The 3-D ice-sheet model has previously been applied to past Antarctic variations in Pollard and DeConto (2009), DeConto et al. (2012) and Pollard et al. (2015). The model predicts ice thickness and temperature distributions, evolving due to slow deformation under its own weight, and to mass addition and removal (precipitation, basal melt and runoff, oceanic melt, and calving of floating ice). Floating ice shelves and grounding-line migration are included. It uses hybrid ice dynamics and an internal condition on ice velocity at the grounding line (Schoof, 2007). The simplified dynamics (compared to full Stokes or higher-order) captures grounding-line migration reasonably well (Pattyn et al., 2013), while still allowing O(10 000's) yr runs to be feasible. As in many long-term ice-sheet models, bedrock deformation is modeled as an elastic lithospheric plate above local isostatic relaxation. Details of the model formulation are described in Pollard and DeConto (2012a, b). The drastic ice-retreat mechanisms of hydrofracturing and ice-cliff failure proposed in Pollard et al. (2015) are only triggered in warmer-than-present climates and so do not play any role in the glacial–deglacial simulations here; in fact, they are switched off in all runs. Tests show that they play no perceptible role in simulations over the last 40 kyr.

The model is applied to a limited area nested domain spanning all of West Antarctica, with a 20 km grid resolution. Lateral boundary conditions on ice thicknesses and velocities are provided by a previous continental-scale run. The model is run over the last 30 000 yr, initialized appropriately at 30 ka (30 000 yr before present, relative to 1950 AD) from a previous longer-term run. Atmospheric forcing is computed using a modern climatological Antarctic data set (ALBMAP: Le Brocq et al., 2010), with uniform cooling perturbations proportional to a deep-sea core δ18O record (as in Pollard and DeConto, 2009, 2012a). Oceanic forcing uses archived ocean temperatures from a global climate model simulation of the last 22 kyr (Liu et al., 2009). Sea-level variations vs. time, which are controlled predominantly by northern hemispheric ice-sheet variations, are prescribed from the ICE-5G data set (Peltier, 2004). Modern bedrock elevations are obtained from the Bedmap2 data set (Fretwell et al., 2013).

The large ensemble analyzed in this study uses full-factorial sampling, i.e., a run for every possible combination of parameter values, with four parameters varied and with each parameter taking five values, requiring 625 (= 54) runs. As discussed above, results are analyzed in two ways: (1) using the relatively advanced statistical techniques (emulators, likelihood functions, MCMC) in Chang et al. (2015, 2016), and (2) using the much simpler averaging method of calculating an aggregate score for each run that measures model–data misfit, and computing results as averages over all runs weighted by their score. Because the second method has no means of interpolating results between sparsely separated points in multi-dimensional parameter space, it is effectively limited to full-factorial sampling with only a few parameters and a small number of values each. The small number of values is nevertheless sufficient to span the full reasonable “prior” range for each parameter, and although the results are very coarse with the second method, they show the basic patterns adequately. Furthermore, envelopes of results of all model runs are compared in Appendix D with corresponding data, and demonstrate that the ensemble results do adequately “span” the data; i.e., there are no serious outliers of data far from the range of model results.

The four parameters and their five values are the following.

OCFAC: sub-ice oceanic melt coefficient. Values are 0.1, 0.3, 1, 3, and 10 (non-dimensional). Corresponds to K in Eq. (17) of Pollard and Deconto (2012a).

Following Whitehouse et al. (2012a, b), Briggs and Tarasov (2013) and Briggs et al. (2013, 2014), we test the model against three types of data for the modern observed state, and five types of geologic data relevant to ice-sheet variations of the last ∼ 20 000 yr, using straightforward mean squared or root-mean-square misfits in most cases. Each misfit (Mi, i=1 to 8) is normalized into individual scores (Si), which are then combined into one aggregate score (S) for each member of the LE. Only data within the domain of the model (West Antarctica) are used in the calculation of the misfits.

One approach to calculating misfits and scores is to borrow from Gaussian error distribution concepts, i.e., individual misfits M of the form [(mod-obs)/σ]2 and overall scores of the form e-M¯/s, where mod is a model quantity, obs is a corresponding observation, σ is an observational or scaling uncertainty, M¯ is an average of individual misfits over data sites and types of measurements, and s is another scaling value (Briggs and Tarasov, 2013; Briggs et al., 2014). However, the choice of these forms is somewhat heuristic, and different choices are also appropriate for complex model–data comparisons with widespread data points, very different types of data, and with many model–data error types not being strictly Gaussian. In order to determine the influence of these choices on the results, we compare two approaches: (a) with formulae adhering closely to Gaussian forms throughout, and (b) with some non-Gaussian aspects attempting to provide more straightforward and interpretable scalings between different data types. Both approaches are described fully below (next section, and Appendix B). They yield very similar results, with no significant differences between the two, as shown in Appendix C. The second more heuristic approach (b) is used for results in the main paper.

The eight individual data types and model–data misfits are listed below, with basic information that applies to both of the above approaches. More details are given in Appendix B, including formulae for the two approaches, and “intra-data-type weighting” that is important for closely spaced sites (Briggs and Tarasov, 2013). The two approaches of combining the individual scores into one aggregate score S for the simple averaging method are described in the following Sect. 2.4. The more advanced statistical techniques (Chang et al., 2015, 2016) use elements of these calculations but differ fundamentally in some aspects, as outlined in Sect. 2.5.

The eight individual data types are the following.

TOTE: modern grounding-line locations. Misfit M1: based on total area of model–data mismatch for grounded ice. Data: Bedmap2 (Fretwell et al., 2013).

Each of the misfits above are first transformed into a normalized individual score for each data type i=1 to 8. The transformations differ for the two approaches mentioned above.

The more advanced statistical techniques (Chang et al., 2015, 2016) consist of an emulation and a calibration stage, involving the same four model parameters and the 625-member LE as above. The aggregate scores S described in Sect. 2.4 are not used at all. The techniques are outlined here; full accounts are given in Chang et al. (2015, 2016).

All scores with the largest CALV value of 1.7 (right-hand column of subpanels) are 0. In these runs, excessive calving results in very little floating ice shelves and far too much grounding-line retreat. Conversely, with the smallest CALV value of 0.3 (left-hand column of subpanels), most runs have too much floating ice and too advanced grounding lines during the runs, so most of this column also has zero scores. However, small CALV can be partially compensated for by large OCFAC (strong ocean melting), so there are some non-zero scores in the upper-left subpanels.

For mid-range CALV and OCFAC (subpanels near the center of the figure), the best scores require high CSHELF (inner x axis) values, i.e., slippery ocean-bed coefficients of 10-6 to 10-5 m a-1 Pa-2. This is the most prominent signal in Fig. 2, and is consistent with the widespread extent of deformable sediments on continental shelves noted above. Ideally the LE should have included CSHELF values greater than 10-5. However, we note that values of 10-5 to 10-6 have been found to well represent active Siple Coast ice-stream beds in model inversions (Pollard and DeConto, 2012b). Subsequent work with wider CSHELF ranges confirms that values around 10-5 are in fact optimal, with unrealistic behavior for larger values (Pollard et al., 2016).

Somewhat lower but still reasonable scores exist for lower CSHELF values of 10-7, but only for higher OCFAC (3 to 10) and smaller TAUAST (1 to 2 kyr). This is of interest because smaller CSHELF values support thicker ice thicknesses at LGM where grounded ice has expanded over continental shelves, producing greater equivalent sea-level lowering and alleviating the LGM “missing-ice” problem (Clark and Tarasov, 2014). In order for the extra ice to be melted by present day, ocean melting needs to be more aggressive (higher OCFAC), and to recover in time from the greater bedrock depression at LGM, TAUAST has to be smaller (more rapid bedrock rebound). This glaciological aspect is explored in Pollard et al. (2016).

Scores are quite insensitive to the TAUAST asthenospheric rebound timescale (inner y axis), although there is a tendency to cluster around 2 to 3 kyr and to disfavor higher values (5 to 7 kyr), especially for high OCFAC.

The main results seen in Fig. 2 are borne out in Fig. 3. The left-hand panels show results using the simple averaging method, i.e., the average score for all runs in the LE with a particular parameter value. Triangles in these panels show the mean parameter value Vm=Σ(S(n)V(n))/ΣS(n), where S(n) is the aggregate score and V(n) is the value of this parameter for run n (1 to 625), and whiskers show the standard deviation. The prominent signal of high CSHELF values (slippery ocean beds) is evident, along with the absence (near absence) of positive scores for the extreme CALV values of 1.7 (0.3), and the more subtle trends for OCFAC and TAUAST.

The right-hand panels of Fig. 3 show the same single-parameter “marginal” probably density functions for this LE, using the advanced statistical techniques described in Chang et al. (2015, 2016) and summarized above. For OCFAC, CSHELF and TAUAST, there is substantial agreement with the simple-averaging results in both the peak “best-fit” values and the width of the ranges. For CALV, the peak values agree quite well, but the simple-averaging distribution has a significant tail for lower CALV values that is not present in the advanced results; this might be due to the discrepancy function in the advanced method (Sect. 2.5), which has no counterpart in the simple averaging method.

Probability densities for pairs of parameter values are useful in evaluating the quality of LE analysis, and can display offsetting physical processes that together maintain realistic results, e.g., greater OCFAC and lesser CALV (Chang et al., 2014, 2015, 2016). In Fig. 4, the left-hand panels show mean scores for pairs of the four parameters, using the simple averaging method and averaged over all LE runs with a particular pair of values. The right-hand panels show corresponding densities for the same parameter pairs using the advanced statistical techniques. Overall the same encouraging agreement is seen as for the single-parameter densities in Fig. 3, with the locations of the main maxima being roughly the same for each parameter pair. There are some differences in the extents of the maxima, notably for CALV, where the zone of high scores with the simple averaging method extends to lower CALV values than with the advanced techniques, as seen for the individual parameters in Fig. 3. In general, though, there is good agreement between the two methods regarding parameter ranges in Figs. 3 and 4, suggesting that the simple averaging method is viable, at least for LEs with full-factorial sampling of parameter space.

Figure 5 illustrates the use of the LE to produce past envelopes of model simulations. Figure 5a, b show equivalent sea-level (ESL) scatterplots for all 625 runs. Early in the runs around LGM (20 to 15 ka), the curves cluster into noticeable groups with the same CSHELF values, due to the relatively weak effects of the other parameters (OCFAC, CALV and TAUAST) for cold climates and ice sheets in near equilibrium. Figure 5c, d show the mean and one-sided standard deviations for the simple method. Most of the retreat and sea-level rise occurs between ∼ 14 and 10 ka. Glaciological aspects of the retreat will be discussed in more detail in Pollard et al. (2016).

Figure 5e, f shows the equivalent mean and standard deviations derived from the advanced statistical techniques. There is substantial agreement with the simple-method curves in Fig. 5c, d, for most of the duration of the runs. The largest difference is around the Last Glacial Maximum ∼ 20 to 15 ka, when mean sea levels are nearly ∼ 2 m lower (larger LGM ice volumes) in the simpler method compared to the advanced. This may be due to the simpler method's scores using past 2-D grounding-line reconstructions (data type GL2D), which are not used in the advanced techniques.

Figure 6 shows probability densities of equivalent sea-level rise at particular times in the runs. Figure 6a–d show results with the simple averaging method, computed using score-weighted densities and 0.2 m wide ESL bins (see caption). The uneven noise in this figure is due to the small number of parameter values in our LE. The separate peaks for LGM (-15 000 yr) in Fig. 6a and b are due to the widely separated CSHELF values and the relatively weak effects of the other parameters (OCFAC, CALV and TAUAST) for cold climates and ice sheets in near equilibrium. Figure 6e shows the equivalent but much smoother probability densities using the advanced statistical techniques. All major aspects agree reasonably well with the simple averaging results, and the separate peaks for -15 000 yr are smoothed into a single broad range.

The code for the ice-sheet model (PSUICE-3D) is available on request from the corresponding author. The post-processing codes for the large-ensemble statistical analyses are highly tailored to specific sets of model output and are not made available; however, modules that compute scores for the individual data types are also available on request.