ICESHEET is an ice-sheet model that assumes steady-state conditions and a simple, perfectly plastic ice-sheet rheology to rapidly generate physically plausible ice-sheet reconstructions from only three 2D model inputs: ice-sheet margins, regional topography, and basal shear stress (based upon the physics first developed by , , and ). Using an iterative process, ICESHEET calculates thickness profiles along flowlines that are generated at regular intervals within the prescribed ice margin. Flowline positions, and thus the ice-sheet thickness profile, are dependent on the 2D input topography and shear-stress maps . The shear-stress map serves as a tuning input that can be calibrated or inverted to produce a target ice-sheet geometry, although significant uncertainties exist in determining basal shear stress (Sect. ).

The model has been applied where large uncertainty in inputs required for dynamic ice-sheet models, such as climate, have reduced the confidence in using the outputs of such models as inputs to sea-level models due to misfits against ice extent and volume distributions that impact GIA, and where large numbers of runs are required making computational efficiency paramount, such as in the exploration of variable global ice-sheet configurations . Limited constraints on climatic conditions, the requirement for large ensemble simulations to explore the range of plausible scenarios, and a need for well-defined sources of uncertainty make ICESHEET an ideal choice for exploring uncertainty in ice-sheet configurations during the PGM.

Two model parameters determine the resolution of a reconstruction with ICESHEET: contour elevation interval and flowline spacing. For our reconstructions, we use values of 20 and 5000 m respectively in order to balance compute time with resolution. The 2D model inputs are defined on a Lambert Azimuthal Equal Area (LAEA) projection centred on lat 90∘, long 0∘, using the WGS84 ellipsoid, and with boundaries defined at -1 265 453 to 4 159 547 m in the x direction and -4 722 734.8 to 1 352 265.2 m in the y direction with no x or y offsets, covering the Eurasian region at a resolution of 5 km. In the following subsections, we describe the set-up and inputs to simulations of the Last Deglaciation and PGM.

ICESHEET, owing to the large uncertainties in the shear-stress input, is capable of producing a wide range of ice-sheet geometries for both the Last Deglaciation and the PGM. While it is useful to retain some of this possible set of geometries for the purpose of uncertainty quantification, not all simulations will fall within our expectations of plausible Eurasian configurations. Existing GIA reconstructions provide constraints on ice-sheet thickness during the Last Deglaciation and it is desirable to transpose this information to the PGM through model calibration. Bayesian uncertainty quantification techniques exist to explore uncertainty and calibrate physical models . However, because the primary input of ICESHEET is the 2D, extremely heterogeneous, and poorly constrained basal shear-stress matrix, “out-of-the-box” methods for sampling uncertain model inputs are unsuitable. Moreover, due to the major simplifications applied within ICESHEET, this 2D input should not only represent ice basal shear stress linked with bedrock geology, but should also encompass the effect of missing ice surface mass balance and the influence of basal processes. Thus, a bespoke framework for quantifying past ice-sheet uncertainty with simple ice-sheet models such as ICESHEET is needed.

We first employ ICESHEET to produce a new simulated history of the Last Deglaciation that we then calibrate against independently derived, regionally aggregated volume metrics for the Last Deglaciation by employing a Bayesian uncertainty quantification method called “history matching”. History matching allows us to identify regions of the ICESHEET input parameter space for which ICESHEET simulations are able to match the regional volume estimates that are expressed in published reconstructions (used here as an “observation”) given the uncertainty in the model and target data . This space is referred to as the “not ruled out yet” (NROY) space and, once identified using the Last Deglaciation constraints, can then also be applied to refine our set of reconstructions for the PGM where empirical constraints on published models are far more limited. This procedure also allows us to identify systematic differences between the geometry simulated by ICESHEET and those reconstructed through GIA modelling, thus testing the capability of our modelling approach in providing meaningful ice geometries for use in sea-level and climate simulators.

We consider two spatiotemporal reconstructions of Eurasian ice-sheet thickness and regional topography during the Last Glacial Period: GLAC-1D and ICE-6G . These reconstructions have been selected as they are widely used, well regarded, and more accessible than others while also representing two contrasting modelling methodologies that are both independent of the ICESHEET methodology . GLAC-1D is the result of a large ensemble of thermodynamic ice-sheet simulations driven by climate reconstructions that have been nudged and selectively refined to fit relative sea-level records. It is provided every 0.1 ka at a spatial resolution of lat 0.25∘, long 0.5∘ . ICE-6G is a solution to the inverse GIA problem and is provided at 0.5 ka after 21 ka, and 1.0 ka before, with spatial resolution of lat 1∘, long 1∘ . ICE-6G provides a better fit to sea-level records than GLAC-1D, but the ice geometry is not compatible with ice-sheet physics , while GLAC-1D provides glaciological consistent ice-sheet geometries that account for ice-flow physics and climate forcing . Both reconstructions account for GIA effects, provide accompanying topography inputs, match against RSL data, and include a range of time slices that span the full deglaciation.

We extract the ice margin from each Last Deglaciation reconstruction, for use as input to ICESHEET, to ensure that we are able to accurately compare difference between thickness slices generated by ICESHEET and those of the reconstruction considered. To do this, we reproject and interpolate each reconstruction onto the same model grid as ICESHEET before applying an algorithm that produces ice-margin geometries from the gridded thickness data (Appendix ).

When using ICESHEET to simulate past ice sheets, the input topography needs to be adjusted for GIA. Since our aim is to reproduce ICE-6G and GLAC-1D volumes, we simply use the topography deformation fields provided by each model, reprojected onto our model grid. We run the ICESHEET model with topography and margins from GLAC-1D and ICE-6G, at 22, 20, 18, and 16 ka. These times are chosen since they capture a range of ice-sheet deglaciation thickness and extent configurations while excluding the very thick slices > 22 ka, which are poorly constrained by sea-level data, and those of small extent after 16 ka which are less relevant for producing the extensive PGM. We label these simulations ICESHEET1D and ICESHEET6G.

In this paper, we adapt a basal shear-stress map, developed for Eurasia during the Last Deglaciation, utilised in and . This map was constructed by dividing the bed of the Eurasian ice sheet into distinct surface geological and geomorphological units, in consultation with geological mapping, sediment thickness maps, and the distribution of glacial landforms observed by satellite imagery and digital elevation models. In the original map, five landscape categories were distinguished: (i) palaeo-ice streams; (ii) marine sediments; (iii) thick and (iv) thin terrestrial Quaternary sediments, as indicated by subglacial bedforms and on sediment maps; and (v) exposed bedrock surface . Due to uncertainties in the identification of such sediment categories during the PGM, we modify the original map in four ways to make it applicable to modelling the Penultimate Deglacial history and to keep our quantification of uncertainties tractable: (i) merging the original continuous sediment and discontinuous sediment categories into a single “onshore” category to reflect the lack of evidence to constrain the location of regions of discontinuous sediment during the Penultimate Glacial Period; (ii) defining the underlying sediment type for each ice-streaming region so that their length may be altered and the underlying sediment revealed (Sect. ); (iii) adding additional ice-streaming regions in the eastern sector and creating a separate ice streaming layer for the PGM (Sect. ); (iv) expanding the southerly and easterly extent of the map to encompass the greater Eurasian PGM ice-sheet extent. Regions within our adapted map are therefore categorised by their underlying sediment in the absence of ice streaming (Fig. a) as well as their potential to ice stream during the PGM and Last Deglaciation (Fig. b).

Each category has an associated shear-stress value uncertainty range, derived from our expert judgement, and described in Table . The larger extent of the Eurasian ice sheet during the Penultimate Glacial Period means that we require a shear-stress map that extends further south, into Continental Europe, and further east towards Siberia. We designed these additional regions based on a digitally compiled maps of geology alongside modern satellite imagery. Our final shear-stress map for Eurasia for the Penultimate Glacial Period consists of 740 categorised regions (Fig. ).

Ice streams are corridors of fast-flowing ice that occur towards the exteriors of ice sheets and significantly reduce local ice thickness . It is important that these regions are represented explicitly in ICESHEET, as the model lacks the dynamic mechanisms needed to generate ice streams on its own , and thus they are instead included as areas of very low shear stress. This enables ICESHEET to capture their main effect for GIA models, to reduce regional ice thickness. Evidence for the configuration of historic ice streaming relies on the identification of flow patterns, shapes, and deformed bed conditions within the geomorphological record . Ice-stream margin features can be dated (e.g. radiocarbon, cosmogenic nuclide, or optically stimulated luminescence) in order to infer the time the associated ice stream was active . Identifying and dating ice-streaming regions during the Penultimate Glacial Period poses a greater challenge compared to the Last Glacial Period as the period pre-dates the application of 14C methods and much of the sediment left behind by streaming has been removed by subsequent glacial activity. This is especially true for the southern margin of the Eurasian ice sheet . By comparison, the extent-limiting influence of the continental shelf break and topography of troughs on the shelf increases confidence that ice streaming in the northern Penultimate Glacial Period Eurasian ice sheet was similar to those of the Last Glacial Period.

We represented ice streams in our shear stress map with a low shear-stress value. To reflect the differences in streaming configurations as well as the disparity in geospatial constraints between the two glacial periods, we produce two separate maps of ice streaming during the Last Glacial Period and the Penultimate Glacial Period. The Penultimate Glacial Period layer is identical to the Last Glacial Period layer in the north, except for the addition of two streaming regions in the northeast, since the Eurasian ice sheet reached a similar extent during both glaciations (Fig. ). However, we completely remove ice streaming along the southern region in the PGM shear-stress map since evidence constraining streaming positions during the Penultimate Glacial Period is sparse, while the larger extent of the PGM ice sheet (Fig. ) means the mapped Last Glacial Period ice streams do not apply as they would terminate within the interior of the ice sheet.

Prescribing shear-stress values based on geological surface type ignores the influence of basal conditions on sliding . However, the basal conditions can influence the effective shear stress and, in turn, affect the geometry of an ice sheet. In order to better capture ice-sheet basal interactions we account for the influence of three such effects: cold based ice, active ice streaming, and hybrid ice streaming. The first approximates the effects on basal conditions when ice becomes frozen to the surface in the central interiors of large ice sheets . Cold-based ice has a high effective shear stress whether the bed is made of hard bedrock or soft sediment. The cold-based ice modification introduces this idea through two parameters. The first controls the size of the cold-based region (modelled as distance of unfrozen region from the margin), ranging from between 300 and 1000 km (Fig. b). The upper limit matches the maximum distance from the margin at the PGM, resulting in no cold-based ice, while the lower limit stops cold-based ice forming at the margin within the range of likely ice streaming. Secondly, we control the shear stress value of the region with a parameter ranging between 120 and 200 kPa.

Ice streaming occurred at different times and locations through the Last Deglaciation of the Eurasian ice sheet (Fig. b). Ice streaming is also likely to occur at periods during ice-sheet advance or retreat of Preceding Glacial , but a limited amount of geomorphological evidence means it is much harder to constrain when and where. As discussed above, it is not sufficient to simply use the LGM ice-stream locations, as these ice streams may not reach the PGM margin and it is not realistic to have an ice stream that terminates within the ice sheet. We therefore introduce an “active ice-streaming distance” parameter for the northern portion of the ice sheet to restrict ice streaming to within a particular distance of the margin ranging between 0 and 1000 km, based upon work by , and thus induce a marginal dependence on the previously static shear-stress input (Fig. a). We also introduce a hybrid ice-streaming modification to represent the shear-stress values that would result from streaming at the southern margin without exact prescription of stream locations. We define a distance from the margin that represents the average length of an ice stream ranging between 0 and 600 km (Fig. c), and also prescribe a shear-stress range whose minimum and maximum values are dependent on the shear-stress values for ice streaming and onshore sediment respectively, acting as a proxy for ice-stream density.

In addition to better capturing the resulting shear-stress implications of basal-driven interactions, the introduction of these three basal modifications allows us to expand the range of ice-sheet geometries and volumes that can be produced by ICESHEET for a given margin; better capture the resulting shear-stress implications of basal-driven interactions; and to improve the physical plausibility of the shear-stress input by increasing the dimension of our parameter space improving our ability to calibrate the model output and widen the range of uncertainty that can be considered. In total, we describe our shear-stress input through nine parameters.

We employ a random Latin hypercube sampling (LHS) design to select a 200-member set of input parameter values from the 7D parameter space controlling the shear-stress input (Table ), after excluding hybrid ice-streaming shear-stress and marginal distance parameters since the position of southern margin ice streams are prescribed for the LGM (Fig. b). LHS is a design method, common in Bayesian uncertainty quantification, that efficiently explores the input parameter space to construct ensembles of model simulations . It is typical to sample a minimum of 10× the number of parameters, but a higher sample density is beneficial, particularly if parameter ranges are wide and poorly constrained, hence our large sample design. For each reconstruction and time period, this parameter set is used in combination with the extracted ice margin to generate a corresponding shear-stress map. We run 200 simulations for each reconstruction and each of the four selected time periods (22, 20, 18, 16 ka), totalling 1600 simulations (Figs.  and ).

GIA models are sensitive to regional distributions of ice-mass loading more so than localised differences in the ice-sheet profile. Since our work is aimed towards developing a GIA ice-sheet input, we choose to assess and calibrate ICESHEET against the ice-sheet volume integrated over three ice-sheet regions, which allows us to assess volume difference at a regional scale, rather than over the whole ice sheet or cell-by-cell: Barents–Kara Sea, British–Irish, and Fennoscandia (Fig. ). To assess the model simulations against ICE-6G and GLAC-1D, we use an implausibility metric routinely used in history matching . The implausibility is akin to a root mean square error normalised by a measure of acceptable discrepancy between a given observation z and modelled value F(p^), where F is the model and p^ is a set of model parameters, for a quantity of interest (e.g. ice volume) given the known uncertainty in the observation and model limitations. The difference between an observation z and the real system y is quantified by the observational error e, such that z=y+e, while the difference between the modelled value at the theoretical best set of input parameters p^* and y is quantified as the structural model discrepancy ϵ, such that F(p^*)+ϵ=y . Additionally, it is often necessary to be able to predict values of F(p^) by training an emulator f(p^), such that F(p^)=f(p^)+ω(p^), where ω(p^) is the emulation uncertainty, to facilitate denser sampling of the model parameter space than is feasible through direct model runs. Here, we emulate multiple quantities i corresponding to volumes of the Eurasian ice sheet over each of the three regions for each time step and margin series. For each quantity i, the implausibility Ii of the model for a given parameter combination p^ is expressed as 1Ii(p^)=(E(fi(p^))-E(ϵi)-zi)2F(Var(ei)+Var(ϵi))+Var(ω(p^)), where E is the expectation (i.e. mean), Var is the variance, and F is a scaling factor for the model and observational uncertainties (see explanation below).

In lay terms, the implausibility represents the discrepancy between the “best guess” (i.e. expectation) of the model emulator E(fi(p^)) and the observation zi, accounting for systematic model bias (represented by the term E(ϵi)) and scaled by the sum of the uncertainties in the observation, model and emulator. Thus, implausibility is large if the discrepancy between model and observation is large relative to the uncertainties.

As explained at the start of this section, the quantities of interest that we emulate and calculate implausibility for are the ice-sheet volumes at each simulated time, region, and margin series (GLAC-1D and ICE-6G), resulting in 24 quantities. We derive the “observed” regional ice volumes zi for each of these quantities from the ICE-6G and GLAC-1D reconstructions; obtain each set of Fi(p^) from the corresponding ICESHEET model ensemble; and train a Gaussian process emulator fi for each quantity, resulting in 24 emulators of ice volume.

E(fi(p^)) and Var(ω(p^)) are calculated as the mean and variance from the emulated volumes fi(p^), where Var(ω(p^))=0 and E(fi(p^))=Fi(p^) for values of input parameters p^ run in the original ICESHEET ensembles. The model bias E(ϵi) and structural uncertainty Var(ϵi) are estimated as the mean and variance of the residuals from the 20 ICESHEET ensemble members with the lowest RMSE against the corresponding GLAC-1D and ICE-6G thickness fields. Since we only have two target reconstructions of ice volume from GLAC-1D and ICE-6G, we choose to estimate Var(ei) as half the difference between the GLAC-1D and ICE-6G volumes for a given region and time, knowing that this quantity underestimates the true uncertainty in the observations. We therefore choose to augment the observation and model structural uncertainties by 20 %, by setting F=1.2. The choice of regional ice-sheet volumes as our metrics, the selection of the F value, and judgement of their impacts of parameter space refinement comprise an iterative process, and other applications may choose different metrics or tolerance for model discrepancy.

Following from Eq. (), we combine our implausibility metrics into a single implausibility I(p^) for a given set of input parameters p^ such that 2I(p^)=1N∑iIi(p^), where N=24 is the total number of implausibility metrics. In other words, the overall implausibility is set as the mean of the implausibilities calculated for each time, region, and margin.

I(p^) is therefore an average measure of how well a particular set of input parameters is able to produce an output via ICESHEET that matches our expectation of ice-sheet volume for each region, time, and margin considered. We restrict our NROY space to parameter values that correspond to model runs with implausibility I(p^) less than 3, following the three-sigma rule typically used in Bayesian history matching .

GLAC-1D and ICE-6G reconstructions have volume estimates of comparable magnitudes for each time considered, with ICE-6G having a volume of 105.0 %, 97.0 %, 112.5 %, and 115.3 % of that of GLAC-1D for 22, 20, 18, and 16 ka respectively. However, the extent of ICE-6G is larger with an area of 120.4 %, 118.6 %, 133.4 % and 143.2 % of that of GLAC-1D for 22, 20, 18, and 16 ka respectively. It appears that producing the smaller ICE-6G area-to-volume ratio is challenging for ICESHEET when used with our shear-stress map. This means that, prior to correcting for an estimate of model bias, nearly all ICESHEET6G ensemble members overestimate the volume of ICE-6G margins, whereas the ICESHEET1D distributions commonly encompass the target GLAC-1D volume. Overall, for most regions and times, the reconstruction target ice volume falls within the distributions of modelled volumes, often towards the lower values. There is a significant lack of overlap between the ICE-6G target ice-sheet volume and the reconstructed volume using ICE-6G margins in the British–Irish sector. This model–data discrepancy is accounted for in the prescription of the model bias correction (Eq. ), which reduces the influence of this misfit on the overall implausibility metric. The algorithm used to extract ice-sheet margins from the target reconstructions leads to some differences in extent, such as an overestimation in the Barents Sea extent at 16 ka (Fig. i, j). This is a result of the smoothing procedure applied during margin creation, which can lead to underestimation where there are small thickness protrusions, and overestimation at some concave margin edges.

Before applying our criteria for implausibility, we find that the 200-member ensemble generated for the Last Deglaciation has a mean implausibility of 4±2. After removing members with implausibility of greater than 3, we find that 116 members have been excluded, leaving 42 % of parameter points within the NROY space, and a new mean implausibility of 2.1±0.4. The NROY space favours reduced ice-sheet volumes with all times considered for ICESHEET1D and ICESHEET6G exhibiting a reduction in average total volume compared with the original distributions (Fig. ). In this work, we express ice-sheet volumes in terms of sea-level equivalent (SLE) volume, which we calculate by dividing a given ice-sheet volume by modern ocean area. We find that the largest mean percentage reduction in volume is for the Fennoscandian region of the ICESHEET6G 18 ka margin at -16 %, while the least reduced is the British–Irish region of the ICESHEET1D 16 ka margin at -4 %. The maximum volume across all margins and times is reduced from 29.7 to 21.1 m after history matching, with the minimum increased slightly from 8.7 to 8.9 m SLE (Fig. ).

Prior to applying the bias correction fields, ice-sheet thickness in the interior of the Barents–Kara Sea region is consistently underestimated over all margins and times, potentially due to the lack of modelled dynamics that are important for marine ice sheets, but shows lower variance than other regions (Fig. ). The largest variance occurs in ice-sheet thickness in the centre of the Fennoscandian region. However, thickness in this region appears to be over-estimated in ICE-6G and under-estimated in GLAC-1D simulations. In addition, both target reconstructions position ice-sheet domes slightly towards the marine margins and exhibit thinner continental marginal ice. This likely reflects the larger accumulation of snow closer to the coast and the influence of a rain shadow in reducing accumulation towards the interior. By contrast, since ICESHEET does not see the effect of climate on the ice-sheet geometry, our simulated position of the ice dome is very central, yet this discrepancy is consistent between model and reconstructions and of a similar order of magnitude to the discrepancy between the two target reconstructions. The ice thickness at the margin is systematically greater in our simulations than in both reconstructions. Because of our choice of metric, history matching against regional volume, we prefer ice sheets that are thinner in the interior and thicker at the edges, but a different target metric would rank simulations differently, such as max thickness, which would likely select thinner overall simulations. Regional differences also exist in post-history matching mean model performance after removal of the model bias field. GLAC-1D and ICE-6G remain respectively under- and overestimated, while the primary misfit is now in the centre of the Fennoscandian ice sheet, likely due to the large disagreement between GLAC-1D and ICE-6G in this region.

To better understand the relationship between implausibility and the shear-stress input parameter values, we generate an optical depth image which reveals the shape of the NROY region within our parameter space (Fig. ). This image shows the density of NROY parameter values, and the minimum implausibility, across each of the 21 faces of the 7D parameter hypercube. Each face is associated with a parameter pair and consists of 1600 (40×40) pixels. For a given face, each pixel represents two fixed values for the two parameters associated with the face, and the pixel NROY density and minimum implausibility values are derived from a 1000-member random sample of the five remaining unfixed parameters. Each 1000-member sample is evaluated using the 24 Gaussian process emulators in order to calculate their associated implausibility values, meaning that each face in the 21-face image is the result of 38.4 million emulator evaluations.

Our analysis reveals that there is a slight preference for lower ice-stream and marine shear-stress values and a relatively strong preference for onshore shear-stress values (Fig. ). This is likely due to the smaller ice-sheet geometries that these lower values result in. Bedrock shear-stress values show no clear relationship, indicating insensitivity of our regional ice-sheet volume metrics to this parameter. This may be due to the small relative area covered by bedrock, in contrast to other types of shear-stress categories, resulting in a limited impact on ice-sheet volume.

We see a strong indication that lower values of onshore sediment shear stress (mean of 55.3±19.6 kPa) and higher values of cold-ice interior distance (mean of 786±138 km) are favourable (Fig. ). A large value of the cold-ice interior distance parameter will produce a smaller area of cold-based ice, since this distance is defined from the margin inwards, and thus such runs will produce smaller ice-sheet volumes that have lower implausibility. In addition, lower overall shear-stress values are shown to be more realistic in most cases but we do not see the same relationship with cold-ice shear stress and bedrock, as this model seems insensitive to these parameters. Finally, a preference for higher values of ice-streaming interior distance (mean of 627±244 km), indicates that longer ice streams, and therefore thinner ice, are preferred. We find that these parameter distributions are common throughout the deglaciation, but with a stronger influence of cold-ice interior distance for smaller ice sheets in the later deglaciation stages. We hypothesise that is a result of the ice sheet being in a state of climate disequilibrium in the later stages of the deglaciation, which may have caused a thick yet narrow ice-sheet geometry due to ice melt at the margins while the thick interior is still present. We also observe a larger influence of marine sediment shear stress on models with a greater margin extent because smaller extents have less ice that covers marine sediments.

In order to model the configuration of the PGM Eurasian ice sheet, and to include new parameters controlling hybrid ice streaming, marginal extent, and topographic deformation, we first generate a new 1000-member, uniform LHS sample of the model parameter ranges as detailed in Table . Our initial ensemble iteration of PGM ice-sheet simulations is run using modern-day topography (initially ignoring the topographic deformation parameter), and the 1000-member set of generated shear-stress map and ice-sheet margin inputs, based on work by (Fig. e), as described in Sect. . While the margin extent parameter was initially sampled as uniform, in order to aid in training of the Gaussian process emulator, the following volume estimates are reported from an emulation-derived sample of 105 parameters using a normal range for the margin extent parameter, centred on 0.5 with a standard deviation of 0.1.

Over the full ensemble, this produces an ice sheet with a volume of 45±15 m SLE (Fig. a), which falls below the ≈70 m SLE value by and within range of the 52.5 m SLE value of and the 33.2 m SLE of , within uncertainty. Next, we apply corrections for glacial isostatic adjusted topography to the ensemble, and utilise our Last Deglaciation history matching (Sect. ) to refine our PGM ice-sheet reconstruction.

Previous research has shown the importance of accounting for GIA when simulating ice sheets with the ICESHEET model and therefore we must account for this in our simulations, using the approach outlined in Sect. . We find that, after applying the simple deformation model, scaling the magnitude of deformation by the topography deformation parameter, our mean deformed topography is depressed by a total volume of 4±1×106 km3 compared to modern-day topography which, if this space were filled with ice, would be equivalent to 9±4 m SLE. On average, the region covered by ice is depressed by 0.1±0.2 km compared with modern day, with areas close to the interior of the ice sheet experiencing the highest levels of deformation, with a maximum depression of 1.2±0.2 km (Fig. ). All topography underneath the ice-sheet mass is depressed by applying ELRA but variation in this depression is minimal at the exterior regions of the ice since the model is less sensitive to the smaller changes in ice thickness at the peripheries of the ice sheet.

Deformed topography has a non-negligible impact on the distribution of ice volume in our ensemble with mean volume increasing from 45±15 to 50±16 m SLE (Fig. ). A single iteration of the ELRA topography, combined with the deformation scaling parameter, allows us to account for the first-order effects of GIA, with our experiments finding that subsequent iterations produce ice volume changes of an order of magnitude less than the first.

We are now able to refine our initial 1000-member LHS ensemble of Eurasian PGM ice-sheet geometries by utilising information gained from our previous history matching procedure against the Last Deglaciation. As in Sect. , we compile our best estimate reconstruction of the PGM with quantified uncertainty by excluding members that have an implausibility of greater than 3. The implausibility values for PGM sample members are derived by utilising the 24 Gaussian process emulators trained on each volume metric, as in Sect. , for the seven common parameters. We account for the presence of a bias term in our initial implausibility by subtracting a scalar bias of 1.76 m from all PGM volumes. This bias was calculated as a result of scaling the NROY PGM volume mean by the mean Last Deglaciation percentage bias at 20 ka.

Applying the NROY constraint acts to reduce the mean of our ice-sheet thickness ensemble from 2.0±0.4 to 1.8±0.3 km SLE (Fig. ). Much of this reduction in volume is from favouring ice sheets with a thinner interior (Fig. ). The pre-history matched mean maximum thickness of 4.8±1.0 km, occurring in the interior, decreases to 4.3±0.9 km, but with slightly thicker ice at the southern margin, compared with the maximum ice thickness over North America at the LGM of 3.38 km, and present-day Greenland and Antarctica at 3.14 and 4.01 km respectively . After history matching, we see the highest variation in thickness in the NROY subset is in the central eastern portion of the ice sheet, except for the Barents Sea region where cold-based ice is present through many of the accepted ensemble members (Fig. ). In addition, we find that history matching favours a reduction in the shear stress value for the interior of the ice sheet, but an increase in the Siberian sector, while the exterior shear-stress values remain similar (Fig. ). Our mean PGM regional ice-sheet volume is 24±8 m SLE for the Barents–Kara Sea (27±9 m SLE pre-history matching), 19±6 m SLE for Fennoscandia (21±7 m SLE pre-history matching), and 1.7±0.2 m SLE for the British–Irish region (1.8±0.2 m SLE pre-history matching). We find the 5th and 95th percentile of our NROY ice-sheet volume distribution for the PGM to be 35 and 62 m SLE respectively. Our lower value is comparable to the Eurasian ice volume simulated with dynamic ice-sheet modelling by of 33.2 m SLE, and our peak probability (48 m SLE) close to the reconstruction by (52.5 m SLE) using GIA inversion methods. The dynamic ice-sheet model output which results in a 70 m SLE PGM Eurasian ice sheet by exceeds our maximum. Similarly, the simulation developed by , and subsequently used in the PMIP protocol , is within the 99th percentile, but greater than our 95th percentile (≈64 m SLE).