Inferring Global Ocean Mass Increase From Tide Gauges Network With Climate Models

Ocean mass increase contributes to global sea level rise, and plays an important role in understanding climate change. Here, we develop a data assimilation approach that enables the inference of ocean mass increase from global tide gauge network. This approach incorporates outputs from climate models and sea level fingerprints caused by water mass changes over land areas. The results suggest a trend of 2.15 ± 0.72 mm/yr for ocean mass increase over the period 1993–2022, which closes the global sea level budget with estimates of thermosteric sea level rise. Furthermore, the inferred ocean mass increase offers an insight into the causes of sea level rise since 1950. These findings emphasize the significance of climate models, in addition to simulating sea level changes, they contribute to understanding causes of sea level rise over the past decades.


Introduction
Ocean mass increase and ocean thermal expansion are identified as the primary contributors to the global mean sea level (GMSL) rise (Barnoud et al., 2021;Chen et al., 2020;Llovel et al., 2023).Sea level observations suggest that, since 1993, the contribution of ocean mass increase exceeds the contribution from ocean thermal expansion (e.g., Chambers et al., 2017), demonstrating its predominant effect on GMSL rise.By nature, the increase in the global ocean mass can be mainly attributed to the water mass loss from global land areas (Yi et al., 2015).This phenomenon involves various processes, such as mass loss in Greenland and Antarctic (Kjeldsen et al., 2015;Riva et al., 2010), glacier melting in mountainous regions (Zemp M. et al., 2016), water storage changes in land areas (Humphrey & Gudmundsson, 2019), and water retention due to dams (e.g., Chao et al., 2008).Recent studies suggest that ice and glacier mass loss dominate the increase in the global ocean mass (Horwath et al., 2022), but changes in terrestrial and groundwater storage also have contributions (Chen et al., 2022).
Despite the importance of ocean mass increase, direct measurements are impossible until the launch of the Gravity Recovery and Climate Experiment (GRACE) satellite gravity mission (Tapley et al., 2019).Since 2002, the GRACE satellites have provided global maps of time-variable gravity, which are further used to determine ocean mass variations at both the global and regional scales (Chen et al., 2020;Watkins et al., 2015).Many investigations using GRACE data have helped elucidate the causes of the observed GMSL rise from satellite altimetry, at least until 2016 (Chambers et al., 2017;Chen et al., 2022;Llovel et al., 2023).The GRACE satellites ceased to measure the gravity field in 2017, but the GRACE Follow-On (FO) mission has continued to monitor global surface mass transport since 2018 (Landerer et al., 2020).
Before the era of the GRACE satellite mission, there were no tools for directly determining global ocean mass increase.An alternative approach is to summarize the contributing sources, such as ice and glacier melting, and surface and groundwater changes (Gregory et al., 2013;WCRP Global Sea Level Budget Group, 2018).This method enables long-term evaluation of ocean mass increase and sea level budget (Church et al., 2011), extending back to as early as 1900 (Frederikse et al., 2020a(Frederikse et al., , 2020b)).It is important to note that many of these contributing sources are not always derived from direct measurements; instead, they are often simulated by models (e.g., Meyssignac et al., 2017).Consequently, these contributing sources may contain various errors and biases.Another indirect approach for estimating global ocean mass increase was proposed by Munk (2002).This approach assumes that a decrease in the global average salinity is associated with the melting of ice sheets and other changes in terrestrial water storage (Wadhams & Munk, 2004).With estimates of global salinity, one can calculate ocean mass increase after correcting for the effect of floating ice melting, which does not contribute to the GMSL rise (Munk, 2003;Ponte et al., 2021).Wadhams and Munk (2004), using such approach, inferred a trend of 0.6 mm/yr for 20th century ocean mass increase.Furthermore, Llovel et al. (2019) obtained an ocean mass increase trend of 1.55 ± 1.20 mm/yr for 2005-2015 by considering salinity changes with the full ocean depth, although this trend is smaller compared to the trend observed by GRACE.
In this study, we propose a different approach to infer the global ocean mass increase.Our method accounts for the contributing factors of sea level rise.Tide gauges (TGs) record long-term relative sea level changes.These changes are linked to the ocean mass increase resulting from ice melting, variations in terrestrial water storage (TWS), steric sea level changes, and ocean circulation.This raises the following question: can we reconcile these contributing sources with measurements from TGs?In other words, is it possible to infer the amount of ocean mass increase from TGs by modeling the pattern of ocean mass increases and other contributing sources?Indeed, TGs provide valuable spatial constraints that can potentially be leveraged to obtain a new estimate of ocean mass increase, offering a new perspective on GMSL rise.Motivated by this question, we develop a data assimilation approach and utilize as many TGs as possible to create a more robust geometry.Beyond estimating the ocean mass increase, our results also carry implications for the global sea level budget.

Data and Methods
In this section, we outline the fundamental idea and its implementations, more descriptions can be found in Supporting Information S1 of this paper as well as Hay et al. (2013) and Mu et al. (2024).TGs measure relative sea level changes along coastlines (Piecuch et al., 2017;Woodworth et al., 2019).These measurements contain global ocean mass increase resulting from glacier and ice mass loss, sterodynamic sea level changes (Gregory et al., 2019), and glacial isostatic adjustment (GIA) effect.The sterodynamic sea level changes encompass variations in steric sea level and ocean circulations (Calafat et al., 2023).In theory, it is highly plausible to determine global ocean mass increase using TGs when both the sterodynamic contribution and the GIA effect are properly evaluated as well as the spatial pattern of ocean mass increase is modeled.Mathematically, we formulate the increment of relative sea level (ΔSL) observed by TGs as follow: where Δt is the time increment, SL  (Griffies et al., 2016).SL • (t) GIA is obtained from a GIA model (Peltier et al., 2015).
The reference rate SL • SLF (Figure 1a) includes mass variations from ice and glacier models/observations and hydrological models (Frederikse et al., 2020a(Frederikse et al., , 2020b)).The SL • SLF serves as an initial estimate.Our data assimilation approach revolves around reconciling this initial estimate with TG measurements, because SL • SLF may not agree with these measurements, considering that its components originate from different models and observations, and contain various errors and biases.The pattern of SL • SLF is computed with the SLF theory (Coulson et al., 2022;Mitrovica et al., 2011;Thompson et al., 2016).Land water mass variations, which can be obtained from both models and observations, inevitably enter into oceans, contributing to ocean mass increase.The SLF theory predicts how water mass variations redistribute over oceans under the effect of gravity, deformation, and rotation (Jeon et al., 2021;Riva et al., 2010).Essentially, the SLF describes relative sea level changes that are caused by land water mass variations.Key features include that sea level falls over the near fields (associated with mass variations) and sea level rises over the far fields (e.g., deep oceans), as depicted in Figure 1a.In this study, we consider the SLF computed by Frederikse et al. (2020aFrederikse et al. ( , 2020b)), which consists of four major mass transport processes over land, including ice mass loss in Greenland and Antarctica, mass loss in global glaciers, and TWS variations.They cover the time span from 1900 to 2018.We calculate the overall trend  for the SLF (see Figure 1a) and employ this trend map as a reference (i.e., the initial estimate) to infer the amount of global ocean mass increase for the period 1950-2022.
To implement our idea, we employ a data assimilation approach.The mathematical framework of the data assimilation process is detailed in Hay et al. (2013) and Mu et al. (2024).In our implementation, we select 794 TGs from the Permanent Service for Mean Sea Level (PSMSL) (Holgate et al., 2013).It is worth noting that the number of TGs varies over time (as shown in Figure S1 in Supporting Information S1), and the number is clearly larger since around 1990 than before, as many TGs are installed during satellite altimetry era.The selection criteria are outlined in Mu et al. (2024).We first choose TGs with data duration larger than 20 years over the period 1950-2022.We then inspect each TG with eyes to eliminate those containing suspicious anomalies (sudden jumps >200 mm).The white dots in Figure 1a illustrate the distribution of these 794 TGs.We consider sterodynamic sea level changes simulated by 33 CMIP6 climate models (see Table S1 in Supporting Information S1).We select these models becaue they provide gridded sterodynamic sea level and global mean thermosteric sea level with positive trends over 1950-2014.Additionally, we utilize sea level pressure data (Kistler et al., 2001) to remove the inverted barometer effect at TGs (Piechch & Ponte, 2015).

Results and Discussion
We first present the global ocean mass increase for the satellite altimetry era , since there are more independent data available for comparison and evaluation.During this period, the global ocean mass increase is estimated to be 2.15 ± 0.72 mm/yr.The estimate and uncertainty are based on the mean and spread (standard deviation), respectively, of the inferred ocean mass increase ensemble, because each climate model allows us to infer an estimate of ocean mass increase, therefore we have 33 estimates of inferred ocean mass increase in total.During 1993-2014, the inferred ocean mass increase, combined with the (ensemble mean) sterodynamic sea level from CMIP6 climate models, agrees excellently with the GMSL observed by satellite altimetry (see Figure 2a), demonstrating a robust closure of the GMSL budget.The satellite altimetry GMSL time series is released by the Archiving, Validation, and Interpretation of Satellite Oceanographic (AVISO), and the GIA effect has been corrected.Throughout 1993-2022, we also observe a good agreement between the satellite altimetry-based  Frederikse et al. (2020aFrederikse et al. ( , 2020b)).The white dots represent 794 tide gauges.Panel (b) shows the SLF contribution to global mean sea level rise (GMSL), black line is the SLF contribution that is inferred by our data assimilation, red line is the original SLF contribution from Frederikse et al. (2020aFrederikse et al. ( , 2020b)), blue line is the SLF contribution due to terrestrial water storage (TWS) variations.The apparent negative trend in SLF TWS is mainly related to water retention by dam construction (Chao et al., 2008).
GMSL and the GMSL derived by combining the inferred global ocean mass increase with the global thermosteric sea level rise estimated by the Institute of Atmospheric Physics (IAP) (Cheng et al., 2017), although we note that the agreement is less strong after 2015.We stress that we must acknowledge the uncertainty in global thermosteric sea level trends.For instance, we find a large trend difference between the IAP thermosteric trend and the Japan Agency for Marine-Earth Science and Technology (JAMSTEC) thermosteric trend (Hosoda et al., 2008) over 2002-2022 (Table 1).Combining our inferred ocean mass increase along with the thermosteric trend from the JAMSTEC offers a better closure of the GMSL budget.However, the evaluation presented in Table 1 has not considered the wet troposphere correction (WTC) effect on altimeter-based GMSL trend, which is shown to contribute to the GMSL rate over the altimeter era.Results by Barnoud et al. (2023) suggested that the GMSL rate increased by about 0.20 mm/yr for 1992-2021 after they used an alternative WTC data set with high quality.Taking into account the WTC effect increases the discrepancy between altimeter-based GMSL rate and the rate of our inferred ocean mass increase + global thermosteric sea level (Table 1).
We particularly highlight the comparison between the inferred ocean mass increase and that observed by GRACE/GRACE-FO.This study adopts six solutions of GRACE/GRACE-FO time variable gravity, including three solutions in terms of spherical harmonic coefficients, and three mascon solutions (Loomis et al., 2019;Save et al., 2016;Watkins et al., 2015).Notably, there is a strong consistency among the GRACE/GRACE-FO ocean mass trends, see Figure S2 in Supporting Information S1.The ensemble mean of these trends is calculated to be 2.04 ± 0.34 mm/yr for the period 2002-2022, which aligns well with the estimate by Barnoud et al. (2021) for 2005-2019 (2.14 ± 0.02 mm/yr) and is slightly larger than the estimate by Royston et al. (2020) for 2005-2015 (1.75 ± 0.10 mm/yr).The trend of the inferred ocean mass increase is larger than the trends of GRACE/GRACE-FO ocean mass variations (Table 1), which yields a smaller sea level discrepancy (i.e., the residual of sea level budget) that is evaluated with altimeter-based GMSL and two thermosteric observations.However, it should be noted that short term (2002-2022) evaluation of the inferred ocean mass could be subject to larger uncertainty, potentially exceeding the spread estimated from the CMIP6 climate models, because the inferred ocean mass increase is expected to be more reliable on a long-term scale, considering the influence of the sterodynamic sea level by the CMIP6 climate models.Additionally, we assume a constant rate for sterodynamic sea level changes after 2015, as CMIP6 climate models only provide historical outputs until 2014 (see assimilation approach in Supporting Information S1).This assumption may introduce additional errors into the inferred ocean mass increase.We compare the ocean mass increase trend for 2003-2014, a shorter period that GRACE provides reliable estimates and is covered by the CMIP6 climate model outputs.The six solutions of GRACE time-variable gravity show a mean trend of 1.98 ± 0.34 mm/yr.The inferred ocean mass increase suggests a trend of 2.42 ± 0.76 mm/yr.Their trend difference is close to the trend difference for the period 2002-2022.
Our results suggest that the inferred ocean mass increase dominates the GMSL rise since 1950 (Figure 2b).During 1950-2014, our data assimilation approach determines a 1.48 ± 0.46 mm/yr trend for global ocean mass increase (Figure 2b and Table 1).This ocean mass trend is substantially larger than the sterodynamic sea level trend (0.53 ± 0.26 mm/yr) estimated by the CMIP6 climate models.Meanwhile, there is a strong agreement between the trend derived from the CMIP6 climate models and the IAP thermosteric sea level trend for 1950-2014 (Figure S3 in Supporting Information S1), but the trend difference is apparent between the CMIP6 climate models and the EN4 (Good et al., 2013) thermosteric sea level (Figure S3 in Supporting Information S1; Table 1).We also observe an evident trend difference between the IAP curve andn the EN4 curve (Figure S3 in Supporting Information S1), indicative of uncertainty in thermosteric sea level observations.

10.1029/2023GL108056
Here, we introduce a new GMSL definition that combines the inferred global ocean mass increase with the sterodynamic sea level changes from the CMIP6 climate models.This definition of GMSL is compatible, given that both components are subject to the data assimilation approach.Remarkably, this newly defined GMSL is excellently consistent with the GMSL reconstructed by Church and White (2011).Notably, the overall trends agree well, and the time series curves demonstrate a high level of agreement (Figure 2b).Given the good consistency between the newly defined GMSL and the GMSL reconstructed by Church and White (2011), one immediate implication is that the new GMSL offers a possible explanation for the causes of the GMSL reconstructed by Church and White (2011).However, it is crucial to note that the new defined GMSL relies, to a large extent, on the performance of CMIP6 climate models, as they play a vital role (i.e., as driver) in the data assimilation approach.For example, from 1950 to 1980, the CMIP6 climate models generate a near-zero trend for the sterodynamic sea level, consequently, the newly defined GMSL is predominantly influenced by the ocean mass increase (see the blue and red lines in Figure 2b).Although our defined GMSL agrees well with the results by Church and White (2011), notable differences emerge when compared to other sea level reconstructions (see Figure S4 in Supporting Information S1).For instance, Jevrejeva et al. (2014) and Ray and Douglas (2011) presented reconstructed GMSL rates of 1.49 ± 0.14 mm/yr and 1.56 ± 0.13 mm/yr for 1950-2007, respectively.These discrepancies can be attributed to differences in the reconstruction methods and selection of TGs.For example, Ray and Douglas (2011) utilized only 89 TGs, a substantially smaller number than our data set.
More insights can be gained by comparing the inferred ocean mass increase to the original SLF by Frederikse et al. (2020aFrederikse et al. ( , 2020b)).The comparison of time series reveals substantial deviations between our inferred ocean mass increase and the original SLF before 1980 (Figure 1b).During 1980-2018, our inferred ocean mass increase exhibits a trend of 1.71 ± 0.53 mm/yr, while the original SLF indicates a trend of 1.40 ± 0.15 mm/yr.Between 1950 and 1980, the original SLF displays a slightly increasing trend (0.19 ± 0.14 mm/yr).In contrast, our inferred ocean  mass increase suggests a larger trend (1.54 ± 0.61 mm/yr).The primary reason for this difference is the significant drop in the contribution of TWS variation ( 0.65 ± 0.09 mm/yr), which is mainly induced by water retention owing to dam construction (Chao et al., 2008).Why does our inferred ocean mass increase not observe or, at the very least, reflect the evident drop?To address this issue, we discuss two factors: the limitation of the SLF reference rate, and the geometry of TGs.
The primary assumptions regarding the use of the SLF (i.e., the pattern of the inferred ocean mass increase) should be acknowledged.First, in contrast to the use of multiple patterns, such as the three patterns employed by Hay et al. (2013), we employ a single pattern of SLF as a reference.This pattern represents the overall trend  of the original SLF and includes all mass variations in global land areas.While this choice offers technical advantages, such as reducing the dimensions of the data matrix, simplifying the computations, and requiring fewer constraints on the parameters, it also introduces limitations.The major limitation is that we can only infer the (total) global ocean mass increase and may not accurately distinguish individual contributions (e.g., TWS variation), because, by merging all four processes into a single pattern, we assume that all processes are either underestimated or overestimated with exactly the same amplitudes at the same time, despite that this underlying assumption is unlikely true in reality.Another limitation is that using the overall trend might mask some temporary changes, such as the significant water retention due to dam construction from 1950 to 1980 (Chao et al., 2008).Although the water retention contributes substantially to the total TWS variation over 1950-1980, its spatial signature might be less distinct in the reference rate of the SLF that covers the time span 1950-2018 and includes other components.This implies that it is challenging to robustly recover an individual component over a particular subperiod.
We highlight that the geometry of TGs is expected to have influence on the inferred ocean mass increase.A dense and evenly distributed network of TGs is advantageous as it captures key features of the SLF (Figure 1a), thereby benefiting the data assimilation process.Conversely, a sparse and uneven distribution could introduce bias in estimating the global ocean mass increase.In our data assimilation, a total of 794 TGs are employed.However, the geometry of TGs is not fixed as illustrated in Figure 1a, it changes over time (Figure S5).During the early 1950s, we only have approximately 260 TGs, increasing to more than 400 since 1966, and remaining between 500 and 600 since 1980.The geometry of TGs before 1980 might create insufficient constraints, failing to recover or reflect the drop in TWS variations.During 1993-2022, the inferred ocean mass increase closes the global sea level budget.This closure certainly demonstrates the good performance of our data assimilation, but it could also be tightly related to the number of TGs and their geometry.Establishing this relation implies that less TGs lead to poorer performance of the data assimilation or larger uncertainty in the inferred ocean mass increase.Note that this uncertainty only considers the spread due to the CMIP6 climate models, and we have not accounted for the geometrical influence of the TGs network.The uncertainty should be larger than the current estimates, especially before 1980.Despite the uncertainty issue, our results prove that the geometry of TGs network, although particularly distributed along coast, is capable of constraining a reasonable global mean, at least during the satellite altimetry era.
We expound the role of CMIP6 climate models, because they are crucial to the determination of global ocean mass increase, and might be related to the apparent difference between the original SLF (red line in Figure 1b) and our inferred global ocean mass increase.It is important to note that CMIP6 climate models do not have an explicit relation to specific components of the SLF, such as TWS variation.The mathematical structure of the data assimilation approach underscores its reliance on the outputs from CMIP6 climate models (Hay et al., 2013;Mu et al., 2024), thus highlighting the critical importance of climate model performance.A notable challenge arises in that the data assimilation is directly driven by local simulations of sterodynamic sea level changes at TGs rather than the global mean, which is typically assumed to be more reliable than the spatial variability.To address the spatial variability along TGs, we introduce a random process at each TG, and include it in the state vector in our data assimilation (Supporting Information S1).This strategy improves the long-term sea level reconstruction at local and regional scales, our analysis reveals that the reconstructed sea level rise at the TGs from 1950 to 2020 is well explained by the sum of contributions, including the inferred ocean mass increase, sterodynamic sea level changes from climate models, and the introduced random process (Figure S6 in Supporting Information S1).However, it is noteworthy that the ensemble mean of random processes at all TGs is surprisingly small, basically, this trend can be negligible over 1950 to 2020.This finding implies that while the random process may enhance the sterodynamic sea level changes at local scale, but it cannot mitigate the mean biases in CMIP6 climate models (if any) for all TGs.For instance, a small trend is observed in the average of the sterodynamic sea level changes at all TGs during 1950-1980 (Figure S3 in Supporting Information S1).This small trend might be one of the reasons that cause the large difference between our inferred ocean mass increase and the original SLF.

Concluding Remark
Based on TG data and climate model simulations, we develop a data assimilation approach to infer ocean mass increase at the global scale.This approach reveals a 2.15 ± 0.72 mm/yr ocean mass trend for the satellite altimetry era.When combined with the ensemble mean sterodynamic sea level from 33 CMIP6 model simulations, the inferred ocean mass trend closes the sea level budget over the satellite altimetry era.Furthermore, the inferred ocean mass increase provides an explanation for the GMSL rise with estimates of ocean thermal expansion (Cheng et al., 2017).The newly proposed GMSL definition, integrating the inferred global ocean mass increase with sterodynamic sea level changes from CMIP6 models, closely aligns with the GMSL reconstructed by Church and White (2011).Beyond estimating ocean mass increase, our results contribute to a deeper understanding of the underlying causes of GMSL rise.However, the excellent agreement between the GMSL defined in this paper and the GMSL reconstructed by Church and White (2011) should be interpreted cautiously, because different sea level reconstructions yield various trends and time series curves due to factors such as methods and the selections of TGs (Church et al., 2004;Dangendorf et al., 2019;Hamlington et al., 2011;Hay et al., 2015;Wenzel & Schröter, 2010).This implies that our defined GMSL may deviate from other sea level reconstructions (Figure S4 in Supporting Information S1).
In this contribution, our data assimilation approach exploits observational constraints (i.e., TGs) to reconcile with the contributing sources of sea level rise.Given the assumptions of this approach, we recognize its limitations and difference from the results by other methods (Figure 1b), and stress the important role of climate models.We anticipate that improved climate models in the future could reduce their spread and provide robust outputs over short time scales, such as a 10-year period, leading to a more accurate estimate of the global ocean mass increase even for a decade.

Figure 1 .
Figure 1.The spatial pattern and time evolution of sea level fingerprint (SLF).Panel (a) shows the linear trend of SLF for 1950-2018, which is computed from the data sets released byFrederikse et al. (2020aFrederikse et al. ( , 2020b)).The white dots represent 794 tide gauges.Panel (b) shows the SLF contribution to global mean sea level rise (GMSL), black line is the SLF contribution that is inferred by our data assimilation, red line is the original SLF contribution fromFrederikse et al. (2020aFrederikse et al. ( , 2020b)), blue line is the SLF contribution due to terrestrial water storage (TWS) variations.The apparent negative trend in SLF TWS is mainly related to water retention by dam construction(Chao et al., 2008).

Figure 2 .
Figure 2. Global mean sea level rise (GMSL).Panel (a) shows GMSL for satellite altimetry era, black line is observed by satellite altimetry, blue line is the sum of the reconstructed mass component plus CMIP6 sterodynamic sea level, red line is the sum of the reconstructed mass component plus IAP thermosteric sea level; panel (b) shows GMSL for 1950-2014, black line is the GMSL reconstructed by Church and White (2011), blue line is the reconstructed global mass component, orange line is the CMIP6 sterodynamic sea level, and red line is the sum of the reconstructed mass component and the CMIP6 sterodynamic sea level.

Note.
Uncertainty estimates: For CMIP6 sterodynamic sea level and the inferred global ocean mass, we consider model spread; for altimeter and GRACE/GRACE-FO, we mainly consider known systemic uncertainty, for example, the drift in altimeter and GIA uncertainty in GRACE/GRACE-FO as suggested byChambers et al. (2017); for the reconstruction byChurch and White (2011), Argo (JAMSTEC), IAP and EN4, we consider twice formal error by least-squares; For the calculation of sum and residuals, we consider uncertainty propagation.a" Rec." = reconstructed.b "Obs."= observed."c "Res." = residual.d "CW" = Church and White (2011).
SLF is then delivered.In Equation1, there are both known and unknown variables.The unknown variables are estimated by our data assimilation.The known variables serve driving roles, and are acquired from models.The known part of SL Stero is computed from the Coupled Model Intercomparison Project Phase 6 (CMIP6) climate models, • (t) • (t)

Table 1
The Reconstructed and Observed Sea Level Rise for Different Periods (Unit: mm/yr)