www.sciencemag.org/content/360/6395/1335/suppl/DC1

Supplementary Materials for

Observed rapid bedrock uplift in Amundsen Sea Embayment promotes ice-sheet stability

Valentina R. Barletta*, Michael Bevis, Benjamin E. Smith, Terry Wilson, Abel Brown, Andrea Bordoni, Michael Willis, Shfaqat Abbas Khan, Marc Rovira-Navarro,

Ian Dalziel, Robert Smalley, Jr., Eric Kendrick, Stephanie Konfal, Dana J. Caccamise II, Richard C. Aster, Andy Nyblade, Douglas A. Wiens

*Corresponding author. Email: vr.barletta@gmail.com

Published 22 June 2018, Science 360, 1335 (2018) DOI: 10.1126/aao1447

This PDF file includes:

Supplementary Text Figs. S1 to S15 Tables S1 to S4 References

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Figure S1: Solid Earth deformation caused by ice changes. Top) During an ice age, mantle underlying an accreting ice

sheet (light blue element) slowly flows away, creating a topographic depression beneath the ice load with peripheral

forebulges. Bottom) When the ice melts, the mantle slowly flows back to re-establish gravitational (isostatic)

equilibrium at a viscosity-controlled rate, causing the lithosphere and surface to rebound. The dashed brown line

represents the state before the deformation. The black arrows represent the measured vertical and the horizontal

component. The red and blue arrows schematically indicate the direction of the bedrock deformation. The

magnitudes of the arrows are not to scale and only indicative.

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Figure S2: Gravity perturbation produced by the Earth deformation. The shrinking ice mass is represented by the

light blue element in the bottom figure and its associated gravity perturbation is represented by the blue dashed line

in the top. The displacement of the bedrock produces the change in volume enclosed between the initial (black

dashed line) and the final topography (black solid line). The red and the blue represent positive and negative

variations respectively. The gravity perturbation caused by the net bedrock displacement is represented with the red

dashed line in the top figure. The green dashed line represents the observed signal, which is the sum of the two

components: ice mass loss and solid Earth deformation (mass redistribution). Independent assessment of the solid

Earth component (as in this study) allows for correcting the net gravity observation to accurately estimating the ice

mass loss.

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Figure S3: Scheme for grounding line migration promoted by bedrock uplift. Left) Initial ice and bedrock

configuration where A indicates the initial grounding line (GL) position. Right) The configuration reached after the ice

has lost some mass causing bedrock uplift and sea level fall.. The dashed lines represent the initial configuration of the

left picture. B indicates the position of the GL if only the reduction in gravitational attraction between ice sheet and

ocean is considered. C indicates the position of the GL when bedrock uplift and the reduction in gravitational

attraction are both considered. This cartoon is based on a similar cartoon in (15) in which the gravitational attraction

was not shown.

S1. GIA in Antarctica

As surface ice masses change, the Earth deforms with both an essentially immediate elastic response and a

much slower time-dependent viscous fluid deformation. During an ice age, mantle material underlying an

accreting ice sheet slowly flows away during the development of isostatic equilibrium to support the load.

This creates a topographic depression beneath the ice load. When the ice melts, the mantle slowly flows

back to reestablish the no load isostatic (gravitational) equilibrium, causing the lithosphere and surface to

rebound (Fig. S1). Fennoscandia and northern North America are today responding to glacial unloading

since the Last Glacial Maximum (LGM), which ended ~21 ka B.P., with ongoing gravity changes associated

with crustal uplift rates higher than 10 mm/yr. This Glacial Isostatic Adjustment (GIA) is especially long-

lived in cratonic regions because of high mantle viscosity (>5x1020 Pa s)(13) associated with them as a result

of having cold upper mantle temperatures. Global Positioning System (GPS) stations located on outcrops

adjacent to the great ice sheets can accurately measure bedrock displacements due to this process. For

example, the elastic motion is particularly well observed in Greenland(24), with uplift rates as high as 30

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mm/yr. Near and beneath the massive ice sheets of Greenland and Antarctica, current bedrock motions

contain contributions from both present and past ice variations(14, 33). Decomposing crustal displacement

into the elastic and longer-term viscoelastic components is thus especially crucial for accurate ice loss

estimates, and requires specific knowledge of regional elastic and viscous properties.

From the pioneering work of Haskell(34), based on two sites in Fennoscandia, to the modern global GIA

models based on much larger data sets(13, 35, 36), the value used for upper mantle viscosity has been

around 1021 Pa s, since the need to fit globally distributed data has led to the use of a mean viscosity profile

representative of a mean Earth value. The performance of these global models is poor in areas where the

actual viscosity structure greatly deviates from this average(37). In Antarctica, these GIA models have

recently been shown to be inaccurate(33, 38). One reason is that they do not incorporate major lateral

variations in Earth structure, particularly the large transition between warm and weak mantle beneath

West Antarctica versus cold and stiff mantle beneath the East Antarctic craton(39, 40). Another reason has

been the lack of glaciological, seismological, and geodetic observational constraints in Antarctica. The

principal reason, however, is that Antarctic ice history is much less known than for the other large Late

Quaternary ice sheets.

More recently, new data sets(33, 38) and recent advances in Antarctic glacial geology(41, 42) have led to

renewed interest in constraining the past extent of the Antarctic Ice Sheet (AIS) and a new generation of

GIA models (14, 35, 42–45) has been developed. These new regional GIA models(14, 44) are a clear

improvement, but still incorporate an upper mantle viscosity that is close to the global average of ~5x1020

Pa se, as ICE6G(13), with an upper bound of 1021 Pa s for W12(14) and a lower bound for IJ05_R2(44)

(~2x1020 Pa s). These models cannot yet predict many of the recently acquired geodetic observations from

Antarctica(12, 33, 46). In the Northern Antarctic Peninsula (NAP), for instance, GPS uplift rates greatly

exceed estimates for the purely elastic response to contemporary ice mass changes, implying a significant

viscous component of uplift and upper mantle viscosities in the range 0.6x1018–2x1018 Pa s (12). Low

viscosity is expected in the NAP considering its tectonic setting, which includes a subduction zone with an

actively extending basin behind an inactive volcanic arc(47) and active volcanism confined to the Bransfield

Strait and the James Ross Island area. A recent joint analysis of time-variable gravity (GRACE) and altimetry

(ICESat) data, clearly showed for the first time a large empirical GIA signal (23.3±7.7 mm/yr) in the

Amundsen Sea coastal region of West Antarctica(46). With similar empirical approach other recent studies

have found a significant GIA signal in ASE(48, 49). This signal is inconsistent with the available GIA models,

which predict negligible(44), or at most 5 mm/yr uplift rates(13, 14, 35).

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S2. GPS measurements

There are abundant indications of geologically recent volcanism and Cenozoic rifting in much of West

Antarctica. There so also limited evidence pointing to rift-related displacements of Oligocene age between

the Thurston Island-Eights Coast and Marie Byrd Land crustal blocks flanking the Amundsen sector(22, 23),

raising the possibility that the mantle viscosity in this region deviates significantly from the global average.

We installed a GPS network on bedrock around the ASE to explore the possibility of a low viscosity shallow

mantle, modeling the horizontal as well as the vertical components of displacement of all Antarctic GPS

Network (ANET) stations in the Amundsen sector.

Table S1: GPS station code (column 1), location (column 2 and 3, latitude and longitude respectively), initial and last

observation times (columns 4 and 5), time span and number of observations (columns 6 and 7).

Station Name Latitude Longitude t_first t_last T_span (yr) n_obs

BACK -74.43 -102.478 2006.04 2015.37 8.036 769

BERP -74.546 -111.885 2003.89 2015.37 10.386 1185

INMN -74.821 -98.88 2013.02 2015.37 1.252 457

LPLY -73.111 -90.299 2006.03 2015.37 8.241 697

THUR -72.53 -97.56 2006.04 2015.37 8.236 1185

TOMO -75.802 -114.662 2012.07 2015.25 2.203 801

The solutions for the six GPS stations in the Amundsen sector (Table S1) were obtained as part of a global

geodetic analysis that included more than 1500 worldwide stations, including the entire Antarctic GPS

Network (ANET), following a recently developed processing strategy(24, 25, 50). Most Amundsen sector

stations are uplifting rapidly (Main Text Figure 1C, Table S2), showing among the largest ice-driven uplift

rates in the world up to 40.6 mm/y. These rates easily exceed the maximum rate of 30 mm/yr observed so

far in Greenland(24), and are comparable to the maximum 41 mm/yr rate recorded by campaign GPS in

the Patagonian ice fields(10).

Amundsen sector GPS displacement time series were computed as part of a global re-analysis (g06c) that

included 4.15 million station-days of observations from a total of 1,854 stations, including all ANET and

GNET stations. The analysis was initially expressed in Ohio State University’s variant of the ITRF2008

reference frame (RF), called OSU2008, and subsequently shifted into a new RF which minimized the RMS

horizontal velocity of 10 stations in East Antarctica and the easternmost portions of the Trans Antarctic

Mountains (the RMS residual velocity of these stations was 0.52 mm/yr) and the RMS vertical velocity of

455 stations, the great majority of which were located at mid- and low-latitudes. This frame transformation

was achieved using the ‘Ensemble of RFs’ approach described by(24).

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Table S2: Table is divided in three sections, for the vertical (Vu) and two horizontal (East – Ve and North - Vn)

velocities. Each section displays the name of the GPS station (column 1), observed GPS velocities (column 2), sigma

(column 3), modeled elastic (column 4), residual (observed minus elastic, column 5). ICE-6G_C (VM5a)(13) GIA model

predicted uplift rate (Column 6), and weight (Column 7).

Station Name

Vu (mm/yr)

Su (mm/yr)

Elastic (mm/yr)

Residual (mm/yr)

ICE6G_C (mm/yr)

Weight

BACK 15.5 1.5 5.43 10.07 6.79 1

BERP 25.6 0.7 6.48 19.12 5.55 1

INMN 35.2 2.4 9.15 26.05 7.05 1

LPLY 6.20 0.5 2.26 3.93 6.32 0.2

THUR -2.2 0.8 1.79 -3.99 5.84 1

TOMO 40.6 3.0 10.70 29.90 4.89 1

Ve Se Elastic Residual

BACK 0.5 0.3 -0.02 0.52 1

BERP 1.6 0.3 -0.12 1.72 1

INMN -3.7 1.2 0.08 -3.78 1

LPLY -1.25 0.3 0.11 -1.35 0.2

THUR -0.9 0.4 0.14 -1.04 1

TOMO -8.1 1.3 -2.13 -5.97 0.5

Vn Sn Elastic Residual

BACK 4.7 0.3 1.71 2.99 1

BERP 8.8 0.3 2.19 6.61 1

INMN 11.3 1.4 2.24 9.06 1

LPLY 0.53 0.5 0.68 -0.15 0.2

THUR -2.3 0.3 0.75 -3.05 1

TOMO -10.4 1.1 -1.39 -9.01 0.5

The Amundsen sector displacement time series were modeled using a standard linear trajectory model(50)

consisting of a constant velocity trend and a 4-term Fourier series (annual and semiannual term).

Uncertainties associated with the velocity estimates were adjusted to take account of correlations in the

times series(51–53). All extant methods for decomposing measurement noise into a combination of white

and flicker noise (or some other combination of stochastic processes) assume that the noise is stationary

and the covariance matrix is Toeplitz. The major source of noise at most of the Amundsen stations,

however, is antenna icing, and this is driven by specific icing events. So the noise source is clearly not

stationary. We use a robust estimation scheme involving iterative reweighting of the observations, so that

data outliers are down-weighted, preventing major icing-related excursions in the time series from biasing

the velocity estimates. The posterior sigmas assigned to the data are therefore unequal, which, again, is

inconsistent with stationarity. We used a Monte-Carlo approach to estimate the formal errors on the

velocity estimates. The post-fit residuals were roughly consistent with a 1/f spectrum. The reality is that

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when the major source of positioning error is antenna icing it is very difficult, and probably impossible to

estimate a reliable confidence interval for the various parameters of the trajectory model.

The GPS time series for the different stations are not equally reliable; to try to reduce the risk of biases in

the results, when performing the grid search, we assign a weight to the data (see below). The sites closest

to the margin of the ice sheet (TOMO and INMN) are the most sensitive to local ice mass variations. Those

local variations cannot always be captured accurately by the temporally sparse satellite altimetry

observations. Moreover, TOMO and INMN also have the shortest time series that is also reflected in their

large uncertainties. The uncertainties associated with GPS velocity estimates in Table S2, reflect the

reliability of the estimate and hence the accuracy. The horizontal velocities for the station TOMO cannot be

considered reliable because of icing during the measurement period, while the vertical component is less

sensitive to icing, therefore we reduce the weight of TOMO horizontal velocities. The station LPLY has icing

problems and is also sensitive to ice mass changes that are not included in our grid. It is reasonable to

assume that the station LPLY is affected by phenomena that we cannot model with the information we

have.

Data Weighting

Note that the in Main Text Figure 3 (left plot), the estimated residual uplift rates for LPLY are, even in the

best cases, the worst among all stations, and all models tested tend to predict similar very small values of

velocities for LPLY (Main Text Figure 3, left plot). The large uplift residual measured at LPLY indicates a

possible effect from ice change outside our grid, and this is an a posteriori confirmation of our assumption

that leads us to choose a lower weight for LPLY. The effect of using weights wj (especially for LPLY)

produces an overall improvement in the and affects the number of cases in the 95% confidence interval.

The pattern and range of viscosity that we find with and without weighting the station are the same, but

the best models can be slightly different without affecting the overall quality of the results for the other

stations. The stations with the largest (weighted) uncertainties (LPLY, TOMO and INMN) are those least

important for constraining the Earth parameters.

S3. Altimetry and Mass changes measurements

We derived estimates of dynamically-driven surface-elevation change using laser-altimetry data from

ICESat(54), Automated Topographic Mapper(55) (updated) Land, Vegetation, Ice Sensor(56) (updated), and

stereo photogrammetry data from Worldview-1 (WV) and Worldview-2 registered to the laser-altimetry

data(57). We treated data from each sensor as point elevation measurements, with a small random error

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(10 cm for the laser-altimetry data, 0.5 m for the WV DEMs) and a correlated error (10 cm for the laser-

laser-altimetry data, 5 m for the WV DEMs) that is constant for each day’s data for the laser altimetry data

and constant for each WV DEM. To account for height variations generated by short-term fluctuations in

accumulation and near-surface density, we subtracted the estimated firn height anomaly relative to the

1979-2014 linear height trend(58) from each measured elevation, as calculated from the RACMO2.3

surface-mass-balance model(59). We used these data to estimate a DEM of the surface for January 1,

2013, and a set of smooth elevation-change maps between this DEM and the surface for annual increments

between 2011 and 2014, by minimizing a residual function that depends on the misfit between the data

and the elevations interpolated from the maps, on the bias estimates, on the roughness of the DEM, and

on the roughness of the elevation-change maps. We generated surface height estimates on overlapping,

100x100-km grids, and merged the grids into a common elevation-change map using a tapered weighting

function to avoid seams at the grid boundaries.

Figure.S4: Map of observed ice mass changes and related error. Left) Map of the trend for ice mass

changes obtained from the 13 yearly grids for the period 2002-2013 measured by altimetry (S3). Right) The

average of the 1-sigma error for the 13 yearly grids for the period 2002-2013. Note that the sigma in the

right figure is not the uncertainty on the trend shown in the left panel.

Because we subtracted the firn model height anomaly from the data before deriving the elevation maps,

these maps give an estimate of the surface elevation for a constant accumulation rate equal to the 1979-

2014 average. We converted these elevation values to mass anomalies for each grid cell by multiplying

them by the cell area (calculated based on the polar-stereographic projection used in our grids) and by the

density of ice, and add the mass anomaly that was used to drive the firn model. The fitting procedure gave

stable estimates for the smooth ice surfaces that characterize most grounded ice in our study area, but

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around mountains, we could not assume that the surface was smooth, and could not fit the same kind of

models. In these regions, we assumed that the ice-dynamic signal was zero, and that the entire mass-

change signal was due to the accumulation anomaly. Because floating ice makes no contribution to loading

signals, we set our mass anomaly to zero over floating ice and ocean, based on an estimate of grounding

line for 2009(8, 60).

We derived estimates of dynamically-driven ice surface-elevation change using laser altimetry, producing a

grid with 1 km resolution for each year between 2002 and 2014, with an average total mass trend for the

region of about -130 Gt/yr. Main Text Figure 1A and Fig. S4 shows the average trend in m/yr for the last 13

years. The ice-mass loss can be observed especially in the drainage basins of Pine Island, Thwaites and

Smith glaciers (PITS).

S4. Earth modeling

Traditional (elastic+global GIA) prediction

Large, and accelerating, uplift rates in Greenland can be explained almost everywhere as an elastic

response(24). This elastic deformation is proportional to the applied forcing (the ice mass load), with the

proportionality decreasing with distance from the load. To compute the elastic component for ASE we used

altimetry-derived ice mass changes as input load for a compressible, spherically symmetric, self-gravitating

Earth model(26) (see more details below). This predicted elastic response (Table S2) shown in Main Text

Figure 1(C, D), can only explain approximately 20% of the recorded signal, at best.

For each site the residual signal after elastic correction (green arrows in Main Text Figure 1) is still very

large. A part of this residual signal could in principle come from LGM deglaciation rebound. We tested this

hypothesis against predictions from existing global and regional GIA models. We note from literature (e.g.

Figure 14 in ref(14)) that in ASE ICE6G(13) predicts the largest uplift compared to other regional GIA models

such as ref(14, 44), therefore we show the ICE-6G_C model(13) uplift predictions (purple arrows in Main

Text Figure 1C). Those predictions are extremely small for three stations (between 16% and 29% of the

residual), rather small for one station (67% of the residual), far too large for one station (161% of the

residual) and even opposite in sign for another (-146% of the residual), clearly completely incompatible

with the GPS residual signal.

Three GPS sites with particularly high uplift rates are located on the eastern edge of the volcanically active

Marie Byrd Land dome (TOMO and BERP) and in the Cenozoic volcanic province of the Hudson Mountains

(INMN). Toney Mountain (TOMO) is a Pleistocene volcano and a possible source for ash layers in ice cores

c. 30,000 years ago(61), and there is evidence for recent volcanism in the Hudson Mountains approximately

2000 years ago(21). The possibility that volcanic activity plays a role in the rapid uplift of these sites

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therefore needs to be considered. However, the distance along the continental margin from Toney

Mountain to the Hudson Mountains is approximately 500 kilometers, whereas recent volcanic uplift (1992-

1997) in the Galapagos Islands covers an area of approximately 135x100 kilometers(62), and in the

Yellowstone caldera (1997-2003) approximately 40x35 kilometers(63). Magmatic or hydrothermal inflation

associated with individual volcanoes in the Andes and Cascades produce even smaller areas of uplift. Hence

although some volcanic contribution to the uplift at TOMO, BERP and INMN cannot be discounted, the fact

that it occurred simultaneously along almost 500 km of a continental margin subject to substantial ice mass

loss points to isostatic adjustment as the main cause.

Modeling elastic time series

Elastic deformations are linearly proportional and instantaneous responses to the loading function and

they can be analyzed as a time series or as a trend. The loading function is given as an ensemble of discs of

different size and different loading history. The elastic uplift is based on a compressible, spherically

symmetric, self-gravitating Earth model. The Green’s functions are spatially convolved with the ice loading

discs using methods presented in ref(26). Load Love numbers, based on the PREM Earth structure(64), are

computed in the center of mass reference frame up to a maximum spherical harmonic degree of 3000

using VE-CL0V3RS v3.5.6 (Visco-Elastic Compressible LOVe numbER Solver) and the degree-1 Love number

was computed as described by(65). By using the assumption that the elastic Love numbers become

asymptotic after the maximum degree, the software implementing the High Resolution technique (VE-

HResV2) allows us to capture the loading concentrated on glaciers a few km wide(26). In this way, the

resolution is limited only by the resolution of the input loading discs.

For the elastic loading we use the high-resolution grids provided for each year from 2002 to 2014 (Section

S3), so for each GPS site we compute the elastic time series that is used as a correction for the observed

GPS uplift, to yield the residual velocities. In fact, given the elastic time series for each GPS site, it's possible

to select the same time frame the site (Table S1) has been operational and compute a coherent trend. We

correct the velocities with the linear trend of the elastic correction computed over the same period of the

GPS time series.

Sensitivity to the load resolution

When a GPS site is very close to a glacier, then the resolution of the loading might become an issue to be

carefully considered. We tested the impact of the grid of our ice history by using a downsampled grid and

the difference is below 2% on the elastic prediction.

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Double-blind test for the elastic code

We tested our solution with a double-blind test for the elastic solution using the same mass grids that we

use for this work, and also using an elementary disk shaped load. At an early stage of this work two of the

coauthors have been separately asked by another coauthor to perform the same computation using their

independent codes. The coauthors performing the computation did not know about each other task. The

two tests yielded nearly the same result, with differences of the order of 2%, or less, depending on the GPS

station.

There is no other similar code to test the code for the viscous modeling against. However, this code has a

very similar structure to the code for the elastic, and the viscous Green function that we are using has been

tested in other ways. We ran the same code with the incompressible approximation (which is a limit of the

compressible that we use) and tested the output against ref. (65): the agreement is perfect.

Uncertainty on the elastic contribution (qualitative assessment)

The uncertainty on the elastic contribution is affected by the uncertainty on the ice mass change in our

yearly grids and the neglected contribution from the ice mass loss outside our grid. We estimated the error

for the neglected mass outside our grid using an independent ICESat-derived ice height change grid

(courtesy Louise S. Sørensen) over the period 2003-2009 computed for another study(66). We cannot use

that estimate to correct the observations because they do not match the GPS observation period and the

mass trend can therefore be different. We can use the value as an indication of the error for not including

the mass changes outside the grids, which is roughly 0.2 mm/yr on all stations except LPLY where it is

slightly over 0.6 mm/yr. This error is roughly uniform over most stations as expected since it is a far field

signal. Including this error in the misfit computation (S11) would not change the patterns from the

parameter grid search (the minima would be the same), but we chose to use this information to define the

weight of the LPLY station (see Data Weighting in S2). The error on our grids due to the ice mass change

uncertainty reflects the distance from the load, as it happens also for the error of the GPS velocity. The

stations closer to the ice changes like INMN and TOMO are more affected. The uncertainty on the rate of

elastic uplift related to the ice mass change is very small (less than 2%). This is not only because of the

distance of the station from the ice load but also because the average uncertainty for the yearly ice mass

grid is less than 10% of the ice mass trend (Fig. S4) so the actual uncertainty on the ice mass trend is much

smaller.

Another source of uncertainty for the elastic uplift comes from the uncertainty on the elastic parameters of

the Earth. As part of some preliminary in depth analysis, we made a test using “realistic” local elastic

parameters derived from the seismic velocities and the difference in the uplift predictions, compared to

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those obtained using PREM, is about 1%. So we assumed that PREM, in terms of the elastic parameters, is a

good start.

In the light of these considerations, we estimated the total uncertainty for the elastic modeling to be less

than 5%. One of the reasons for this small value is related to the specific nature of our loading. Significant

ice mass changes are happening several km away from the GPS, therefore the impact of the crustal elastic

parameter variability on the GPS is not significant as in some of the hydrology-related example discussed in

Dill et al. 2015(67). The impact is small also because in ASE the crust is thinner than the global one used in

Dill et al. 2015(67), and this further reduces the distance for sensitivity to the crustal structure details.

All in all, we estimated the total uncertainty on the elastic model to be very small (compared to the error

on the GPS measurement) and rather uniform over the six stations, so that this effect would not add useful

information to constrain the grid search. Therefore we decided that it was safe not to include it, and use

instead the insight coming from this analysis when choosing the weight for the stations.

Viscoelastic modeling

Earth model structure:

We investigate the hypothesis of a geodynamically important low viscosity shallow upper mantle for this

region using a compressible, spherically symmetric, self-gravitating Earth model with a viscous Maxwell

rheology. The viscosity structure (Table S3) consists of an elastic lithosphere (L) with a variable thickness

between 20 and 90 km, a viscoelastic Shallow Upper Mantle (SUM=1018-1020 Pa s) down to 200 km, a Deep

Upper Mantle (DUM=1019-1020 Pa s) down to 400 km, a Transition Zone (TZ=1019-1020 Pa s) down to 670 km,

and a Lower Mantle (LM=1023 Pa s) down to the core-mantle boundary. The structure for density and

elasticity is PREM-based(64) and consists of 31 finer layers. We computed 807 models, obtained by varying

the viscosity of three layers of the upper mantle (SUM, DUM and TZ) and the lithospheric thickness (LT)

over the range of values in Table S3. Based on those 807 models we computed approximately 10,000

model variations by linearly combining different segments of ice history (S5).

Table S3: Layered Earth viscosity structure. The viscosity is constant in each layer R_outer ≥ R > R_inner.

Layer Name R_outer (km) R_inner (km) Viscosity (Pa s)

LT 6371 6351 – 6281 1051

SUM 6351 – 6281 6206 – 6136 1x1018 - 1x1020

DUM 6206 – 6136 5971 1x1019 - 1x1020

TZ 5971 5701 1x1019 - 1x1020

LM 5701 3480 1022

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Modeling viscoelastic trends

The viscoelastic response at any given time t depends on the entire loading history occurring before the

time t. However the sensitivity to a specific time window is strictly related to the viscosity structure of the

Earth model. The viscous deformation of the Earth in response to the ice-mass loss is based on a

compressible, spherically symmetric, self-gravitating Earth model with a viscous Maxwell rheology. With

VE-CL0V3RS v3.5.6 we compute the elementary viscoelastic time-dependent Green's functions (convoluted

with the Heaviside function) up to degree 1195 and thereafter we assume they don't change with time so

that the contribution of the higher degrees to the combined Green's function is negligible. This was verified

when choosing the maximum degree so that the results do not suffer from effects of truncation.

S5. Ice history

GIA modeling requires a suitable glaciation history. Low viscosities, in the range of the model (Table S3),

produce a response that is largely sensitive only to the last couple of centuries of ice history and is,

conversely, highly insensitive to millennial-scale ice mass changes. Considering that the ice mass was likely

having small fluctuation during the Little Ice Age (LIA), a colder period that ended 200 years ago(68, 69), it

is safe to assume that, for low mantle viscosity, ice mass changes before the end of the LIA have a

negligible contribution to the bedrock deformation prediction. The large residual signal, therefore, has to

be solely explained as the viscoelastic response to recent (approximately last century) ice history.

In more detail, deglacial histories of the Amundsen Sea sector derived from seafloor and terrestrial records

suggest that ice sheet grounding line retreat from its position on the shelf edge to near the modern

configuration was largely completed by early Holocene (~10,000 years ago), but records are particularly

sparse during late Holocene(70). Cosmogenic surface exposure age results suggest that in the interior of

WAIS the ice thinning during the Holocene was no more than 45 m(70). For the last 40 years there are

instead records of a significant grounding line retreat in PIG(8, 71, 72) and increase of ice discharge in

ASE(4). Therefore during the Holocene and until the last century there are no evidences of ice changes

comparable to those of the last decades in ASE. The global sea level record indicates that before the 1850

there was a global sea level drop(73). This indicate a global trend of ice accumulation during a cold period

known as Little Ice Age (LIA), which is mostly attributed to the northern hemisphere and started about in

1300 ended in 1850. Ice cores indicate that the LIA in Antarctica most likely ended 200 years ago(68, 69).

We started from these few evidences to build simplified ice history for ASE, to be used as a starting guess.

We assume that before 200 years before present the ice was in balance (no significant ice mass changes),

and we consider different variations consistent with the lack of evidences of significant ice changes in ASE

before the end of the LIA, and with at most moderate variations between the end of LIA (200 years ago)

15

and the onset of the PIG grounding line retreat 40 years ago(8, 71, 72). For this reason, in the first 3

experiments we assume that between 100 and 200 years ago the ice mass change was smaller than in the

last 100 years. Later on, we tested the sensitivity to ice history and we extended the period up to 200 years

to assess the effect of a longer ice mass change history. Therefore, we set up a first estimate for ice mass

change history for the last 100 years, with the ice mass change history before that period is assumed to

have been zero. By inspecting the observed horizontal displacements (Main Text Figure 1D) and the

modeled elastic displacements we can see that they have similar directions. This is compatible with the

hypotheses that the pattern of ice changes responsible for the observed displacements could have been

very similar to the pattern used to compute the elastic displacements due to the present-day ice mass

changes. As a first estimate we assume that the spatial distribution of the recent past ice history was the

same as the modern one. The quality of the fit (section S6) supports, a posteriori, this simplistic

assumption, and in absence of any independent evidences pointing towards other more complex

hypotheses, in this first study we preferred to stick to this assumption. We use 13 yearly grids from 2002 to

2014 containing height variations, dH, of ice with respect to the DEM of 2010, as described in section S3

(Altimetry data on ice mass changes) as a base to build our ice history. For the period modeled, the

observed ice changes are used for each element (disc) of the input grid, and this ice mass change history

segment will be referred to as H0. This segment accounts for the observed variability, and the residual

between H0 predictions and the observed trend between 2010 and 2014 is very small. For the ice mass

changes between 1902 and 2002 we used the pattern of the trend of the ice mass changes between 2010

and 2014 from H0 (Main Text Figure 1A) rescaled by a factor to account for a slower mass loss rate in the

past. As an initial estimate, 25% of the trend of ice changes recorded between 2002 and 2014, H0, has been

used to build this second segment of ice history (1902-2002) that will be referred to as H1(=0.25 x trend of

H0).

The 25% assumption of the present-day ice loss rate is compatible with studies on ice flow(4). A slower ice

flow between 1970 and 2002 relative to the last 10 years has been found. We assume that a lower flow

rate is linked with a lower ice mass loss rate. Of course, there are differences between basins and for

example in the Thwaites basin (basin 19 in basin definition(74)) there is a less significant flow rate

increase(4). We used this information to build a variant of our ice history where, for the period between

1902 and 2002, we use the same thinning rate in Thwaites basin as the present day one. We name this new

ice history contribution b19 (basin 19 in basin definition(74)).

The segment H0 accounts for the observed ice mass changes since 2002, as given in the available grid of

measured ice mass changes. The segment H1 accounts for our estimate of ice mass changes between 1900

and 2002 for the whole grid with a 25% of present-day ice mass loss rate. However, if for instance the ice

mass loss rate in the Thwaites basin did not change in the recent past, then H1 underestimates the

16

Thwaites basin contribution by exactly a factor of 4. Restoring Thwaites basin to the steady state, would

add 3 times segment b19 (because one is already accounted in H1). We use an additional segment of ice

history that we name H2 that accounts for the interval between the years 1850 and 1902, for which we use

the same trend as H1.

Strategy for testing different ice history configurations

We divide the ice history in multiple independent segments (H0, H1, H2 ... Hn) covering different periods

(and different regions as the case of the segment b19) and we use the linearity property of GIA to compute

the contribution for each segment and combine them afterwards using different weights.

For example, given the first segment of ice history H0 (between 2002 and 2014) and the second H1

(between 1902 and 2002) we compute the GIA contribution for each segment separately and sum them to

obtain the total response. Linearity allows us to vary the weights of the segments so we can better fit our

prediction without redoing the full computation every time. If we want to double the ice melting between

1900 and 2002, we must multiply by 2 the GIA signal produced by H1 and sum it with the GIA signal

produced by H0. The segment H2 (between 1850 and 1902) can be added and weighted in the same way as

H1. We also consider a variant in which we extract the ice history for the Thwaites basin only, the

contribution b19, and we compute the GIA signal just for that basin and only for the segment between

1900 and 2002 (using the previous assumption of the 25% of present day ice melting). We sum this b19

contribution with H0 and H1 to obtain the result for this variant.

S6. Inversion details

For the Earth model of section S4 and ice history assumptions of section S5, we perform a grid search over

values of the physical parameters by comparing the modeled predictions with the observations (corrected

for the elastic contribution). We examine a L2-norm misfit function (defined in S11) and its 95%

confidence interval to assess the performance of models as a function of the free parameters. We keep the

ice history fixed at our first estimate for the ice mass changes H0+H1 (described in S5) to perform three

numerical experiments to show the effect of the different parameters (LT, SUM, DUM, TZ). We then assess

the effect due to the uncertainty of the ice history.

Experiment 1

In the first experiment (E1) in Fig. S5, we assess the suitable range of lithospheric thickness (LT) and SUM

viscosities, by keeping the viscosity of deeper layers (DUM and TZ) at a fixed value of 7x1019 Pa s. This value

is 6 times lower than the AP study(12), but larger than what we expect to find for the SUM viscosity. The

misfit in Fig. S5 rapidly increases for SUM viscosity lower than 3x1017 Pa s (175 on the 10xLog10 in the scale

17

used for convenience in the figures), and it also increases when LT is less than 30 km. We find a suitable

range of viscosity for the SUM between 1018 and 1019 Pa s. For a stiff mantle, >3x1019 Pa s (>195 10xLog10),

the misfit is insensitive to the LT, while for a softer SUM an LT greater than 50 km is always preferred.

Further investigations on the effect of an extra load on Thwaites basin show an improvement in fitting the

horizontal residual motions, but the overall fit worsens (see below Sensitivity to load geometry).

Figure.S5: Lithospheric thickness and shallow upper mantle viscosity. In both grids the misfit (color scale from 0 to

30) as a function of lithospheric thickness (x axis) and SUM viscosity (y axis) is shown for fixed DUM and TZ viscosity of

7x1019 Pa s (solid squares in the left grid and hollow squares in the right) for the experiment E1 and for ice history

H0+H1. Note that the viscosity is represented in 10xLog10 Pa s. (right) the case where the DUM has same viscosity as

the SUM is shown (solid circles). The areas outlined in green represent the 95% confidence intervals for each case. The

best fit for the E1 is LT 80 and SUM 182 with a =3.1 (left plot, panel a). For the variant (right plot) the best fit is LT 50,

SUM=DUM=188, (6.3x1018 Pa s) and the =2.34.

Sensitivity to load geometry

We explore another way to get a better fit for thin lithosphere. We change the geometry of the load, which

mostly affects the horizontal displacements and has little effect on the vertical. By inspecting the pattern of

the horizontal displacements (Fig. S6, left plot), for the 50 km LT model, we note that the directions for the

modeled horizontal displacements for stations BACK, BERP, TOMO and INMN are rotated toward the

Thwaites basin (enclosed in the purple line in Fig. S6B, right plot) relative to the measured displacements.

That suggests that there could be some mass loss missing in our ice history in that basin.

Therefore, we want to restore Thwaites basin to a steady state mass loss during the whole period (1902-

2002), adding 3 times the segment b19 (one is already accounted in H1, S5) with the same setup for the

Earth parameters as the experiment E1 (Fig. S6B, left plot). We find a shift of the values within the 95%

confidence interval toward thinner lithospheres. The best result in this case is for 50 km; we show the

deformation with and without the additional b19 ice load segment for this result (Fig. S6B, right plot). By

18

changing the mass loss in Thwaites basin we improve the fit of the horizontal displacement but not the

overall fit. To effectively improve the geometry of the problem we need more reliable horizontal time

series and more information about the ice mass changes in the last two centuries, information that is not

available at present.

Figure.S6: Changing ice mass in Thwaites. As in Fig.S5, for experiment E1, for ice history (H0+H1+3xb19) with the

variant with more mass on Thwaites basin we show misfit as a function of LT and SUM. The squares filled with solid

color (left plot) represent the cases within the 95% confidence interval. In the right plot we show the horizontal for

the best case with and without the Thwaites basin (the area with purple contour line in the right plot) variant applied.

The error bars used for the ellipse in the right plot do not represent the original ones but instead account for the

weight (error_bar = sigma/weight)

Experiment 2

We find a better fit and better constrained viscosity by lowering the viscosity of the deeper mantle layers.

Using E1 as reference we vary the depth of the SUM down to 400 km (right plot, Fig. S5) and observe a

narrower range of suitable viscosities and a shift in the best-fit viscosity toward higher values. In this case

(right plot, Fig. S5), the LT is still not well constrained, but for the best fit (188 10xLog10 Pa s) we find a

minimum corresponding to 50 km-thick lithosphere. Improvement in the misfit and the shift and narrowing

of the best-fit viscosity range indicates that the problem (ice + GPS geometry) is clearly sensitive to mantle

viscosity variations to at least 400 km depth. This provides the impetus to explore the effect of the viscosity

variation in deeper layers, with two additional experiments (Main Text Figure 2).

Experiment E2 shows the misfit as a function of SUM and DUM variation for fixed LT (Main Text Figure 2A)

and it confirms the range of suitable viscosities already found for SUM. We find a range of viscosity slightly

19

higher for DUM but most importantly we find a relation between the SUM and DUM viscosities. This

indicates that these model features are correlated, and that the inversion is sensitive to DUM viscosity. The

region within the 95% confidence in the two-dimensional viscosity space gives fits with comparable quality

if an increase in SUM viscosity is compensated by a decrease in DUM viscosity that is linear in the

exponents, with a ratio of -0.86. This indicates a slightly larger sensitivity to DUM than SUM viscosity,

although as the ratio is close to -1 they have almost the same importance. From E2 we also see that the LT

lower than 40 km gives much larger misfit. However, it is not clear that 70 km lithosphere gives worse

misfit than a 50 or 60 km lithosphere, as they are within the 95% confidence interval (Main Text Figure 2A).

Experiment 3

In a third experiment E3 (Main Text Figure 2B), we explore the misfit while varying the TZ and the DUM for

the three values of SUM viscosity that give the best fit in E2 (SUM=184, 186 and 188 10xLog10 Pa s) and we

find further improvements in the misfit function (> 10%). Even more interesting is that we find an inverse

relationship also between the TZ and DUM viscosities. The ratio of -2 between the exponents of the

viscosity means that TZ and DUM tend to trade off against each other, but with a wider range of good

values for TZ, so the problem is less sensitive to TZ. However, it is important to stress that the results are

sensitive to the variation in TZ viscosity (see below: test for TZ sensitivity). The relation we find shows that

the TZ layer has a minor but significant influence on the bedrock motion. The spatial scale of the ice

geometry affects the whole upper mantle, which is captured sufficiently with the aperture of the GPS array

and the sensitivity of the GPS measurements.

Figure.S7: Misfit as a function of the DUM and TZ computed with 4 degree of freedom (instead of 5). In the left

panel the misfit is computed with respect to one of our best models corresponding to the black square at the

coordinate (in 10xLog10) SUM=186, DUM=192 and TZ=194. In the right panel the misfit is computed with respect to

20

the vertical observations only. All the squares shown are within the 95% confidence interval (below =3), we chose

not to show all the others.

Two additional tests for TZ sensitivity

We also made 2 other different tests to prove the sensitivity to TZ viscosity (Fig. S7). In one test, we use

only the vertical velocities, that have more reliable observations, and in the other test we use one of our

preferred models instead of the observations to compute the misfit . Both these tests give a sharper 95%

confidence interval that clearly divide good values for TZ viscosity from those which give poor fit

Inversion results

Figure.S8: Correlation between observation and best-fit models. The observation (after the elastic correction) and

the error are those of Table S2 column 5 and 3 respectively. The modeled deformation are those of our three best

models in Main Text Figure 3A, the color code is the same. R2 indicates in the graph is the correlation index. The error

bars represent the observational error Table S2 column 3.

Our experiments E2 and E3, accounting for uncertainties on ice history (±50% H1 variations), give a strong

indication of a relationship between the three mantle layers and the existence of a rather narrow range of

viscosity (Table S4) that gives a prediction capable of explaining most of the observed signal (with R2=0.96,

see Fig S8). Our preferred model derived with our initial estimate H1 has 60 km lithosphere, the shallow

21

upper mantle of 3.98x1018 Pa s, the deeper upper mantle of 1.58x1019 Pa s and the transition zone of

2.5x1019 Pa s (Main Text Figure 3A, “Best 2” model, purple dots and arrows).

Table S4: Range of best viscosity for the whole upper mantle and transition zone.

Layer Depth of the

bottom (km)

Viscosity (Pa s) Viscosity

(10xLog10)

Preferred

model

LT 50 – 60 - - 60 km

SUM 200 2.5x1018 – 6.3x1018 184 - 188 186 (10xLog10)

DUM 400 1x1019 – 2.5x1019 190 - 194 192 (10xLog10)

TZ 670 1.58x1019 – 3.98x1019 192 - 196 194 (10xLog10)

Sensitivity to ice history uncertainties

The results of the grid search are obtained under assumptions about the ice history (S5), which remains

quite uncertain. However, there are two main characteristics of the theory of the viscoelastic relaxation

that, combined with the present day measurement and the sparse information we have about the local ASE

ice, help to narrow the discussion substantially. We leave the technical details to a more specialized

context, but we provide a qualitative sketch, for the readers’ convenience.

Figure.S9: Effect of viscosity on spatial pattern of the deformation rates. The curves are the deformation rates

produced by a disk of 50 km radius losing mass at 1 m/yr in water equivalent. The blue, red and green colors represent

models with 1020, 1019 and 4x1018 Pa s shallow upper mantle viscosities respectively. The solid lines represent the

deformation rates caused by the unloading occurred between -10 and -110 years from now. The dashed lines

represent the rescaled deformation caused by the unloading occurred between -200 and -300 years from now. The

scale factor is indicated in the label and it is computed to match the maximum of the corresponding solid line. The

dot-dashed blue line is the same as the dashed blue line but for the unloading occurred between -1900 and -2000

years from now.

22

The first characteristic is that the viscoelastic structure of the mantle acts as a filter in the spatial frequency

domain, when the uplift rates are considered: the high spatial frequency contributions decay much faster,

meaning that the pattern of the uplift rates due to recent mass variations show more spatial details, while

the old mass variations give a smoother pattern (Figure.S9). “Recent” and “old” (i.e. the time scale) are

relative to the viscosity, of course. The second characteristic is that the present day uplift rate is most

sensitive to the most recent cumulative amount of mass variation, and the sensitivity decreases

monotonically with time. This is the reason why in Figure.S9 the curves produced by older unloading events

must be rescaled to be comparable with the most recent ones. Moreover the cumulative mass variation in

the same time interval in the recent past (last hundred years) needed to produce a given present day uplift

rate is systematically larger for high viscosity models than for low viscosity ones (Figure.S9). However for

high viscosity the sensitivity to mass changes of several hundreds to thousand years ago is much higher

than for low viscosity. These two characteristics combined allow us to exclude form this discussion a wide

range of models, either because they would require post-LIA mass changes contradicting the local

evidences, or because they would require additional unreasonable assumptions on the previous ancient ice

history. This leaves us with a potential narrow range of scenarios to discuss.

In detail, the presence of high frequency spatial features, visible for example in the large difference

between INMN and BACK (Main text Figure 1), is compatible with a signal produced by recent ice mass

variations, specifically mass variations with a pattern very similar to the present day one (almost colocation

of maximum uplift rate with maximum present day mass loss). Regardless the viscosity and the cumulative

mass changes, ice changes with a different pattern would change the signature of the present day signal.

Simplifying the arguments for the sake of clarity, this leaves two options. The first is to consider only Earth

models with low viscosity that makes them insensitive to the ice history beyond the last few centuries. The

second is to consider also high viscosity models, and correspondingly make additional restrictive

assumption on the pre-LIA ice history, assuming a substantial constancy of the mass loss pattern. This

assumption, however, becomes more and more unlikely, the longer the time span. But even under this

unlikely assumption, in order to produce the present day uplift rates and their pattern, high viscosity

models require large mass changes in the post-LIA period also, and the higher the viscosity, the larger the

mass changes need to be. Large post-LIA ice changes are inconsistent with the observations(4). Assuming

instead that the post-LIA scenario cannot contradict those observations, the only option would be to

assume a LIA and pre LIA scenario with very large (due to the reduced sensitivity) ice mass loss, with a

pattern consistent with the present-day one. This scenario, however, is partially in contrast with indicators

of relatively small ice changes such as the cosmogenic surface exposure age(70) or the global sea level fall

before 1850(73). Moreover, for models with very high viscosity (like that used in the traditional global GIA

23

models), we simply notice that the maximum values of the uplift rates (after removing the elastic signal) in

ASE are 3 times larger than what is currently measured in North America. This implies that locally the

amount of cumulative mass lost needed to produce those rates considering only an LGM ice history, would

be unreasonably large, at least comparable (or larger) than the 3 kilometer-thick Laurentide ice sheet.

Based on these general considerations, our preliminary study leads us to investigate the possibility of

moderate variations of our initial ice history (S5) only. Results compatible with the same spatial pattern can

be found combining models with a viscosity slightly higher than our best low viscosity model, and a

proportionally larger mass changes for the ice history as we show below.

Excluding the unrealistic scenarios as discussed above, we are still left with assessing the sensitivity to ice

uncertainty (S5) by exploring the effect of a very wide range of possible total mass variations (from ±50%

up to ±100%). Nonetheless, the results for the upper mantle viscosity remain confined to a fairly narrow

range. Each of the experiments (and related variations) performed has a good minimum with a misfit

between 2.28 and 2.36; we show three examples in Main Text Figure 3A. We note the very good fit for the

vertical residual, with almost all predictions within the errors (Main Text Figure 3A, left plot) and a RMSD

(root mean squared deviation) comparable with the RMSE (root mean squared error). The horizontal (Main

Text Figure 3A, right plot) predictions are quite close to the error but not as good as the vertical, leading to

a total misfit function that is more than twice the one obtained for the vertical. The quality of the three

model predictions is clearly the same.

As discussed above and in section S5, we do not have evidence that suggests a substantially different

spatial ice mass loss pattern from the present-day one. However, differences in the ice mass change rate,

and therefore the total amount of mass loss, could produce a response with a pattern that is still consistent

with the GPS measurements. To investigate this, for the models of the experiments E2 and E3, we test the

misfit by varying the segment H1 up to ±100% and by extending and weighting the ice history up to 200

years ago (by adding new segments). Here we show only few significant results that allowed us to

understand the relationship between the uncertainties of ice mass loss rates and viscosity, which lead us to

determine the range of viscosities compatible with the reasonable ensemble of ice history scenarios.

We vary the trend of the H1 segment by ±50% (Fig. S10) and we find a shift of ±0.2 in the exponent for the

viscosity of all 3 layers (SUM, DUM and TZ), still following the same relationship we have found in Main

Text Figure 2. For each variation of H1 we find the best fit when the 3 layers satisfy that relationship and

the minima all have the same quality. The more we increase the weight of H1 (i.e. we increase the mass

loss) the stiffer the mantle needs to be to produce a good fit. And the more we decrease the weight of H1

(i.e. we decrease the mass loss) the weaker the mantle needs to be. Our E3 grid (Fig. S10B, left plot) covers

a range wide enough to see the effect of ±50% variation of H1, while the E2 grid (Fig. S10A, right plot)

24

covers a range wide enough to see both the effect of ±50% and ±100% variation of H1. The relationship we

find for ±100% H1 is again shifted by 0.2 in the exponent with respect to the ±50% H1.

Figure.S10: Effect of loading history variations. In panel a) we show the grid search for experiment E2, using 4

different variations of the ice history by changing the ratio of H1. Two cases lower than H1 are plotted with squares

for 0.5xH1 and 0xH1 in the left and right plot respectively. Two cases larger than H1 are plotted with circles for 1.5xH1

and 2xH1 in the left and right plot respectively. The color scale for the accounts only for the range within the 95%

confidence interval. In panel b) we show the grid search for experiment E3 using 2 different variations of the ice

history by changing the ratio of H1 (left plot) and 2 different variations of the longer ice history H1+H2. Circles and

squares represent the case of ice history lower and higher than H1, 0.5xH1 and 1.5xH1 respectively (in the left plot). In

the right plot circles and squares represent 0.4x(H1+H2) and 0.8x(H1+H2) respectively. The color scale for the in this

case does NOT accounts for the range within the 95% confidence interval. The upper bound of color scale of the misfit

is lower than the value for the 95% confidence interval and it is chosen to make the relationship between DUM and

TZ more evident.

In this way, by varying the mass loss rate we are varying the cumulative ice mass variation in the history of

the region. However, the cumulative mass variation cannot be completely arbitrary and must have a

reasonable upper and lower bound as well as a reasonable time span. Another way to increase the

25

cumulative mass is to extend the ice history back into the past. However for models with low viscosity

(around our best model) even very large ice changes before 200 years ago (comparable to present day ice

loss) would not produce any significant changes in the misfit (the sensitivity is too low). Extending the ice

history back in time would be relevant for models with much stiffer viscosity with respect our best model,

yet the mass loss required in the last 200 years to get a good fit would be much larger than our initial ice

history estimate. We don’t have enough information to set an upper boundary, however H1 is a reasonable

estimate based on a study on ice flow(4), so we can assume that an upper boundary is a cumulative mass

loss equivalent to +100% H1 starting 200 years ago (about 6000 Gt of cumulative mass loss). On the other

hand, we cannot imagine the mass loss to be completely zero before 12 years ago and that gives an

extreme lower bound (that corresponds to 0xH1 or -100% H1).

With the variation up to ±100% of H1 segment, we explored the effect of possible total mass variations

covering a very wide range. Across this range, the results for the upper mantle viscosity remain confined to

a fairly narrow range. However, for the models with stiffer mantle there is an increased sensitivity to the

past history, especially relevant in the case of very large mass variations. Therefore, here we show the

setup of experiment E3 where we extend the ice history into the past up to 150 years adding a further

segment (H2) for the case of 60 km lithosphere (Fig. S10B, right plot). This further segment of ice history

accounts for the interval between the years 1850 and 1902, and we use the same trend as H1 for the

computation. We explored the effect of varying H2 in the same way we did for H1 but the sensitivity to H2

alone is too low. However we found good fit by varying the mass loss rate during the whole period from 12

to 152 years ago (H1+H2). By using 80% of the reference ice mass changes for H1+H2, that is 0.8x(H1+H2),

we find a result that is very similar to the one from our initial estimate H1. And when we halve that using

40% of H1+H2, the result is even closer to what we find using half (0.5xH1) of our initial estimate H1. The

cumulative total mass is different in the 2 cases, about 3000 Gt and 4000 Gt for H1 and 0.8x(H1+H2)

respectively. However, less mass in the first segment H1 is compensated by the mass added in the second

segment H2.

What is most relevant is that adding H2 yields results for the viscosity that remain in the same range,

showing that the sensitivity of the problem to the mass distribution, both in space and time, is small

compared to the sensitivity to the viscosity. Our best Earth model (SUM=186, DUM=192, TZ=194 10xLog10

Pa s) for H0+H1 is still within the 95% confidence interval if we add ±50% of H1 (Fig. S10). That means that a

variation of 50% in the ice history (before 2002) produces a variation in the uplift rate that is still within 2

sigma of the observations.

26

S7. Sensitivity to complexity of modelling (i.e. Is it possible to

improve the Ice History+Earth model for ASE?)

The quality of the fit for three of our best models is very good (Fig. S8) with the residual discrepancy

comparable to the observation errors. The model that we used is simple, nonetheless the quality of the fit

is a strong indication of the fact that though simple, it is able to catch all the major feature of the

phenomenon. And given that we are solving an inverse problem, the paucity of GPS constraints (we have 6

new GPS time series) sets an upper limit to the complexity of the viscosity model. This prevents the

development of complex 3D models at this point. Such models could be used only if most of the

parameters were assigned a priori.

Assumptions were made concerning the Elastic structure, and the local ice history, and therefore we were

careful in trying to assess the stability of our results with respect to all these uncertainties. Here we expand

the discussion on this topic, for the interested readers.

Background signal

BERP and THUR (Main Text Figure 3) have the smallest error and longest time series, so together with BACK

they provide the strongest constraints on the inversion. Most importantly BERP and THUR have almost

opposite directions and they are influenced in a very different way by the load and the Earth model. THUR

captures the far field effect of the ice mass changes and BERP captures the medium-/near-field effect.

Therefore, the good fit in BERP and THUR for both the vertical and the horizontal components gives very

good a posteriori indications regarding the absence of a significant uniform background signal, and the

suitability of the elastic correction. The 1D symmetric Earth model, that neglects the effects of potential

lateral heterogeneity in Earth structure at the scale of the ASE, gives very good predictions over the large

distance between BERP and THUR stations. If a significant uniform background signal (the far-field GIA for

the rest of Antarctica, for example) were present, its associated horizontal component would be uniform

(in magnitude and direction) for all the stations, and that would worsen the fit on one or both BERP and

THUR. Here for significant signal we mean a signal larger than the observational error (of the order of 1

mm/yr).

Elastic response

The elastic response at THUR is positive, because the local accumulation is not enough to counteract the

signal from nearby Pine Island Glacier. Therefore, if we had underestimated the elastic correction, the

residual in THUR would be much more negative, while BERP and other stations would have a much smaller

residual. A small residual could hypothetically be explained by global or regional GIA(13, 14, 35). However,

27

in this case THUR would show an extremely strong negative signal that could be explained only by much

stronger local accumulation than is observed.

Ancient GIA contributions

Except for LPLY (Main Text Figure 3A) there is no need for a significant background signal either coming

from the far field or the far distant past. A background signal from older ice history due to the deglaciation

since the LGM, 21,000 to 10,000 years ago(70), would only be relevant in the presence of high viscosity

deep mantle layers (below 200 and down to 670 km), because in the presence of low viscosity in deeper

layers (DUM and TZ) the deformation produced by the LGM deglaciation would already be relaxed.

Therefore, the absence of a clear background signal is compatible with low viscosity in deeper layers (DUM

and TZ).

Horizontal displacement fit

The fit for the horizontal displacement can be improved for some stations, but we need longer GPS time

series to have reliable motion estimates.

The vertical deformation at the GPS site is mostly sensitive to the cumulative mass loss of the nearby area,

regardless of its spatial distribution, while the horizontal motion is sensitive to the location of the mass

change. We investigated the effect of changing the geometry of the ice history for the grounding line of

Pine Island and Thwaites glacier(8, 71, 72) on the fit to the horizontal motions of BACK, INMN and BERP

(see Sensitivity to the load geometry in S6). The retreat of the grounding line in our ice history implies a

different distribution of the load but not necessarily a difference in the cumulative mass loss. In fact, while

there is no mass-change signal on the floating ice today, over the 1950-2000 period there would have been

large mass changes in that area(72). At the same time, the ice mass loss upstream would have been much

lower than present-day. However, the improvement in the misfit for the horizontal motion was

accompanied by a worsening of the overall misfit.

Lithosphere thickness

Elastic lithosphere thicknesses of 50-60 km are consistent with seismological studies of this region, which

show seismic high velocity lithosphere extending to depths of 60-80 km(18). A recent inversion of seismic

data to obtain the best fitting geotherm suggests a lithospheric thickness of about 60 km, as delineated by

the intersection of the lithospheric geotherm with the mantle adiabat(75). Lithosphere thicker than 70 km

can give a reasonably good fit (at least to the vertical motions) if the viscosity of deeper layers (DUM and

TZ) is higher than 7x1019 Pa s (3 times higher than our best fit), and the viscosity of SUM is 2.5 times lower

28

than our best fit (Fig. S4). In fact, a lithosphere thicker than 70 km provides the long wavelength content in

the uplift as well as low viscosity in deeper layers does. However the inversion shows a preference for the

long wavelength content being the result of lower viscosity in deeper layers.

Figure.S11: Example of poor fit for Vertical and Horizontal bedrock motion predictions. The vertical (left) and

horizontal (right) observed GPS minus elastic modeled residual velocities (green dot in left plot, green arrows in right

plot) as in column 5 of Table S2. Blue, pink, and orange symbols denote vertical and horizontal prediction from three

viscosity models providing poor fit. The error bars are the weighted errors (error bar = sigma/weight). Note that the

weights are 1 except for LPLY and the horizontal for TOMO. In the legends, Lxx indicates the lithosphere thickness in

km, and Sxxx the SUM viscosity in 10xLog10 Pa s.

When varying the ice history within ±50% of H1, a stiff shallow upper mantle (> 1019 Pa s) gives a poor fit to

the vertical displacements. This happens because the signal is too smooth (in other words dampened) even

in the presence of a thin lithosphere (Fig. S11, left). The fit in this case is even worse for the horizontal

motions, which are almost invisible in Fig. S11 (right) for the model L30S194 (blue symbols). A preference is

evident (in all cases) for a lithosphere thicker than 50 km, with a clear rapid degradation of the fit for a

thinner lithosphere (Fig. S11). This occurs because the measure bedrock deformation signal has a

significant, long-wavelength component. Far from the ice, the bedrock response in the presence of a thin

lithosphere would be smaller or of the opposite sign of that observed (Fig S11).

29

Mantle viscosity and cumulative mass loss

Results so far show that both the vertical and horizontal deformations measured by the GPS (corrected by

the elastic contribution) can be explained by the rapid response of a low viscosity mantle to a simple ice

history based on the present-day pattern of ice mass changes. The hypothesis of an ice loss pattern that

remains constant for several centuries or millennia is not very likely, and it is unnecessary for low viscosity

models, that are mostly sensitive to the last centuries. This is a clear a posteriori indication in support, that

the problem is dominated by recent (century to decadal-scale) changes. The clear response to the pattern

of recent ice mass changes also puts an upper bound to the cumulative ice mass changes since LIA. We find

that with a large cumulative mass loss, only models with a stiffer mantle would give a good fit, the stiffness

being necessary to sustain the large mass without producing uplift rates that are too large (S6). However, a

stiff model would be much more sensitive to the remote past ice history (up to some thousands of years

ago), and any additional contribution from this remote past would only make the misfit worse both in

amplitude and in pattern. So a stiff mantle is compatible only with an ice history with negligible changes

before the LIA, which is, however, very unlikely(41). Therefore, it is reasonable to assume an upper limit to

the total mass loss in the recent past.

S8. Geological framework and implications

Mantle viscosity is one of the least understood physical properties of the Earth but it is also one of the most

geodynamically important. Tectonic velocities and stresses in subduction and rift zones are strongly

dependent on the time-scales and patterns of mantle convective flow that are, in turn, strongly affected by

the mantle viscosity structure.

Global and regional GIA models typically assume a fairly stiff mantle, which produces a smooth uniform

vertical response at the spatial scale of the ASE, in sharp contrast to the more spatially variable observed

uplift. This is also true for the horizontal predictions. From Main Text Figure 1D we can see that the

observed (and residual) pattern of horizontal displacements is non-uniform for both direction and

magnitude. This non-uniform spatial pattern of bedrock motion also indicates that a stiff mantle is not

appropriate for this region.

There are abundant indications of geologically recent volcanism and Cenozoic rifting in much of West

Antarctica, and limited evidence points to rift-related displacements of Oligocene age between the

Thurston Island-Eights Coast and Marie Byrd Land crustal blocks flanking the Amundsen sector(22, 23),

raising the possibility that the mantle viscosity in this region deviates significantly from the global average.

This was one of the reasons for the placement of GPS and seismic stations on bedrock around the ASE,

indeed.

30

Low or very low viscosity upper mantle is consistent with other indications of elevated mantle

temperatures in this region, including thin crust and lithosphere indicating extensional tectonics during the

Mesozoic and Cenozoic(18, 75, 76), as well as slow upper mantle velocities below the lithosphere(18).

The existence of slow velocity, high temperature shallow upper mantle is indicated by recent seismological

studies(18, 75, 76), suggesting low mantle viscosities (<1020 Pa s) are appropriate. Seismic high velocity

lithosphere extending to depths of 60-80 km are shown in ref. (18).

Low viscosities below the ASE are associated with high geothermal heat flux estimates(28), active

volcanism(21), and possible Oligocene rifting(23).

The low viscosity that we find in deeper layers could be an indication of a plume-like thermal structure such

as the one hypothesized nearby Marie Byrd Land(19, 30), and suggested by global tomography(77)

However it be can more confidently be attributed to mantle chemically altered by hydrous fluids and

volatile silicate melts generated by protracted, and well documented, pre-Cretaceous subduction of the

Phoenix plate along the West Antarctic margin(31).

S9. Implications of low viscosity in ASE

GRACE correction (The gravitational signature of the preferred

model)

Along with the deformation we can also compute the gravitational signature in water equivalent (w.e),

which represents the apparent surface mass change produced by the mantle displacement.

The observed bedrock uplift rates (Main Text Figure 3A) are higher than those expected in ASE(13, 14) due

to the low-viscosity mantle and resultant short time-scale Earth response to ice changes. An effect of low

viscosity is that the uplift will accelerate with time (Main Text Figure 3B, c), with an increasing impact on

the gravity field. GRACE mission detects this apparent mass change together with the ice mass change and

it cannot distinguish between the two (Fig. S12). So the accuracy of GRACE-derived ice mass changes is

affected by the knowledge of the GIA component in the region of interest(78).

Here, for one of our preferred models we compute the gravitational signature of GIA uplift in water

equivalent (w.e). By computing the volume over basins 21 and 22 (in the ref. (74) basin definition) we

derive a GIA correction of +16.7 Gt/yr for the GRACE derived measurements valid for the observational

period of GRACE (2002-2014). And for the whole area under study the correction is +19 Gt/yr. That means

that if the ice mass changes measured with altimetry indicate 130 Gt/yr, then GRACE-derived

measurements should detect only 111 Gt/yr. In ref. (2) the GIA correction for the basins 21 and 22 using

31

traditional and empirical GIA was between +3.5 and +5.5 Gt/yrs. Published GRACE-derived ice mass loss

estimates(1, 2, 66) are therefore systematically underestimated by a value between 11.2 and 13.2 Gt/yr,

about 10% of the total ice mass loss for ASE. Some studies(46, 48, 49) have derived a strong empirical GIA

signal in PITS basins. We obtain a very localized signal that cannot be too different or more widespread if

we want to fit the data. In the variant that accounts for more mass loss in Thwaites (Sensitivity to load

geometry in S6) we obtain up to 21 Gt/yr of GIA correction, and we can consider that an upper bound for

our estimate.

Figure.S12: Gravitational signature as surface density (in water equivalent) for our preferred GIA model in

Amundsen Sector. This can be compared with the ice mass changes and used as correction for GRACE-derived mass

balances. Note the scale of 500 km to be compared with the GRACE resolution of 300 km.

Under the assumption that the ice mass loss rate does not increase the uplift rate and associated gravity

perturbation will increase (Main Text Figure 3B). The maxima of the uplift rate, as the prediction in TOMO,

which is close to the maximum, in 100 years, will be 2.5 times higher than present-day. The medium field

deformation rate in 100 years will be 3.5 times higher than present-day. The apparent mass is proportional

to the volume of deformation, and since the area (of the basins 21 and 22) is constant, then in 100 years

the volume and so the apparent mass rate for the GIA correction will be between 2 and 3 times larger

(300%), which means a conservative increment >2% per year. If the ice mass loss will increase the

correction increment will be higher. Future estimates of gravity-derived ice mass changes must take into

account the accelerating solid Earth effect, especially for time series covering more than a decade. For the

32

same reason, the GRACE-derived mass changes over the period 2003-2009 used for inter comparison with

ICESat-derived ice changes, should be adjusted with a lower GIA correction of 15 Gt/yr. Considering the GIA

acceleration, the GIA correction ranges from 15 Gt/yr (for 2002-2010) to 17.4 Gt/yr (for 2002-2016). Taking

into account the GIA acceleration, the range of the systematic error on the published GRACE-derived ice

mass loss estimates (computed on different time span) is wider: between 10.0 and 13.9 Gt/yr.

A variable GIA correction that, in turn, depends on the recent ice mass changes will be a new circular

problem to solve when deriving accurate ice mass changes from the future gravity missions. Note that on

long time series (several decades) the acceleration on GIA signal if not corrected might be mistaken by

acceleration in ice mass loss.

The uplift pattern of the preferred model

For one of our preferred models we compute the vertical displacement not only at the GPS station

locations, but also on a regular grid (1x1 degree) so that we can show the pattern of uplift for the whole

region (Fig. S13). Two forebulges are noticeable and one is where the THUR station is located (only

predictions for THUR are negative, Main Text Figure 3). The distance of the forebulge from the main uplift

area (which corresponds to the area where the mass changes are occurring) is affected by the lithospheric

thickness. With thinner lithosphere, the forebulge is closer to the maximum uplift and that affects stations

like BACK, which have negative predictions when the lithosphere is thinner than 40 km.

Figure.S13: Uplift pattern for our preferred GIA model in Amundsen Sector. Left) the GIA uplift for the delayed

(viscous) component. Right) the elastic uplift. Note the spatial scale is 500 km.

33

In case of comparison with low resolution data, such as empirical GIA derived from GRACE, we note that

the signal is very localized, i.e. the total volume of the deformation rate is 4.98 km3/yr and the average

uplift over the region covered by the Amundsen sector (basins 21 and 22 in the (74) basin definition) is 12

mm/yr. The average uplift rate over the Amundsen sector is much lower than the signal predicted and

observed close to the maxima (e.g., in TOMO 41 mm/yr). The empirical GIA (46, 48) are obtained from the

difference of GRACE-derived and the Altimetry-derived ice volume changes under the assumption that the

apparent mass change produced by GIA is equivalent to the volume of the deformation produced by GIA

multiplied by a constant density of the mantle (3700 kg/m3). This overestimates the apparent mass, based

on the same assumption, using only the volume of our uplift prediction we would obtain 18.4 Gt/yr to be

compared with the +16.7 Gt/yr that we found in the previous subsection.

Ice sheet stability

GIA driven by low viscosity is a stabilizing factor for the marine ice-sheet dynamics of West Antarctica(16,

17). Even assuming that the current rate of ice thinning is constant in the ASE throughout the next century,

we find that the uplift rate will increase (Main Text Figure 3B, C). In 100 years, uplift rates at the GPS sites

will be between 2.5 and 3.5 times more rapid than currently observed (Main Text Figure 3B) and the

bedrock at the Pine Island Glacier grounding line will have risen by about 8 meters compared to the present

(Main Text Figure 3B, C). This cumulative deformation is about three times higher than values shown to

reduce run-away ice surface velocities within 100 years(15). Changes in bedrock topography driven by a

medium viscosity model (~1020 Pa s)(44) under realistic climate forcing, can provide a stabilizing effect on

the ice sheet in 500 years(15). For ASE we find almost 2 orders of magnitude lower viscosity, so the timing

required to produce a stabilizing effect is reduced considerably (see S10), from 500 years to a few decades.

The lower viscosity that we measure in ASE implies therefore a faster stabilization effect than expected in

published studies (15–17). In light of the discussion of section S10, considering the range of viscosity that

we find for the whole upper mantle (from the base of the lithosphere down to 670 km), and what we know

from the literature, the onset of this stabilization mechanism (the threshold), from a purely viscoelastic

perspective, can be predicted to occur with a timing 40% shorter in a conservative scenario and more likely

2 times shorter in a realistic scenario.

Under strong climate forcing, the ice mass loss rate is expected to be faster than in the other scenarios, and

the bedrock would become ice free at a faster rate. So in this scenario, the difference between a model as

in ref(16, 17) (see reference model m190 in S10) and a model with lower viscosity like our best model (see

the comparison model m186 in S10) would be more pronounced and the deformation more than 2 times

faster. According to ref(16, 17), for those weaker model the collapse is expected to be delayed, and

34

eventually the condition for stabilization could be possibly reached before the complete collapse of WAIS,

but to verify this a new coupled ice-Earth simulation is necessary.

In fact, while the timing of the build-up of the critical deformation depends on the Earth structure alone,

the exact timing when the ice sheet actually reaches the equilibrium, and the “degree of stability” depend

of course on the other details of the ice-Earth coupling and the climate forcing scenarios, but they will not

contradict the systematic behavior clearly shown in (16, 17). So we cannot predict the stability of the

Antarctic ice sheet whichever ice model is used, and indeed different ice models predict quite different

scenarios independently of the coupling to the solid Earth. But we can say that the solid Earth feedback,

which is recognized as a relevant stabilization feedback, will start to be effective with approximately the

timing provided above.

Antarctica GIA modeling and sea level

We find that ASE is characterized by a rheological structure that is very different to that commonly used in

global and regional GIA models(13, 14, 35, 44), with upper mantle viscosity of 2 order of magnitude lower.

This strengthens the possibility that low viscosity could characterize other areas of West Antarctica where

the presence of a hot mantle is suggested(19, 28, 30). However the only region were low viscosity has been

dynamically constrained is the North Antarctic Peninsula(12). Other areas, e.g. Palmer Land(79) and Siple

coast(80) show a viscosity around 1020 Pa s, which cannot be considered low. While this viscosity is much

lower than the average under East Antarctica, it is still 2 orders of magnitude higher than ASE or NAP.

Low viscosity in West Antarctica would have strong impact on the reconstruction of the paleo-ice-sheet

history and the associated LGM GIA models(13, 14, 35, 44), each of which need to be visited in light of our

results. In fact, low viscosity around ~4x1019 Pa s, one order of magnitude higher than our best model for

ASE, implies low sensitivity to ice history before 3000 years ago. This implies a much smaller contribution of

the LGM deglaciation to the present day pattern of uplift rates measured by GPS(33, 38) and the empirical

GIA(48) observed in West Antarctica. The observations could be better explained with ice reconstruction of

the last several centuries and their associated GIA models. Many of the paleo-ice reconstructions since

LGM are tuned or constrained with GIA models (where glaciological evidences are lacking). Low sensitivity

to LGM deglaciation for West Antarctica results in a less constrained cumulative mass loss since LGM,

possibly different than previously thought(44). That, in turn, results in a different Antarctic contribution to

eustatic sea level (ESL) history (by meters), and different interpretations of relative sea level (RSL) histories

since LGM. The budget for the ESL history reflects how well the global ice reconstruction since LGM has

been captured. A different ESL contribution from Antarctica will mandate a revision of the other

contributions to ESL history, i.e. a revision of global GIA models(13).

35

GPS sensitivity and depth

The spatial scale of the geometry of ice changes affects the whole upper mantle (down to 670 km), which is

captured sufficiently with the aperture of the local GPS network and the sensitivity of the GPS

measurements. GIA modeling, especially on the spatial and temporal scales of Fennoscandia, is sensitive

only to the uppermost 400 km of the mantle(81). We believe that the limited sensitivity to depth for such

GIA studies arises from the high viscosity (long time scale), the large-scale geometry of observations (in

space and time), and the precision of the observations (the relative sea level history). In the presence of a

high viscosity mantle, the time scale of its response is of the order of millennia, and to fully resolve the

signal it would be necessary to have precise measurements extending over a very long time span, including

the beginning of the phenomenon. In this study, the low viscosity of the region reduces the time span over

which the phenomenon is observable with precision, and the GPS stations have a good spatial distribution

to resolve a large range of spatial wavelengths, from tens to several hundreds of km. This is the reason why

the problem is sensitive to properties in the whole upper mantle and transition zone down to 670 km.

GIA sensitivity to mantle viscosity down to 400 km has been noted for regions that are of smaller spatial

scale than ASE, such as the Northern Antarctic Peninsula (NAP)(12). The configuration of the GPS sites in

the NAP and the related uplift predictions show sensitivity to changes in Earth model parameters down to

350 km but no sensitivity is found below 400 km(12). For NAP, varying the TZ by one order of magnitude

produces changes in the misfit of 1%, while our misfit for ASE improves by more than 10% for a decrease in

TZ viscosity by a factor of 2.8. In NAP a lithosphere thicker than 100 km is found even with very low

viscosity between 200 km and 400 km depth(12). That means that the long wavelength component in NAP

uplift rates is much higher than what we find in ASE.

Complex mantle structure

In our study, GPS geodesy provides insight into the weak rheology of the upper mantle of the ASE region

down to 670 km. We show that a rapid bedrock response to ice changes can be controlled by a viscosity

structure that is more complex than one single mantle layer as assumed in previous studies(9–11). This

possibility must be taken into account in areas experiencing rapid uplift caused by ice melting at similar

spatial scales to the ASE, such as Iceland(9), Patagonia(10) and Alaska(11). Investigating the effect of a

complex mantle structure in those regions may result in a revision of viscosities and better explanation of

the observed bedrock displacements.

36

Geophysical characteristic of ASE associated with Low viscosity

The low-viscosity mantle that we find below the ASE supports elevated geothermal heat flux(28), which is a

non-climate-related boundary condition important for the long-term evolution of the ice sheet especially in

warm climate scenarios(7, 29). In fact, the characteristics of the subglacial geology in ASE impart important

boundary conditions for the evolution of the ice sheet. The high geothermal heat flux associated with the

low-viscosity mantle keeps the base of the ice sheet at temperatures close to the melting point(82). This

could be important when attempting to decipher ocean-ice interactions at the grounding-line. The low-

viscosity is also compatible with the narrow rifts of West Antarctic Rift System, that channel warm oceanic

ingress beneath the ice, hence promoting ice thinning(7). Deglacial unloading influences seismicity(83) and

volcanic activity(84, 85), and our results provide estimates of ASE bedrock deformations and stresses that

can be used to assess the effect of increasing ice mass loss on seismic and volcanic activity(21, 84).

Notice, however, that our determination of the viscosity is based on purely geodynamical methods,

without any geologic or tectonic assumption. This makes our estimate of the viscosity even more relevant,

because completely independent from other techniques. But on the other hand, the fact that the

characteristic times of the dynamic effect of viscosity that we measure are incomparably shorter than those

of any geologic or tectonic phenomena, there is nothing that we can say about the possible origin of the

low viscosity mantle.

S10. Timing of the onset of the stabilization feedback

Evolving bed topography driven by elastic rebound and time-dependent mantle flow can produce a

negative feedback mechanism that stabilize the grounding line and reduces dynamic ice loss, via falling

relative sea level, rising pinning points, and flattening of inland-tilting subglacial slopes(15). Where low

mantle viscosity is simulated, the rapid uplift localized near the grounding zone has the largest stabilizing

influence on ice sheet behavior(16, 17).

As shown in (15–17), the solid Earth response has a stabilizing effect, and the stabilization effect is

systematically stronger, the lower the viscosity(16, 17). This is a purely geodynamical effect, irrespective of

the actual geologic of tectonic phenomenon responsible for the low viscosity. The mechanism through

which the Earth response, that is basically the bedrock uplift and the changes in the slope at the grounding

line together with the combined gravity variation, affects the ice sheet evolution is discussed in detail by

Adhikari et al ref(15) and (16, 17). The solid Earth response acts as a sort of threshold-mechanism(15, 17): if

the Earth response reaches a certain threshold (amount of deformation) then a stabilization process comes

into play (Konrad at al. (17) say that this can take 1000s yrs, depending on the Earth viscosity). This

37

stabilization contribution can be strong enough to effectively stabilize the WAIS or not, depending on the

other model parameters (ice modeling, climate forcing scenarios) (16, 17).

In all the recent studies (16, 17) the coupling of the ice dynamics with the solid Earth (16, 17) is actually

“weak”, because the characteristic times of the ice dynamics and of the Earth response are very different,

the time of the Earth response being much larger. As a direct consequence, the ice sheet evolution follows

almost exactly the same path at the beginning of the simulations for the first hundreds/thousands years,

even when the Earth model differ dramatically in the upper mantle viscosity structure and lithospheric

thickness (16, 17). The paths start to diverge only when a critical amount of bedrock uplift and gravity

change, which is clearly dependent on the specific topography and geology of the region and all the ice-

model parameters, is reached by one of the models. This is the time when the solid Earth response,

mediated by the sea, can initiate its negative feedback that favors stabilization, and the models start to

diverge. What is relevant is if and when this “trigger point” is reached.

Estimate of the Earth response time

Is there a way to estimate the how much faster the Earth response is, and therefore how much faster the

critical uplift is reached, comparing the Earth structures only?

Thanks to the weak coupling, the answer is positive: this timing is almost completely dependent on the

solid Earth response to the average cumulative mass alone (not the details of the ice model). If this were

not the case, the models with different viscosity would give visible differences from the beginning, and not

only after a certain time (16, 17). In high viscosity models this “trigger point” is not reached before the ice

sheet collapse, and so this is not visible. But given enough time, the Earth will always relax completely.

Depending on the forcing scenario and the details of the ice modeling, ice sheet stability can eventually be

reached at a subsequent time, and the latter is instead dependent on all the model parameters, and that

cannot be predicted on the basis of the Earth structure alone, of course.

In the case of a homogeneous Earth, the response time can be obtained analytically by the single viscosity

value since the characteristic time of the response of the Earth are inversely proportional to the viscosity.

In a multilayered model there is a spectrum of characteristic times, related to the layers’ viscosity at depth,

the shallower layers being the more relevant at local spatial scale. In case of layered Earth models, the

response time can only be estimated numerically, with a suitable numerical experiment. Please notice that

(15–17) make use of a solid Earth model that is laterally homogeneous (1D model), with a radial

stratification simpler than ours, and therefore our general consideration can safely be used to understand

their results.

38

We assume, as in the first centuries of simulation time in Refs (16, 17), that the cumulative load (ice

history) is the same for two different Earth models.

Numerical experiments

Figure.S14: Cumulative uplift at the center of the load for model m190 (red) and m186 (blue). See Table S4

for the models details. In green the ratio between the time when the same uplift is reached in model m186

and m190. The load is a disk of 50 km radius losing mass at a rate of -1 m/yr water equivalent for the first

100 years. The uplift is always larger for m186, and as indicated by the vertical red dashed arrows the ratio

is already around two at 200 years, 100 years after the load disappearance, and around 4 at 800 years, 700

years after the load disappeared, and more than 5 after 2000 years. This situation exemplifies what

happens close to the border of the ice sheet, and at the Grounding Line.

We perform two simplified experiments, representing the two extreme condition encountered in the ice

sheet evolution. We assume the same loading history for two Earth models with 50 km lithospheric

thickness and different upper mantle structure (down to 670 km) and we compute the cumulative

deformation at different times (Fig. S14 and S15). The reference model (label m190, red curve in Fig. S14

and S15) is exactly as the one used by Gomez et al.(16), the upper mantle is 1019 Pa s down to 200 km,

2x1020 Pa s down to 670 km and 3x1021 Pa s for the lower mantle. For the comparison model (label m186,

blue curve in Fig. S14 and S15) we use the best viscosity parameters we find for ASE, an upper mantle of

39

3.8x1018 Pa s down to 200 km, 2x1019 Pa s down to 400 km and 2.5x1019 Pa s down to 670 km. In both

experiments the loading is a disk of 50 km radius and the total time span of the simulation is 2000 years.

Experiment 1 (Fig. S14) simulates a load that loses mass for a fixed period (-1 m/yr of water equivalent for

100 years). This exemplifies what happens close to the grounding line (GL) where the bedrock is ice free or

is about to become ice free. Experiment 2 (Fig. S15) simulates a load that loses mass constantly (-1 m/yr of

water equivalent) for the whole duration of the simulation and that represents what happens in the

interior of the ice sheet.

Figure.S15: Cumulative uplift at the center of the load for model m190 (red) and m186 (blue). See Table S4

for the models details. In green the ratio between the time when the same uplift is reached in model m186

and m190. The load is a disk of 50 km radius loosing mass at a rate of -1 m/yr water equivalent for 2000

years. The uplift is always larger for m186, and the ratio is decreasing from 18 at the beginning to 1.4 after

400 years, and after 2000 years is it still around 1.2. This situation exemplifies what happens in the interior

of the ice sheet.

Revised timing for onset of stabilization feedback

With these numerical experiments we can shed some light on the effect of lowering the viscosity on the

timing of the onset of this stabilization feedback.

40

The first result is that the ratio between the response times is not constant in time, as qualitatively

expected. The second is that it also depends on whether the observation point lies within or outside the

region of the load.

The response of the comparison model (m186, blue line in Fig. S14 and S15) is always faster. Around the GL

(Fig. S14), in the first 200 years the response starts less than 2 times faster, then the ratio rapidly grows and

at 800 years the comparison model (m186) is already 4 times faster, and at 1800 years it is 5 times faster. In

the interior of the ice sheet (Fig. S15), in the first 400 years the comparison model is from 80% to 40%

faster, and after that it slowly becomes 20% faster.

Combining the results of these two experiments, we can say that the time ratio changes as soon as the

bedrock becomes ice free or reaches floatation (moving from the scenario of Fig. S15 to that of Fig. S14), a

transition that depends on the ice thickness and the ice mass loss rate. The fastest ice mass loss rates are

recorded at the margin of the ice sheet where the ice is also usually thinner than the interior. So around

the margin of the ice sheet and the GL the deformation produced by the comparison model is likely to be

about 2 times faster. Extrapolating the results from Fig. S14 and S15, it is easy to understand what happens

close to the GL: approaching the coastal line from the ice sheet interior, the ratio increases, and in the

region around the GL the response becomes between 2 and 4 times faster. This confirms that in presence

of lower viscosity, the critical uplift necessary to trigger the stabilization feedback is reached faster

everywhere, but it also shows that the difference in the uplift rates between the region still covered in ice

and the ones which became recently free is largely enhanced. This leads to the build-up of a negative slope

that enhances the stabilization effect (15) (see also Figure S3). Due to the linearity of the problem this

behavior is expected to be preserved even in presence of more realistic load histories, up to the point

where the ice histories starts to diverge.

S11. The misfit definition

We perform a grid search over different physical parameters and using the following reduced 𝜒𝑟𝑒𝑑2 :

𝜒𝑟𝑒𝑑2 =

1

𝑁 − 1∑ (𝑤𝑗

𝐺𝑃𝑆𝑟𝑒𝑠,𝑗 − 𝑀𝑗

𝜎𝐺𝑃𝑆𝑗

)

2𝑁

𝑗=1

Equation.S1

Where GPSres are the GPS residual (column 5 Table S2), Mj are the modeled viscous deformations, GPS are

the errors on the GPS observation (column 3 Table S2), wj are the weights (column 7 of Table S2), and N is

the number of observations.

41

We computed and analyzed separately the 2red for the vertical and the horizontal displacement. The

deformation detected by the GPS is a 3-dimensional deformation and therefore the three components (Vu,

Ve and Vn) have the same weight when we compute a combined 2red, where

𝜒 = √(𝜒𝑢

2 + 𝜒𝑒2 + 𝜒𝑛

2)

3

Equation.S2

and 𝜒𝑢2, 𝜒𝑒

2 and 𝜒𝑛2 are the reduced values for the vertical, east and north components respectively.

We computed 807 different basic models (for H0 and H1) by varying the lithosphere thickness (LT) and the

viscosity of the three viscous layers (SUM, DUM and TZ). On two subsets of those basic models we

computed also the segment b19 (variation in Thwaites basin) and H2 (ice variation between 1850 and

1902). By applying our strategy (S5) for combining the segments of the ice history we could analyze many

thousands of combinations (Earth model and ice history). We are aware that the limited number of GPS

stations and their spatial distribution determine the resolving power of the data set. This means that we

expect that only a limited number of free parameters can be constrained. Allowing for too many free

parameters would lead to an underdetermined problem. Therefore, we prefer a simple approach in which

the effect of the variation of the individual parameters is investigated step by step.

42

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