests are intended to assess the stability and internal consistency of the released TED product. They should not be interpreted as an independent validation of the FEMFL model itself, because the same modelling framework is used in the comparisons. All evaluations in this section use the updated station set after applying the screening criteria described in Sect. 2.1.

Figure 8 examines the effect of using SoilGrids as the global source of shallow subsurface information. For the 2612 stations with documented foundation-depth information, FEMFL solutions generated with SoilGrids-based parameters are compared with solutions constrained by the available site information. These stations are mainly distributed in North America, Europe, Australia, and parts of South America, where foundation records are relatively complete. The resulting differences are small: annual-amplitude differences are mostly about 0.01–0.03 mm, and annual-phase differences are generally within about 1–3°. Stations with shallow foundations show slightly larger differences than those with deeper foundations, which is expected because they are more directly affected by near-surface temperature variations. Overall, these differences are much smaller than the typical annual TED signal, suggesting that the SoilGrids-based parameterization introduces limited uncertainty in the modelled annual cycle for these stations.

The parameter-sensitivity results are summarized in Fig. 9. The linear thermal expansion coefficient α, thermal diffusivity κ, and Young's modulus E are each perturbed by ± 10 %, and the resulting changes in the annual TED amplitude and phase are examined. The amplitude response is generally limited. The largest sensitivity occurs mainly in dry inland regions, where stronger seasonal temperature forcing also produces larger TED signals. In these regions, most annual-amplitude changes remain below about 0.15 mm, although a small number of stations show larger responses. Humid coastal regions usually show weaker sensitivity, mostly below 0.1 mm, while high-latitude regions show intermediate behaviour. Among the three tested parameters, α and κ have a more direct influence on the annual amplitude, whereas the effect of E is generally smaller. The phase response is also moderate, with most annual-phase changes within about 1–3° and only a limited number of stations reaching several degrees. These results indicate that the modelled annual TED cycle is relatively stable under reasonable perturbations of the main thermal and elastic parameters, although larger sensitivity can occur in regions with stronger near-surface thermal contrasts.

The spectral behaviour of TED series provides an additional check on the temporal structure of the dataset (Figs. 10 and 11). The latitudinally stacked spectra show a dominant annual peak at 1 cpy and a weaker semiannual peak near 2 cpy in all latitude bands, consistent with temperature-driven origin of the modelled displacement. Spectral power is also present at higher frequencies, indicating that the released series retain nonseasonal variability rather than representing only fitted annual and semiannual terms. The annual peak is stronger in mid- and high-latitude bands than in low-latitude bands, reflecting the stronger seasonal temperature cycle outside the tropics. The regional spectra show similar features, but with clear differences in spectral power among regions. North America, Europe, and East Asia generally show stronger annual and semiannual peaks, whereas Australia and some lower-latitude or maritime regions show weaker seasonal power. The spectra decrease smoothly toward higher frequencies, with no clear artificial high-frequency artefacts. Taken together, the SoilGrids comparison, parameter perturbation tests, and spectral diagnostics suggest that the TED dataset is internally consistent and reasonably stable for the main annual and subannual signals. Larger uncertainties may still occur at stations with complex monument structures, poorly documented foundations, strong seasonal snow cover, time-varying soil moisture, or freeze–thaw conditions.

The effect of TED on GNSS height time series is first examined by applying the TED correction alone, without removing NTAL, NTOL, or HYDL (Fig. 12). Positive WRMS reductions are found at about 79 % of the assessed stations, indicating that removing TED reduces the scatter of GNSS height series at most sites. The improvement is generally modest, mostly within 3 %–5 %, and locally reaches about 5 %–10 %. Some stations show negative WRMS reductions, locally around -2 % to -5 %, indicating that the TED-only correction does not improve every GNSS height series. This behaviour is expected because TED is only one part of the non-tectonic vertical displacement observed by GNSS, and its amplitude is usually smaller than hydrological loading and many local site-dependent signals. Even so, the spatial pattern suggests that TED explains part of the GNSS height variability, especially in continental regions with stronger seasonal temperature forcing.

Figure 13 compares the WRMS changes obtained from the FEMFL correction and the traditional homogeneous FEM correction. The difference between the two corrections is smaller than the TED-only WRMS reduction itself, mostly within about ± 1 %–2 % and locally approaching ± 3 %. Positive values are more common in parts of North America and Europe, where FEMFL gives slightly larger WRMS reductions than the homogeneous FEM solution. This pattern is broadly consistent with the annual-amplitude and annual-phase differences shown in Fig. 6. The improvement from FEMFL should therefore be interpreted as a refinement of the homogeneous FEM correction through depth-dependent subsurface parameterization, rather than as a large change in the overall scatter of the GNSS residuals.

The representative stations in Fig. 14 help explain why the TED-only improvement is limited at some sites. At AARS and KSCO, HYDL and NTAL can reach several millimetres, and individual NTAL fluctuations can exceed 10 mm. At SHAO and BRO1, the non-tidal loading displacements are generally smaller, mostly within a few millimetres, but hydrological and ocean loading still contribute clear seasonal variations. By comparison, TED is smoother and usually remains around the 1–2 mm level. The lower panels show the GNSS height series before and after removing the combined TED, HYDL, NTAL, and NTOL corrections, only to illustrate the relative timing and amplitude of these displacement components. The direct assessment of the TED product is based on the TED-only WRMS reduction in Fig. 12 and the FEMFL–homogeneous FEM comparison in Fig. 13.