Ice loss from the Antarctic ice sheet (AIS) is expected to become the major contributor to sea level in the next centuries. Projections of the AIS response to climate change based on numerical ice-sheet models remain challenging due to the complexity of physical processes involved in ice-sheet dynamics, including instability mechanisms that can destabilise marine basins with retrograde slopes. Moreover, uncertainties in ice-sheet models limit the ability to provide accurate sea-level rise projections. Here, we apply probabilistic methods to a hybrid ice-sheet model to investigate the influence of several sources of uncertainty, namely sources of uncertainty in atmospheric forcing, basal sliding, grounding-line flux parameterisation, calving, sub-shelf melting, ice-shelf rheology and bedrock relaxation, on the continental response of the Antarctic ice sheet to climate change over the next millennium. We provide probabilistic projections of sea-level rise and grounding-line retreat, and we carry out stochastic sensitivity analysis to determine the most influential sources of uncertainty. We find that all investigated sources of uncertainty, except bedrock relaxation time, contribute to the uncertainty in the projections. We show that the sensitivity of the projections to uncertainties increases and the contribution of the uncertainty in sub-shelf melting to the uncertainty in the projections becomes more and more dominant as atmospheric and oceanic temperatures rise, with a contribution to the uncertainty in sea-level rise projections that goes from 5 % to 25 % in RCP 2.6 to more than 90 % in RCP 8.5. We show that the significance of the AIS contribution to sea level is controlled by the marine ice-sheet instability (MISI) in marine basins, with the biggest contribution stemming from the more vulnerable West Antarctic ice sheet. We find that, irrespective of parametric uncertainty, the strongly mitigated RCP 2.6 scenario prevents the collapse of the West Antarctic ice sheet, that in both the RCP 4.5 and RCP 6.0 scenarios the occurrence of MISI in marine basins is more sensitive to parametric uncertainty, and that, almost irrespective of parametric uncertainty, RCP 8.5 triggers the collapse of the West Antarctic ice sheet.

The Antarctic ice sheet (AIS) is the largest reservoir of freshwater on Earth
(

Assessing the future response of the Antarctic ice sheet requires numerical
ice-sheet models amenable to large-scale and long-term simulations and
quantification of the impact of modelling hypotheses and parametric
uncertainty. So far, there exist only a limited number of projections

Despite recent progress in the numerical modelling of ice-sheet dynamics

Uncertainties in the ice-sheet initial state, climate forcing and parameters
in numerical ice-sheet models are another major limitation for accurate
projections. To date, the impact of such parametric uncertainty is assessed
most often by using large ensemble analysis; that is, the model is run for
different values of the parameters and the uncertainty in the projections is
estimated from the spread in the model runs. For example,

The field of uncertainty quantification (UQ) develops theory and methods to
describe quantitatively the origin, propagation and interplay of sources of
uncertainty in the analysis and projection of the behaviour of complex
systems in science and engineering; see, for instance,

In this paper, we apply probabilistic methods to assess the impact of
uncertainties on the continental AIS response over the next millennium. We
use the fast Elementary Thermomechanical Ice Sheet (f.ETISh)
model

On the one hand, our study adds to previous studies

This paper is organised as follows. First, Sect.

We perform simulations of the response of the Antarctic ice sheet
(Fig.

Antarctic bedrock topography and sub-shelf melting.

We perform simulations at a spatial resolution of 20 km while accounting for
grounding-line migration at coarse resolution with a flux condition derived
from a boundary layer theory at steady state based on either a Weertman (or
power) friction law

The main changes in f.ETISh version 1.2 as compared to version 1.0

We drive our simulations with both atmospheric and oceanic forcings.
Present-day mean surface air temperature and precipitation are obtained from

Basal melting underneath ice shelves is determined from the PICO ocean-model
coupler

The calibration of the basal sliding coefficient follows the data
assimilation method of ice-sheet geometry by

We aim at quantifying the response of the Antarctic ice sheet to climate
change while accounting for uncertainty in atmospheric forcing, basal
sliding, grounding-line flux parameterisation, calving, sub-shelf melting,
ice-shelf rheology and bedrock relaxation. We account for uncertainty in
these physical processes by introducing uncertainty in parameters in the
f.ETISh model (see Table

List of parameters and parameter ranges used in the uncertainty analysis.

Alongside oceanic forcing, atmospheric forcing is generally considered to be
the primary driver of future changes in the AIS mass balance

Long-term RCP temperature scenarios

The scenario plays a significant role in the amplitude and speed of the AIS
retreat. Recent studies

Basal sliding controls the motion of fast-flowing ice streams, which drain
about 90 % of the total Antarctic ice flux

We introduce basal sliding as a Weertman sliding law, that is,

In addition to the usual exponents

The f.ETISh model employs a parameterisation of the grounding-line flux based
on a boundary layer theory at steady state by either

We applied the TGL parameterisation under the weakly non-linear sliding law.
There is no consensus on the compatibility between the TGL parameterisation
and the Weertman sliding law, as the TGL parameterisation was derived from
the Coulomb friction law near the grounding line

Ice loss due to ice calving at the edges of ice shelves is responsible for
almost half of the present-day ice mass loss of the Antarctic ice sheet

The nominal calving rate

We introduce uncertainty in calving by controlling the magnitude of
the calving rate with a scalar multiplier factor

Sub-shelf melting is mainly controlled by sub-shelf ocean circulation, which
can be affected by atmospheric changes. Ice-shelf thinning caused by
increased sub-shelf melting leads to a reduction in ice-shelf buttressing.
West Antarctica, where the bedrock lies mainly below sea level, is
particularly vulnerable, as suggested by observational

High melt rates at the base of ice shelves result from the inflow of
relatively warm Circumpolar Deep Water in ice-shelf cavities

Here, we capture the basic overturning circulation in ice-shelf cavities with
the PICO box model. In the PICO model, the strength of the overturning flux
is represented by a single parameter that depends on the density difference,
or equivalently on both the salinity and temperature differences, between the
incoming water masses on the continental shelf and the water masses near the
deep grounding line of the ice shelf. An increase in the ocean temperature on
the continental shelf leads to a stronger overturning flux and higher melt
rates at the base of ice shelves. The ocean temperature on the continental
shelf is determined from the present-day ocean temperature

We introduce uncertainty in sub-shelf melting by controlling the strength of
the overturning flux through uncertainty in the ocean temperature on the
continental shelf. Hence, we consider the ocean melt factor

Ice rheology in large-scale ice-sheet models is usually described as an
isotropic material obeying Glen's flow law

We introduce an ice-shelf tune parameter

Bedrock relaxation due to deglaciation may induce a negative feedback that
promotes stability in marine portions and mitigates the effect of a marine
ice-sheet instability

We account for the differences between East and West Antarctica by
introducing two characteristic relaxation times

To quantify the impact of uncertainties on the AIS response, we adopt a
probabilistic framework. Here, we assume, in the absence of any prior
information other than the aforementioned nominal values and minimal and
maximal values of the uncertainty ranges, that the parameters

Given the probabilistic characterisation of the uncertainty in the parameters, the propagation of uncertainty serves to assess the impact of the uncertainty on the global mean sea-level change. In particular, its intent is to estimate the probability density functions for the change in GMSL as well as some of its statistical descriptors such as its mean, variance and quantiles. Various methods have been developed in UQ to estimate these statistical descriptors in a non-intrusive manner treating the ice-sheet model as a black box. Here, we use emulation methods based on a polynomial chaos (PC) expansion.

An emulator, also known as a surrogate model, is a computational model that
mimics the ice-sheet model at low computational cost. Although emulators can
also be obtained by Gaussian process regression

The emulator of the relationship between the parameters and the projection is
then used as a substitute for the ice-sheet model in a Monte Carlo method

The use of an emulator has the following advantages: (i) it provides an
inexpensive approximation of the ice-sheet model that accelerates UQ; (ii) it
provides an explicit view of the relationship between the parameters and the
projection, highlighting potential linear or non-linear dependences and
interactions between the parameters; (iii) it allows efficient interpolation
of the projections in the parameter space; (iv) it can be used to carry out
stochastic sensitivity analysis to assess the influence of each parameter on
the projections; (v) under certain conditions, the same emulator can be
reused between UQ analyses with different probability density functions for
the parameters; and (vi) it can be used for Bayesian calibration

We consider an ensemble of 20 distinct model configurations given by each
combination of RCP scenario with a sliding law (

Stochastic sensitivity analysis serves to identify which uncertain parameters
and their associated physical phenomenon are most influential in inducing
uncertainty in the ice-sheet response. Here, we adopt the variance-based
sensitivity indices

We compute the Sobol indices directly from the PC coefficients

To gain insight into the impact of the uncertainty in determining which
regions of Antarctica are most at risk of ungrounding, we construct
confidence regions for grounded ice for several probability levels. We define
a confidence region for grounded ice for a given probability level as a
region of Antarctica that remains covered everywhere with grounded ice with a
probability of at least the given probability level under the uncertainty
introduced in the ice-sheet model (see Appendix

We present nominal and probabilistic projections (relative to 2000 CE) for short-term (2100), medium-term (2300) and long-term (3000) timescales under different RCP scenarios and sliding laws.

We first present nominal projections obtained using the nominal values of the
parameters given in Table

Nominal projections (in metres) of AIS contribution to sea level on short-term (2100), medium-term (2300) and long-term (3000) timescales in different RCP scenarios.

One of the advantages of a polynomial chaos expansion is that it provides an explicit approximation to the parameters-to-projection relationship, which can be visualised to gain insight into the relationship between the parameters and the projections.

In Figs.

Representation of the parameters-to-projection relationships in
RCP 2.6 under the weakly non-linear sliding law as given by a PC expansion.
The PC expansion is shown for each parameter individually, with the other
parameters fixed at their nominal value. PC expansions at

Same as Fig.

Figure

We find in RCP 8.5 similar trends. However, whereas

Additionally, Fig. S2 shows the emulators for several pairs of parameters
with the other parameters fixed at their nominal values in RCP 2.6 and
RCP 8.5. These figures show essentially the same trends as those identified
in Figs.

Under parametric uncertainties, we find (Table

AIS contribution to sea level.

In RCP 2.6, we find (Table

Figure

Probability density functions for the AIS contribution to sea level
at

Figure

Probability of exceeding some characteristic threshold sea-level
rise values as a function of time, evaluated from the complementary
cumulative distribution functions of the probability density functions for

Probabilistic projections (in metres) of the AIS contribution to sea
level on short-term (2100), medium-term (2300) and long-term (3000)
timescales in different RCP scenarios and under different sliding conditions
with Schoof's grounding-line parameterisation.

Figure

Sobol sensitivity indices for the AIS contribution to sea level in
different RCP scenarios and for different values of the sliding exponent

By contrast, in warmer RCP scenarios and for longer timescales, the dominant
source of uncertainty becomes the uncertainty in sub-shelf melting, which
accounts in RCP 8.5 for more than 90 % of the uncertainty in sea-level rise
projections. As shown in Fig.

Finally, we find that, in all RCP scenarios and under all sliding laws, the uncertainty in the bedrock relaxation times for West and East Antarctica has a limited impact (Sobol index smaller than 1 %), which is a direct consequence of the very limited dependence of the projections on the bedrock relaxation times. Moreover, the interactions between the parameters have a negligible impact as the sums of the individual Sobol indices account almost entirely for the variances of the projections.

Figure

Confidence regions for grounded ice under the weakly non-linear
sliding law. Confidence regions are shown at

We find that ice remains grounded in regions above sea level. By contrast, in
all RCP scenarios, the risk of ungrounding is highest in marine sectors of
West Antarctica with fast-flowing ice streams, especially in Siple Coast, in
the Ronne basin, notably Ellsworth Land, and in the Amundsen Sea sector. In
warm RCP scenarios and at longer timescales, we also observe a risk of
grounding-line retreat in the Wilkes marine basin in East Antarctica, where
the grounding line could retreat between 100 km (with a risk of 95 %) and
500 km (with a risk of 5 %) from its present-day position. The risk of
retreat in Wilkes basin may partially explain the acceleration in sea-level
rise that we observed in Fig.

In RCP 2.6, we observe that the grounding line is quite stable over the next millennium, with the 100 % confidence region for grounded ice being almost unchanged from the present-day grounded ice region (the 100 % confidence region for grounded ice by 3000 only differs from the present-day grounded ice region by a few tens of kilometres). In RCP 4.5, ice remains grounded in most of the West Antarctic ice sheet over the next centuries, but our results also suggest a risk of retreat of the grounding line in some sectors of West Antarctica on longer timescales. In RCP 6.0, we find that the West Antarctic ice sheet belongs by 3000 to the 66 % confidence region for grounded ice, while in RCP 8.5, it belongs by 3000 only to the 5 % confidence region. This suggests a risk of 33 % that a major collapse of the West Antarctic ice sheet might occur in RCP 6.0 by 3000 and a risk of 95 % that a major collapse of the West Antarctic ice sheet might occur in RCP 8.5 by 3000.

As compared with the nominal projections of a limited retreat of the
grounding line in West Antarctica in RCP 6.0 by 3000, we find that the impact
of the parametric uncertainty is that a complete collapse of the West
Antarctic ice sheet may occur with a risk of 33 % in RCP 6.0 by 3000.
Moreover, Fig.

Additionally, we compared the projections under the weakly non-linear sliding
law with projections under the other sliding laws, that is, the viscous
sliding law (Fig. S7) and the strongly non-linear sliding law (Fig. S8). We
find a lower risk of retreat of the grounding line under the viscous sliding
law with a slower disintegration of the West Antarctic ice sheet compared to
the other sliding laws, especially in the drainage basins of Thwaites and
Pine Island glaciers, which belong to the 50 % confidence region for
grounded ice by 3000 in RCP 8.5, while they belong to the 5 % confidence
region by 3000 in RCP 8.5 for the other sliding laws. The strongly non-linear
sliding law seems to favour a faster and deeper retreat of the grounding
line, especially in the marine sectors of East Antarctica and the drainage
basins of Thwaites and Pine Island glaciers. However,
Figs.

So far, we have represented the uncertain parameters

For the long-term projections, Fig.

Robustness of probabilistic projections with respect to the scaling
factor

We ran the same ensemble of simulations under a more plastic sliding law
(

Same as Fig.

We find that overall the AIS contribution to sea level is an increasing
function of the sliding exponent, with the differences between successive
exponents getting smaller as

Same as Table

We ran the same ensemble of simulations under the weakly non-linear sliding
law using, this time, the TGL parameterisation instead of the SGL
parameterisation. Under the TGL parameterisation, Table

Same as Fig.

Figure

Same as Table

Regarding the short-term AIS contribution to sea level, we projected in the
RCP 2.6 scenario a median of 0.02–0.03 m under the SGL parameterisation and
5 %–95 % probability intervals from

Regarding the long-term AIS contribution to sea level, our projections under
the SGL parameterisation are similar to other estimates by

Similarly to

Consistent with our results,

The significance of the contribution of the Antarctic ice sheet to sea level
under climate change is primarily controlled by the sensitivity, the response
time and the vulnerability of its marine drainage basins, with the West
Antarctic ice sheet more sensitive and vulnerable than the East Antarctic ice
sheet. The instability of marine drainage basins and their ability to trigger
accelerated ice loss and significant grounding-line retreat is determined by
bedrock topography and ice-shelf buttressing which depends on the importance
of ice-shelf thinning. Our nominal projections showed that the AIS
contribution to sea level by 3000 is rather limited (less than 1 m) in
RCP 2.6, RCP 4.5 and RCP 6.0, while an accelerated ice loss that leads to a
contribution of several metres is triggered in RCP 8.5
(Fig.

Our probabilistic results provide insight into the impact of parametric
uncertainty on these projections. In RCP 2.6, the projections hold
irrespective of parametric uncertainty: the AIS contribution to sea level
by 3000 has a 95 % quantile of up to 0.91 m (Table

In conclusion, the projections hold irrespective of parametric uncertainty in the strongly mitigated RCP 2.6 scenario: accommodating parametric uncertainty in the ice-sheet model leads to projections in agreement with the nominal projections of limited ice loss and limited grounding-line retreat in RCP 2.6. However, the projections are more sensitive to parametric uncertainty for intermediate scenarios such as RCP 4.5 and RCP 6.0: accommodating parametric uncertainty in the ice-sheet model leads to projections in disagreement with the nominal projections and indicates instead some risk of triggering accelerated ice loss and significant grounding-line retreat for intermediate scenarios such as RCP 4.5 and RCP 6.0. Finally, the warm RCP 8.5 scenario triggers the collapse of the West Antarctic ice sheet, almost irrespective of parametric uncertainty.

A first limitation of our study is associated with the modelling hypotheses
inherent to our ice-sheet model. The f.ETISh model is an ice-sheet model that
focuses on essential marine ice-sheet mechanisms, similarly to the ice-sheet
model by

We studied the multi-centennial response of the Antarctic ice sheet under uncertainty using methods from the field of UQ. We investigated uncertainties in atmospheric forcing, basal sliding, grounding-line parameterisation, sub-shelf melting, calving, ice-shelf rheology and bedrock relaxation. We used emulation-based methods to represent the parameters-to-projection relationship, stochastic sensitivity analysis to assess the significance of each source of uncertainty in inducing uncertainty in the projections and confidence regions for excursion sets to assess the risk of grounding-line retreat. We found that all investigated sources of uncertainty, except bedrock relaxation time, contribute to the uncertainty in the projections. We showed that the sensitivity of the projections to uncertainties increases and the contribution of the uncertainty in sub-shelf melting to the uncertainty in the projections becomes more and more dominant as atmospheric and oceanic temperatures rise, with a contribution to the uncertainty in sea-level rise projections that goes from 5 % to 25 % in RCP 2.6 to more than 90 % in RCP 8.5. We showed that the significance of the AIS contribution to sea level is controlled by MISI in marine basins, with the biggest contribution stemming from the more vulnerable West Antarctic ice sheet. We found that, irrespective of parametric uncertainty, the strongly mitigated RCP 2.6 scenario prevents the collapse of the West Antarctic ice sheet, that in both the RCP 4.5 and RCP 6.0 scenarios the occurrence of MISI in marine basins is more sensitive to parametric uncertainty, and that, almost irrespective of parametric uncertainty, RCP 8.5 triggers the collapse of the West Antarctic ice sheet.

All datasets used in this paper are publicly available, including Bedmap2

In this Appendix, we concisely provide further details about how we used PC
expansions in our study; we refer the reader to, for instance,

Let us represent the ice-sheet model as an abstract model

A polynomial chaos expansion is an approximation of the response function

In order to fit the PC expansion in Eq. (

Validation tests for the PC expansion.

We solve Eq. (

We estimate statistical descriptors of the uncertain model response with
Monte Carlo simulation in which the PC expansion is used as a computationally
efficient substitute for the ice-sheet model. For instance, we estimate the
probability density function of the response through kernel density
estimation

The accuracy of the PC expansion has to be assessed with respect to the
degree of the PC expansion and the number of training points. We validate the
accuracy of the PC expansion using cross-validation and convergence tests. We
generate a new set of samples in the parameter space and we compare the exact
response of the ice-sheet model with the approximate response of the PC
expansion. Figure

In this Appendix, we concisely provide further details about how we used
stochastic sensitivity analysis; we refer the reader to, for instance,

We first assume that the uncertain parameters are statistically independent.
Sobol indices are based on the decomposition of the response function

As a consequence of the orthonormality of the functions

Here, we substituted the ice-sheet model by a PC expansion based on
orthonormal polynomials and estimated the Sobol indices directly from the PC
coefficients

In this Appendix, we concisely provide further details about how we defined
and computed confidence regions for grounded ice. To distinguish between
grounded ice and floating ice at a given time, the f.ETISh model

We compute the confidence regions using an adaptation of a thresholding
algorithm by

The supplement related to this article is available online at:

All the authors discussed the results presented in this paper. KB conducted the design, execution and UQ analysis of the experiments, with relevant inputs from MA for the UQ methodology and SS and FP for the physical interpretation of the results. The paper was written by KB and MA with relevant comments from all the co-authors.

The authors declare that they have no conflict of interest.

We would like to thank Andy Aschwanden and one anonymous referee for their very helpful comments that helped improve the overall quality and readability of the manuscript. Kevin Bulthuis would like to acknowledge Andreas Wernecke for personal communication about the manuscript. Kevin Bulthuis would like to acknowledge the Fonds de la Recherche Scientifique de Belgique (F.R.S.-FNRS) for its financial support (F.R.S.-FNRS Research Fellowship). Computational resources have been provided by the Consortium des Équipements de Calcul Intensif (CÉCI), funded by the Fonds de la Recherche Scientifique de Belgique (F.R.S-FNRS) under grant no. 2.5020.11.

This paper was edited by Olivier Gagliardini and reviewed by Andy Aschwanden and one anonymous referee.