The Gravity Recovery and Climate Experiment (GRACE) has contributed to major improvements in understanding ice mass balance of major ice sheets (Antarctica and Greenland)1,2 and mountain glaciers3,4 and changes in terrestrial water storage4,5,6, all important contributors to sea level rise7. GRACE has also been able to observe the resulting global mean ocean mass (GMOM) changes of approximately 2.07 mm/yr during 2005–20156. An independent GMOM rate from the difference between satellite Altimeter data and Argo float data (Altimeter-Argo) shows general agreement with the GRACE rate (e.g., Chen et al.8; Kim et al.6). This difference (Altimeter-Argo) corrects total sea level change (from altimetry) by removing steric sea level change due to density variations (from Argo). The GRACE Follow-On (GFO) mission extends the GRACE time series of surface mass redistribution, although there is a gap (July 2017–April 2018) between GRACE and GFO. The continued growth of the GRACE/GFO time series is expected to provide increasing understanding of contemporary climate change.

In the late phase of the GRACE mission (after August 2016) and into the GFO mission (since June 2018) there have been growing differences between the two GMOM estimates (GRACE/GFO and Altimeter-Argo)9. Barnoud et al.10 examined both Altimeter and Argo data and showed that uncertainty in Argo salinity data may explain about 40% of the discrepancy, but further study is required to explain the remainder. The discrepancy may be due to multiple causes. First, previous GMOM rates from GRACE were estimated over the global oceans6,11 but values may differ over the smaller region sampled by Altimeter-Argo12. Second, a GMOM estimate from GRACE/GFO is dependent on choice of Glacial Isostatic Adjustment (GIA) model6. Gravity field changes observed by GRACE/GFO include contributions from both contemporary surface mass loads and from GIA, the viscoelastic mantle response to past ice mass changes since the last glaciation. GIA effects are removed using one of a number of available GIA models, but there are differences among models, indicating that their deficiencies likely contribute to GMOM rate uncertainty10. Furthermore this uncertainty ought to be assessed over the Altimeter-Argo region, rather than the global oceans, when comparing GRACE and Altimeter-Argo estimates. In this study, we examine uncertainties in GRACE/GFO GMOM and find that varying GIA model choice can yield improved agreement between GMOM estimates from GRACE and Altimeter-Argo. We consider both uncertainty in GIA models over the Altimeter-Argo regions, and in degree-1 terms that must be estimated to obtain an GRACE/GFO GMOM.


To estimate GMOM using GRACE/GFO, we obtained leakage-corrected terrestrial mass loads using forward modeling13 (FM) and subsequently estimated ocean mass redistribution conforming to the shape of geoid considering self-attraction and loading (SAL)14. The data are CSR RL06 GRACE/GFO spherical harmonic (SH) coefficients with \(\Delta\) C20 and \(\Delta\) C30 replaced by SLR values15,16. GIA signals were removed using the model of Peltier et al.17, but we consider other GIA models18,19,20 to understand the consequences of model choice. We applied 300 km Gaussian smoothing to the SH coefficients but no decorrelation filter21. A decorrelation filter suppresses north–south noise patterns (longitudinal stripes) in GRACE/GFO data, but is also likely to attenuate longitudinally oriented signals (e.g., ice mass loss along Greenland coast lines). Using the smoothed SH coefficients, we estimated leakage corrected mass fields via FM, and then applied SAL to determine ocean water distribution. FM iteratively updates mass changes on land and the negative of this is used to compute changes in mean ocean mass (see Chen et al.13 for the detail of the FM method). SAL is then used to compute the spatial distribution of ocean mass changes22. At each iteration, degree-1 coefficients were generated, and then removed so that the converged FM solution does not include them. Then, using the converged FM solution, we estimated degree-1 coefficients using equations (1)–(6) in Supplementary Information. Our method to estimate degree-1 terms differs from previous studies in that leakage effects were corrected by FM rather than by using an empirical ocean buffer zone23 and ocean mass distribution was realized using SAL rather than by assuming a uniform sea level change24.

GRACE/GFO GMOM changes were obtained from the SAL estimation, and we refer to the estimate as GMOMSAL. Ocean dynamics and atmospheric loading signals, including the inverted barometer effect, were removed in GRACE processing using GRACE/GFO dealiasing fields, so these fields were added to GMOMSAL25. Global mean atmospheric mass change was corrected to ensure global mass conservation between the terrestrial surface and atmosphere8. The correction mainly affects seasonal GMOMSAL variations.

Altimeter-Argo GMOM is based on data from various altimetry data centers that provide monthly gridded total sea level change (including steric effects and ocean mass changes). These include observations from a number of altimetry satellites. We used ensemble means of two datasets: Copernicus Marine Environment Monitoring Service (CMEMS) and Commonwealth Scientific and Industrial Research Organization (CSIRO). Ocean bottom deformation associated with GIA was corrected using the same GIA model17 as in GRACE/GFO data processing. Further ocean bottom deformation resulting from contemporary surface mass load redistribution was estimated using leakage corrected GRACE/GFO mass fields over land and associated SAL water distribution over the oceans as discussed above. Altimetry observations were corrected for this26, unlike previous studies6,7,10,27.

GMOM is estimated from altimetry observations by subtracting steric sea level changes. Steric change is caused by temperature (T) and salinity (S) variations. Prior to about the year 2000, T and S were obtained from shipboard measurements (e.g., XBT) with limited spatial and depth sampling (typically to depths near 700 m). With the advent of the Argo float network, spatial coverage is now near-global and depths extend to 2000 m. Several data centers provide gridded T and S estimates from Argo. We used results from three groups: Scripps Institution of Oceanography28, International Pacific Research Center (IPRC) and Japan Agency for Marine-Earth Science and Technology (JAMSTEC)29. We used ensemble means of the three data sets which should suppress some errors7,27. There may also be steric variations in the abyssal oceans30,31,32, not sampled by Argo. This contribution to global mean sea level is highly uncertain. Estimates for the past decade include −0.13 ± 0.34 mm/yr31 and 0.29 ± 0.21 mm/yr32. The deep ocean steric effect is not considered in this study but is potentially a source of differences between GRACE/GFO and Altimeter-Argo estimates. Our Altimeter-Argo GMOM estimate is denoted as GMOMA-A hereafter.

Argo float spatial sampling became nearly global since about 2005, so we estimate GMOMA-A after 2005. Neither Altimetry nor Argo floats sample high latitude oceans, nor are all coastal areas sampled. The Altimeter-Argo ocean region (see Fig. S1), is used to estimate GMOM for both Altimeter-Argo (GMOMA-A) and GRACE/GFO (GMOMSAL). Previous studies (e.g.,Kim et al.6,Chen et al.8) did not consider a common region when forming the two estimates of GMOM.

We compare GMOMSAL and GMOMA-A in Fig. 1. As examined previously9,10, the GMOMA-A trend in black (Altimeter-Argo) and GMOMSAL trend in red differ significantly after August 2016. However, evident differences between the two are already found during Northern Hemisphere Spring starting in 2013. The trend from the black line is 3.01 mm/yr, which is 0.11 mm/yr larger than the previous estimate for the same period10. This is mainly because the previous study did not consider seafloor load deformation. The correction for seafloor deformation is 0.08 mm/yr for the global oceans and 0.17 mm/yr for the Altimeter-Argo region.

Figure 1
figure 1

GMOMSAL from GRACE/GFO (red and blue) and GMOMA-A from Altimeter-Argo (black and gray). The black line is based upon an ensemble average of Argo data including both thermosteric and halosteric contributions. The gray line is similar to the black but only includes a thermosteric contribution. Red and blue lines were based on GIA models of Peltier et al.17 and Caron et al.20, respectively. Numbers are associated linear trends with 95% confidence intervals.

We can compare GMOMSAL with Mascon solutions from CSR33 and JPL34 over the same Altimeter-Argo ocean region. Table 1 shows the three different GMOM trends from our estimate, based on CSR RL06 SH coefficients, and CSR and JPL Mascons. Our estimate (2.33 ± 0.11 mm/yr) is close to Mascon values, particularly to CSR (2.24 ± 0.12 mm/yr). Barnoud et al.10 also estimated a GMOM trend from the same CSR Mascon solutions, finding a value of 2.08 mm/yr, somewhat smaller than our estimate. The difference is due to the ocean regions used. Like Barnoud et al.10 we excluded latitudes above 66° but here we also exclude coastal areas not sampled by Altimeter-Argo (see Fig. S1). We obtained the same value (2.08 ± 0.09 mm/yr) by including these coastal regions.

Table 1 GMOM comparisons between this study and Mascons.

Barnoud et al.10 showed that spurious trends in halosteric sea level from Argo data partly explained the discrepancy between GRACE/GFO and Altimeter-Argo estimates (called here GMOMSAL and GMOMA-A) after 2016. To test this, we estimated GMOMA-A using ensemble-averaged thermosteric-only sea level (gray line in Fig. 1). The trend difference between gray (2.75 mm/yr) and red (2.33 mm/yr) lines is smaller than between black and red lines.

The remaining trend difference between GMOMA-A and GMOMSAL is likely due to deficiencies in GIA models. We estimated apparent water mass loads using the four GIA models of A et al.18, Purcell et al.19, Caron et al.20 and Peltier et al.17. This GIA mass load change should be removed from GRACE/GFO data to estimate GMOMSAL. However, there are differences among the four GIA model estimates (denoted by OMA, OMPurcell, OMCaron and OMPeltier hereafter). The four models provide GIA mass changes over the global oceans shown by gray bars in Fig. 2. OMCaron is the largest, −1.03 mm/yr, about −0.19 mm/yr larger than OMPeltier. In general, differences among GIA models over the global oceans are less than remaining difference between gray (2.75 mm/yr) and red (2.33 mm/yr) lines in Fig. 1. However, over the Altimeter-Argo ocean region (red bars in Fig. 2), differences among models are greater. In particular, OMCaron yields −1.43 mm/yr, about −0.40 mm/yr larger than over the global oceans. This is mainly because large values in OMCaron over the southern oceans are excluded in the Atlimeter-Argo region (Fig. S1). As a result, GRACE/GFO estimates (using the GIA model of Caron et al.20) show an increased GMOMSAL rate over the Altimeter-Argo ocean region, 2.75 mm/yr (blue line in Fig. 1), very near that of GMOMA-A with thermosteric-only corrections (gray line in Fig. 1).

Figure 2
figure 2

Apparent ocean mass rates due to GIA from GIA models of A et al.18,Purcell et al.19,Peltier et al.17, and Caron et al.20. Gray and red bars represent apparent water mass loads over global oceans and coverage of Altimeter and Argo, respectively.

However trend differences between GMOMSAL and GMOMA-A depend on the selected portion of the time series. For example, trends of blue and black lines for 2005–2015 are 2.72 ± 0.14 mm/yr and 2.51 ± 0.16 mm/yr, respectively. For this period, the GMOMSAL rate (2.42 ± 0.14 mm/yr) using the GIA model of A et al.18 is closer to GMOMA-A (2.51 ± 0.16 mm/yr). Varying trend differences between GMOMSAL and GMOMA-A are likely assocated with data uncertainties. This suggests that it is difficult to judge the best GIA model choice among the four by comparing GMOMSAL and GMOMA-A.

The trend difference between blue and red lines in Fig. 1 (GMOMSAL based respectively on the GIA models of Caron et al.20 and Peltier et al.17) is −0.43 mm/yr. Different estimates of SH degree-1 terms explain about −0.12 mm/yr of the trend difference, −0.43 mm/yr. Figure 3 shows estimated SH degree-1 terms from GRACE/GFO mass fields using GIA models of Caron et al.20 (blue) and Peltier et al.17 (red). There are evident trend differences in the two estimates, particularly in \(\Delta\) C10. Degree-1 terms from Sun et al. (TN13)23 (black) are close to the red lines because the same GIA model of Peltier et al.17 was used. Because SH degree-1 terms depend upon the adopted GIA model, TN13 can be used when adopting the Peltier model, but must be estimated for other models.

Figure 3
figure 3

Degree-1 SH Coefficients. Gray lines are from TN13, and red and blue lines are estimated SH coefficients after GIA correction with Peltier et al.17 and Caron et al.20, respectively. Numbers in each panel are linear trends with associated 95% confidence intervals.

In summary, variations among GIA models over the Altimeter-Argo region are large enough to explain differences in GMOMSAL and GMOMA-A estimates, and associated degree-1 SH coefficients, are also affected by GIA model and play an important role in these differences.


Even though GIA models exclude SH degree-1 terms, residual GIA effects (GIA model errors evident from differences among models) contaminate degree-1 estimates derived from other harmonics via equations (1)–(6) in Supplementary Information. Similarly, residual ocean dynamics (ROD) in GRACE/GFO after the correction of ocean dynamic effects during GRACE processing may contribute to errors in degree-1 terms. ROD should have a small effect when averaged over a large area, including the global oceans or the Altimeter-Argo region, but they might still contaminate estimates of degree-1 terms with related effects on GMOM estimates. We can estimate sensitivity to ROD using synthetic GRACE-like data (see “Method” and Fig. S2), along with ocean dynamics fields used in GRACE processing (GAD) and ocean bottom pressure data that provide a measure of ROD effects.

Here we examine two measures of ROD effects. One is the difference between GAD for CSR RL06 and that for CSR RL05 (hereafter ROD1), and the other is the difference between CSR RL06 GAD and bottom pressure data from the German contribution to Estimating the Circulation and Climate of the Ocean (GECCO2)35 (hereafter ROD2) (Fig. S3). ROD2 is much larger than ROD1, so the two choices are useful to understand possible ROD contamination in degree-1 terms. Long-term ROD variations, particularly from GECCO2, cause a spurious trend in global mean ROD. To suppress this, ROD values at grid points were adjusted to have zero global mean every month. The same 300 km Gaussian smoothing was applied to the synthetic data, as for GRACE/GFO.

Using the synthetic data (see “Method”) with either ROD1 or ROD2, we estimated degree-1 terms and compared them with known true values for the synthetic data. Because the negative of total terrestrial mass change and SAL were used to compute ocean mass loads, our ocean mass fields do not include ROD. However, when estimating degree-1 terms, ROD is inevitably included (see Supplementary Information and Fig. S3) and would affect GMOM estimation.

Figure 4 shows synthetic true degree-1 terms and estimates contaminated by estimated ROD effects (ROD1 and ROD2). Gray lines in Fig. 4(a)–(c) are true variations and red lines are estimated with ROD1. Because of the small magnitude of ROD1, gray and red lines are almost identical. The larger magnitude ROD2 (blue lines), shows much larger differences between true and estimated degree-1 SH coefficients. Figure 4(d) shows the degree-1 contributions to GMOM. ROD1 (red) is close to true (gray), while ROD2 (blue) causes much larger differences. The trend difference between blue and gray is 0.12 mm/yr, mostly due to the trend difference in \(\Delta\) C10. In practice it is difficult to assess the importance of ROD, due to sparse in-situ observations of ocean bottom pressure. However this experiment shows that it may be possibly a source of significant error in degree-1 estimates.

Figure 4
figure 4

Estimates of degree-1 terms and their effects on GMOM from synthetic GRACE-like data. (a)–(c) Gray lines are true degree-1 SH coefficients. Red and blue lines are estimated degree-1 SH coefficients from synthetic GRACE-like synthetic data including ROD1 and ROD2, respectively. (d) Gray line is true GMOM only considering degree-1 terms over Altimeter- Argo regions. Red and blue lines are GMOM using estimated degree-1 SH coefficients with ROD1 and ROD2, respectively. Numbers in each panel are linear trends with associated 95% confidence intervals.


Estimates of global mean ocean mass (GMOM) variations are provided by observations of GRACE/GFO and Altimeter-Argo. The two GMOM estimates are in good agreement during the early period (2005–2015), but differences increase beyond this period. A spurious halosteric sea level trend was suggested as the possible cause of the disagreement, but only explained the abrupt difference after August 2016.

The remaining difference is possibly due to GIA model error. GMOM trends from GRACE/GFO are heavily affected by GIA model choice. Four GIA models yield variable rates: 2.75 ± 0.11 mm/yr (model of Caron et al.20) for January 2005 to December 2019, is close to the Altimeter-Argo trend (using thermosteric-only sea level changes) of 2.75 ± 0.11 mm/yr for the same period. The trend associated with the model of Peltier et al.17 is much smaller, 2.33 ± 0.11 mm/yr. Different estimates of degree-1 terms partly explain the GMOM trend differences. Even though GIA models do not include degree-1 SH coefficients, they alter other GRACE/GFO SH coefficients used to estimate degree-1 terms. Degree-1 uncertainty may also be caused by residual ocean dynamic (ROD) effects, as indicated by experiments with synthetic GRACE-like data. Sparse observations of ocean bottom pressure make it difficult to be more precise.

GFO continues to extend the time series begun by GRACE, and further understanding of contemporary climate change, including ice mass variations, should follow in the future. However, this study shows that further progress is needed in development of GIA models. These models estimate SH coefficient rates of change, so associated errors will grow as the time span increases. Current methods of estimating degree-1 SH terms are also a limitation in estimates of secular trends in surface mass distribution.


Synthetic GRACE data

Synthetic data were used to understand uncertainty in estimates of degree-1 SH coefficients. We used synthetic GRACE-like data to compare estimated degree-1 coefficients with true values from the synthetic data. Synthetic GRACE-like data were computed using surface mass fields from numerical climate models and other sources in the literature. The global land data assimilation system (GLDAS)36 was used to represent terrestrial water storage change. Ice mass variations in Greenland and Antarctica were taken from Kim et al.6, and mountain glaciers from Zemp et al.3. Figure S2(a) shows an example of synthetic surface mass loads. We estimated GRACE errors as the difference between GRACE and Gaussian smoothed GRACE solutions37,38. This error estimate would also include residual surface mass signals. To remove these residual signals, we employed an empirical orthogonal functions (EOF) method which decomposes spatially and temporally correlated fields38. The EOF method should distinguish and allow separation of residual signals within the difference between GRACE and Gaussian smoothed GRACE data. Residual surface mass load signals ought to be both spatially and temporally distinct from actual errors. Figure S2(b) shows an example of GRACE error estimated here and Fig. S2(c) shows synthetic GRACE-like data (sum of synthetic GRACE-like signal (Fig. S2(a)) and errors (Fig. S2(b))).