Preliminary assessment of model parametric uncertainty in projections of Greenland Ice Sheet behavior

Introduction Conclusions References

125 000 yr using geological forcings, and subsequently driven into the future using an asymptotically increasing air temperature anomaly curve.All modeled ice sheets lose mass after 2005 AD.After culling the ensemble to include only members that give reasonable ice volumes in 2005 AD, the range of projected sea level rise values in 2100 AD is 30 % or more of the median.Data on past ice sheet behavior can help reduce this uncertainty, but none of our ensemble members produces a reasonable ice volume change during the mid-Holocene, relative to the present.This problem suggests that the model's exponential relation between temperature and precipitation does not hold during the Holocene, or that the central-Greenland temperature forcing curve used to drive the model is not representative of conditions around the ice margin at this time (among other possibilities).Our simulations also lack certain observed physical processes that may tend to enhance the real ice sheet's response.Regardless, this work has implications for other studies that use ice sheet models to project or hindcast the behavior of the Greenland ice sheet.

Introduction
The Greenland Ice Sheet is projected to contribute to sea level change by 2100 AD (Meehl et al., 2007) and beyond, but both the rate of ice mass loss and its eventual magnitude are unknown.The ice sheet contains enough ice to raise mean sea level worldwide by about 7.2 m (Bamber et al., 2001), if totally melted.Satellite measurements suggest that the ice sheet's mass balance is negative and perhaps becoming more so with time (Velicogna, 2009;Alley et al., 2010, and references therein;Zwally et al., 2011).Scaling arguments suggest ∼0.1-0.5 m of mean sea level rise due to Greenland ice loss by 2100 AD (Pfeffer et al., 2008).Pfeffer et al. (2008) argue that the lower end of this range is more plausible than the higher end, because the larger number requires a rapid factor-of-10 increase in ice velocities.However, the difference between the ends of this range is economically important (e.g., Sugiyama et al., 2008), indicating a need for further investigation.
Ice sheet models provide an additional way of assessing future sea level change due to Greenland ice loss (e.g., Huybrechts and de Wolde, 1999;Greve, 2000;Gregory and Huybrechts, 2006;Price et al., 2011).These models typically include simplified treatments of ice flow, basal sliding, snowfall, and surface melting.The ice sheet modeling community has developed advanced treatments of all these processes, plus enhanced basal flow due to surface melting, ice shelf growth, calving, and sub-shelf melt (e.g., Parizek and Alley, 2004;Alley et al., 2008;Pollard and DeConto, 2009;Walker et al., 2009;Bueler and Brown, 2009;Robinson et al., 2010;Price et al., 2011).These new treatments are not implemented in all models at the present time.However, the models show remarkable success in simulating many aspects of ice sheet behavior over millennial time scales and longer (e.g., van Tatenhove et al., 1995van Tatenhove et al., , 1996;;Greve, 1997;Simpson et al., 2009;Pollard and DeConto, 2009).
Remaining challenges in assessing future Greenland Ice Sheet changes include  , 2002;Heimbach et al., 2008;Aschwanden et al., 2009;Stone et al., 2010;Greve et al., 2011).Ice sheet models have many uncertain parameters, and the choice of parameter values has a strong influence on modeled behavior (Stone et al., 2010;Greve et al., 2011).Because the thermal field within the ice sheet is mostly unknown (cf.Greve, 2005), ice sheet models are "spun up" to the present using reconstructed former surface temperatures and sea levels.Achieving a good match between the modeled and observed ice thickness distributions at the end of this spinup is challenging (Aschwanden et al., 2009;Greve et al., 2011).In general, simulated ice volumes at the ends of spin-up runs are larger than expected (e.g., Heimbach et al., 2008;Stone et al., 2010;Robinson et al., 2010;Vizcaino et al., 2010;Greve et al., 2011;cf. Bamber et al., 2001).Finally, data on past ice sheet variations (e.g., Alley et al., 2010, and references therein) provide a check on ice sheet models: if a model reproduces past changes well, then we can have more confidence in its projections of future changes (cf.Oreskes et al., 1994).Perturbed physics tuning exercises may help address these challenges.In a perturbed physics ensemble, the model is run many times with different parameter combinations to identify a group of runs that provide a reasonable fit to observations, usually the modern geometry of the ice sheet (e.g., Ritz et al., 1997;Stone et al., 2010;Greve et al., 2011; for fits to paleo-data, see Tarasov and Peltier, 2003;Lhomme et al., 2005;Simpson et al., 2009).These "good" ensemble members are likely more reliable estimators of future behavior than the ensemble as a whole (cf.Weigel et al., 2010).This approach is well established in climate modeling (e.g., climateprediction.net;Stainforth et al., 2005;Piani et al., 2005), and a small but growing number of ice sheet modeling studies use ensemble methods (e.g., Tarasov and Peltier, 2004;Napieralski et al., 2007;Hebeler et al., 2008;Stone et al., 2010).
In this paper, we present results from a small perturbed-physics ensemble with the ice sheet model SICOPOLIS (Greve, 1997;Greve et al., 2011;sicopolis.greveweb.net).Our approach builds on existing work by Stone et al. (2010)  modeling community (seaRISE partners, 2008;Greve et al., 2011).The results indicate that our present uncertainty about the best values of model parameters translates to a large spread among model-based projections of future Greenland Ice Sheet behavior.
The paper proceeds as follows.We describe the ice sheet model, ensemble design, climate forcing time series, and ensemble culling in Sect.2, Methods.Section 3, Results, describes similarities and differences among the ensemble members during different parts of the model spinup period, then discusses the effects of ensemble culling on the range of model-projected future sea level increases from the Greenland ice sheet.Section 4, Discussion, treats the success of our ensemble in addressing the three challenges in ice sheet modeling identified above.Section 5, Conclusions, emphasizes the main outcomes of the study and provides some caveats that should be borne in mind when interpreting our model output.

Methods
Briefly, we applied the Latin hypercube ensemble methods of Stone et al. (2010) to the SICOPOLIS ice sheet model, as set up by Greve et al. (2011).The Latin hypercube method provides a quasi-random sampling of parameter space that is more even than that produced by Monte Carlo methods (Bevington and Robinson, 2003;Saltelli et al., 2008) and avoids wasting model evaluations on uninfluential parameters, as can happen with a grid design (Urban and Fricker, 2009).We can thus make a reasonable exploration of parameter space with a relatively small number of model evaluations.

Model description
As noted above, we carried out our simulations with the SICOPOLIS ice sheet model.SICOPOLIS has been previously described by Greve (1997) and Greve et al. (2011), and we refer interested readers to those papers for more information.The model is broadly comparable to most other large-scale ice sheet models, such as Glimmer (Rutt et al., 2009).3179 The model setup that we use is specifically intended for the problem of projecting future sea level change (Greve et al., 2011).It includes a horizontal resolution of 10 km, with 81 grid points in the vertical direction.These points are concentrated near the base, where the bulk of ice deformation occurs.The model time step is 1 yr.
Use of the SICOPOLIS model allows us to incorporate many thousands of years of geological data into model spinup (Sect. 2.3,below).As with most Greenland ice sheet models, SICOPOLIS calculates stresses within the ice body using the shallowice approximation (e.g., Hutter, 1983;Greve and Blatter, 2009).This approximation is reasonable over the bulk of the ice sheet (perhaps 70-80 % by area, and a greater percentage by volume), but fails in areas where the real ice sheet exhibits fast flow, such as in ice streams (see Joughin et al., 2010, for surface velocity maps).Higher-order models provide improved representations of ice flow where the shallow-ice approximation fails (Pattyn et al., 2008;Hindmarsh, 2009), but typically require more computing time than shallow-ice models.To our knowledge, higher-order models have not yet been used to carry out long integrations like those we undertake here with SICOPOLIS.

Ensemble design
We vary five model parameters among 100 ensemble members (Fig. 1; Supplementary Material).The number of model runs was chosen to achieve a reasonable tradeoff between covering parameter space and minimizing computation time.For comparison, our total number of evaluated model time steps (12.65 × 10 6 ) is slightly larger than that of Stone et al. (2010), who performed a larger number of shorter model runs.
The free parameters (and their ranges) include the ice flow enhancement factor (1-5, dimensionless), ice and snow positive degree-day factors ("PDD factors"; 5-20 mm day −1 • C and 1-5 mm day −1 • C, respectively); the geothermal heat flux (30-70 mW m −2 ), and the basal sliding factor (0-20 m yr −1 /Pa).The ice flow enhancement factor corrects for differences between the rheology of ice as it is measured in the laboratory and that observed on an ice sheet scale; these differences are likely due to impurities and anisotropic fabric within the ice (Greve, 1997, his Eqs. 3 and 4; see also  et al., 2009, their Eq. 9).The positive degree-day factors describe a statistical relationship between surface temperatures and the rate of surface lowering (Braithwaite, 1995;Calov and Greve, 2005).The geothermal heat flux varies over the Earth's surface, but is difficult to measure under the ice sheet (see discussion in Stone et al., 2010); thus, it is often taken to be constant for purposes of ice sheet modeling (e.g., Ritz, 1997).Finally, the basal sliding factor determines how rapidly the ice slides over its bed where the interface is not frozen (Greve and Otsu, 2007, their Eq. 2).

Rutt
For the first four parameters, the ranges are roughly the same as those investigated by Stone et al. (2010), who based their ranges on data-based studies from the literature (e.g., Dahl-Jensen and Gundestrup, 1987;Braithwaite, 1995).We expanded the ranges for the positive degree-day factors so that the EISMINT-3 preferred values (Huybrechts et al., 1998) lie well within the investigated range, instead of at one end, as in Stone et al. (2010).We also expanded the range of the geothermal heat flux parameter; Stone et al. (2010) found that this parameter was relatively uninfluential, and we hypothesized that a larger range might show an effect.The range we investigate is still within previous estimates (Greve, 2005;Buchardt and Dahl-Jensen, 2007;Stone, 2010).The basal sliding parameter ranges from 0 to about double the best value identified by Greve and Otsu (2007).
This list of free parameters is somewhat different from that used by Stone et al. ( 2010), but consistent with Ritz et al. (1997), in that we fix the atmospheric temperature lapse rates (Fausto et al., 2009) and include the basal sliding factor as a free parameter.In effect, we take the surface temperature and precipitation as given, even though these data sets contribute additional uncertainty to projected ice volumes (van der Veen, 2002;Stone et al., 2010).

Initial condition and climate forcing time series
All runs were driven by the same surface temperature, precipitation, and sea level forcings.The paleoclimate spinup (Fig. 2) closely resembles that of Greve et al. (2011).We began with the observed modern ice thickness and bedrock elevation grid (Bamber 3181 Discussion the Eemian (Alley et al., 2010, and references therein).However, the errors in the initial condition should average out over the spinup period (see discussion in Rogozhina et al., 2011).
From 125 ka onward, we drove the model using a temperature anomaly curve based on the GRIP oxygen isotope record (Dansgaard et al., 1993;Johnsen et al., 1997) and background sea levels from the SPECMAP compilation of oxygen isotope measurements in deep-sea sediment cores (Imbrie et al., 1984).Precipitation changes by ∼7 % for each degree of temperature change relative to the present (Greve et al., 2011, their Eq. 6;cf. van der Veen, 2002).The transfer functions for converting oxygen isotope measurements to surface temperatures and sea levels are given in Greve et al. (2011).Modern-day surface temperatures and precipitation values are from Fausto et al. (2009) and Ettema et al. (2009); these patterns are scaled in the model according to the calculated temperature and precipitation anomalies.
Near the end of the paleoclimate spinup, we substituted an instrumental record of southwestern Greenland mean annual temperature anomalies (Vinther et al., 2006) for the GRIP-based temperatures (Fig. 2).The Vinther et al. (2006) compilation covers the years 1784-2005 AD; we chose to begin the instrumental period in 1840, when the number of missing temperature records per year becomes noticeably smaller.Use of the Vinther et al. (2006) temperatures helps us to capture the interannual variability that could be important in explaining modern mass balance trends (Alley et al., 2010;Zwally et al., 2011).
After 2005, the surface temperature anomaly increases according to In this expression, ∆T is the final temperature anomaly (6 this relation and the time scale come from Greve (2000).The final temperature rise is reasonable, given that Greenland is expected to warm ∼1.5-2 times as much as the global average (Church et al., 2001).We used this temperature forcing curve instead of one produced by a climate model because our goal is to highlight parametric uncertainties within the ice sheet model; differences among surface temperature change projections (Stainforth et al., 2005;Meehl et al., 2007) represent another layer of uncertainty.
We hold the sea level anomaly constant between 1840 AD and the end of the simulations, consistent with some earlier simulations of future Greenland evolution (Huybrechts, 1998).In effect, this assumption is conservative; allowing sea level to rise over the future would tend to increase mass loss from the ice sheet.However, the real ice sheet is likely much less sensitive to changes in sea level than temperature, even considering the large-amplitude sea level changes that take place over glacial-interglacial cycles (Alley et al., 2010; see also Gomez et al., 2010).In any case, the science of sea level rise is evolving rapidly, so we chose to neglect this effect for the present study.

Culling the ensemble
We evaluate the trustworthiness of each run by comparing the simulated total ice volume in 2005 AD to the modern ice volume (∼7.2 m sea level equivalent; Bamber et al., 2001), which is somewhat uncertain.The modern ice volume was estimated by Bamber et al. ( 2001) from kriging of geographically distributed ice thickness measurements, made from airborne radar units.Adjacent measurements do not always agree exactly, reflecting some aggregate of bed roughness and measurement uncertainty.Moreover, observation density varies over the ice sheet; flight lines lie close to one another near airfields and become sparse farther south.Thus, the true modern ice volume could be either larger or smaller than the central estimate, but how wide the range of possible values might be is difficult to estimate.
To account for uncertainty in the modern ice volume, we used a simple windowing approach.All runs that fell within a certain distance of the modern ice volume AD (dashed line in Fig. 3) were kept, whereas the others were discarded.We investigated window widths of ±20 %, 10 %, 5 %, and 2.5 % of the modern ice volume.These estimates of uncertainty in modern ice volume are somewhat ad hoc, but as we show later, they have little influence on our projection uncertainty.Total ice volume is a reasonable comparison metric because we are interested in future sea level change, and because inferences about the past state of the ice sheet are usually stated in terms of volume changes relative to the present (e.g., Alley et al., 2010, their Fig. 13).Other metrics for comparing simulated ice sheets to the observed one exist, including the ice-covered area, maximum ice thickness (Ritz et al., 1997;Stone et al., 2010), the root mean squared ice thickness (e.g., Greve and Otsu, 2007), and matching the distribution of ice surface velocities (e.g., Aschwanden et al., 2009).
Using an aggregate measure such as total ice volume helps avoid nontrivial statistical issues with autocorrelation, in which adjacent residuals between observations and model predictions are not independent of one another (e.g., Bloomfield and Nychka, 1992).
As noted above, data on the ice sheet's past behavior (Alley et al., 2010, and references therein) also provide constraints on model behavior.Consistent with most earlier Greenland ice sheet modeling studies, we neglect this information in our model tuning; instead, we use it to evaluate the reasonableness of the culled ensemble (Sect.4.1, below).

Results
In this section, we make some general observations on the behavior of the full ensemble (Fig. 3, gray and blue curves) before treating those runs that reproduce the modern ice volume well (Fig. 3, blue curves).Because the initial condition and forcings are identical for all model runs (Fig. 2), variability among runs (Figs. 3, 4) is due solely to parameter choice (Fig. 1).As noted above, all simulations start from the modern ice geometry, with an ice volume of 7.2 m sea level equivalent (Fig. 3).Temperatures begin a few degrees below modern values (Fig. 2), before rising twice to much warmer values at about −120 and −115 ka.

Discussion
Although apparently reasonable for the Eemian, the temperature values and structure of the GRIP record during this time is suspect (Chappellaz et al., 1997).From this maximum, temperature and sea level generally decline.Ice volumes increase over the same period, stabilizing sometime before −75 ka (Fig. 3).Despite the general resemblance of the model curves to one another (Fig. 3), there is substantial divergence among ensemble members.The spread among model runs is most noticeable during the Eemian warmth; some model realizations produce a nearly ice-free Greenland, whereas in others the ice volume changes only slightly.

Early glacial period through the early Holocene (−75 ka to −10 ka)
Between −75 and −10 ka, simulated ice volumes are remarkably stable and consistent among runs (Fig. 3).This result seems counterintuitive given the large temperature fluctuations seen in the GRIP record (up to ∼15 • C; Fig. 2).However, these fluctuations are only a few ka long, and the resulting ice losses due to mass balance changes are small compared to simulated ice volumes.

Early Holocene to the beginning of the preindustrial period
(−10 ka to −0.16 ka, or 1840 AD) Despite relatively stable Holocene temperatures (Fig. 2), simulated ice volumes generally decrease between −10 ka and −0.16 ka (Fig. 3).Many runs show a very slight growth near the end of this period, reflecting the "Little Ice Age" as seen in the central Greenland ice cores (Alley et al., 2010, and references therein).As for the Eemian, the Holocene warmth produces considerable spread among the individual model runs.One run even grows for a few ka before slowly shrinking.This realization (#95, Supplementary Material) has exceptionally low values of both the ice and snow positive degree-day factors, which allow increased precipitation (Sect.2.3; Greve et al., 2011) to overtake ablation temporarily.

1840 AD to 3500 AD
The Vinther et al. (2006) instrumental temperatures begin the climb out of the "Little Ice Age", and the assumed future climate trajectory builds off the end of this record.Ice volumes decline very slightly between 1840 AD and 2100 AD, with larger decreases thereafter.As during the Eemian and earlier in the Holocene, the spread among model realizations is large.

The culled ensemble
Culling the ensemble reduces the divergence among model runs during warm periods (Figs. 3, 4).The 26 ensemble members that lie within 10 % of the estimated modern ice volume in 2005 AD change by comparable amounts between warm and cold periods.Curiously, the spread among these runs during cold periods is almost as large as that of the full ensemble.Depending on the culling window width (Sect.2.2, above), we have different numbers of ensemble members remaining (Fig. 4, bottom).An assumed 20 % uncertainty in modern estimated ice volume leaves 52 ensemble members out a possible 100.For 10 %, 5 %, and 2.5 % uncertainty in modern ice volume, the ensemble culling leaves 26, 11, and 6 model runs, respectively.

Future sea level change in the culled ensemble
The spread in projected future sea level change among model runs is large, even after ensemble culling (Fig. 4).For 2100 AD and an assumed 10 % uncertainty in Greenland mass loss is ∼0.14 m, and the range among the 26 model runs that meet the modern ice volume criterion is ∼0.1 m, for a fractional uncertainty of ∼70 %.Much of this uncertainty persists even if we use stricter culling criteria -for example, assuming that we know the modern ice volume to within 2.5 % gives a fractional uncertainty of 30 % in 2100 AD.

Discussion
Our results highlight the challenge presented by parametric uncertainty for projections of future Greenland ice sheet behavior, building on previous work in this area (e.g., van der Veen, 2002;Stone et al., 2010).In the Introduction, we identified three additional modeling problems that we hoped to address with our perturbed-physics ensemble.
These were (1) characterizing model response to parameter choice, (2) establishing an initial state for prognostic simulations, and (3) matching data on the ice sheet's past behavior.We address the ensemble's success in meeting these goals here.

Parameter choice, simulated modern ice volume, and ice sheet sensitivity
Given these model results (Figs. 3,4), we might ask which parameter values are most consistent with the modern observed ice volume.If there are one or more parameter combinations that match the modern condition well, these values can be used in other modeling experiments.
This question was previously posed by Stone et al. (2010), who noted that high values of the ice flow and ice positive degree day factors yielded the best matches with observed total ice volume; the other parameters played smaller roles.Our results are consistent with Stone et al. (2010) in that the ice PDD factor has a dominant influence on simulated ice volumes at the end of the spinup (Fig. 5; cf. ).However, we find that treating the basal sliding factor as a free parameter reduces the influence of the ice flow factor.
We can make few general statements about optimal parameter combinations for ice sheet spinup (Fig. 5).None of our "best" runs, those that fall within 10 % of the modern ice volume in 2005 AD, have an ice positive degree-day factor greater than ∼15 mm day −1 • C.This value is well within the range identified by Braithwaite (1995).Otherwise, the "best" runs span the entire free range of the four remaining parameters, indicating that any value of these parameters is potentially consistent with the modern ice volume.There is some suggestion that large basal sliding factors are most compatible with smaller values of the ice flow factor and vice versa (Fig. 1), but this apparent tradeoff might disappear given more model runs.
In our ensemble, the ice positive degree-day factor largely determines the near-term future ice sheet response (Fig. 6).As for modern ice volumes (Fig. 5), there is a strong relation between the ice PDD factor and 2005-2100 AD ice volume change, but the other parameters appear to be relatively unimportant (Fig. 6).

Simulated modern ice thicknesses
By construction, our culled ensemble matches the modern ice volume.However, problems persist in the modeled ice thicknesses (Fig. 7).Consistent with earlier results using a similar model setup (Greve et al., 2011, their Fig. 2), the ice is generally too extensive in the south and has large gaps in the north.Our tuning exercise did cover part of the falsely ice-free area in northern Greenland noted by Greve et al. (2011).Ice thickness errors of 1.5 km or more are present around the edges of the ice sheet; in particular, the ice is too thin upflow from Jakobshavn, about a third of the way northward on the west coast.It is possible that an appropriately-tuned higher-order model would produce a better fit to the observed ice thickness grid than we have yet achieved with SICOPOLIS.However, the large-scale shape of the ice sheet is more strongly controlled by surface mass balance than ice flow; for thin ice, changes in ice thickness due to flow are small compared to those caused by surface mass imbalances (Greve, 1997).We attribute the bulk of the remaining errors in geographically distributed ice thickness values to problems with the modeled mass balance.

Comparison of modeled results to assessed past volume changes
As noted in the Introduction, the ice sheet's past behavior provides a check on ice sheet model results.Alley et al. (2010) recently surveyed the literature and gave assessments of ice volume changes, relative to the present, for three time slices covered by our runs, the Eemian, Last Glacial Maximum, and mid-Holocene.These assessments are based on a combination of isostatic rebound (Peltier, 2004;Fleming and Lambeck, 2004) and data-constrained ice sheet modeling (Cuffey and Marshall, 2000;Lhomme et al., 2005) studies.Because some of these studies use ice sheet models to translate observed ice core oxygen isotope values into ice sheet changes, these assessed ice volume changes are not truly independent of our results.However, they do provide a firstorder check on our model output.
The culled ensemble simulates assessed ice volume changes well during the Eemian (−115 ka; Fig. 4), but has problems during the mid-Holocene and Last Glacial Maximum (−5 ka and −20 ka, respectively).In particular, none of the model runs produce a smaller-than-modern ice sheet during the mid-Holocene (−5 ka), as expected from paleo-data.The overlap between our "good" model runs and the estimated Last Glacial Maximum (−20 ka) ice volume change is also minimal, but this discrepancy could be reduced by a different weighting function for evaluating the model runs.Moreover, the agreement between our simulated late-Eemian ice volume changes and previously estimated values may be fortuitous; the peak in the GRIP temperature record at −115 ka could be due to flow disturbances in the ice core (Chappellaz et al., 1997;Cuffey and Marshall, 2000).
The apparent disagreement between simulated and assessed ice volume changes in the mid-Holocene (Fig. 4) could be due to the temperature forcing curve used to drive the model, or to the assumed-constant exponential relation between surfacetemperature anomaly and precipitation (among other possibilities).Air temperatures over some parts of the Greenland Ice Sheet were likely warmer during the mid-Holocene than geographic scaling of the GRIP oxygen isotope record would indicate (Y.Axford, personal communication, 2011;Dahl-Jensen et al., 1998;Young et al., 2011; see also Vinther et al., 2009); if accounted for in a model simulation, these warmer regional temperatures might bring modeled ice volumes closer to the estimated values.Further, our model runs assume a constant ∼7 % increase in precipitation per degree Celsius temperature increase (Greve et al., 2011).However, this relation has long been controversial (see discussion in van der Veen, 2002), and it may be especially poor for the mid-to late-Holocene (Cuffey and Clow, 1997).Further research is needed to reduce the discrepancies between model-and data-based reconstructions of past Greenland Ice Sheet configurations.

Conclusions
Our results (Fig. 4) suggest that parametric uncertainty in ice sheet model-based projections of Greenland Ice Sheet behavior is on the order of 30-70 %, expressed as the range of plausible model outcomes divided by the median.This outcome is not sensitive to the parameter ranges we investigated, but rather depends on the uncertainty in modern ice volumes.Our results do not provide a probabilistic assessment of future ice sheet changes; we make no statement about the relative plausibility of different model runs within our culled ensemble.
Our analysis neglects several sources of uncertainty that will tend to increase the ranges shown in Figure 4.In particular, we assume that the climate and subglacial topography boundary conditions are well known.Estimating the effects of errors in these data sets on modeled ice volumes is complex, but Stone et al. (2010) found that updating older, EISMINT-3 data sets (Huybrechts, 1998)  best estimates for most model parameters (Huybrechts, 1998;Stone et al., 2010), except the basal sliding factor, which comes from Greve and Otsu (2007)  oxygen isotopes in ocean sediment cores (Imbrie et al., 1984) and in central-Greenland ice cores (Dansgaard et al., 1993;Johnsen et al., 1997).1840-2005 AD temperatures come from southwestern Greenland observations (Vinther et al., 2006).Future temperatures assume an asymptotic increase to ∼5 degrees C above 1976-2005 levels, with a time scale of 100 yr (Greve, 2000; see text).Background sea level is held constant from 1840-3500 AD.Labeled Y -axis scaling is the same for all panels in the top row, but differs among panels in the bottom row.Color coding is the same as in Fig. 2; gray, all model runs; dark blue, model runs that lie within 10 % of the estimated modern ice volume in 2005 AD; red line, "paleoclimate spinup with additional tuning" model run from Greve et al. (2011).The green points with error bars in the top panels indicate assessed changes in the ice sheet, relative to the modern, from Alley et al. (2010, their Fig. 13).None of our model runs produce a smaller-than-today ice volume during the mid-Holocene (−5 ka).For the future time slices (2100, 2300, and 3000 AD), the range of potential future ice volume changes is always at least 30 % of the median, regardless of how strictly the ensemble is culled.(Huybrechts, 1998;Stone et al., 2010), except the basal sliding factor, which comes from Greve and Otsu (2007).Influential parameters are indicated by points that are tightly arranged about a curve that dips steeply from one side of the plot to the other (Saltelli et al., 2008).In our ensemble, the ice positive degree-day factor has the greatest influence on simulated modern ice volume, with the snow PDD factor and the basal sliding factor taking second and third places.However, ice PDD factors greater than ∼15 mm day −1 • C appear inconsistent with the modern ice volume constraint.3).Color coding is the same as in Fig. 5.The ice positive degree-day factor appears to dominate the short-term future behavior of the model.

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Fig. 6 .
Figure 6 . Blue crosses indicate parameter combinations that are consistent with the modern ice volume after model spinup (within 10 % of the estimated modern ice volume in 2005 AD; Sect.2.2).