## Introduction

Changes in global mean sea-level (GMSL) provide an important measure of the warming climate1,2,3, but regional relative sea-level (RSL) is most relevant for planners and decision-makers4,5. Since no process that affects sea-level causes a globally uniform sea-level change, regional sea-level changes usually differ from GMSL rise. Over the modern satellite era (1993-present) regional rates of geocentric sea-level (GSL) (i.e., sea-level relative to a reference geoid) can be more than double that of the GMSL in some locations while being near zero at other locations6,7,8. Local vertical land motion (VLM) adds to these deviations from the global mean9,10,11,12,13. Projections of future regional RSL changes typically rely on a thorough understanding of the underlying processes and how they contribute to sea-level change on differing temporal and spatial scales14,15. However, the diversity of these processes, the temporal and spatial scales over which they vary, and the available observations of individual processes make such a process-based assessment challenging.

The coastlines of the U.S. provide a particularly good example of the regional variability of recent RSL changes. During the modern altimeter era both tide gauges and altimetry observe a faster sea-level rise along the U.S. East coast than along the West coast (Fig. 1). In addition, there are local variations in the rate of RSL along each coastline. Many studies have investigated recent sea-level trends along coastlines of the U.S. and found that the high rates along the U.S. East coast have been associated with VLM and glacial isostatic adjustment16,17,18, as well as ocean sterodynamic processes, with different processes acting North and South of Cape Hatteras where the Gulf Stream separates from the coast19,20,21,22,23,24. The highest rates of RSL rise in the continental U.S. are found in the western Gulf of Mexico, driven in large part by high rates of subsidence associated with subsurface fluid withdrawal (e.g.,25). In contrast to elevated rates seen along the Atlantic and Gulf coasts, the Pacific coast of the U.S. has seen rates lower than the global average during the satellite-altimeter era (e.g.,26,27,28). Decadal climate variability that is partly represented by the Pacific Decadal Oscillation (PDO) has played a role in suppressing recent RSL rise along this coastline over the satellite era, although similar decadal variability has also led to periods of substantially higher rates of rise over the course of the 20th century29,30,31.

Understanding of the processes contributing to RSL along the U.S. coastlines has improved in recent years due partly to the maintenance and expansion of the sea-level observing network over this time period. Despite this rich era of observations, gaps or limitations in our understanding persist (e.g.,32) that impact our ability to provide assessments of future RSL14. In this study, we attempt to explain the RSL trends observed along U.S. coastlines during the satellite-altimeter era by accounting for and combining contributions from individual processes. Specifically, we investigate whether observed VLM, sterodynamic effects, and ocean mass changes can explain local sea-level changes over 1993–2018 at 55 tide-gauge locations around the contiguous U.S. (Fig. 1). The goal of this work is to understand whether the identified processes fully account for RSL trends measured by tide gauges. We estimate the contributions of the different drivers of RSL rise and compare their sum to the observed sea-level trends over 1993–2018. Similar local sea-level budget exercises have been performed for some parts of the U.S. coastline and for different periods (e.g.,16,18,33,34). Our focus here is on the satellite-altimeter time period and the entirety of the coastal U.S. By only covering the time period from 1993 to 2018, the satellite-altimeter data provides an additional check on our process-based understanding. In cases where the budget “closes”, there may be increased confidence that sea-level reconstructions and projections are accounting for the relevant processes, as this demonstrates a more complete understanding of why sea-level is changing. For locations where the process contributions do not match the observed RSL trends, we attempt to identify where our understanding is lacking and how this might be connected to the available observations.

## Results

### Drivers of regional relative sea-level change

For the assessment of sea-level at location r, we compare regional RSL trends (RSLTG) to the effects of contemporary mass redistribution (CMR), sterodynamic effects (SD), glacial isostatic adjustment (GIA), and VLM. The difference between RSLTG and the sum of these components gives a residual (RES), which, along with its uncertainties, we use as an indicator for our ability to fully explain observed RSL trends. We write the relationship between these terms using the following budget equation:

$$RS{L}_{TG}(r)\,=\,\, CMR(r)\,+\,SD(r)\,+\,GI{A}_{RSL}(r)\,\\ -\,(VLM(r)\,-\,GI{A}_{VLM}(r))\,+\,RES(r)$$
(1)

GIA causes both sea-level changes and VLM, and therefore we separate these components to avoid double-counting:

$$GI{A}_{RSL}(r)\,=\,GI{A}_{GSL}(r)\,-\,GI{A}_{VLM}(r).$$
(2)

CMR is the GSL trend associated with contemporary mass redistribution resulting from changes in land ice mass and terrestrial water storage (Fig. 2a). SD is the trend in sterodynamic sea-level (see Gregory et al.35 for extended definition) that encompasses both global-mean steric changes and regional sea-level changes associated with ocean dynamics (Fig. 2b). GIARSL is the RSL trend associated with glacial isostatic adjustment that encompasses both the effects on GSL (GIAGSL, Fig. S1b) and VLM (GIAVLM, Fig. 3a). VLM is the VLM trend observed by GPS (Fig. 3b). Since VLM encompasses the VLM associated with GIA, we subtract GIAVLM from VLM in Eq. 1 to avoid counting this contribution twice, and the difference between the two is shown in Fig. 3c. Subsidence is defined as negative VLM, and leads to a positive contribution to RSL. We correct RSLTG for the inverted barometer effect, which only has a small effect on the trends (see Fig. S1a and the Methods Section). We evaluate all quantities over 1993 to 2018. For altimetry-derived GSL trends (GSLALT), Eq. 1 reduces to the contemporary mass redistribution, sterodynamic variability, GIA, and residual terms:

$$GS{L}_{ALT}(r)\,=\,CMR(r)\,+\,SD(r)\,+\,GI{A}_{GSL}(r)\,+\,RES(r)$$
(3)

By evaluating Eqs. 1, 3 along the coastlines of the U.S. with adequate attention to the uncertainty in the relevant processes, we assess our ability to explain recent RSL trends and the degree to which these limitations impact our understanding of them.

### Contemporary mass redistribution

Due to gravitation, rotation, and deformation (GRD) effects, exchange of water mass between land and ocean, such as melting of glaciers and ice sheets or changes to the hydrological cycle, results in sea-level changes that vary from place to place36,37,38. In general, mass loss causes sea-level to drop near the source of the loss, to rise at a reduced rate compared to the global average at intermediate distances from the source, and to rise at a rate exceeding the global average at larger distances from the source.

Here, we use the GRD patterns from Frederikse et al.39 (Fig. 2a) who computed mass changes and the GRD response from four sources: The Antarctic Ice Sheet, Greenland Ice Sheet, glaciers, and land hydrology (which includes groundwater withdrawal, dam retention and natural water storage variability). The global-mean contribution from these four processes is 1.97 ± 0.35 mm/y over 1993–2018 (90% CI). Ice mass loss from the Antarctic ice sheet causes a roughly uniform trend along the coastlines of the U.S., while ice mass loss from Greenland and Arctic glaciers leads to a gradient along the U.S. east coast with increasing RSL trend contributions from north to south. Along the Pacific coast of the U.S., ice mass loss from the Alaskan glaciers leads to a similar north–south gradient.

### Sterodynamic sea level variability

The sterodynamic contribution cannot be observed directly: from hydrographic observations, including Argo profiling floats, steric changes can be computed, but because sterodynamic effects also contain bottom pressure changes, especially on continental shelves, steric changes alone cannot be used to estimate the sterodynamic signal32,40. In terms of satellite observations, the footprint of satellite altimeters, uncertainty in the corrections applied to the data41, and coarse resolution of the GRACE satellites42 further inhibit the assessment of coastal trends. To estimate the sterodynamic contribution, we use the Estimating the Circulation & Climate of the Ocean43 framework. ECCO is a data-constrained ocean circulation model that has been used for a wide range of investigations into sterodynamic sea-level across different timescales (e.g.,44,45,46). Figure 2b depicts the spatial pattern of the sterodynamic sea-level trends around the U.S., showing the different regimes along both coastlines with trends on the order of 2 mm/yr along the East coast, while the trends along the West coast are substantially smaller and often close to zero.

### Glacial isostatic adjustment and vertical land motion

Vertical movement of land plays a key role in local RSL changes at many locations, including along U.S. coastlines9,10,11,12,13,17,25,47. One process that causes substantial VLM along the U.S. coastlines is glacial isostatic adjustment (GIA)35,48, which is the ongoing solid-Earth response to the retreat of ice sheets after the last ice age. The impact of GIA on coastal VLM is especially noteworthy along the U.S. East Coast, where the ongoing collapse of the peripheral forebulge causes subsidence and aggravated rates of RSL rise18,49,50,51. To assess the GIA contribution at each tide gauge, we use the ensemble of GIA models from Caron et al.52. In addition to GIA, works such as Burgette et al.53 have shown that interseismic strain associated with tectonic activity can cause uplift of up to several millimeters per year along parts of the western U.S. coastline. Apart from large-scale patterns of land motion due to GIA and tectonics, VLM occurs on much smaller spatial scales54,55. Compaction of sediments due to subsurface fluid extraction is an important driver of these local VLM patterns. Subsidence related to groundwater withdrawal can be especially pronounced in river deltas with large populations and extensive agriculture (e.g.,56,57,58,59,60). These effects are visible along the Gulf Coast25 and Atlantic coast61, with subsidence rates of up to several millimeters per year.

To estimate VLM unrelated to GIA, we use Global Positioning System (GPS) observations. GPS-based trend estimates to assess VLM at tide gauges have been used extensively in recent studies (Woppelmann and Marcos, 201662,63,64. Many GPS records are much shorter than the altimeter record65,66, and most tide gauges do not have a collocated GPS station. To address these limitations, we use a GPS imaging technique67 to estimate VLM at the tide-gauge location, which involves computing the average of all GPS observations around the tide gauge, and weighting by the record length and distance from the tide gauge (see Methods).

We separate the VLM contributions from GIA from those arising from other processes (Eq. 1). While fully attributing the rate of VLM to particular processes at each tide-gauge location is beyond the scope of this study, it is possible to assess the extent to which VLM may be occurring as a result of local processes. The GIA VLM estimates (GIAVLM in Eqs. 1, 2) at each tide gauge are shown in Fig. 3b, while the GPS-based VLM rates (VLM in Eq. 1) are shown in Fig. 3a. Note that, along the U.S. Atlantic coast, the GIA estimates used here differ from those obtained in other studies (see Fig. S2 for comparison). Since the VLM contribution is tied here solely to the GPS-based estimate (Eq. 1), the choice of GIA model affects the partitioning of VLM into GIA and non-GIA components. Also note that the GIA-induced geoid changes differ among individual estimates. While also absorbing any errors in the chosen GIA model, the differences between the GPS-measured rates and the VLM associated with GIA provides an estimate of the VLM associated with more local processes like groundwater withdrawal and hydrocarbon extraction (Fig. 3c). As an example, there are high rates of non-GIA subsidence along the Gulf Coast, consistent with subsurface fluid extraction that has been ongoing in the area25. For more detailed attribution of VLM rates along the east coast in terms of GIA and other processes, the reader is referred to past studies dedicated to this topic (e.g.,17,18,51).

### Regional sea-level evaluation using tide gauges

We separate the U.S. coastlines into three regions: (1) the northeast coast (the Atlantic coast north of Cape Hatteras), (2) the southeast coast (the Atlantic coast south of Cape Hatteras combined with the Gulf coast), and (3) the west coast (the entire U.S. Pacific coast). This separation follows naturally from the spatial covariance structure of coastal sea-level variability (e.g.,22,68,69) and provides a structure for presentation of results.

The RSL trend assessment for the 18 tide gauges used along the northeast (ordered from south to north) is shown in Fig. 4 (numerical values tabulated in Table S2). The majority of the tide gauges have trends near or above 4 mm/yr. These high rates result in part from GIA-induced subsidence, particularly for mid-Atlantic locations, which is consistent with previous studies in the region16,18,49. The GPS-based VLM trends indicate that some locations have higher subsidence rates than predicted by GIA models (e.g., Solomon’s Island), suggesting additional VLM not related to GIA17,70,71. For 15 of the 18 stations, the residual is not statistically different from zero at the two-standard deviation level. However, Fig. 5 shows that despite the non-significance of their difference from zero, the majority of the residuals are negative, which may point to biases in one or more of the terms on the right-hand side of Eq. 1. The sterodynamic rates in this region have magnitudes around 2.5 mm/y, and do not vary significantly across the region. The coastal sterodynamic trends estimated from ECCO North of Cape Hatteras are higher than in the surrounding areas (Fig. 2b), whereas a similar gradient is not observed in the satellite-altimeter trend pattern (Fig. 1).

Along the Southeast (Fig. 6) 16 out of 17 tide gauges show a trend exceeding 4 mm/yr. The rate associated with contemporary mass redistribution is near the component’s global mean of 1.97 mm/yr, while the average contribution from sterodynamic effects approaches 2.5 mm/yr in the region. The large tide-gauge trends in Rockport and Grand Isle are tied to residual VLM, associated with subsurface fluid extraction25. For the other stations, the residual VLM trend is not significantly different from zero at the two-standard deviation level. In total, 13 of the 17 southeast tide gauges have a RSL residual not statistically different from zero. Similar to the northeast, despite their non-significance the majority of residuals in this region are negative (Fig. 5).

Finally, we consider 20 tide gauges on the west coast (Fig. 7). Consistent with altimetry measurements, the tide-gauge trends are lower than those in the other two regions. Except for the San Diego tide gauge, which is undergoing a high rate of non-GIA subsidence, all of the tide gauges show rates below 3 mm/yr. For the west coast, the sterodynamic contribution is much smaller than the other regions, with rates only slightly above zero. This lower rate over the altimetry era is part of a seesaw pattern in the North Pacific Ocean, and likely caused by internal variability such as the PDO28. The lower sterodynamic rate is the primary driver of the lower RSL rates when compared to trends observed in the other regions. In this region, 19 locations have residuals not statistically different from zero, with 4 of these residuals greater than zero. Large positive residuals are found at Santa Monica and Port San Luis, and a large negative residual is estimated at several gauges—mostly concentrated in the Northwest (Fig. 5). These high-magnitude residuals are part of a larger gradient decreasing from south to north along the west coast. RSL trend comparisons on the west coast are more susceptible to uncertainties in GIA and VLM measurements. The model ensemble from Caron et al.52 shows a large inter-model spread, especially along the Alaskan coastline and around Seattle. Furthermore, GPS stations in this region are generally more sparsely distributed which makes VLM trend estimation with tide gauges more uncertain.

### Regional sea-level evaluation using satellite altimetry

Based on comparisons around the U.S. coastlines, we are able to explain RSL trends within the uncertainty estimates at 47 of the 55 tide gauges considered here. This leaves eight locations where trends cannot be explained in terms of the contributing processes. Additionally, even for residuals smaller than our uncertainty estimates, regional offsets or biases (e.g., northeast region) have been identified that point to potential systematic issues that need further investigation. The satellite-altimetry data provides the opportunity for an additional observation-based assessment. As discussed above, the satellite altimeters measure GSL, which is independent of VLM. As an initial comparison, the altimeter-measured trend at the point nearest the tide gauge is compared to the trend measured at the tide gauge with the VLM trend removed (Fig. 8a). The comparison leads to a substantial spread, particularly for the southeast region where the estimated VLM trends are high.

This evaluation suggests that the VLM estimates are playing a role in driving locations’ larger residuals. To investigate this potential disagreement, we evaluate how well the sterodynamic, contemporary mass redistribution, and non-VLM GIA trend contributions explain the altimeter-measured trend (Eq. 3). For the west coast, the sum of the contributions generally agrees with the altimeter trends (Fig. 8b). Examining the distribution of the associated residuals, however, we see that the locations in the northwest have consistently negative residuals. Because of the proximity to the Laurentide Ice Sheet, trends in this region are more susceptible to uncertainties in GIA models, potentially providing an explanation for the negative residuals.

For the northeast, the reconstructed trends from the combined contributions are all higher than the altimeter-measured trends (Fig. 8b). When combined with the negative residuals obtained with the tide gauges (Fig. 5), it appears that the trend contributions from one of the processes is being overestimated in the northeast. While it is difficult to quantitatively assess the source of overestimation, the trends associated with sterodynamic effects in the region are particularly large and likely contributing to the negative biases in this region. Based on the bias present in both this reconstructed sea level comparison with the altimetry and tide gauges, however, it is possible to rule out VLM as the primary contributor the negative offset in the residuals.

For the southeast region, the reconstructed trends generally agree with the trends measured by altimetry, at least relative to the northeast region. As shown in Fig. 5, while there is no coherent bias in the RSL residuals in this region, the reconstructed GSL residuals with respect to the altimetry trends show a small positive bias (with the exceptions of Rockport and Port Isabel) (Fig. 8d). These results point toward a long-wavelength signal (possibly associated with processes responsible for residuals in the northeast) playing a role in driving a smaller regional bias, and non-GIA VLM driving larger errors on a local scale.

### Assessment of vertical land motion estimates

Based on the large spread in Fig. 8a relative to the spread in Fig. 8b, it is likely that the local VLM processes are playing a role in driving the disagreement between the tide-gauge trends and the satellite-altimeter trends for all regions. While GPS observations theoretically capture the relevant VLM processes, the lack of collocated GPS observations, the short and inconsistent time periods of the GPS records, and the presence of discontinuities and low-frequency noise in GPS time series contribute to additional uncertainties that are impossible to quantify. To evaluate the impact of the lack of collocation, Fig. 9 shows the residuals in Fig. 5 compared to the distance of the tide gauge to the nearest GPS station. There is no clear relationship between the distance from the tide gauge to the nearest GPS station and the magnitude of the residual. Thus, we cannot conclude that a lack of collocation causes larger residuals. Even though a shorter horizontal distance between a tide gauge and GPS receiver should provide a more accurate estimate of VLM experienced by the tide gauge, works such as Keogh and Törnqvist72 suggest that it does not necessarily guarantee it, for example, when the receiver and tide gauge have foundations at different depths.

## Discussion

We have sought to explain and delineate the contributors of the RSL trends observed along the coastlines of the U.S over the time period from 1993 to 2018. Similar investigations have been conducted in other regions and/or other periods16,18,33,34,73,74, but the entirety of the coastal U.S. provides diversity in terms of the combination of processes affecting regional RSL rise. Additionally, the chosen time period for this investigation offers the opportunity to examine regional RSL change with both in situ measurements and satellite observations.

In 47 of the 55 tide gauges used in this investigation, the difference between the combined estimated contributions from contemporary mass redistribution, sterodynamic effects and VLM is not statistically different at the 95% confidence level from the trends estimated directly from the tide-gauge locations. We do find that there are still multiple sources of uncertainties that hinder the complete attribution of observed sea-level changes to the underlying processes. On larger spatial scales, clear patterns can be seen and explained: observed trends on the west coast are driven primarily by contemporary water mass redistribution and minimally by sterodynamic variability, and many locations around the Gulf coast have experienced particularly high rates of subsidence due to subsurface fluid withdrawal. On the Atlantic coast, sterodynamic and contemporary mass redistribution rates have similar magnitudes with GIA playing a larger role as tide gauges approach the forebulge to the north.

While the high number of locations with residuals not statistically different from zero demonstrates our ability to account for the trend contributions from the relevant processes, the uncertainty margins are still large, and—especially in the northeast—one or more of the components are possibly biased or not being assessed correctly from our process-based approach. Through additional testing, we have sought to understand the residuals from our analysis and have highlighted both the VLM and sterodynamic trend contributions as likely contributors at locations with larger or regionally biased residuals. Moving forward, assessing the spatial scales of the processes impacting VLM at a particular tide gauge would allow for a determination of how close a GPS station needs to be to account for VLM at the tide-gauge location. A key takeaway, however, is that few of the available tide gauges in the U.S. have a truly collocated GPS station, which would ultimately be a solution to many challenges covered here75. Another challenge remains the estimation of sterodynamic effects on coastal sea-level given the limitations of available observations and particular lack of information at the coast. The misfits found in the Northeast region are likely tied to issues with our sterodynamic estimate.

While these needed improvements are generally understood by the scientific community (e.g.,14,32), the results contained underscore the importance of maintaining and improving our observing network. As additions to the sea-level observing network are made and available sea-level records continue to lengthen, these gaps in knowledge can be filled and greater confidence can be placed in our understanding of past and ongoing sea-level changes. By demonstrating and communicating a process-based understanding of ongoing RSL rise, increased confidence can be placed in assessments that are used to inform planning efforts. Of additional importance is the knowledge of where uncertainty remains in our understanding of the relevant processes, which can be factored in when planning for future sea-level rise.

## Methods

### Sea-level observations from tide gauges and altimetry

Monthly tide-gauge records were retrieved from the Permanent Service for Mean Sea-Level76 (PSMSL, 2019). The global set was first reduced to just those along the coastlines of the contiguous U.S. To ensure the tide gauges considered provide representative data over the time period of interest, records that were less than 70% complete during the time period from 1993 to 2018 were removed. This led to a total of 55 tide gauges along the U.S. coastlines considered in this study (see Table S1). These tide gauges were corrected for the inverse barometer effect using the UK Met Office’s HadSLP2 dataset77 and the inverse barometer relation of Ponte et al.78. For altimetry, we use the gridded NASA MEaSUREs Gridded Sea Surface Height Anomalies Version 181279. To compute the altimeter-derived sea-level trend, we take the average trend of all grid cells within a 300 km radius of the tide-gauge location. The 300 km radius was chosen to match the averaging radius used for ECCO data and to maintain enough averaged grid points while capturing local sea-level signals.

### Contemporary mass redistribution estimates and glacial isostatic adjustment

Both GIA and contemporary mass redistribution cause gravitational, rotational, and deformation (GRD) effects, which affect tide-gauge, altimetry, and GPS observations. To consistently treat VLM and sea-level observations and since we have separated out the VLM terms in Eqs. 1, 2, we express all changes as GSL changes, and thus only use the geocentric GRD patterns to assess the effects of GIA and contemporary mass redistribution35,65,80.

We correct all tide-gauge and altimetry observations for the geocentric GRD effects associated with GIA (GIAGSL in Eq. 2) using the estimates from Caron et al.52 which comes with an estimate of the associated uncertainties.

For the GRD effects from global mass redistribution, we use the estimates from Frederikse et al.39. This study provides annual-mean GRD effects resulting from mass changes of ice sheets, glaciers, and land hydrology from groundwater depletion, natural variability, and dam retention. These estimates are based on in situ observations and models for 1993–2003, while for 2003–2019, the estimates are based on GRACE observations81,82. Both the in situ and GRACE mass estimates come with an estimate of the uncertainties, based on the individual in situ mass estimate uncertainties and the uncertainties in GRACE processing and GIA, which we propagate into GRD uncertainties. See Frederikse et al.39 for more details of the computation of these uncertainties.

### Sterodynamic sea level

To estimate coastal sterodynamic effects, we use the ECCO state estimate version 4 release 483,84. ECCO is an ocean state estimate, which combines an ocean model with a wide range of observations to compute a physically consistent best estimate of the state of the ocean and provides sterodynamic sea-level changes on an ~1-degree resolution grid. We use the dynamic sea-level variable (SSHDYN) from the model. To avoid the possible influence of small-scale features that lead to trends in single coastal grid cells, we average all ECCO grid points within a 300 km radius around each tide gauge. This 300 km is chosen as a trade-off between having multiple grid cells for each tide gauge and avoiding the inclusion of uncorrelated open-ocean signals.

Since ECCO is constrained to altimetry, the GMSL trend in ECCO does not just represent sterodynamic effects, but also contemporary mass redistribution effects. To avoid double-counting, we remove GMSL from ECCO and add back the observation-based global-mean thermosteric sea-level rise estimate of 1.16 ± 0.4 mm/yr from Cheng et al.85.

### Vertical land motion from GPS observations

To estimate VLM at each tide gauge, we use the GPS imaging method67, in which all available GPS observations around each tide gauge are averaged, weighted by the distance to the tide gauge and the uncertainty of the VLM trend at the GPS station (see Table S1 for list of GPS stations by tide gauge). We refer to Hammond et al.67 for a detailed description of the GPS imaging method. All VLM velocities are expressed in the ITRF2014 reference frame86. The origin of ITRF2014 tracks the secular changes in the position of the Earth center of mass (CM), which is consistent with our GIA and contemporary GRD estimates.

### Linear trend and uncertainty estimation

For the altimetry, tide-gauge, and sterodynamic term, linear trends from 1993 to 2018 were computed via least squares, and the uncertainty was computed as the standard error from the least squares estimate. To account for serial correlation, we reduce the degrees of freedom following the approach of Haigh et al.87. The uncertainty on the VLM trend is given by the imaging method. The GIA trend estimates used here contribute additional uncertainty, which is included in the uncertainty estimates.

To estimate uncertainties on the other values in Figs. 4, 6, 7, additional considerations are required. The uncertainties associated with the individual contributors are not strictly independent and must be correctly represented to obtain uncertainties on the combined (sum) contributions and residual terms. Specifically, we use the following equations to estimate the uncertainties on the sum of the contributors and the residual:

$${U}_{{SUM}}\,=\,\sqrt{{U}_{{CMR},P}^{2}\,+\,{U}_{{SD},P}^{2}\,+\,{U}_{{VLM}}^{2}\,+\,{U}_{{GIA}}^{2}\,+\,{U}_{{SUM},T}^{2}}$$
(4)
$${U}_{{RES}}\,=\,\sqrt{{U}_{{CMR},P}^{2}\,+\,{U}_{{SD},P}^{2}\,+\,{U}_{{VLM}}^{2}\,+\,{U}_{{GIA}}^{2}\,+\,{U}_{{RES},T}^{2}}$$
(5)

where CMR refers to the contemporary mass redistribution trend, SD is the sterodynamic trend, VLM is the VLM trend, and P and T refer to the process-based uncertainty of data and formal uncertainty from trend estimation (accounting for serial correlation88), respectively. USUM,T and URESID,T are estimated from the linear trend computation. The process-based uncertainty for contemporary mass redistribution (UCMR,P) is computed following Frederikse et al.39 and is based on the spread among multiple estimates of each individual component (glaciers, ice sheets, and terrestrial water storage), as well as uncertainties in GRACE observations after 2003.

The process-based uncertainty on the sterodynamic trend (USD,P) is provided by the uncertainty in the global-mean thermosteric trend from 1993 to 2018 that is added to the ECCO data (see Sterodynamic section above). Throughout the analysis, all uncertainties are addressed at the 1-sigma level until assessing budget closure and presenting results, at which point we double the values to become a 2-sigma estimate.