GIA causes changes in the gravitational potential observed by GRACE satellites, causes SED which affects VLM observed by GNSS stations, and causes local RSL changes, which are observed by tide gauges . To correct all these observations in a consistent way and to derive robust uncertainties, we use the ensemble of GIA estimates from C18, which provides a large ensemble of predictions, computed from varying the solid-Earth parameters and the ice-sheet histories. Each ensemble member comes with a likelihood that reflects its fitness to a global dataset of vertical GNSS velocities and sea-level paleorecords. Note that some circular reasoning is introduced here, as GNSS data were used to benchmark the GIA estimates. However, since the vast majority of the benchmark data comes from paleoindicators, and the cost function used to compute the likelihood is much more sensitive to paleodata rather than GNSS data, the impact of this at first glance circular reasoning on the results is limited. From this ensemble, we derive the mean and associated uncertainties for GIA-induced changes. We use a subset of 5000 ensemble members from the original set that contains 128 000 GIA estimates. This number is sufficient to approach the original ensemble set with a maximum deviation of 2.5 % from the original covariance matrix. Each ensemble member provides an estimate of the GIA effect on the gravitational potential, which we express in units of equivalent water height (EWH), an estimate of the solid-Earth deformation, and an estimate of the RSL changes.

To derive mean values and associated uncertainties from the ensemble, we use the likelihood L of each ensemble member. We can then compute the mean value Tμ and its uncertainty Tσ for the process of interest as follows: Tμ=∑n=1NL(n)∑n=1NL(n)T(n),Tσ=∑n=1NL(n)∑n=1NL(n)T(n)-Tμ2, where N is the number of ensemble members and T(n) the value of the individual ensemble member. All uncertainties are on the 1σ level, unless otherwise specified. Note that the underlying probability density function (PDF) does not have to be Gaussian or symmetric. We can also derive an empirical PDF from the ensemble, from which confidence intervals at any level can be computed. To compute the empirical PDF, we first sort the values from low to high. Then we define 20 bins between the 1st and 99th percentiles of these values and compute the sum of the likelihood of all ensemble members that fall within each bin. The number of 20 bins is chosen as a trade-off between resolution and sample size. The linear trends in the ensemble-mean EWH, SED, and RSL with their uncertainty estimates are shown in Fig. .

The regions with the largest trends and uncertainties are around the former ice sheets, while the far-field trends and uncertainties are smaller.

To estimate contemporary mass redistribution, we use the JPL GRACE Release 6 (RL06) mass concentration (mascon) solution . This solution provides monthly-mean estimates of EWH anomalies from March 2002 until June 2017, with some gaps at the beginning and end of the GRACE record and has a nominal spatial resolution of 3∘ by 3∘. We only look into mass changes on land, and we do not take ocean-bottom pressure changes driven by ocean dynamics into account, since its spatial variations are much smaller than the land mass changes e.g.,. For mascons that contain a coastline, a coastline resolution improvement (CRI) filter has been used to prevent the leakage of gravity signals between land and ocean, which leads to a better spatial representation of mass changes in coastal regions . The RL06 solution comes with an estimate of the measurement uncertainty in the EWH changes in each mascon.

We first restore the original GIA correction applied to the GRACE solution and then we generate a 5000-member ensemble of GRACE EWH estimates, each perturbed randomly using the measurement uncertainty estimate in each mascon. These uncertainties are based on the formal error covariance matrix of the GRACE solutions; see for details. This measurement uncertainty is given for each individual mascon and time step, and we assume that these uncertainties are uncorrelated in space and time. We then correct each perturbed mascon for GIA using one of the GIA ensemble members. Following , we adjust the degree-2 order-1 terms of the GIA estimates to account for the fact that GRACE observations are taken from an inertial reference frame and not from the rotating Earth.

With this procedure, we now have 5000 estimates of contemporary mass redistribution, from which we derive the expected contemporary mass redistribution and the associated uncertainty, as well as the total land mass change. Note that the uncertainty estimates of these trends are only based on the spread in the ensemble and not on the uncertainty that arises from fitting a linear trend to the data. Since geophysical time series often exhibit serial correlation, this assumption results in an underestimation of the uncertainty in the resulting trends . Figure  shows the trends and associated uncertainties in the GRACE estimates. The uncertainty in the trend is dominated by the GIA uncertainty, as the spatial pattern is almost identical to the pattern shown in Fig. b, and the measurement uncertainties only reach values of about 1 mmyr-1 EWH.

To isolate the role of cryospheric and hydrologic processes, we separate the observed EWH changes into changes from the Greenland Ice Sheet, the Antarctic Ice Sheet, glaciers, and TWS. The EWH changes on both ice sheets can be isolated by only selecting the mascons that cover ice sheets. Glaciers tend to be smaller in size than ice sheets and glaciers often do not form the dominant source of mass redistribution in the mascons that contain glaciated regions. Therefore we cannot directly apply the separation per mascon. Instead, we use a similar approach as . First, we determine those mascons that contain glaciated regions, based on the Randolph Glacier Inventory . For mascons that do not overlap with a glaciated region, all mass changes are attributed to TWS changes. For the mascons that do contain glaciers, we distinguish three cases:

Mass changes from the peripheral glaciers of Greenland and Antarctica are part of the mass balance of both ice sheets.

The aforementioned procedure results in an ensemble of 5000 estimates of monthly land mass changes. To compute the resulting SED and RSL changes, we solve the sea-level equation for each ensemble member. We use the pseudo-spectral approach , which requires the land mass changes to be transformed into spherical harmonics. We apply this transformation using the SHTns library up to degree and order 180. We solve the sea-level equation in the center-of-mass reference frame and apply the rotational feedback . We assume that the solid-Earth response to contemporary mass redistribution is purely elastic and thus differs from viscoelastic GIA. We use the elastic Love numbers from , which are based on the Preliminary Referenced Earth Model PREM;. This procedure results in an ensemble of 5000 estimates of local SED and RSL at each GRACE time step.

We use the GNSS dataset from the University of Nevada, Reno http://geodesy.unr.edu, last access: 25 July 2019, which provides processed daily time series of over 14 000 permanent GNSS receivers in the ITRF2008 reference frame. For consistency, we only use observations that overlap with the GRACE era (2002–2017). We remove stations for which fewer than 1825 daily observations that overlap with the GRACE era (2002–2017) are available, which corresponds to a minimum record length of 5 years. This requirement, which is a trade-off between accuracy and data availability, results in a total of 8228 stations. Figure  shows a histogram of the record length per station, both before and after removing the observations that fall outside of the GRACE era.

The histogram shows that most GNSS stations have a record length of around 8 to 10 years, or about half the length of the GRACE time span. Only a small fraction of the GNSS stations cover the full GRACE era. Since we only consider the observations within the GRACE era, not all data can be used, which results in a decrease in the record length that is available for some stations. We interpolate the monthly VLM that results from SED due to contemporary mass redistribution and GIA on the GNSS time steps, which enables us to separate the full GNSS time series into its components and the residual VLM following Eq. ().

To estimate linear trends in the observed and residual height anomalies, we apply the MIDAS robust trend estimator to the observed and residual height anomalies. The MIDAS estimator does not compute the linear trend using linear least squares but is based on the median of the height difference of each pair of anomalies that are separated by 1 year, while the uncertainty estimate is based on the standard deviation of the 1-year-separated anomaly differences. This approach has two advantages over the classical approach: it is less sensitive to discontinuities in the time series, which are omnipresent in GNSS records , and it is computationally less expensive than least-squares estimation using an appropriate serially correlated noise model. To propagate the SED uncertainty into the residual VLM trends, we estimate the MIDAS trends for each ensemble member and GNSS station, which results in two sources of uncertainty: the spread in the residual VLM trends due to SED and the uncertainty that results from estimating linear trends from noisy GNSS data. We assume that the uncertainty from the trend estimator and the ensemble spread are independent, and these terms are added up in quadrature to obtain the final uncertainty.

The global-mean land mass changes due to contemporary mass redistribution are shown in Fig.  and Table , which show that all cryospheric processes cause a net land mass loss while TWS does not show a significant positive or negative linear trend. Over the first decade of the GRACE era, a positive TWS term has been observed by and , although the land mass loss over the last few years of the GRACE record has resulted in a trend that does not deviate significantly from zero. For all cryospheric processes, the total mass changes are dominated by the trend, and the variability around the longer-term trend is relatively small compared to TWS, whose interannual variability around the mean trend is substantial, especially after 2010. It is known that TWS exhibits substantial decadal and multidecadal variability , and a large part of this variability corresponds to the strong El Niño–Southern Oscillation cycles during this period, which are known to have a large effect on water stored on land .

The uncertainties in the trend and time series for Greenland and glaciers are smaller than the uncertainties for Antarctica and TWS. This difference is caused by the uncertainty in the GIA contribution, which is small for Greenland and most glaciated regions, while the GIA contribution is larger and more uncertain for the Antarctic Ice Sheet. The uncertainty in the TWS term is largely caused by the uncertainty in the GIA contribution from the former Laurentide Ice Sheet, which covered large parts of North America (see Fig. ). Due to this uncertainty, the partitioning of the observed EWH changes over North America between GIA and contemporary changes is uncertain, which leads to a large spread in the possible TWS contribution from this region. The distributions shown in Fig.  are not symmetric, which causes slightly skewed confidence intervals in Table . The total land mass loss over the second half of the record is larger than the loss over the first half, which is caused by the accelerating contributions from Greenland and Antarctica and the slowdown of the TWS contribution. The trends in the individual terms are consistent with recent studies , although quantifying TWS changes without GRACE is still challenging , which still hinders the validation of GRACE-based TWS changes.

As discussed in Sect. , the mass changes will lead to local RSL and SED patterns. Figure  shows the resulting trends in local RSL with the accompanying uncertainty.

As expected from the barystatic contribution (Fig. ), the RSL trends are dominated by cryospheric processes, while the TWS-induced RSL trend is generally no more than a few tenths of millimeters per year and has a less pronounced spatial pattern. The local uncertainties are dominated by the TWS contribution. For large parts of the ocean, the uncertainty in the TWS contribution has a similar magnitude as the trend. The uncertainty in cryosphere-induced local RSL changes is limited to about 0.1 mmyr-1, except for regions very close to glaciers and the ice sheets. The uncertainty in the total signal is again on the order of 0.1–0.2 mmyr-1, which is substantially smaller than the signal itself.

Figure  shows that, next to uplift at locations where the ice mass loss takes place, mass changes in the cryosphere result in some far-field SED signals, including subsidence of about 0.5 mmyr-1 around Australia, and an uplift signal with a similar magnitude in Europe and northern Asia. TWS changes cause considerable near-field SED trends, e.g., in South America and Asia. Both uplift and subsidence occur, which shows that the barystatic trend consists of the sum of regions of land mass loss and land mass gain, and as such the TWS-induced SED signals, which have a predominantly near-field signature, will likely not follow the variability in the barystatic contribution from TWS. The SED uncertainty is large close to former glaciated regions due to the local impact of GIA. For other regions, the uncertainty is below the 0.1 mmyr-1 level, also for the locations for which the local trend is large.

The TWS-induced total land mass change shows substantial temporal variability, and the mass loss at both the Greenland and Antarctic ice sheets is accelerating. Therefore, linear RSL and SED trends derived over a subset of the GRACE period are likely to deviate from the trends derived over the full period. As an example of the size of these deviations, Fig.  shows the linear RSL and SED trends over the first and second halves of the GRACE era, both covering 7.5 years.

As a result of the accelerating barystatic sea-level trends, the local RSL trends are overall larger during the second half of the GRACE era, although the spatial pattern only shows limited changes between the different periods. In contrast, local SED trends, mostly associated with TWS changes, show vast differences between both periods. Notable regions include the Amazon Basin and southern Africa, which subside over the first half of the period and show uplift over the second half of the GRACE period. The opposite occurs in the Rio de la Plata basin and northwest Australia. Linear trends in SED thus do depend on the time span over which these trends are computed, which implies that due to SED extrapolating a linear trend in VLM will cause biases.

The linear trends in GNSS records (Fig. a) do show substantial spatial variability, even for nearby stations, although the uncertainty, which generally exceeds 0.5 mmyr-1 due to the short GNSS records, noisy data, and the presence of jumps in the data , should be considered when comparing nearby stations. Nevertheless, the observed VLM trends show many well-known large-scale features, mostly associated with GIA and uplift associated with contemporary ice mass loss. To quantify the role of SED in actual VLM trends as observed by GNSS stations, we computed the linear SED trends over the time span of each GNSS station (Fig. ).

The cryosphere-driven SED trends mostly show smooth spatial variations, and compared to the trend the interannual variations are small, which implies that the specific time span of the GNSS record has a limited impact on the observed linear trend in SED. Cryosphere-induced SED results in linear trends of a few tenths of a millimeter per year or larger at many GNSS stations. These trends not only reflect the well-known near-field uplift signals at or near the ice mass loss locations, which dominate the VLM signal for many regions where ice mass loss occurs, but also in the far field, with notable uplift in large parts of Europe and the US and subsidence in Australia. The uncertainty in these trends is negligibly small, except for the stations close to the locations of ice mass loss.

In contrast, the TWS-induced SED trends show a less smooth pattern: sharp contrasts in the trends between nearby stations can, for example, be seen in North America and South America, owing to the aforementioned difference in the time span of the GNSS records and the TWS trends that depend on the period over which these trends are computed. SED trends at coastal locations can be substantial, which points at possible biases when VLM estimates used to correct tide-gauge observations do not account for TWS-induced SED. For many regions, the SED trend is dominated by GIA, especially for the northeastern parts of North America and northern Europe, while for South America and Australia contemporary mass redistribution is the dominant SED component. Not only the SED trends but also the associated uncertainties are mostly driven by the GIA. Outside regions with a notable GIA signal, the uncertainties in the trends are small.

The removal of all modeled SED signals (Fig. c) highlights regional differences: for Europe, Australia, and South America, SED explains a large part of the observed large-scale VLM features, while for North America a different pattern emerges, with an uplift signal over large parts of the United States. This uplift could have its cause in the GIA estimates: the ensemble mean predicts a substantial subsidence pattern over large parts of North America, associated with the collapse of the Laurentide forebulge (Fig. e), which is stronger than projected from some other models, such as ICE6G-VM5a . For most cryospheric regions, the trends change, but a substantial residual signal remains. A possible explanation for this large residual is the fact that both the GIA estimates and the GRACE observations provide contemporary mass redistribution estimates at relatively coarse spatial resolutions, while it is known that SED induced by GIA and contemporary ice mass loss can have a very local character in these regions due to localized mass changes and complex Earth structures e.g.,. Some large-scale VLM features visible in the GNSS trends cannot be explained by the modeled SED. For some regions, such as Japan and Chile, tectonic activity is a likely candidate, but the uplift in southern Africa is unlikely to be tectonic in nature. Nevertheless, the scatter plot in Fig.  shows that the SED trend for most stations is close to the observed VLM trend, although there are multiple stations that show large trends that are not explained by the model. Given the fact that many relevant local processes, such as tectonics and local subsidence, are still present in the residual VLM trend, these outliers are not surprising.

The mean linear trend for all 8166 stations is 0.34 mmyr-1 with a standard deviation of 4.46 mmyr-1, while the mean residual trend is 0.44 mmyr-1 with a standard deviation of 4.28 mmyr-1. The coefficient of determination (R2) of the modeled trends is just 7.5 %. The full list of stations also contains many stations for which the MIDAS estimates very large uncertainties, sometimes exceeding 10 mmyr-1. If we limit our station selection to those stations for which the uncertainty in the observed VLM trend is smaller than 1 mmyr-1, which is the case for 4252 out of 8166 stations, we obtain a coefficient of determination of 34 % and a reduction of the standard deviation from 2.30 to 1.86 mmyr-1.

To demonstrate the impact of SED on tide-gauge observations, we revisit the analysis by T16;. In this study, linear trends from a set of 15 long-term tide-gauge records are compared to multiple recent global-mean sea-level reconstructions. A discrepancy was found between the long tide-gauge records and most reconstructions, including the recent reconstructions from and . This study did not use GNSS observations to correct tide gauges for local VLM. Here, we determine whether observed and residual VLM explain a part of this discrepancy.

For many of these stations, long VLM records from nearby GNSS stations are available. Some stations from that study are not in the vicinity (i.e., more than 50 km) of any GNSS station with a long record, or the combination of VLM and the tide-gauge trend results in an unrealistically low sea-level trend. We have removed these stations and added nearby stations as a replacement where possible. Text S1 in the Supplement gives an overview of the stations that differ from T16 with an explanation of the reason, and Table S1 in the Supplement lists all tide-gauge stations together with the GNSS stations used to determine the local VLM trends. The linear trends in local sea level have been computed from annual tide-gauge data obtained from the Permanent Service for Mean Sea Level using the Hector software while assuming a power-law spectrum for the temporally correlated noise in each tide-gauge record. To stay consistent with T16, we only use annual tide-gauge data from 1901 to 2000.

We compute linear trends in the uncorrected tide-gauge record, and we compute the trend by applying three different correction models. In the “GIA-removed” model, the ensemble-mean GIA RSL trend is removed from the tide-gauge trend. For the “VLM-removed” model, the observed VLM trend is removed from the tide-gauge trend. Both the GIA-removed and the VLM-removed models are commonly used as a correction to tide-gauge observations e.g.,. In the full model, we remove both the residual VLM trend and the GIA RSL trend. The full model removes the spatial variations due to GIA and local VLM not related to SED, while it avoids the extrapolation of the nonlinear SED signal due to contemporary mass redistribution. In the full model, sea-floor deformation signals due to contemporary mass redistribution and GIA are also retained in tide-gauge records. The uncertainties in the VLM and GIA estimates are derived from the ensemble and subsequently added in quadrature to the uncertainty in the linear trend from the tide-gauge record. Figure  shows the resulting linear trends at each station. For comparison, we also computed the trends for the GIA-removed and full models using the ICE6G-D VM5a model , which is an updated version of the ICE6G-C VM5a GIA model used in T16.

From the individual trends, we computed the mean and the inter-station spread, which we express as the standard deviation between the trends from each tide-gauge station (Fig. ). All correction models reduce the spread between the stations, compared to the uncorrected trends. Removing the observed VLM trends from the tide-gauge trends also reduces the inter-station spread in the trends, although to a lesser extent than the GIA-removed model. All three correction models do reduce the station-mean trend from the 1.7 mmyr-1 in the uncorrected model. The VLM-removed model has the lowest mean trend (1.1 mmyr-1), but this correction suffers from the two aforementioned problems: the linear trend due to SED over the GNSS record is not representative of the full tide-gauge record and the sea-floor deformation signal is removed from the tide-gauge data. The latter effect explains the low mean trend for this method as over the last few decades, due to contemporary mass redistribution, the sea floor has subsided by about 0.1 mmyr-1 , which would result in an underestimation of GMSL if sea-floor deformation is not taken into account. The full model results in a station-average trend of 1.3 mmyr-1, which lies in between the GIA-removed and VLM-removed models, but overall with the lowest inter-station spread. The mean trend of the full model is close to the trend found by and , who find a mean trend of 1.3 and 1.2 mmyr-1, respectively, over the 20th century. When we correct the tide-gauge records for GIA using the ICE6G-D model, we find a station-mean trend of 1.5 mmyr-1, which is close to the trend found by T16. The difference between C18 and ICE6G-D points at the impact of the uncertainty in GIA models on tide-gauge trends. When we compare the full correction models, the difference between C18 and ICE6G-D again becomes much smaller, which shows that the residual VLM trend partially offsets the uncertainty in GIA estimates. Note that T16 also argues that averaging the linear trend in sea level from these long tide-gauge data likely underestimates the global-mean sea-level trend due to the spatial patterns associated with ice mass loss and ocean dynamics. Here, we do not consider this spatial sampling bias, so the gap between the long tide-gauge records and global-mean sea-level reconstructions discussed by T16 cannot yet be fully reconciled from these results.