Antarctica: ice dynamics  A scenario-dependent projection based on the results of Levermann et al. (2014). GMSL rise is modeled as a quadratic function of time, beginning with the observational rate of dynamic mass loss in 2006 and reaching Lex at 2100, where x is chosen by the Monte Carlo from a normal distribution with zero mean and standard deviation λ. The parameters L and λ are scenario-dependent; for instance, RCP2.6 has L  =  56 mm and λ  =  0.92, RCP8.5 91 mm and 0.86. The 2100 rate is held constant between 2100 and 2300.
Greenland: surface mass balance    GMSL rise is projected from GMST change (Church et al., 2013) as the time integral of EFG(T(t)), where G gives the change in Greenland SMB as a cubic function of GMST change according to Equation (2) of Fettweis et al. (2013), derived from regional climate model projections. F is a factor representing systematic uncertainty in G, and E is a factor in the range 1.00–1.15 representing the enhancement of mass loss due to reduction of surface elevation. The Monte Carlo chooses E and F independently; E is uniformly distributed, and F  =  eN, where N is normally distributed with zero mean and standard deviation of 0.4. Projections of GMST change are the same as for Antarctic surface mass balance.     The 2100 rate is held constant between 2100 and 2300.
Greenland: ice dynamics     Scenario-dependent projection based on the literature at the time of AR5 (Church et al., 2013). GMSL rise is modeled as a quadratic function of time, beginning with the observational rate of dynamic mass loss in 2006 and reaching 0.020–0.085 m for RCP8.5 and 0.014–0.063 m for the other RCPs at 2100. The Monte Carlo chooses the final amount uniformly within the ranges given.        The 2100 rate is held constant between 2100 and 2300.
Glaciers   GMSL rise is projected from GMST change (Church et al., 2013) as mfI(t)p, where I(t) is the time integral of GMST change (in °C yr) since 2006. Four glacier models are represented by different f,p pairs, with f in the range 3–5 mm and p  ≈  0.7. The Monte Carlo gives equal probability to the four glacier models and chooses the random normally distributed factor m with a standard deviation of 20% representing systematic uncertainty. Projections of GMST change are the same as for Antarctic surface mass balance.      The same relationship with global surface temperature change is applied out to 2300 (Church et al., 2013) with a cap on the total sea level equivalent of 0.32 m to reflect current estimates of global glacier volume (Farinotti et al., 2019; Grinsted, 2013). Projections of global surface temperature change come from the CMIP5-based 16-member emulator ensemble of Palmer, Howard, et al. (2018; Table S2).
Land water storage        Scenario-independent projection based on the literature at the time of AR5 (Church et al., 2013). GMSL rise is modeled as a quadratic function of time, beginning with the estimated rate for 2006 and having its time-mean for 2081–2100 uniformly distributed within the range  − 10 to  + 90 mm by the Monte Carlo.  The 2100 rate is held constant between 2100 and 2300.
For the scenario-dependent terms, the ensemble spread arises from differences among the underlying CMIP5 (or emulator) simulations of GTE and GMST change and from any additional methodological uncertainties (Church et al., 2013). For each scenario, the climate model ensemble (CMIP5 or emulator) was treated as a normal distribution, with time-dependent ensemble mean QM(t) and standard deviation QS(t), where Q is GTE or GMST, both with respect to the time mean of 1986–2005, and t is time. Larger Monte Carlo ensembles were constructed with members Qi(t)  =  QM(t)  +  riQS(t), where {ri} is a set of normal random numbers (with zero mean and unit standard deviation). The {ri} are time-independent, and the same {ri} were used for GTE and GMST, so that variations within the ensemble were correlated over time and between the two quantities.
The glacier contribution to GMSL is based on a relationship between the global glacier contribution and GMST change (Church et al., 2013), which is also applied post-2100. The total contribution is capped at 0.32 m, based on current estimates of total glacier mass (Farinotti et al., 2019; Grinsted, 2013). However, we note that this is a simplistic assumption. It is possible that remaining glaciers might reach a new steady state under a stable future climate following preferential loss of low-altitude ablation areas, a possibility that was not accounted for in the AR5 projections, or here.
The different GMSL components are combined using a 450,000-member Monte Carlo simulation that samples from the underlying distributions. The procedure preserves the correlation between GTE and GMST change in the underlying CMIP5 model simulations (or the emulator ensemble for the period post-2100). As a result, many of the GMSL components are correlated, as discussed further in section 4. In addition, the effect of increased accumulation on the dynamics of the Antarctic ice sheet is represented in the same way as described in AR5 (Church et al., 2013; SM1.5), resulting in these terms also being weakly correlated. The sampled distributions are based on the 5th to 95th percentile ranges of the climate model simulations and literature-based assessed ranges for the scenario-independent terms. Each member of the Monte Carlo simulation is composed of a time series for each of the seven GMSL contributions listed in Table 1 with the correlations between terms preserved.
Local Sea-Level Projections
As we move to local MSL projections, a number of additional processes are taken into account. First, the spatial patterns of MSL change associated with each of the barystatic GMSL contributions (Table 1, ii–vii) are incorporated using estimates of the effects on Earth’s gravity, rotation, and solid earth deformation (GRD, Figure 3). Following previous studies (Bilbao et al., 2015; Palmer, Howard, et al., 2018; Perrette et al., 2013), the effects of local changes in ocean density and circulation are included by establishing regression relationships between global thermal expansion and local sterodynamic sea-level change in CMIP5 climate model simulations (see supporting information, Figures S1–S4). Finally, the spatial pattern of local MSL change from ongoing glacial isostatic adjustment (GIA, Figure 4) is included in our local MSL projections.
Figure 4: (a)–(c) three estimates of the effect of glacial isostatic adjustment (GIA) on sea-level change. The zero line is indicated by the dotted contours. (d) the standard deviation of the three GIA estimates. Units for all panels are mm yr^{−1}.
The projections of local MSL change for specific tide gauge locations (Figure 1, Table S1) are derived directly from the GMSL Monte Carlo projections described in the previous section. This represents an advance over the local MSL projections presented in AR5 (Church et al., 2013), which combined the different components post hoc using statistical approximations (see supplementary materials of Cannaby et al., 2016; Church et al., 2013). These approximations break the correlation structure among sea-level components and compromise the traceability of the local projections, including our understanding of how the different variances combine for total sea-level change locally.


Local Sea-Level Projections
As we move to local MSL projections, a number of additional processes are taken into account. First, the spatial patterns of MSL change associated with each of the barystatic GMSL contributions (Table 1, ii–vii) are incorporated using estimates of the effects on Earth’s gravity, rotation, and solid earth deformation (GRD, Figure 3). Following previous studies (Bilbao et al., 2015; Palmer, Howard, et al., 2018; Perrette et al., 2013), the effects of local changes in ocean density and circulation are included by establishing regression relationships between global thermal expansion and local sterodynamic sea-level change in CMIP5 climate model simulations (see supporting information, Figures S1–S4). Finally, the spatial pattern of local MSL change from ongoing glacial isostatic adjustment (GIA, Figure 4) is included in our local MSL projections.
Figure 4: (a)–(c) three estimates of the effect of glacial isostatic adjustment (GIA) on sea-level change. The zero line is indicated by the dotted contours. (d) the standard deviation of the three GIA estimates. Units for all panels are mm yr^{−1}.
The projections of local MSL change for specific tide gauge locations (Figure 1, Table S1) are derived directly from the GMSL Monte Carlo projections described in the previous section. This represents an advance over the local MSL projections presented in AR5 (Church et al., 2013), which combined the different components post hoc using statistical approximations (see supplementary materials of Cannaby et al., 2016; Church et al., 2013). These approximations break the correlation structure among sea-level components and compromise the traceability of the local projections, including our understanding of how the different variances combine for total sea-level change locally.
The local MSL projection Monte Carlo simulations presented here are computed as follows. For a given RCP scenario, a single instance of the 450,000-member Monte Carlo of GMSL is randomly drawn. Each instance includes a time series for the seven GMSL components that preserves the underlying correlations among them. The barystatic timeseries (Table 1, ii–vii) are combined with the corresponding GRD estimates (Figure 3) from one of the three sets at the tide gauge latitude and longitude. This selection is made at random with all GRD patterns based on the same model, in order to preserve any correlated errors. The only exception is for land water, for which only a single GRD estimate is available (Slangen et al., 2014). The timeseries of global thermal expansion is combined with a randomly drawn regression coefficient from one of the 21 CMIP5 models in order to estimate the sterodynamic sea-level change at the tide gauge location. The resulting seven timeseries of local MSL change are then combined with an estimate of the rate of MSL change associated with GIA using one of the three estimates (Figure 4) drawn at random. This procedure (shown schematically in Figure 5) is repeated 100,000 times for each tide gauge location to build up a distribution of MSL projections under each RCP scenario. Following the approach of AR5, we take the 5th and 95th percentiles of this distribution to indicate the spread of projections for individual components and the total MSL change.
Figure 5: A schematic representation of the Monte Carlo simulation performed for the local mean sea level (MSL) projections. The above process is repeated 100,000 times to build up a distribution of sea level projections for each tide gauge location for each RCP scenario.
Results
Global-Mean Sea-Level Projections
Our projections of GMSL change show good agreement with recent observations based on satellite altimeter measurements (Figure 6). For the overlapping period of 2007–2015 the 50th percentile of the RCP4.5 projection gives the same rate as the altimeter observations of 3.8 mm per year. The observed rate of GMSL for the entire 1993–2015 period is 3.0 mm per year, indicating an acceleration over time (Nerem et al., 2018) that is also seen in the projections. For the period out to 2030 there is little difference among the projected rates of across the three RCP scenarios.
Figure 6: Comparison of satellite altimeter observations (black) with our projections of global-mean sea-level change for the period 1993 to 2030. The 50th percentile and 5th to 95th percentile range for RCP4.5 are shown by the solid line and shaded region, respectively. Also shown are the 5th and 95th percentile projections for RCP2.6 and RCP8.5 (dotted lines) as indicated in the figure legend. The satellite altimeter timeseries has been adjusted so that the mean value matches the 50th percentile of the RCP4.5 projections over the period 2007–2015.
Our projections of GMSL change over the 21st century (Figure 7; Table 2) yield similar ranges to those presented in AR5 (Church et al., 2013) and SROCC (Oppenheimer et al., 2019). The inclusion of an updated Antarctic ice dynamics component following Levermann et al. (2014) in the present study increases the overall uncertainties and the skewness of the distribution and results in a slightly higher central estimate for RCP8.5 compared to AR5. The SROCC projections were also based on AR5, but with an updated estimate of the contribution from Antarctica based on several process-based studies (including Levermann et al., 2014). The SROCC projected ranges at 2100 are very similar to AR5, except for the RCP8.5 scenario, which is systematically higher and shows a larger uncertainty. Our extended GMSL projections show a high degree of consistency with the CMIP5-based 21st century projections evaluated at 2100 (Table 2), with all ranges agreeing to within a few centimeters.
Figure 7: Projections of global-mean sea-level change for RCP2.6 (left), RCP4.5 (middle), and RCP8.5 (right) based on the 21st century methods (a–c) and the extended 2300 methods (d–f) (Table 1). Sea-level components are shown as indicated in the figure legend. The shaded regions show the 5th to 95th percentile range from the 450,000-member Monte Carlo simulation for global thermal expansion (red) and the total (gray). The dashed and dotted lines indicate the 50th percentile and 5th to 95th percentile range from the Monte Carlo simulation presented in IPCC AR5 (Church et al., 2013). The gray shaded bars on the right-hand side of each plot indicates the 5th to 95th percentile range at 2100 or 2300 from the IPCC SROCC (Oppenheimer et al., 2019). All projections are plotted relative to a baseline period of 1986–2005. Note the change of y-axis scale for for panel (f).
Table 2: Comparison of Projected Ranges of Global-Mean Sea-Level Rise
Projection      Year    RCP2.6  RCP4.5  RCP8.5
IPCC AR5        2100    0.28–0.61 m     0.36–0.71 m     0.52–0.98 m
IPCC SROCC      2100    0.28–0.59 m     0.38–0.72 m     0.61–1.11 m
This study (21st century)       2100    0.28–0.66 m     0.37–0.78 m     0.55–1.11 m
This study (extended 2300)      2100    0.28–0.67 m     0.35–0.78 m     0.52–1.11 m
This study (extended 2300)      2200a   0.5–1.5 m       0.7–1.8 m       1.3–2.9 m
This study (extended 2300)      2300a   0.6–2.2 m       0.9–2.6 m       1.7–4.5 m
IPCC SROCC      2300a   0.6–1.1 m       —       2.3–5.4 m
SROCC expert elicitation        2300a   0.5–2.3 m       —       2.0–5.4 m
Nauels et al. (2017)    2300a   0.8–1.4 m       1.3–2.3 m       3.4–6.8 m
Note. All projections are expressed relative to a baseline period of 1986–2005.
^a Due to large uncertainties associated with post-2100 projections, these values are reported to the nearest 0.1 m.


Table 2: Comparison of Projected Ranges of Global-Mean Sea-Level Rise
Projection      Year    RCP2.6  RCP4.5  RCP8.5
IPCC AR5        2100    0.28–0.61 m     0.36–0.71 m     0.52–0.98 m
IPCC SROCC      2100    0.28–0.59 m     0.38–0.72 m     0.61–1.11 m
This study (21st century)       2100    0.28–0.66 m     0.37–0.78 m     0.55–1.11 m
This study (extended 2300)      2100    0.28–0.67 m     0.35–0.78 m     0.52–1.11 m
This study (extended 2300)      2200a   0.5–1.5 m       0.7–1.8 m       1.3–2.9 m
This study (extended 2300)      2300a   0.6–2.2 m       0.9–2.6 m       1.7–4.5 m
IPCC SROCC      2300a   0.6–1.1 m       —       2.3–5.4 m
SROCC expert elicitation        2300a   0.5–2.3 m       —       2.0–5.4 m
Nauels et al. (2017)    2300a   0.8–1.4 m       1.3–2.3 m       3.4–6.8 m
Note. All projections are expressed relative to a baseline period of 1986–2005.
^a Due to large uncertainties associated with post-2100 projections, these values are reported to the nearest 0.1 m.
The extended 2300 projections illustrate the long-term committed sea-level rise under all RCP scenarios and the large uncertainties associated with these time horizons. At these extended time horizons, there is a greater distinction between scenarios than for the 21st century, and the benefits of reduced greenhouse gas emissions on the potential magnitude of committed future sea-level rise are clear (cf. RCP2.6 and RCP8.5 at 2300, Table 2). For the extended 2300 projections, the total glacier ice mass becomes exhausted between 2200 and 2300 under RCP4.5 and between 2100 and 2300 under RCP8.5 (Figure 7).
Given the different methods, and the inherently large uncertainty associated with projections on multicentury time horizons, our projected values at 2300 are broadly consistent with the estimates presented in IPCC SROCC (Oppenheimer et al., 2019) and Nauels et al. (2017) (Table 2). Our results show a substantially larger projected range for RCP2.6 (0.6–2.2 m) than the SROCC likely range (0.6–1.1 m) and Nauels et al. (2017; 0.8–1.4 m). This larger range arises primarily from the Antarctica ice dynamics term (Figure 8; Figure S5) and may have important implications for adaptation planning. Both SROCC (2.3–5.4 m) and Nauels et al. (2017; 3.4–6.8 m) show higher projected ranges under RCP8.5 than the present study (1.7–4.5 m). For SROCC, these larger values arise primarily from the Antarctic component (Figure 8). For Nauels et al. (2017) the difference seems to arise from larger contributions and greater uncertainties in both global thermal expansion and Greenland surface mass balance (Figure S5).
Figure 8: Components of projected global-mean sea-level (GMSL) change at 2100 (a–c, based on 21st century methods) and 2300 (d–f, based on extended 2300 methods) for RCP2.6 (left), RCP4.5 (middle), and RCP8.5 (right). The horizontal lines and shaded regions indicate the 50th percentile and the 5th to 95th percentile range, respectively, from the 450,000-member Monte Carlo simulation. Scenario-independent projections are shown in gray. For reference, the corresponding global-mean surface temperature (GMST) change is shown in the final column of each panel, with secondary y-axis on the right-hand side. All projections are expressed relative to the 1986–2005 average. Corresponding projected ranges from IPCC SROCC (Oppenheimer et al., 2019) are indicated by the dashed rectangles, based on the supplementary data files (GMSL and Antarctica) and table 13.SM.1/table 13.8 (other components) of Church et al. (2013).
In order to gain some initial insights into the drivers of GMSL change, we present the breakdown of components at 2100 and 2300 based on the 5th to 95th percentile range (Figure 8). For all scenarios and both time horizons, the single largest component of uncertainty is that associated with the contribution from Antarctica (combined effects of changes in surface mass balance and ice dynamics). The 5th to 95th percentile range for Antarctica includes negative values, which arises from positive surface mass balance owing to a warmer atmosphere transporting more moisture. The components and their uncertainties generally increase under the higher emissions scenarios for both time horizons. At 2100, the RCP8.5 scenario induces substantial increases in the contribution ranges for Greenland and worldwide glaciers. The exhaustion of glacier mass for the extended projections under RCP4.5 and RCP8.5 results in reduced uncertainty for this term at 2300.
Since the 21st century projections in both SROCC and the current study use AR5 methods with updates only for Antarctica, the GMSL-component projections at 2100 are identical to SROCC except for that term (Figures 8a–8c). For all three RCP scenarios our projections show substantially larger uncertainties in the Antarctica component with higher 95th percentiles that translate into more modest differences in GMSL. For the projections on extended time horizons, the methods differ to a greater extent. The SROCC 2300 projections are based on Table 13.8 of AR5 (Church et al., 2013), which drew upon a diverse set of model simulations that were broadly categorized as “Low,” “Medium,” and “High” scenarios. The extended 2300 projections presented here are based on the RCP scenarios, using a physical framework that is consistent with the 21st century projections and traceable to CMIP5 climate model simulations.
For 2300, we see substantial differences between SROCC and the present study for the available GMSL components (Figures 8d–8f). No estimate of post-2100 land water changes were made for AR5/SROCC, and our methods use a simple assumption of applying the 2100 rates over the period 2100–2300 (Table 1). The magnitude and relative importance of GMSL components at 2300 show strong scenario dependence. For RCP8.5 the dominant terms become thermal expansion, Greenland and Antarctica with the scenario-independent land water changes and mass-limited glacier contribution becoming less important compared to RCP2.6 or RCP4.5. RCP8.5 also shows the largest difference between the projected ranges for the present study and SROCC, with substantial differences for all three of the leading component terms.
The 5th to 95th percentile component ranges combine nonlinearly to the overall projected ranges for GMSL (Figure 8). The reason for this is illustrated in Figure 9, which shows the correlation between components evaluated across the 450,000-member Monte Carlo set at 2100. Global thermal expansion, Greenland surface mass balance, and worldwide glaciers are all positively correlated: Stronger warming promotes an increased contribution to GMSL from all of these terms.
Figure 9: Correlation matrices of the different GMSL components for each RCP scenario based on the Monte Carlo spread at 2100. The matrices illustrate the relationships between GMSL components and explain why the total variance is not identical to the sum of the variances of the components.


For 2300, we see substantial differences between SROCC and the present study for the available GMSL components (Figures 8d–8f). No estimate of post-2100 land water changes were made for AR5/SROCC, and our methods use a simple assumption of applying the 2100 rates over the period 2100–2300 (Table 1). The magnitude and relative importance of GMSL components at 2300 show strong scenario dependence. For RCP8.5 the dominant terms become thermal expansion, Greenland and Antarctica with the scenario-independent land water changes and mass-limited glacier contribution becoming less important compared to RCP2.6 or RCP4.5. RCP8.5 also shows the largest difference between the projected ranges for the present study and SROCC, with substantial differences for all three of the leading component terms.
The 5th to 95th percentile component ranges combine nonlinearly to the overall projected ranges for GMSL (Figure 8). The reason for this is illustrated in Figure 9, which shows the correlation between components evaluated across the 450,000-member Monte Carlo set at 2100. Global thermal expansion, Greenland surface mass balance, and worldwide glaciers are all positively correlated: Stronger warming promotes an increased contribution to GMSL from all of these terms.
Figure 9: Correlation matrices of the different GMSL components for each RCP scenario based on the Monte Carlo spread at 2100. The matrices illustrate the relationships between GMSL components and explain why the total variance is not identical to the sum of the variances of the components.
Conversely, Antarctic surface mass balance is strongly anticorrelated with these terms because a warmer atmosphere tends to promote greater snowfall on Antarctica and reduce GMSL. As discussed in section 3.1, the AR5 methods resulted in a weak correlation between the surface mass balance and ice dynamics terms for Antarctica, which is also included here. We find similar correlations among components for all RCPs (Figure 9), although these tend to be slightly reduced for the higher emissions scenarios. Analysis of correlations at 2300 in the extended projections yields similar results (Figure S6), except for the glacier term that shows weaker correlations for RCP4.5 and RCP8.5 owing to the cap on total ice mass (illustrated in Figures 7e and 7f).
Local Sea-Level Projections
In this section we present our local MSL projections for 16 example tide gauge sites (Figure 1; Table S1). We focus our presentation on the highest (RCP8.5) and lowest (RCP2.6) emissions scenarios and include annual-mean tide gauge and satellite altimeter timeseries to illustrate the observed trends and local sea-level variability (Figure 10). In general, there is good agreement between the observed decadal rates of MSL change and the early part of the projections, noting that the satellite altimeter timeseries do not account for vertical land motion processes associated with, for example, GIA, local subsidence, or tectonic activity. Locations of poorer agreement between observed and projected MSL trends include Lima and Port Louis. However, the high degree of consistency between altimeter and tide gauge observations at these locations suggests the discrepancy arises from climatic variability rather than non-GIA vertical land-motion processes. There is an apparent jump in the Pago Pago tide gauge timeseries towards the end of the record that could be related to a nearby earthquake in 2009 that resulted in several tsunami waves hitting the island. This jump is not seen in the satellite altimeter timeseries, confirming the likely role of substantial vertical land motion at this location. The observed interannual sea-level variability varies considerably by location and demonstrates that the reality of future sea-level change will be a combination of the climate response and unforced variability (e.g., Roberts et al., 2016).
Figure 10: Local sea-level projections for RCP2.6 (blue) and RCP8.5 (red). Shaded regions indicate the 5th to 95th percentile range of the 100,000-member Monte Carlo simulation. The dotted lines indicate the 5th and 95th percentile projections from the IPCC SROCC (Oppenheimer et al., 2019). Local annual-mean tide gauge data are indicated by the solid black line. Local annual-mean satellite altimeter data are indicated by the solid gray line. All timeseries are shown relative to the 1986–2005 average. Note the different y-axis for Barentsburg.
As with GMSL, local MSL projections for the 21st century are similar to those reported in the IPCC SROCC (Figure 10) with agreement varying somewhat across tide gauge sites. Differences are thought to arise primarily from (i) the methods for estimating sterodynamic sea level and/or (ii) the methods used to combine sea-level components. It is apparent that the SROCC projections include some residual variability that originates from the underlying CMIP5 climate model simulations of sterodynamic sea-level change, which may also be present in the 1986–2005 reference state. Our regression-based approach to local sterodynamic sea-level change is designed to better isolate the climate change signal, resulting in smoother projections that do not include this simulated variability. This regression approach makes the approximation of a linear relationship between local sterodynamic sea-level change and global thermal expansion, which may also result in some differences with SROCC projections. Statistical approximations were used to combine the different local sea-level components for AR5 outside of the GMSL Monte Carlo framework that assumed terms were either perfectly correlated or perfectly uncorrelated, as described in equation 13.SM.1 of Church et al. (2013). This breaks the correlation structure among the GMSL components (Figure 9) and likely results in differences in the SROCC projected ranges for some locations. Re-running our local projections using only the GIA estimates used for AR5/SROCC (i.e., Lambeck and ICE5G, Figure 4) makes a negligible difference to the results shown in Figure 10. Analysis of the differences among our GRD fingerprints suggests that any differences in this regard are also likely to be negligible (see section 4.3).
Most tide gauge locations show that MSL is currently rising and that this rise will accelerate over the 21st century under the RCP8.5 scenario. The 21st century rates of sea-level change under RCP2.6 are relatively stable and most locations show the scenarios diverging from the mid-21st century. For most locations, the change in sea level over the 21st century is large compared to the tide gauge variability and implies that adaptation measures will be necessary to preserve current levels of coastal flood protection.
Barentsburg (Svalbard) and Reykjavik (Iceland) show atypical MSL projections. In both cases, the proximity to Greenland results in negative sea-level rise from this component (see Figure 3), which largely cancels out the positive contributions from the other climatic components, resulting in small scenario dependency at these locations. Barentsburg has a substantial rate of MSL fall associated with GIA, which accounts for the more negative values seen at this location compared to Reykjavik. While Oslo retains substantive scenario dependency, the negative GIA signal results in a much-reduced rates of rise under RCP8.5 and the expectation of a sea-level fall under RCP2.6. Barentsburg, Reykjavik, and Oslo clearly illustrate that projections of GMSL cannot necessarily be taken as indicative of local MSL change.
Excluding these atypical tide gauge locations, we still see substantive variations in future sea-level rise across the remaining tide gauge sites. The range of behavior is spanned by New York and Stanley II with ranges at 2100 under RCP2.6 (RCP8.5) of 0.27–0.84 m (0.57–1.35 m) and 0.21–0.51 m (0.45–0.91 m), respectively. New York has a large spread in sterodynamic sea-level change and also a substantial positive contribution from GIA. The relative proximity of Stanley II (the Falkland Islands) to Antarctica results in a strong attenuation of the MSL change associated with Antarctic ice dynamics, which reduces both the overall magnitude and the spread of uncertainty in future projections.


Most tide gauge locations show that MSL is currently rising and that this rise will accelerate over the 21st century under the RCP8.5 scenario. The 21st century rates of sea-level change under RCP2.6 are relatively stable and most locations show the scenarios diverging from the mid-21st century. For most locations, the change in sea level over the 21st century is large compared to the tide gauge variability and implies that adaptation measures will be necessary to preserve current levels of coastal flood protection.
Barentsburg (Svalbard) and Reykjavik (Iceland) show atypical MSL projections. In both cases, the proximity to Greenland results in negative sea-level rise from this component (see Figure 3), which largely cancels out the positive contributions from the other climatic components, resulting in small scenario dependency at these locations. Barentsburg has a substantial rate of MSL fall associated with GIA, which accounts for the more negative values seen at this location compared to Reykjavik. While Oslo retains substantive scenario dependency, the negative GIA signal results in a much-reduced rates of rise under RCP8.5 and the expectation of a sea-level fall under RCP2.6. Barentsburg, Reykjavik, and Oslo clearly illustrate that projections of GMSL cannot necessarily be taken as indicative of local MSL change.
Excluding these atypical tide gauge locations, we still see substantive variations in future sea-level rise across the remaining tide gauge sites. The range of behavior is spanned by New York and Stanley II with ranges at 2100 under RCP2.6 (RCP8.5) of 0.27–0.84 m (0.57–1.35 m) and 0.21–0.51 m (0.45–0.91 m), respectively. New York has a large spread in sterodynamic sea-level change and also a substantial positive contribution from GIA. The relative proximity of Stanley II (the Falkland Islands) to Antarctica results in a strong attenuation of the MSL change associated with Antarctic ice dynamics, which reduces both the overall magnitude and the spread of uncertainty in future projections.
We combine the tide gauge data with our local MSL projections to explore the relative importance of variability, scenario, and model uncertainty over the 21st century following Hawkins and Sutton (2009). Local sea-level variability is estimated by de-trending the tide gauge records and computing the standard deviation of the residual timeseries. The scenario standard deviation is estimated using the central estimates under RCP2.6, RCP4.5, and RCP8.5. Finally, the model uncertainty is estimated by computing the average standard deviation of the Monte Carlo simulation across the three RCP scenarios. Our analysis suggests that sea-level variability is likely to be a key driver of MSL change for the coming decades at all tide gauge locations (Figure 11). Conversely, the impact of RCP scenario will only make a substantive contribution towards the end of the 21st century. At Barentsburg and Reykjavik, differences across scenarios explain 10–20% of the projected variance, which is related to the negative contribution from Greenland canceling out other terms (as discussed above). At all locations, model uncertainty explains a large share of the overall variance and is particularly important for Barentsburg and Reykjavik.
Figure 11: Assessment of the fraction of total variance of sea-level change explained by model, scenario, and variability, following Hawkins and Sutton (2009) as indicated in the figure legend for Auckland.
The extended 2300 projections again illustrate the large levels of committed sea-level change associated with both RCP2.6 and RCP8.5 for the coming centuries (Figure 12). The projections show greater separation between these two scenarios post-2100 and the large degree of uncertainty on these time horizons. For several sites, the projected range at 2300 for RCP8.5 exceeds 5 m. Even under RCP2.6 central estimates of sea-level rise are in excess of 1 m for most locations, and the projected range exceeds 2 m at many locations. The geographical variations in projections seen over the 21st century (Figure 10) are essentially preserved (as a proportion of the signal size), resulting in differences in the projected ranges up to several meters. At these extended time horizons, the projected sea-level changes are an order of magnitude greater than the interannual tide gauge variability. At most locations, the magnitude of MSL rise and the projected range is larger than for the corresponding projection of GMSL (Figure 7). A large part of the increased spread comes from the amplification of the Antarctic ice sheet signals (the greatest source of uncertainty, Figure 8) by the GRD patterns, which have local values greater than unity for most tide gauge sites.
Figure 12: Local sea-level projections for RCP2.6 (blue) and RCP8.5 (red). Shaded regions indicate the 5th to 95th percentile range of the 100,000-member Monte Carlo simulation. Annual tide gauge data are indicated by the black line. All timeseries are shown relative to the 1986–2005 average. Note the different y-axis for Barentsburg, Oslo, and Reykjavik.
Analysis of Model Uncertainty
In this section, we further explore the contributions to the model uncertainty that is represented by the projected ranges of sea-level change for a given scenario. In particular, we consider which MSL components are dominant in determining the total variance in projected ranges as a function of geographic location, time horizon, and RCP scenario. We compute the total variance for the GMSL timeseries and several example tide gauge locations for both RCP2.6 and RCP8.5. We also compute the covariance matrix across the Monte Carlo ensembles as a function of time. The off-diagonal elements of the matrix are combined into an additional term that we label “interactions”—with this contribution arising from correlations among the components. While this analysis was performed on both, the 21st century and extended 2300 projections, the results up to 2100 are similar (Figure S7). For this reason, we focus our presentation on the extended 2300 results so that we can look across all relevant time scales. In some instances, the anticorrelation between terms leads to a reduction of total variance. For simplicity of the graphical representation and to focus discussion on the relative importance of contributions to variance in general, our analysis is based on the absolute variances.
The total variance at 2300 under RCP8.5 is more than double that for RCP2.6, both globally and at all tide gauge locations (Figures 13 and 14; left column), indicating the inherently larger uncertainties under high emissions scenarios, related to the uncertainty in model climate sensitivity. For GMSL (Figure 13, top row) the ensemble spread is initially dominated by global thermal expansion, but uncertainty in Antarctic ice dynamics becomes the dominant term from the latter half of the 21st century. Prior to 2100, there is little difference in the breakdown of variance by RCP scenario. Post-2100 we see a much larger contribution from Greenland surface mass balance under RCP8.5, becoming the second largest source of variance after Antarctic ice dynamics.
Figure 13: Time evolution of variance associated with model uncertainty for GMSL and three example tide gauge sites under RCP2.6 and RCP8.5 based on the extended projections to 2300. The left column shows the time evolution of total variance. The central and right columns show the time-evolution fraction of variance explained for RCP2.6 and RCP8.5, respectively, as indicated in the bottom-right panel legend. For GMSL, “exp” refers to global thermal expansion. For tide gauge sites, “exp” refers to sterodynamic sea-level change. Estimates for each variance term come from the diagonal of the covariance matrix. The off-diagonal contributions are presented as a combined term labeled “interactions.”
Figure 14: Time evolution of variance associated with model uncertainty for four example tide gauge sites under RCP2.6 and RCP8.5 based on the extended projections to 2300. The left column shows the time evolution of total variance. The central and right columns show the time-evolution fraction of variance explained for RCP2.6 and RCP8.5, respectively, as indicated in the bottom-right panel legend. Note that “exp” refers to sterodynamic sea-level change. Estimates for each variance term come from the diagonal of the covariance matrix. The off-diagonal contributions are presented as a combined term labeled “interactions.”
The breakdown of variance for Barentsburg (Svalbard; Figure 13) shows a dominant contribution from glaciers over the 21st century, and this term remains important out to 2300. This feature is due to the large glacier mass on Svalbard, which leads to large negative values in the GRD estimates associated primarily with vertical land uplift. We see strong scenario dependence on the contributions to variance post-2100, with Greenland surface mass balance and sterodynamic sea-level change becoming much more important under RCP8.5 than RCP2.6. Variance arising from GIA estimates makes only a minor contribution at this location.


Figure 13: Time evolution of variance associated with model uncertainty for GMSL and three example tide gauge sites under RCP2.6 and RCP8.5 based on the extended projections to 2300. The left column shows the time evolution of total variance. The central and right columns show the time-evolution fraction of variance explained for RCP2.6 and RCP8.5, respectively, as indicated in the bottom-right panel legend. For GMSL, “exp” refers to global thermal expansion. For tide gauge sites, “exp” refers to sterodynamic sea-level change. Estimates for each variance term come from the diagonal of the covariance matrix. The off-diagonal contributions are presented as a combined term labeled “interactions.”
Figure 14: Time evolution of variance associated with model uncertainty for four example tide gauge sites under RCP2.6 and RCP8.5 based on the extended projections to 2300. The left column shows the time evolution of total variance. The central and right columns show the time-evolution fraction of variance explained for RCP2.6 and RCP8.5, respectively, as indicated in the bottom-right panel legend. Note that “exp” refers to sterodynamic sea-level change. Estimates for each variance term come from the diagonal of the covariance matrix. The off-diagonal contributions are presented as a combined term labeled “interactions.”
The breakdown of variance for Barentsburg (Svalbard; Figure 13) shows a dominant contribution from glaciers over the 21st century, and this term remains important out to 2300. This feature is due to the large glacier mass on Svalbard, which leads to large negative values in the GRD estimates associated primarily with vertical land uplift. We see strong scenario dependence on the contributions to variance post-2100, with Greenland surface mass balance and sterodynamic sea-level change becoming much more important under RCP8.5 than RCP2.6. Variance arising from GIA estimates makes only a minor contribution at this location.
In contrast, Reykjavik (Iceland; Figure 13) has a large contribution to variance from the different GIA estimates, and this is the dominant source of variance over most of the 21st century. Post-2100, Antarctic ice dynamics dominates the variance under RCP2.6, with GIA remaining an important contribution for this scenario. For RCP8.5, Greenland surface mass balance and Antarctic ice dynamics contribute similarly to total variance post-2100.
As discussed in the previous section, New York (USA; Figure 13) has a particularly large uncertainty associated with future changes in ocean circulation, and this is reflected in the dominance of sterodynamic sea-level change over the 21st century. GIA is also relatively important at this location, particularly under RCP2.6. Again, it is ultimately the Antarctic ice dynamics that becomes the dominant source of variance post-2100, but sterodynamic sea-level change remains a sizeable contribution under RCP8.5.
The breakdown of variance for Mera (Japan; Figure 14) and Diamond Harbour (India; Figure 14) is typical of many lower latitude locations with characteristics that are very similar to that of GMSL (Figure 13, top row). Mera has a substantive contribution from land water under RCP2.6 that is absent from Diamond Harbour. At both locations, the contribution from GIA is very small, indicating that future changes will be dominated by contemporary climate-driven changes. As with all tide gauge locations, we cannot rule out the potential importance of nonclimatic processes, particularly those associated with vertical motion (e.g., subsidence and tectonic activity).
Palermo (Argentina; Figure 14) shows a marked reduction in the contribution from Antarctic ice dynamics, which is associated with its proximity to West Antarctica (Figure 3b). This is also reflected in the reduced total variance seen at this tide gauge location for both RCP scenarios compared to GMSL. This results in a larger relative importance of many of the other MSL components. Variance arising from different scenarios of land water change is the second largest post-2100 term (after Antarctic ice dynamics) under RCP2.6. For RCP8.5, post-2100 variance is dominated by Greenland surface mass balance.
Like Palermo, Stanley II (Falkland Islands; Figure 14) also sees a reduced total variance compared to GMSL resulting from the proximity of Antarctica. In this case, Antarctic ice dynamics makes a negligible contribution to total variance because the tide gauge location is close to the zero contour of the associated GRD pattern of MSL change (Figure 3b). The relative importance of the other terms over the 21st century is similar to Palermo. Post-2100, land water dominates the variance under the RCP2.6 scenario, and Greenland surface mass balance dominates for the RCP8.5 scenario. In the absence of substantive signals from Antarctic ice dynamics, the anticorrelated contributions of Greenland and Antarctic surface mass balance lead to a particularly large “interactions” term that reduces the overall variance compared to GMSL.
The importance of the land water contribution for both Palermo and Stanley II under RCP2.6 arises for a number of reasons. First, the land water projections are scenario-independent, so its relative importance increases under RCP2.6 compared to RCP8.5. Second, the relative importance of land water is further increased at these locations due to the strong attenuation of the Antarctica signals, in relation to GRD (Figure 3), as discussed above. Third, both Palermo and Stanley II are in a region where the global land water contribution (and its uncertainty) is amplified at regional scales by the GRD patterns (Figure 3).
Overall, we see a strong geographic and time dependence of the contribution to total variance. There is little scenario dependence until after 2100, and this is typified by a substantially increased contribution from Greenland surface mass balance under RCP8.5. Antarctic ice dynamics tend to dominate the total variance from the latter half of the 21st century out to 2300 under both RCP2.6 and RCP8.5. However, this is not the case for locations in proximity to West Antarctica (e.g., southern South America), where the GRD pattern of MSL results in greatly attenuated signals for Antarctic ice dynamics. The contrasting results presented here make a clear case for the need for site-specific local sea-level projections. In addition, the reduction of variance (uncertainty) in future projections implies different research priorities, depending on geographic location and time horizon.
As part of our analysis of variance, we also investigate the contribution from uncertainty in the GRD estimates presented in section 2.4 (Figure 3). We choose the three tide gauge sites with the largest spread in one or more GRD components, that is, Barentsburg, Reykjavik, and Stanley II, and conduct the following simple analysis. We run an additional instance of the MSL Monte Carlo for each location using the average value of the GRD fingerprints at each site for RCP8.5 (i.e., the scenario with the largest signal). The projected MSL and component ranges are then compared to the full Monte Carlo simulations that include a random choice of GRD estimate. The results demonstrate that using multiple GRD estimates has an essentially negligible contribution to the total variance—the differences at 2100 for RCP8.5 are just about perceptible for Barentsburg (Figure 15). It is therefore reasonable to use only a single set of GRD estimates when computing local MSL projections. However, our analysis does not account for uncertainty in the associated space-time mass distributions that is used to compute the GRD patterns. This is an area that may benefit from further research.
Figure 15: Projected ranges of MSL change at 2100 for three example tide gauge locations under RCP8.5. The dashed lines indicate the results when the average of three GRD estimates (rather than random selection) is used in the Monte Carlo simulations. All projections are expressed relative to a baseline period of 1986–2005. Scenario-independent components are indicated in gray.
Summary and Conclusions


As part of our analysis of variance, we also investigate the contribution from uncertainty in the GRD estimates presented in section 2.4 (Figure 3). We choose the three tide gauge sites with the largest spread in one or more GRD components, that is, Barentsburg, Reykjavik, and Stanley II, and conduct the following simple analysis. We run an additional instance of the MSL Monte Carlo for each location using the average value of the GRD fingerprints at each site for RCP8.5 (i.e., the scenario with the largest signal). The projected MSL and component ranges are then compared to the full Monte Carlo simulations that include a random choice of GRD estimate. The results demonstrate that using multiple GRD estimates has an essentially negligible contribution to the total variance—the differences at 2100 for RCP8.5 are just about perceptible for Barentsburg (Figure 15). It is therefore reasonable to use only a single set of GRD estimates when computing local MSL projections. However, our analysis does not account for uncertainty in the associated space-time mass distributions that is used to compute the GRD patterns. This is an area that may benefit from further research.
Figure 15: Projected ranges of MSL change at 2100 for three example tide gauge locations under RCP8.5. The dashed lines indicate the results when the average of three GRD estimates (rather than random selection) is used in the Monte Carlo simulations. All projections are expressed relative to a baseline period of 1986–2005. Scenario-independent components are indicated in gray.
Summary and Conclusions
We have presented MSL projections under the RCP2.6, RCP4.5, and RCP8.5 climate change scenarios for both global-mean sea level and for a number of example tide gauge locations around the world. Our 21st century projections are directly traceable to the CMIP5-based sea-level projections presented in AR5 (Church et al., 2013) with updated treatment of the contribution from Antarctic ice dynamics following Levermann et al. (2014) and show similar results to projections presented in the SROCC (Oppenheimer et al., 2019). Our regression approach for sterodynamic sea-level change more cleanly isolates the forced signal and enables us to characterize the relative importance of variance arising from scenario, model spread, and observed sea-level variability over the 21st century, following Hawkins and Sutton (2009). A key aspect of the study is the use of the same Monte Carlo framework for both global and local sea-level projections. This means that global and local projections are entirely consistent, preserving the correlations among components and allowing us to quantify the contributions to total variance (uncertainty) by geographic location, time horizon, and scenario. We introduced a new set of exploratory extended projections to 2300 that are rooted in CMIP5 projections through the use of a physically based emulator to extend individual CMIP5 climate model simulations (Palmer, Harris, et al., 2018). These emulator-based projections are designed for maximum consistency with the 21st century projections and show a high level of agreement at 2100. Our main findings are summarized as follows:
We have developed a consistent set of local sea-level projections that are directly traceable to the GMSL projections developed for AR5 and preserve the relationships between components (an important factor in determining the fraction of variance explained).
Our projections of GMSL at both 2100 and 2300 yield similar numbers to those recently reported in the IPCC Special Report on the Ocean and Cryosphere in a Changing Climate (IPCC, 2019), noting the large inherent uncertainties associated with multicentury time horizons. For RCP2.6, our projected ranges at 2300 are larger than some recent studies (including the SROCC likely range) and may have important implications for long-term adaptation planning.
Combined analysis of local MSL projections and tide gauge records suggests that sea-level variability dominates the total variance in the coming decades, with climate change scenario becoming increasingly important over the 21st century. Model uncertainty tends to be the largest source of variance from the mid-21st century.
Local MSL projections can show large departures from the GMSL response and are typically associated with substantially larger uncertainties. A few locations will see MSL decrease in future owing to spatial patterns of GRD and the local contribution from GIA.
On century time scales, the projected MSL changes are large compared to observations of local sea-level variability. This indicates that many places will be exposed to greater coastal flood risk unless effective adaptation measures are taken.
The extended projections to 2300 illustrate the large degree of committed sea-level rise even in strong mitigation scenarios. They also illustrate the substantially increased risk for the highest emissions scenarios, which are associated with several meters of MSL rise at most tide gauge locations.
Correlations between component terms mean that the variance of the total MSL change is not identical to the sum of the variances of the components. Moreover, the breakdown of variance depends on both, geographic location and time horizon, with differences in scenario post-2100. Antarctic ice dynamics dominate the total variance post-2100 except at locations where GRD patterns strongly attenuate this signal.
This study highlights the need for development of site-specific MSL projections for effective planning (GMSL projections of cannot be reliably used to indicate the local changes). The time-space-scenario dependence in the contributions to total variance suggests that research priorities for reducing uncertainty in sea-level projections are likely to vary by geographic location and planning time horizon.


Hay et al. (2015)
Title:
Probabilistic reanalysis of twentieth-century sea-level rise
Key Points:
Reconstruction of the global mean sea-level changes since 1900, from combining the probability distributions of a set of tide gauge records, refines estimates and is more consitent with estimates based on the sum of contributions
The global mean sea level rose by about 1.2 ± 0.2 mm/year between 1901 and 1990 and by 3.0 ± 0.7 mm/year between 1993 and 2010.
The rate of sea-level rise has been accelerating
Keywords:
sea level reconstruction, tide gauge, global mean sea level rise, probabilistic framework
Corresponding author:
Carling C. Hay.
Citation:
Hay, C. C., Morrow, E., Kopp, R. E., & Mitrovica, J. X. (2015). Probabilistic reanalysis of twentieth-century sea-level rise. Nature, 517(7535), 481–484. doi:10.1038/nature14093
URL:
https://www.nature.com/articles/nature14093
Abstract
Estimating and accounting for twentieth-century global mean sea-level (GMSL) rise is critical to characterizing current and future human-induced sea-level change. Several previous analyses of tide gauge records [1,2,3,4,5,6] — employing different methods to accommodate the spatial sparsity and temporal incompleteness of the data and to constrain the geometry of long-term sea-level change — have concluded that GMSL rose over the twentieth century at a mean rate of 1.6 to 1.9 millimetres per year. Efforts to account for this rate by summing estimates of individual contributions from glacier and ice-sheet mass loss, ocean thermal expansion, and changes in land water storage fall significantly short in the period before 1990 [7]. The failure to close the budget of GMSL during this period has led to suggestions that several contributions may have been systematically underestimated [8]. However, the extent to which the limitations of tide gauge analyses have affected estimates of the GMSL rate of change is unclear. Here we revisit estimates of twentieth-century GMSL rise using probabilistic techniques [9,10] and find a rate of GMSL rise from 1901 to 1990 of 1.2 ± 0.2 millimetres per year (90% confidence interval). Based on individual contributions tabulated in the Fifth Assessment Report [7] of the Intergovernmental Panel on Climate Change, this estimate closes the twentieth-century sea-level budget. Our analysis, which combines tide gauge records with physics-based and model-derived geometries of the various contributing signals, also indicates that GMSL rose at a rate of 3.0 ± 0.7 millimetres per year between 1993 and 2010, consistent with prior estimates from tide gauge records4. The increase in rate relative to the 1901–90 trend is accordingly larger than previously thought; this revision may affect some projections11 of future sea-level rise.
Editorial Summary: Twentieth century sea levels revisited
Rates of sea-level rise calculated from tide gauge data tend to exceed bottom-up estimates derived from summing loss of ice mass, thermal expansion and changes in land storage. Carling Hay et al. provide a statistical reassessment of the tide gauge record — which is subject to bias due to sparse and non-uniform geographic coverage and other uncertainties — and conclude that sea-level rose by about 1.2 millimetres per year from 1901 to 1990. This is slightly lower than prior estimates and is consistent with the bottom-up estimates. The same analysis applied to the period 1993–2010, however, indicates a sea-level rise of about three millimetres per year, consistent with other work and suggesting that the recent acceleration in sea-level rise has been greater than previously thought.
Main
Tide gauges provide records of local sea-level changes that, in the case of some sites, extend back to the eighteenth century [12,13,14]. However, using the database of tide gauge records [15] to estimate historical GMSL rise (defined as the increase in ocean volume normalized by ocean area) is challenging. Tide gauges sample the ocean sparsely and non-uniformly, with a bias towards coastal sites and the Northern Hemisphere, and with few sites at latitudes greater than 60° (see, for example, refs 4, 9). In addition, tide gauge time series show significant inter-annual to decadal variability, and they are characterized by missing data (that is, intervals without observations at the start, middle or end of a time series). From the perspective of estimating GMSL changes, the data are contaminated by local and regional signals due to ongoing glacial isostatic adjustment (GIA) associated with past ice ages [16,17], the spatially non-uniform pattern of sea-level rise associated with changes in contemporary land ice sources[18,19,20,21], ocean/atmosphere dynamics [22], and other local factors including tectonics, sediment compaction, groundwater pumping and harbour development.
Different approaches have been used to address these complexities in efforts to estimate twentieth-century GMSL rise [23]. These include averaging rates at sites with the longest records [1,2], averaging rates determined from regional binning of records [3], incorporating shorter records into the analysis to distinguish between secular trends and decadal-scale variability [3], and using altimetry records to determine dominant sea-level geometries and then using tide gauge records to estimate the time-varying amplitudes of these geometries [4,5]. In most cases, other criteria were applied to cull the tide gauge sites adopted in the analysis (for example, excluding sites near tectonic activity or major urban centres).
Estimates of twentieth-century GMSL rise from these previous analyses range from 1.6 to 1.9 mm yr−1 (refs 1, 2, 3, 4, 5, 6) and define an important enigma. Independent model- and data-based estimates of the individual sources of GMSL, including mass flux from glaciers and ice sheets, thermal expansion of oceans, and changes in land water storage, are insufficient to account for the GMSL rise estimated from tide gauge records8, particularly before 19907. For example, a tabulation of contributions to GMSL rise from 1901 to 1990 in the Fifth Assessment Report (AR5; ref. 7) of the Intergovernmental Panel of Climate Change (IPCC) total 0.5 ± 0.4 mm yr−1 (90% confidence interval, CI) less than a recent tide gauge derived rate of 1.5 ± 0.2 mm yr−1 (90% CI) estimated by Church and White4 for the same period (the confidence range for this estimate is taken from AR5; refs 7 and 23). Using IPCC terminology, the latter suggests that it is ‘extremely likely’ (probability P = 95%) that GMSL rise from 1901 to 1990 was greater than 1.3 mm yr−1, although the bottom-up sum of contributions is ‘likely’ (P > 67%) below this level. The above discrepancy has been attributed to underestimation of almost all possible sources: thermal expansion, glacier mass balance, and Greenland or Antarctic ice sheet mass balance7,8.
In this Letter, we revisit the analysis of GMSL since the start of the twentieth century using Kalman smoothing9 (KS; see Methods). This statistical technique naturally accommodates spatially sparse and temporally incomplete sampling of a global sea-level field, provides a rigorous, probabilistic framework for uncertainty propagation, and can correct for a distribution of GIA and ocean models. We applied the approach to analyse annual records from 622 tide gauges included in the Permanent Service for Mean Sea Level (PSMSL) Revised Local Reference database15,24 and reconstruct the global field of sea-level change for each year from 1900 to 2010.
To examine the skill with which the KS reconstruction reproduces the tide gauge observations, we compute the time series of residuals at each tide gauge site and examine the distribution of the mean residual (that is, bias) for each site (Fig. 1a). The mean of the mean residuals across all 622 observations is 0.3 mm, with a standard deviation of 5.1 mm, indicating minimal systemic bias.
Figure 1: Fit of the KS-based reconstruction of sea level to the tide gauge record. a, Histogram of mean residuals (mm) between the sea-level reconstruction and the tide gauge observations at all 622 sites. The mean of all mean residuals is 0.3 ± 5.1 mm (±1 s.d.). b–f, Time series of reconstructed annual sea level (black lines, KS mean estimate; grey shading, 1σ uncertainty) at New York, USA (b), Fremantle, Australia (c), Zemlia Bunge, Russia (d), Vaasa, Finland (e), and Champlain, Canada (f), together with the associated annual mean tide gauge observations (red lines).


In this Letter, we revisit the analysis of GMSL since the start of the twentieth century using Kalman smoothing9 (KS; see Methods). This statistical technique naturally accommodates spatially sparse and temporally incomplete sampling of a global sea-level field, provides a rigorous, probabilistic framework for uncertainty propagation, and can correct for a distribution of GIA and ocean models. We applied the approach to analyse annual records from 622 tide gauges included in the Permanent Service for Mean Sea Level (PSMSL) Revised Local Reference database15,24 and reconstruct the global field of sea-level change for each year from 1900 to 2010.
To examine the skill with which the KS reconstruction reproduces the tide gauge observations, we compute the time series of residuals at each tide gauge site and examine the distribution of the mean residual (that is, bias) for each site (Fig. 1a). The mean of the mean residuals across all 622 observations is 0.3 mm, with a standard deviation of 5.1 mm, indicating minimal systemic bias.
Figure 1: Fit of the KS-based reconstruction of sea level to the tide gauge record. a, Histogram of mean residuals (mm) between the sea-level reconstruction and the tide gauge observations at all 622 sites. The mean of all mean residuals is 0.3 ± 5.1 mm (±1 s.d.). b–f, Time series of reconstructed annual sea level (black lines, KS mean estimate; grey shading, 1σ uncertainty) at New York, USA (b), Fremantle, Australia (c), Zemlia Bunge, Russia (d), Vaasa, Finland (e), and Champlain, Canada (f), together with the associated annual mean tide gauge observations (red lines).
Comparing reconstructions and tide gauge observations at a selection of individual sites (Fig. 1b–f) shows generally excellent agreement, although there are a small number of outliers. An example outlier is the Champlain tide gauge (Fig. 1f), which has a mean residual of 52 mm. This particular misfit (also evident at other sites in the vicinity) can be attributed to the St Lawrence being a regulated water system where flow is dominated by anthropogenic control rather than global-scale climate dynamics25. The eight sites that have mean residuals greater than ±3σ (15 mm) from the mean exhibit an average interannual sea-level variability (estimated as the standard deviation after detrending the tide gauge observations) of ±130 mm, more than triple the mean inter-annual variability of ±40 mm across all sites. Although these outliers have large inter-annual variability, the site-specific variability is incorporated into the covariances computed in the probabilistic reconstruction, and the uncertainties in the estimated sea-level trends at these sites reflect this.
The sum of the KS-estimated GMSL changes associated with the mass balance of the Greenland and Antarctic ice sheets, the mass balance of 18 mountain glacier regions, and thermal expansion (Fig. 2, blue line and shading; see Methods) is characterized by an average GMSL rate of 1.2 ± 0.2 mm yr−1 (90% CI) for 1901–90. As shown in Fig. 3, this is significantly lower than the estimates of 1.5 ± 0.2 mm yr−1 from Church and White4 (magenta line in Fig. 2) and 1.9 mm yr−1 from Jevrejeva et al.3 (red line in Fig. 2). The KS-estimated acceleration is 0.017 ± 0.003 mm yr−2, larger than our estimates based on the Church and White4 (0.009 ± 0.002 mm yr−2) and Jevrejeva et al.3 (0.011 ± 0.006 mm yr−2) time series (see Methods).
Figure 2: Time series of GMSL for the period 1900–2010. Shown are estimates of GMSL based on KS (blue line), GPR (black line), Church and White4 (magenta line) and Jevrejeva et al.3 (red line). Shaded regions show ±1σ pointwise uncertainty. Inset, trends for 1901–90 and 1993–2010, and accelerations, all with 90% CI. Confidence intervals for Church and White4 are from refs 7 and 23. Confidence intervals were not available for Jevrejeva et al.3; data in this reference ends in 2002, so the rate quoted here for 1993–2010 is actually for 1993–2002. Since the GPR methodology outputs decadal sea level, no trend is estimated for 1993–2010. Accelerations are consistently estimated from the KS, GPR, and GMSL time series in refs 3 and 4 (see Methods) from 1901 to the end of each reconstruction.
Figure 3: Comparison of mean GMSL rates for 1901–90. Shown are estimates of GMSL rise for the period 1901–90 obtained from six different sampling methods along with previously published rates (see main text for description of each). The box covers the 1σ uncertainty range, while the bars represent the 90% CI. In the case of the Jevrejeva et al.3 estimate, uncertainties and confidence intervals were not available.
Church and White [4] combined stationary empirical orthogonal functions (EOFs), computed from ∼20 years of satellite altimetry data spanning latitudes up to about ±60°, with amplitudes estimated from sparse tide gauge observations. Given the relatively short duration of the altimeter record, the EOFs may be dominated by patterns due to interannual variability rather than the geometry associated with long-term sea-level change26,27. Jevrejeva et al.3 used tide gauge records to compute regional sea-level means and from these computed a global average. Both methodologies involve spatially sparse, temporally incomplete sampling of the global sea-level field, which introduces a potentially significant bias into estimates of GMSL. The KS technique differs from these approaches by using the spatial information inherent in the observations to infer the weights associated with the individual, underlying contributions to the sea-level change. The method extracts global information from the sparse field by taking advantage of the physics-based and model-derived geometry of the contributing processes, thereby reducing the potential for sampling bias.
To understand the origin of the differences between the KS estimate and the higher values of refs 3 and 4, and in particular to quantify the impact of regional binning, spatial sparsity and missing data, we performed several tests.
First, we applied to the KS global sea-level reconstruction a regional binning algorithm similar to that of Jevrejeva et al.3. In particular, we sampled the reconstruction at the locations of the 622 tide gauge sites, imposed sections of missing data consistent with the PSMSL data availability15, binned the tide gauges into 12 ocean regions, and averaged across these regions to compute a GMSL curve. The resulting estimate of the mean GMSL rate from 1901 to 1990 (Fig. 3; ‘KS PSMSL sampling’), 1.6 ± 0.4 mm yr−1 (90% CI), is significantly closer to the estimate of Jevrejeva et al.3, indicating that combined spatial sparsity and missing data generate an upward bias in estimates of GMSL rates (Fig. 3). Second, we performed a bootstrapping test that repeated the above algorithm for tide gauge subsets ranging from 25 to 600 sites that confirmed this result (see Methods and Extended Data Fig. 3). We also implemented a test to estimate the possible bias in the estimate of GMSL rate introduced in the EOF analysis of Church and White4 (see Methods; Fig. 3; ‘KS EOF’); the result was consistent with the difference between the KS and Church and White4 results in Fig. 2.
We performed several other tests to explore the impact of sparsity and missing data on the estimates. Specifically, we applied the binning algorithm as described above but without imposing sections of missing data. The resulting mean GMSL rate estimate for 1901–90 was 1.0 ± 0.4 mm yr−1, close to the KS result (Fig. 3; ‘KS 622 sites, no missing data’). Third, we sampled the full reconstruction at a large number of globally distributed sites—that is, the sampling was not confined to the tide gauge sites and no sections of missing data were imposed on the time series—and performed the same regional binning and averaging (‘KS global reconstruction’). The resulting rate estimate, 1.2 ± 0.1 mm yr−1, was identical to the KS result (Fig. 3). This indicates that regional binning of estimates, in the absence of sparsity and missing data, does not introduce a significant bias.


First, we applied to the KS global sea-level reconstruction a regional binning algorithm similar to that of Jevrejeva et al.3. In particular, we sampled the reconstruction at the locations of the 622 tide gauge sites, imposed sections of missing data consistent with the PSMSL data availability15, binned the tide gauges into 12 ocean regions, and averaged across these regions to compute a GMSL curve. The resulting estimate of the mean GMSL rate from 1901 to 1990 (Fig. 3; ‘KS PSMSL sampling’), 1.6 ± 0.4 mm yr−1 (90% CI), is significantly closer to the estimate of Jevrejeva et al.3, indicating that combined spatial sparsity and missing data generate an upward bias in estimates of GMSL rates (Fig. 3). Second, we performed a bootstrapping test that repeated the above algorithm for tide gauge subsets ranging from 25 to 600 sites that confirmed this result (see Methods and Extended Data Fig. 3). We also implemented a test to estimate the possible bias in the estimate of GMSL rate introduced in the EOF analysis of Church and White4 (see Methods; Fig. 3; ‘KS EOF’); the result was consistent with the difference between the KS and Church and White4 results in Fig. 2.
We performed several other tests to explore the impact of sparsity and missing data on the estimates. Specifically, we applied the binning algorithm as described above but without imposing sections of missing data. The resulting mean GMSL rate estimate for 1901–90 was 1.0 ± 0.4 mm yr−1, close to the KS result (Fig. 3; ‘KS 622 sites, no missing data’). Third, we sampled the full reconstruction at a large number of globally distributed sites—that is, the sampling was not confined to the tide gauge sites and no sections of missing data were imposed on the time series—and performed the same regional binning and averaging (‘KS global reconstruction’). The resulting rate estimate, 1.2 ± 0.1 mm yr−1, was identical to the KS result (Fig. 3). This indicates that regional binning of estimates, in the absence of sparsity and missing data, does not introduce a significant bias.
To assess the robustness of our probabilistic reanalysis, we also performed a second, independent statistical analysis based on Gaussian process regression28 (GPR), a technique that also naturally accommodates data sparsity and gaps, and incorporates a suite of GIA and ocean models (see Methods; black line in Fig. 2). The mean GMSL rate for 1901–90 estimated from the GPR analysis, 1.1 ± 0.4 mm yr−1, is consistent with the results of the KS analysis (Fig. 3).
Previous analyses appear to have overestimated the mean GMSL rate over the twentieth century. The KS estimate for the period 1901–90 indicates that it is ‘very likely’ (probability P = 90%) that the rate of GMSL rise during this period was between 1.0 and 1.4 mm yr−1. This estimate closes the sea-level budget for 1901–90 estimated in AR5 (ref. 7) without appealing to an underestimation of individual contributions from ocean thermal expansion, glacier melting, or ice sheet mass balance. Moreover, it may contribute to the ultimate resolution of Munk’s sea-level enigma28 (defined by the argument that Earth rotation measurements and bounds on ocean warming are inconsistent with a rate of sea-level rise beginning in the late nineteenth century of 1.5–2.0 mm yr−1), since it may lower the signal of twentieth century ice melting in Earth rotation measurements.
In contrast, for the period 1993–2010—which coincides with the era of satellite altimetry measurements of sea surface height changes29—the KS estimate is consistent with previous results (Fig. 2). The KS estimate, 3.0 ± 0.7 mm yr−1 (90% CI), is essentially identical to the tide gauge analysis of Church and White4 (2.8 ± 0.5 mm yr−1; ref. 23). It is also consistent with the estimate based on TOPEX and Jason altimeter measurements (3.2 ± 0.4 mm yr−1; ref. 29 as cited by ref. 23 for the period 1993–2010, see also ref. 7).
To assess the anomalous nature of recent sea-level change, we compute 15-year rates through the KS-derived GMSL time series in Fig. 2 from 1901 to 2010. Figure 4 shows both the time series and distribution of these 96 rates, where the 5 most recent time windows are shown in red. The former is in qualitative agreement with a previous inference of multi-decadal trends in acceleration during the twentieth century30. While the rates show significant variability, the rate for the 1996–2010 time window, 3.1 mm yr−1, is the largest of all computed 15-year rates.
Figure 4: Moving 15-year averages of GMSL rate estimated using the KS reconstruction of sea level across the entire interval 1901–2010. The x-axis represents the mid-point of each 15-year averaging window, and the shading gives the 1σ uncertainty range. Inset, histogram of 15-year mean GMSL rate estimates (mm yr−1) for all time windows. The five most recent windows are shown in red.
We have revisited twentieth century GMSL rise using probabilistic techniques that combine sea-level records with physics-based and model-derived geometries of the contributing processes. Our estimated GMSL trend for the period 1901–90 (1.2 ± 0.2 mm yr−1) is lower than previous estimates, indicating that the rate of GMSL rise during the last two decades represents a more significant increase than previously recognized. Projections of future sea-level rise based on the time series of historical GMSL, notably semi-empirical approaches11, should accordingly be revisited.
Methods
Probabilistic estimation methods
Kalman smoothing (KS) and Gaussian process regression (GPR), both discussed in detail below, share three advantages over the approaches taken in traditional tide gauge analyses. First, the Bayesian nature of both approaches naturally accommodates the spatiotemporal changes in the availability of the sea-level records (that is, sparsity and missing data). Second, the probabilistic approaches correct for a distribution of GIA and ocean models rather than adopting a specific model for each process, and they thus reduce a potentially important bias in previous estimates of the GMSL change17,31. Last, as both methods are fully probabilistic, they allow for the propagation of measurement and inferential uncertainties and correlations throughout the complete analysis time period. Despite these commonalities, the implementations of KS and GPR differ significantly.
Kalman smoother
The KS methodology is divided into four steps9, the first three of which are repeated by employing the spatial fields of GIA and ocean dynamic models from all possible combinations of 161 different Earth rheological models and 6 global climate model (GCM) simulations from CMIP5 (ref. 32) (see below for details of the rheological and climate models). First, a priori model estimates of both local sea level and the individual mass contributions from the Greenland, West Antarctic and East Antarctic ice sheets, as well as 18 major mountain glacier regions, are recursively corrected by tide gauge observations as the estimates are propagated forward through time. The local sea level is linked to the individual mass contributions through the unique spatial patterns, or ‘fingerprints,’ of sea-level change associated with rapid mass loss from land-based ice18,19,20,21. The forward step yields an estimate of local sea level and land ice contributions at each time slice, conditional on all earlier observations and a particular combination of GIA and GCM models. Second, the procedure is run backward in time, with the initial state estimate being the last estimate from the first step. The third, smoothing step optimally combines the results of the first two passes based upon the uncertainties of the respective estimates. The result is an estimate of local sea levels and land ice contributions conditional upon the entire set of observations and specific pairings of GIA and GCM models. Finally, the results from different GIA/GCM combinations are linearly combined, weighted by their likelihood, to yield an a posteriori probability distribution for local sea levels and land ice contributions, conditional upon the tide gauge observations.
A comprehensive discussion of our application of the KS technique to the analysis of tide gauge measurements is given in ref. 9, which also includes synthetic tests to assess the performance of the procedure. Several subsequent refinements of this approach are summarized below.
Reference 9 defined the state vector to include estimates of sea level at every tide gauge site, the mass loss rates of three ice sheets, and the temporally correlated noise in the sea-level observations. Using only tide gauge observations limits our ability to separate estimates of sea level from estimates of the temporally correlated noise. This led us to modify the KS approach in two ways.
First, the state vector includes only an estimate of total sea level at every tide gauge site in addition to the desired mass loss rates. This yields the following state vector, xk, at every time step, k:


A comprehensive discussion of our application of the KS technique to the analysis of tide gauge measurements is given in ref. 9, which also includes synthetic tests to assess the performance of the procedure. Several subsequent refinements of this approach are summarized below.
Reference 9 defined the state vector to include estimates of sea level at every tide gauge site, the mass loss rates of three ice sheets, and the temporally correlated noise in the sea-level observations. Using only tide gauge observations limits our ability to separate estimates of sea level from estimates of the temporally correlated noise. This led us to modify the KS approach in two ways.
First, the state vector includes only an estimate of total sea level at every tide gauge site in addition to the desired mass loss rates. This yields the following state vector, xk, at every time step, k:
xk = [hk Bk]^T
where hk is a vector of sea level at the 622 tide gauge sites, and Bk is a vector containing the scalar weightings of 3 ice sheets and 18 mountain glacier regions (see below), as well as a uniform component that accounts for global mean thermal expansion and any additional mass contributions from smaller mountain glaciers.
Second, while in ref. 9 the observation model consisted of the sum of the estimated sea level, correlated noise, and white noise, here, the observation model consists only of the estimated sea level and white noise at each tide gauge site. Temporal correlations due to ocean dynamics are now modelled by the annual, spatial, CMIP5 ocean model fields (see below for a more detailed description of the CMIP5 model fields).
Sea level is modelled as the Euler integration of the contributions from melt sources, Bk-1y, (with y being the matrix of sea-level fingerprints associated with rapid land-ice mass loss), the ongoing rate of sea-level change due to GIA, G, and the rate of change of sea level due to ocean dynamics, Sdot_{k-1} , from the spatial fields in the CMIP5 model outputs:
hk = h_{k-1} + ∆t(B_{k-1} y + G + Sdot_{k-1}) + wh
where wh represents a zero-mean, white noise term associated with sea level.
The scalar weightings of the fingerprints are modelled as a random walk:
Bk = B_{k-1} + wB
where wB represents a zero-mean, white noise term associated with the melt contributions. The forward filtering pass of the Kalman smoother follows the steps outlined in ref. 9. A final departure from the methodology presented in ref. 9 is that we implemented a three-pass fixed-interval smoother33 in place of a Rauch-Tung-Stiebel two-pass smoother [34].
Gaussian process regression
The GPR approach, in contrast, models sea level as a multivariate Gaussian field defined by spatiotemporal mean and covariance functions that describe the underlying processes responsible for sea-level variability. Specifically, Gaussian process priors describing the contributions from land ice, GIA, and ocean models are conditioned simultaneously upon the available observations to produce the conditional, posterior distribution of sea level at decadal intervals throughout the twentieth century. In contrast to the KS, the GPR approach directly estimates the intertemporal covariance of the posterior; the associated computational demands require the use of decadal rather than annual means. Rather than being based upon discrete GIA and GCM models as in the KS approach, the GPR approach employs Gaussian process priors for the GIA and ocean dynamics contributions that are estimated, respectively, from the 161 GIA model predictions and 6 GCM outputs (see below). The distribution describing each land ice mass contribution is modelled assuming a prior spatio-temporal covariance, with the temporal component estimated from previous, non-sea level based estimates of land ice melt and the spatial component from the sea-level fingerprints associated with the melt source.
We model decadal-average sea level as a spatiotemporal field:
f(x,t) = fGIA(x,t) + fM(x,t) + fLSL(x,t)
where fGIA, fM, and fLSL are respectively the components of sea level due to ongoing GIA, land ice mass loss, and local effects associated with ocean dynamics, tectonics and other non-climatic factors, each as a function of location, x, and time, t. Each sea-level component is modelled as a Gaussian process with a prior mean function, μi(x,t), and covariance function, Ki(x,t,x′,t′).
The total field can be partitioned into observed sites, f1, and unobserved sites, f2, and subsequently written as a joint, multivariate distribution, such that:
[f1, f2] ~ N([μ1, μ2],[K11 K12, K^T12 K22])
Observations, y, are modelled as the underlying sea-level field with additive white noise characterized by zero mean and a covariance Σp, such that the joint distribution becomes:
[y, f2] ~ N([μ1, f2],[K11+Σp K12, K^T12 K22])
Using standard statistical results (see, for example, ref. 35), the posterior mean and covariance, f2 and V2, of the unobserved field conditioned upon the observations are:
f2 = f2 + K^T12 [K11+Σp]^{-1} y
and
V2 = K22 - K^T12 [K11+Σp]^{-1} K12
To estimate the underlying constituents of the total sea-level field, the prior mean and covariance of the unobserved field (that is, μ2, K12, K22) are set to the distribution of the desired quantity alone. For example, setting μ2, K12, and K22 equal to μ2M, K12M, K22M, returns the posterior mean and covariance of sea-level change due to the melt contributions. Once all the underlying constituent sea-level fields are determined, the global mean of those components can be computed and added to estimate GMSL.
The elements of the prior covariance matrix of the melt contribution, KM, are defined as:
K^Mi,j = Σ^n_{a=1}(A^{M,L}_{i,j,a} + A^{M,RQ}_{i,j,a})(B^M_{i,j,α})
where the subscripts indicate the ith row and jth column element of the ath ice sheet or mountain glacier. The time dependence of the covariance matrix is taken to be the sum of a linear component, AM,L, which accounts for secular changes in the melt contributions, and a rational quadratic term, AM,RQ, that represents a smoothly-varying function of variability:
A^{M,L}(tq,tp) = k1 tq tp
and
A^{M,RQ}(tq,tp) = k2 (1 + ∆t^2q,p / 2 α τs^2)^{-α}
Here, tq and tp represent the time at the qth and pth time step, Δtq,p represents the time difference between these steps, and k1, k2, α, and τs are hyperparameters that define the linear amplitude, rational quadratic amplitude, roughness, and characteristic timescale of the covariance functions35. To estimate the hyperparameters we adopt an empirical Bayesian approach where we compute the parameters that maximize the likelihood of reconstructed time series of previous mountain glacier estimates36 and ice sheet estimates37.
The spatial weighting of the prior covariance, BM, is computed as the outer product of the unique fingerprint associated with melt from the corresponding land-based ice source.
The prior spatiotemporal mean and covariance for the GIA contribution to sea-level change, μGIA(x,t) and KGIA(x,t), respectively, are taken as the sample mean and covariance of the 161 predictions of sea-level change described below.
The distribution of the contribution to sea-level changes from thermosteric and ocean dynamic effects is partially modelled as the sample mean and covariance of the CMIP5 model outputs32. However, since a small number of models are used to compute the distribution statistics, the estimated distribution may not be representative of the parent distribution. Consequently, we augment the sample covariance with a space-time separable covariance structure consisting of the product of two Matérn functions [35], C: one representing the temporal distribution and the other representing the spatial, such that the total prior covariance describing local sea-level change is given by:
KLSL(x,t) = KCMIP5(x,t) + C(t,ν1,τ) C(x,ν2,L)


where the subscripts indicate the ith row and jth column element of the ath ice sheet or mountain glacier. The time dependence of the covariance matrix is taken to be the sum of a linear component, AM,L, which accounts for secular changes in the melt contributions, and a rational quadratic term, AM,RQ, that represents a smoothly-varying function of variability:
A^{M,L}(tq,tp) = k1 tq tp
and
A^{M,RQ}(tq,tp) = k2 (1 + ∆t^2q,p / 2 α τs^2)^{-α}
Here, tq and tp represent the time at the qth and pth time step, Δtq,p represents the time difference between these steps, and k1, k2, α, and τs are hyperparameters that define the linear amplitude, rational quadratic amplitude, roughness, and characteristic timescale of the covariance functions35. To estimate the hyperparameters we adopt an empirical Bayesian approach where we compute the parameters that maximize the likelihood of reconstructed time series of previous mountain glacier estimates36 and ice sheet estimates37.
The spatial weighting of the prior covariance, BM, is computed as the outer product of the unique fingerprint associated with melt from the corresponding land-based ice source.
The prior spatiotemporal mean and covariance for the GIA contribution to sea-level change, μGIA(x,t) and KGIA(x,t), respectively, are taken as the sample mean and covariance of the 161 predictions of sea-level change described below.
The distribution of the contribution to sea-level changes from thermosteric and ocean dynamic effects is partially modelled as the sample mean and covariance of the CMIP5 model outputs32. However, since a small number of models are used to compute the distribution statistics, the estimated distribution may not be representative of the parent distribution. Consequently, we augment the sample covariance with a space-time separable covariance structure consisting of the product of two Matérn functions [35], C: one representing the temporal distribution and the other representing the spatial, such that the total prior covariance describing local sea-level change is given by:
KLSL(x,t) = KCMIP5(x,t) + C(t,ν1,τ) C(x,ν2,L)
where KCMIP5 is the sample covariance of the CMIP5 model outputs, ν1 and τ are the smoothness parameter and characteristic timescale of the temporal Matérn function, respectively, and ν2 and L are the smoothness parameter and characteristic length scale of the spatial Matérn function, respectively. For the exponents within the Matérn functions we follow ref. 10 and set the exponent on the spatial component to ν2 = 5/2 (reflecting a relatively smooth, twice-differentiable field) and the exponent on the temporal component to ν1 = 3/2 (reflecting a once-differentiable time series, in which rate is always defined but can change abruptly). As with the melt covariance hyperparameters, we use an empirical Bayesian approach to estimate the maximum-likelihood time and length scales of the Matérn functions to be 46 years and 90 km, respectively. Note that there is some trade-off between the Matérn exponent values and the hyperparameter characteristic scales: the selection of, say, a lower exponent (giving rise to a less smooth functional form) would result in a longer length scale.
In addition to capturing the inaccuracies of the ocean dynamics distribution, the Matérn functions also model local tectonic, geomorphological and other non-climatic contributions to local sea-level change. These hyperparameters, and a white-noise variance, are computed by finding the parameters that maximize the likelihood of the available tide gauge observations given the complete sea-level model.
Sea-level fingerprints
Extended Data Fig. 1a and b shows global maps of sea-level change, known as sea-level fingerprints, associated with rapid, uniform mass loss across the Greenland Ice Sheet (GIS) and the West Antarctic Ice Sheet (WAIS), respectively. The sea-level changes are normalized by the equivalent GMSL change. Both fingerprints are characterized by a large amplitude sea-level fall in the region adjacent to the melting ice sheet with a gradual rise in sea level moving away from the ice sheet. The computation of the fingerprints is based upon a gravitationally self-consistent sea-level theory that takes into account shoreline migration and changes in grounded, marine-based ice cover as well as the impact on sea level of perturbations in the Earth’s rotation axis38,39,40.
In addition to the GIS and WAIS, fingerprints were computed for the East Antarctic Ice Sheet (EAIS) and glaciers of Alaska, the Alps, Baffin Island, the Caucasus, Ellesmere Island, Franz Josef Land, High Mountain Asia, Altai, Iceland, Kamchatka, the low-latitude Andes, New Zealand, Novaya Zemlya, Patagonia, Scandinavia, Severnaya Zemlya, Svalbard, and Western Canada/US.
We also include a spatially uniform pattern to account for changes in GMSL due to land ice sources not included in the above set of glaciers. In the Kalman smoother, this uniform ‘fingerprint’ also captures changes in GMSL due to globally uniform thermal expansion and terrestrial water storage variations9.
GIA models
The first step when analysing tide gauge records is to correct for sea-level contributions due to the ongoing GIA of the Earth in response to the ice age cycles. Predictions of GIA are dependent on the geometry and deglaciation history of the Late Pleistocene ice sheets and the Earth’s viscoelastic structure. In this study, we computed 160 different GIA predictions distinguished on the basis of the adopted lower-mantle viscosity, upper-mantle viscosity, and thickness of a high-viscosity (effectively elastic) lithosphere. Additionally, we computed a GIA prediction using the VM2 viscosity profile41. These were combined with the ICE-5G (Ref. 41) global ice sheet reconstruction for the last glacial cycle. A detailed description of physical processes that contribute to the total GIA signal can be found in ref. 42.
We adopted values for the three rheological model parameters that encompass all recent estimates of the Earth’s structure. The lower-mantle viscosity was varied in the range (2–100) × 1021 Pa s, upper-mantle viscosity in the range (0.3–1) × 1021 Pa s, and lithospheric thickness in the range 72–150 km. Extended Data Fig. 2a and b shows the mean and standard deviation of the model predictions. The largest variance is seen in the region within the near field of the former ice sheets, including areas of ancient ice cover and the so-called peripheral bulges.
Ocean dynamics models
We treat the thermosteric and ocean dynamic contributions to sea level using the historical experiment output from 6 global climate models of the World Climate Research Programme’s (WCRP) Coupled Model Intercomparison Project phase 5 (CMIP5) data set32. Following ref. 9, the models we use are: bcc-csm1-1 from the Beijing Climate Center, CanESM2 from Environment Canada, the NOAA-GFDL model GFSL-ESM2M, the Institut Pierre Simone Laplace IPSL-CM5A-LR model, MRI-CGCM3 from the Japanese Meteorological Institute, and NorESM1-M from the Norwegian Climate Centre. For the KS methodology, we use the zero-mean spatial field ‘zos’ that is supplied by all the models. In the GPR, we add to ‘zos’ each model’s estimated globally averaged sea-level change due to thermal expansion: ‘zossga’.
While the CMIP5 model outputs are provided as global ocean grids, the field values at the specific locations of tide gauges are required, as input, to both the KS and GPR analyses. Where the tide gauges are coincident with model grid points, the associated value of the model output is used. Otherwise, an inverse distance weighting interpolation scheme is used to estimate the field at the desired location.
We examined three alternative interpolation schemes to assess the sensitivity of the KS GMSL estimate to this choice: (1) a nearest-neighbour approach, selecting the value on the CMIP5 grid that is closest to the tide gauge site; (2) a Delaunay interpolant, computing a linear interpolation between the irregularly spaced model cells along the coastlines; and (3) a Gaussian process (or simple kriging) methodology. For the Gaussian process interpolation, we employed a Gaussian process prior with a mean equal to the mean of the model grid values within a 200 km radius of the tide gauge location and a Matérn covariance function with smoothness parameter equal to 5/2. Since we are interested in the variability of the ocean models immediately surrounding each tide gauge site, the length scale of the Matérn covariance function was set to 1° (∼110 km). Neither the nearest-neighbour approach nor the Delaunay interpolated altered the estimate of the GMSL rate over the time period 1901–90. The Gaussian process interpolation scheme changed the GMSL estimate by less than 2%, significantly smaller than the estimated ±0.2 mm yr−1 90% CI on the estimate.
Computation of GMSL rates and accelerations