Probabilistic predictions of the sea level contribution from Antarctica often have large uncertainty intervals. Calibration of model simulations with observations can reduce uncertainties and improve confidence in projections, particularly if this exploits as much of the available information as possible (such as spatial characteristics), but the necessary statistical treatment is often challenging and can be computationally prohibitive. Ice sheet models with sufficient spatial resolution to resolve grounding line evolution are also computationally expensive.

Here we address these challenges by adopting and comparing dimension-reduced calibration approaches based on a principal component decomposition of the adaptive mesh model BISICLES. The effects model parameters have on these principal components are then gathered in statistical emulators to allow for smooth probability density estimates. With the help of a published perturbed parameter ice sheet model ensemble of the Amundsen Sea Embayment (ASE), we show how the use of principal components in combination with spatially resolved observations can improve probabilistic calibrations. In synthetic model experiments (calibrating the model with altered model results) we can identify the correct basal traction and ice viscosity scaling parameters as well as the bedrock map with spatial calibrations. In comparison a simpler calibration against an aggregated observation, the net sea level contribution, imposes only weaker constraints by allowing a wide range of basal traction and viscosity scaling factors.

Uncertainties in sea level rise contribution of 50-year simulations from the current state of the ASE can be reduced with satellite observations of recent ice thickness change by nearly 90 %; median and 90 % confidence intervals are 18.9 [13.9, 24.8] mm SLE (sea level equivalent) for the proposed spatial calibration approach, 16.8 [7.7, 25.6] mm SLE for the net sea level calibration and 23.1 [

The ASE dominates the current Antarctic sea level contribution, but other regions have the potential to become more important on centennial scales. These larger spatial and temporal scales would benefit even more from methods of fast but exhaustive model calibration. Applied to projections of the whole Antarctic ice sheet, our approach has therefore the potential to efficiently improve our understanding of model behaviour, as well as substantiating and reducing projection uncertainties.

The Antarctic ice sheet is currently losing mass at a rate of around 0.5 to 0.6 mm global mean sea level equivalent per year (mm SLE a

In Antarctic ice sheet model ensemble studies, the projected sea level contribution for high emission scenarios by the end of the century typically ranges from about zero to about 40 cm; i.e. the ensemble spread (

However, the formal comparison of model simulations with two-dimensional observations, such as satellite measurements of Antarctica, poses statistical challenges.
Measurements of the earth system typically show coherent spatial patterns, meaning that nearby observations are highly correlated due to the continuity of physical quantities. Model-to-observation comparisons on a grid-cell-by-grid-cell basis can therefore not be treated as statistically independent. On the other hand, appropriate treatment of these correlations with the inclusion of a co-variance matrix in the statistical framework for calibration can be computationally prohibitive

A further difficulty is the computational expense of Antarctic ice sheet models that have sufficient spatial resolution to resolve grounding line migration. This can be overcome by building an “emulator”, which is a statistical model of the response of a physically based computer model. Emulation allows a small ensemble of the original ice sheet model to be extended to a much larger number. This approach has recently been applied in projections of the Antarctic ice sheet contribution to sea level rise by interpolation in the input parameter space in general

The aim of this study is to develop a practical, yet comprehensive calibration approach for data from the high-resolution ice sheet model BISICLES. This approach is compared to more traditional methods by means of a synthetic model test and the impact on probability density functions for the dynamic sea level contribution from 50-year simulations of the Amundsen Sea Embayment. We derive principal components of ice thickness change estimates with a singular value decomposition, thus exploiting more of the available information of satellite observations than previous studies. The statistical independence of those PCs aids the use of Bayesian (probabilistic) inference. We use emulation of the ice sheet model to ensure dense sampling of the input space and therefore smooth probability density functions.

In Sect.

We use the ice sheet model ensemble published in

The scaling parameters are subsequently perturbed between half and double the default values in a Latin hypercube design by

The ensemble covers a wide range of sea level rise contributions for the 50-year period, with the most extreme members reaching

We allow for a short spin-up phase of 3 years (selected by manual inspection) for the model to adjust to the perturbations. The following 7 years are used as the calibration period; therefore the temporal mean of the ice thickness change from year 4 to year 10 (inclusive) of the simulations will be compared with satellite observations which also span a 7-year period.

Other spin-up and calibration periods have been tested and show small impact on the results for calibrations in basis representation. For example the median for the basis calibration of the sea level contribution at the end of the simulations is 18.9 mm SLE with the described 3-year spin-up and 7-year calibration period and 19.1 mm SLE for a 7-year spin-up followed by a short 3-year calibration period. We further tested the 3-year spin-up with a 4-year calibration period and other calibration approaches (see Supplement).

We regrid the simulated ice thickness change fields for this period to the same spatial resolution as the observations (

The simulations used here are not intended to be predictions of the future but instead project the current state of the ASE glacial system with a constant recent-past climate forcing and perturbed parameters into the future. No changes in the climate are represented in the ensemble. End-of-simulation sea level contribution distributions are presented to illustrate and compare the value of calibrations and should not be understood as best estimates of future sea level contribution. For a full description of the model ensemble see

The calibration target is based on a compilation of five satellite altimeter datasets of surface elevation changes from 1992 to 2015 by

There is no exact start date of the simulations, which makes a dating of the calibration period difficult. However, the ice flow observations from

In the following we propose a new ice sheet model calibration approach, as outlined in Fig.

Flow diagram of the proposed calibration procedure. Horizontal boxes represent steps in the analysis, diamonds represent observations and numbers refer to corresponding sections in this study.

Let

The fraction of ensemble variance represented by a principal component is proportional to the corresponding eigenvalue of

This truncation limits the rank of

The first five normalised PCs of the model ice thickness change fields, building an orthogonal basis. They represent the main modes of variation in the model ensemble and are unitless since normalised. The lower left graph shows the fraction of total variance represented by each PC individually (grey) and cumulatively (red), based on squared singular values.

One of the calibration approaches we investigate uses the PCs derived before for both the model and observations (see Sect.

We perform the transformation as in Eq. (

Figure

It is only the part of the observations which can be represented by five PCs (right of Fig.

For a probabilistic assessment we need to consider the probability density in the full, five-dimensional parameter space. This exploration can require very dense sampling of probabilities in the input space to ensure appropriate representation of all probable parameter combinations. This is especially the case if the calibration is favouring only small subsets of the original input space. In our case more than 90 % of the calibrated distribution would be based on just five BISICLES ensemble members. For computationally expensive models sufficient sampling can be achieved by statistical emulation, as laid out in the following.

A row of

We use Gaussian process (GP) models, which are a common choice for their high level of flexibility and inherent emulation uncertainty representation

Due to the statistical independence of the principal components we can combine the

Given the emulator in basis representation, a calibration can be performed either after reprojecting the emulator output back to the original spatial field

We assume the existence of a parameter configuration

We assume the model discrepancy to be multivariate Gaussian distributed with zero mean;

We simplify the probabilistic inference by assuming the model error/discrepancy

Note that the existence of

The inclusion of model discrepancy can at the same time lead to identifiability issues where the model signal cannot be distinguished from the imposed systematic model error. Constraints on the spatial shape of the discrepancy have been used to overcome such issues

For these reasons we choose a rather heuristic method which considers the impact of discrepancy on the calibration directly and independently for each PC. Therefore

We thereby force the observations to fulfil the three-sigma rule by considering them as part of the model distribution

Probabilistic calibrations search for the best input parameters, but stand-alone probabilistic calibrations cannot guarantee that those are also good input parameters in an absolute sense. While “good” is subjective, it is possible to define and rule out implausible input parameters. The implausibility parameter is commonly defined as

For all

The calibration distribution in Eq. (

In this section we test our calibration approach on synthetic observations to see whether our method is capable of finding known-correct parameter values. We select one member of the BISICLES model ensemble at a time and add 14 different realisations of noise to it. The noise is added to see how the calibration performs if the observations cannot be fully represented by the ice sheet model.

We use spatially independent, zero-mean, normally distributed, random noise with variance equal to the local variance from the 14 periods of satellite observations. This way the variance incorporates dynamic changes (acceleration/deceleration of the ice thickness change) and technical errors (e.g. measurement and sampling errors). For each selected model run we generate 14 noise fields and add them to the single model ice thickness change field. These 14 realisations replace the 14 periods of satellite observations for the synthetic model tests.

For Fig.

Likelihood of parameter combinations of the synthetic test case (evaluations of Eq.

Figure

As can be seen from Fig.

The signal of friction law and ocean melt is not strong enough to adequately constrain the calibration, even though both parameters are known to have a strong impact on the ice sheet

From this test we conclude that basal friction law and ocean melt scaling cannot be inferred with this calibration approach and calibration period. We will therefore only calibrate the bedrock as well as basal traction and viscosity scaling factors. Several studies used the observed dynamical changes of parts of the ASE to test different friction laws.

To put the likelihood distribution from Fig.

Another approach is to use the net yearly sea level contribution (SLC) from the observations SLC

Again, we set the model discrepancy to twice the observational uncertainty which we find from the variance of the yearly sea level contributions for the 14 biyearly satellite intervals.

Likelihood of parameter combinations of the synthetic test case for reprojected emulator estimates (top,

Results for the synthetic model test for the calibration in (

Moving on to using satellite data, the basis calibration finds that the modified bedrock from

For updated probability distributions of sea level contribution after 50 years in Fig.

Total sea level contribution after 50 years in mm SLE: (weighted) mean, most likely contribution and percentiles, with and without calibrations.

In general, previous Antarctic ice sheet model uncertainty studies have either focused on parameter inference

The theoretical basis for most of the methodology used here has been laid out in

The modified bedrock removes a topographic rise near the initial grounding line of Pine Island Glacier, which could be caused by erroneous observations

The non-spatial calibration on total sea level contribution alone cannot distinguish between the two bedrocks (Fig.

The extremely small area of likely input parameters for the reprojected (

The average sea level contribution from the observations used here is 0.36 mm SLE a

The ice sheet model data used here are not based on a specific climate scenario but instead project the state of the ice sheet under current conditions into the future (with imposed perturbations).

Relating climate scenarios to local ice shelf melt rates is associated with deep uncertainties itself. CMIP5 climate models are inconsistent in predicting Antarctic shelf water temperatures so that the model choice can make a substantial (

The truncation of a principal component decomposition can cause or worsen problems related to the observations not being in the analysed model output space (see difference in Fig.

The model perturbation has been done by amplitude scaling of the optimised input fields alone; other types of variations to the basal traction coefficient fields could potentially produce model setups with better agreement to the observations

It should also be noted that for a given ice geometry the surface speed (used for initialisation) and ice thickness change (used for calibration) are not fully independent (conversation of mass). Finding the unperturbed traction and viscosity fields to show good agreement with ice thickness change observations is not surprising, yet it is a good test of the initialisation process, initialisation data and the quality of the initial ice geometry. For the same reasons, the optimised fields cannot be considered without uncertainty. This uncertainty can be quantified by ice thickness change observations, as has been shown here. A combined temporal and spatial calibration could help to use even more of the available information captured by observations in regions like the ASE where dynamic changes in the ice sheet took place within the observation period. The temporal component could in particular help to constrain the basal friction law exponent and ocean melt scaling.

We present probabilistic estimates of the dynamic contribution to sea level of unforced 50-year simulations of the Amundsen Sea Embayment in West Antarctica from a grounding-line-resolving ice sheet model. The Bayesian calibration of a published ice sheet model ensemble with satellite estimates of changes in ice thickness from 2008 to 2015 involves spatial decomposition to increase the amount of available information from the observations and emulation techniques to search the parameter space more thoroughly.

The calibration has been tested on synthetic test cases and can reliably constrain the bedrock, basal traction and ice viscosity amplitudes. Identifying the most successful basal friction law and ocean melt rate is more challenging, and interference of those parameters could benefit from a temporally resolved calibration approach and a longer calibration period. The use of net sea level contribution alone allows a wide range of parameter setups, which share the initial net mass loss. This ambiguity (weak constraint) also results in relatively wide sea level contribution probability distributions. The extra information from the use of two-dimensional calibrations adds stronger parameter constraints, showing that this method has the potential to reduce uncertainties in ice sheet model projections. We compare and discuss spatial calibrations in both basis and reprojected representation.

Using satellite observations we find the modified bedrock topography derived by

Code can be accessed at

The supplement related to this article is available online at:

AW led this study, with TLE, PBH and NRE giving valuable advice on the study design and IJN on the model data processing and interpretation. All authors contributed to the interpretation of the study results. AW prepared the initial manuscript with contributions from all co-authors.

The authors declare that they have no conflict of interest.

We would like to thank Hannes Konrad for sharing and advising on the satellite observations and Mark Brandon for general advice. We also thank the anonymous reviewers which helped to significantly improve this work.

This research has been supported by the EPSRC Research of Variability and Environmental Risk (ReCoVER: EP/M008495/1) under the Quantifying Uncertainty in Antarctic Ice Sheet Instability (QUAntIS) project (RFFLP 006) (via the work of Tamsin L. Edwards, Neil R. Edwards and Philip B. Holden). It has further been supported by LC3M, a Leverhulme Trust Research Centre Award (RC-2015-029) (via the work of Neil R. Edwards and Philip B. Holden); The Open University Faculty of Science, Technology, Engineering and Mathematics; and the University of Bristol Advanced Computing Research Centre (via the work of Andreas Wernecke).

This paper was edited by Olaf Eisen and reviewed by four anonymous referees.