On the Antarctic and Greenland ice sheets (AIS and GrIS), snow falling at the surface progressively compacts into ice, passing through an intermediary stage called firn. The process of firn densification depends on local conditions, primarily the temperature, the melt rate and the snow accumulation rate, and accurate modelling of densification is key to several applications in glaciology. Firstly, variability in firn densification affects altimetry measurements of ice sheet surface elevation changes. Consequently, uncertainties in modelled densification rates have a direct impact on mass balance estimates, which rely on a correct conversion from measured volume changes to mass changes (Li and Zwally, 2011; McMillan et al., 2016; Shepherd et al., 2019). Errors in the firn-related correction can lead to over- or underestimation of mass changes related to surface processes and also lead to misinterpreting elevation change signals as changes in mass balance and in ice flow dynamics. Secondly, firn models are used to estimate the partitioning of surface meltwater into runoff off the ice sheet, and refreezing within the firn column, which strongly influences mass loss rates (van den Broeke et al., 2016). Model estimates of current and future surface mass balance of the AIS and GrIS are thus dependent on accurate models of firn evolution. And finally, the densification rate determines the firn age at which air bubbles are trapped in the ice matrix. Knowing this age is crucial for precisely linking samples of past atmospheric composition, which are preserved in these bubbles, to paleo-temperature indicators, which come from the water isotopes in the ice (Buizert et al., 2014).

Firn densification has been the subject of numerous modelling studies over the last decades (e.g. Herron and Langway, 1980; Goujon et al., 2003; Helsen et al., 2008; Arthern et al., 2010; Ligtenberg et al., 2011; Simonsen et al., 2013; Morris and Wingham, 2014; Kuipers Munneke et al., 2015). However, there is no consensus on the precise formulation that such models should use. Most models adopt a two-stage densification process with the first stage characterizing faster densification for firn with density less than a critical value and then slower densification in the second stage. The firn model intercomparison of Lundin et al. (2017) demonstrated that, even for idealized simulations, inter-model disagreements are large in both stages. Firn compaction is driven by the pressure exerted by the overlying firn layers. Dry firn densification depends on numerous microphysical mechanisms acting at the scale of individual grains, such as grain-boundary sliding, vapour transport, dislocation creep and lattice diffusion (Maeno and Ebinuma, 1983; Alley, 1987; Wilkinson, 1988). Deriving formulations closely describing the densification of firn at the macroscale as a function of these mechanisms is challenging. Consequently, most models rely on simplified governing formulations that are calibrated to agree with observations. The final model formulations have usually been tuned to data either from the AIS (Helsen et al., 2008; Arthern et al., 2010; Ligtenberg et al., 2011) or from the GrIS (Simonsen et al., 2013; Morris and Wingham, 2014; Kuipers Munneke et al., 2015), consisting of drilled firn cores from which depth–density profiles are measured. However, the calibration of firn densification rates to firn depth–density profiles requires the assumption of a firn layer in steady state. To overcome this limitation, some models have been calibrated against other types of data such as strain rate measurements (Arthern et al., 2010; Morris and Wingham, 2014) or annual layering detected by radar reflection (Simonsen et al., 2013), but such measurements remain scarce and do not extend to firn at great depths below the surface. Ultimately, firn model calibration is an inverse problem that relies on using observational data to infer parameter values.

In this study, we adopt a Bayesian approach in order to address firn model calibration. This provides a rigorous mathematical framework for estimating distributions of the model parameters (Aster et al., 2005; Berliner et al., 2008). Bayesian inversion has been applied in several glaciological studies, and it has been demonstrated that this methodology improves our ability to constrain poorly known factors such as basal topography (Gudmundsson, 2006; Raymond and Gudmundsson, 2009; Brinkerhoff et al., 2016a), basal friction coefficients (Gudmundsson, 2006; Berliner et al., 2008; Raymond and Gudmundsson, 2009), ice viscosity (Berliner et al., 2008) and the role of the subglacial hydrology systems on ice dynamics (Brinkerhoff et al., 2016b). In the Bayesian framework, model parameters are considered as random variables for which we seek an a posteriori probability distribution that captures the probability density over the entire parameter space. This distribution allows us not only to identify the most likely parameter combination, but also to set confidence limits on the range of values in each parameter that is statistically reasonable. This enables us to quantify uncertainty in model results, to challenge the assumptions inherent to the model itself and to assess correlation between different parameters. Calculations rely on Bayes' theorem (see Sect. 2.4 and Eq. 7), but because of the high-dimensional parameter space and the non-linearity of firn models, solutions cannot be computed in closed form. As such, we apply rigorously designed Monte Carlo methods to approximate the target probability distributions efficiently. By exploiting the complementarity between the Bayesian framework and Monte Carlo techniques, we recalibrate three benchmark firn models and improve our understanding of their associated uncertainty.

We present the results of the calibration process after 15 000 algorithm iterations and compare the MAP and original models' performances against the 22 evaluation cores. We also evaluate the uncertainty of the posterior distributions and compare performances between the different MAP models. All the evaluation simulations are performed without climatic and surface density noise in order to make the evaluation fully deterministic.

For HL and even more so for Ar, the posterior distributions for the parameters demonstrate some strong disagreements with the original values (Fig. 3a, b). The 95 % credible intervals for each parameter (Table 1) incorporate 95 % of the marginal probability density in the posterior. Two original parameter values of HL (a, b) and three of Ar (Eg, α, β) lie in the tails of the posterior distributions (Fig. 3a, b) and even outside these intervals in the case of b, Eg, α and β. This indicates that our analysis provides strong evidence against these original values. The strongest disagreements relate to the accumulation exponents of both models (a, b, α, β). In contrast, the original LZ values agree better with the posterior distribution and all lie within the 95 % credible intervals (Table 1 and Fig. 3c). The posterior distributions show some strong correlation between certain pairs of parameters (Fig. 3). Notable examples are the pre-exponential factors and their corresponding activation energy in HL and Ar, for which the posterior correlations are even stronger than in the prior distributions. The complete correlation matrices and a detailed analysis of all posterior correlation features are provided in the Supplement.

We use the original models and the MAP estimates to simulate firn profiles at the evaluation sites and we compare DIP results with the observed values. This is an effective way to assess possible improvements in parameter estimates reached through our method since the evaluation sites were not used in the calibration process. The match between observations and the model is improved for MAPHL (Fig. 4a) and even more for MAPAr (Fig. 4b), with the original Ar strongly underestimating DIP values. These improvements translate into significantly reduced root-mean-squared errors (RMSEs) in modelled values of both DIP15 (-24 % for HL and -45 % for Ar) and DIPpc (-22 % and -61 %) (Table 2).

For LZ, the relative performance of the MAPLZ model for both DIP15 and DIPpc is worse (+2 % and +24 % in RMSE), but differences are of smaller magnitude (Table 2 and Fig. 4c). Parameter values of MAPLZ and the original LZ are closer, which explains more moderate differences in RMSE compared to HL and Ar. Comparing modelled and observed depth–density profiles of evaluation data illustrates the differences in performance visually (e.g. Fig. 5). Profiles of the original models of HL and Ar frequently lie outside the credible intervals of their respective MAP models. In contrast, profiles of MAPLZ and of the original LZ tend to be close together. At the climatic outlier of our evaluation data (DML in Fig. 5), improvements are reached for the three MAP models (Fig. 5g, h, i). This demonstrates benefits of this method even at the limits of the calibration range. However, at a majority of the evaluation sites, the 95 % credible intervals computed for the three models do not include the observed value (Fig. 4). This highlights that the governing equations of the models, which intend to capture densification physics, require improvement and that parameter calibration in itself cannot overcome this shortcoming.

Compared to the original HL, MAPHL reaches improvements in DIP15 for 12 of the 22 evaluation cores and in DIPpc for 5 of the 11 evaluation cores (Fig. 6a). Generally, MAPHL performs better at AIS sites and worse at GrIS sites. An analysis of the improvement of MAPHL as a function of climatic variables (Fig. 6a) shows that the original HL gives better results in a narrow range of Tav: from -30 to -25∘. As such, the better performance at the GrIS evaluation sites of the original HL is likely due to its parameterization being better suited for the particular temperature range corresponding to the conditions of the latter sites. In contrast, MAPHL seems more appropriate for covering a wider range of climatic conditions. For Ar, the original model shows better performance than MAPAr at few evaluation sites (six for DIP15 and two for DIPpc) which are only in AIS and confined to low-accumulation conditions (Fig. 6b). This is counterintuitive given that Arthern et al. (2010) tuned the original Ar to measurements from high-accumulation sites of the AIS. Finally, the original LZ performs better than MAPLZ at most GrIS sites (Fig. 6c), which is unsurprising given that its original calibration was GrIS-specific. Again, this seems related to the original LZ performing significantly better in the same narrow range of temperatures as for HL. In total, MAPLZ performs better for 10 of the 22 DIP15 and 4 of the 11 DIPpc evaluation measurements.

As explained in Sect. 2.3, the original LZ model was developed for GrIS firn only (Li and Zwally, 2011) and later complemented by an AIS-specific model (Li and Zwally, 2015). We compute results at the AIS and GrIS evaluation sites using the Li and Zwally (2015) model for the AIS and the Li and Zwally (2011) model for the GrIS, so that both models are applied to the ice sheet for which they were originally developed. We call this pairing of models LZ dual and evaluate its general performance. The RMSE for DIP15 of LZ dual is slightly larger (+8 %) than that of MAPLZ and significantly larger (+38 %) for DIPpc (Table 2). We note that the higher RMSE values of LZ dual are strongly affected by its densification scheme performing very poorly at the climatic outlier of the evaluation data, with conditions that are outside of the calibration range of Li and Zwally (2015).

We also compare MAP results with the IMAU firn densification model (IMAU-FDM), which has been used frequently in recent mass balance assessments from altimetry (Pritchard et al., 2012; Babonis et al., 2016; McMillan et al., 2016; Shepherd et al., 2019). IMAU-FDM was developed by adding two tuning parameters to both densification stages of Ar. All four extra parameters are different for the AIS (Ligtenberg et al., 2011) and for the GrIS (Kuipers Munneke et al., 2015), thus also resulting in two separate models. For the evaluation data, the performance of IMAU-FDM for DIP15 is slightly better than MAPAr and MAPLZ but worse than MAPHL, and its performance for DIPpc is significantly worse than all three MAP models (Table 2).

To assess the uncertainty captured by the Bayesian posterior distributions, we compute results on the evaluation data with the 500 parameter sets randomly selected from each of the three posterior ensembles. For all three models, the average performance of their random sample is similar to the corresponding MAP performance, with a maximum RMSE change of 6 % (Table 2). This demonstrates a low uncertainty in the optimal parameter combinations identified by calibration. Furthermore, the best-performing 95th percentile of the random selection allows the construction of the uncertainty intervals shown in Figs. 4 and 5. Of the original models, LZ reaches the lowest RMSE values. Of all models, MAPHL performs best in DIP15 and MAPAr in DIPpc (Table 2). MAPLZ performs worse than the other MAP models even when accounting for uncertainty by using the 500-sample random selections (Table 2).

This calibration method is potentially applicable to models of similar complexity in a broad range of research fields. We exploit it here to investigate the parameter space of HL, Ar and LZ and to re-estimate optimal parameter values conditioned on observed calibration data; no further complexity is introduced since the number of empirical parameters remains the same. We treat the accumulation exponents of Ar (α, β) as free parameters, whereas Arthern et al. (2010) decided to fix their values to 1. Analogous to a and b in HL, these exponents capture the mathematical relationship between densification rates and the accumulation rate, used as a proxy for load increase on any specific firn layer. No physical argument favours a linear proportionality between densification and load increase, and any prescribed value for these exponents is a choice of the model designer. Unlike Arthern et al. (2010), Herron and Langway (1980) previously inferred a=1 and b=0.5. Our calibration data show strong evidence against both these pairs of values; all four are in the extreme tails of the posterior distributions (Fig. 3a, b). Our results of stage-1 exponents (a, α) smaller than 1 indicate a weaker increase in densification rates with pressure than assumed in the original versions of Ar and HL. In firn, the load is supported at the contact area between the grains, which increases on average due to grain rearrangement (in stage 1) and grain growth. As such, firn strengthens in time and the actual stress on ice grains increases more slowly than the total load (Anderson and Benson, 1963). Morris and Wingham (2014) incorporated this by including a temperature-history function, causing slower densification of firn previously exposed to higher temperatures. This is consistent with both grain rearrangement and grain growth because these processes are enhanced at higher temperatures (Alley, 1987; Gow et al., 2004). Lower values of the stage-2 exponents (b, β) illustrate the larger strength of high-density firn with larger contact areas between grains. The difference in sensitivities of stage-1 and stage-2 densification to accumulation also holds in the LZ model, as illustrated by the posterior correlation between its free parameters. The correlation coefficient between the accumulation-related parameters of both stages, lz12 and lz22, is significantly positive (0.74, Fig. S5 in the Supplement). High values of lz12 make β0 more sensitive to b˙ (Eq. 6). However, β0 appears in the numerator of the β1 calculation (Eq. 6), and higher values of lz22 thus moderate the sensitivity of stage-2 densification to b˙. As such, positively correlated lz12 and lz22 provide further evidence that stage-1 densification rates are more sensitive to accumulation rates. This example demonstrates how posterior correlations provide insights into model behaviour. The posterior correlations of all three models are further discussed in the Supplement.

In the IMAU model introduced in Sect. 3, tuning parameters have been added to Ar in order to reduce its sensitivity to accumulation rates (Ligtenberg et al., 2011; Kuipers Munneke et al., 2015). The calibration method presented in this study detects and adjusts for this over-sensitivity in Ar without the need for more tuning parameters in the governing densification equations. The sensitivity of stage-1 densification to b˙ can be computed from the derivative of the rate coefficient: 10∂c0∂b˙=ρwk0Argexp⁡-EcRT+EgRTavαb˙α-1. Similarly, the derivative ∂c1∂b˙ is obtained by replacing k0Ar and α with k1Ar and β. Our calibration process strongly favours smaller values of α, β and Eg with respect to the original values (Fig. 3b). We can compare the magnitudes of the derivatives under the original Ar parameterization and under the MAP parameterization. The magnitudes vary for particular combinations of Tav and b˙. Under all the annual mean climatic regimes of our dataset, the MAP parameters result in a decreased sensitivity of both stage-1 and stage-2 densification rates to b˙.

HL, Ar and LZ only use temperature and accumulation rates as input variables. Other models use additional variables hypothesized to affect densification rates. These include the temperature history mentioned above (Morris and Wingham, 2014), firn grain size (Arthern et al., 2010), impurity content (Freitag et al., 2013), and a transition region between stage-1 and stage-2 densification (Morris, 2018). Other models are explicitly based on micro-scale deformation mechanisms (Alley, 1987; Arthern and Wingham, 1998; Arnaud et al., 2000). These efforts undoubtedly contribute to progressing towards physically based models. A potential problem with such approaches is overfitting calibration data by adding parameters to model formulations while detailed firn data remain scarce. As long as more firn data are not available to appropriately constrain the role of each variable in model formulations, we favour the use of parsimonious models relying on few input variables. It is noteworthy that MAPLZ, which relies on eight free parameters, performs worse on the evaluation data than MAPHL and MAPAr with two fewer free parameters. This highlights that gains in model accuracy should rely not only on better calibration of parameters but also on a reconsideration of the governing densification equations. Additionally, firn core data invoke the assumption of a steady-state depth–density profile. As such, parameter calibration poorly captures seasonal climatic effects on densification. Comprehensive datasets of depth–density profiles (Koenig and Montgomery, 2019) are very valuable to model development. Efforts in collecting and publishing strain rate measurements from the field (Hawley and Waddington, 2011; Medley et al., 2015; Morris et al., 2017), and possibly from laboratory experiments (Schleef and Löwe, 2013), can further benefit model calibration and the progress towards more representative equations.

In order to quantify the consequences of our calibration, we investigate two aspects for which firn models are of common use: calculating firn compaction rates and predicting the age of firn at zpc depth, agepc (years). At every site i of our dataset, we compute the 2000–2017 total compaction anomaly, cmpan,i (m) and the agepc,i value with each of the 500 parameter sets randomly drawn from the posterior ensembles of the three different models (HL, Ar, LZ). This allows evaluation of both parameter-related and model-related uncertainty. Total compaction anomaly (cmpan) – calculated as the cumulative anomaly in surface elevation change due only to firn compaction changes during the 2000–2017 period with respect to the climatic reference period – is given by 11cmpan,i=cmptot,i00–17-17cmpref,iyr, where cmptot00–17 (m) is the total firn compaction over 2000–2017, and cmprefyr (m yr-1) is the annual mean compaction over the reference period (see Sect. 2.2). At all sites, we compute the coefficients of variation (CVs) for both cmpan and agepc from the 500 simulations with each model, and we average the CVs across all sites. CV is the ratio of the standard deviation to the mean and provides an effective assessment of relative dispersion of model results. Because low mean values of cmpan can inflate its CV, we consider only half of the sites at which the mean computed cmpan is highest. For all three models, the CV values for both cmpan and agepc lie between 5.5 % and 7.5 % (Table 3). These values give typical uncertainty in firn model output related to uncertain parameter values. Proceeding to the same calculations but using all three models, i.e. an inter-model ensemble of 1500 simulations at each site, gives an overview of the combined parameter- and model-related uncertainty. The CVs are 19.5 % for cmpan and 7.5 % for agepc, demonstrating larger inter-model disagreement on cmpan calculations (Table 3). By using the CV values, we can calculate reasonable uncertainty estimates for cmpan and agepc. For instance, in the dry snow zone of GrIS, simulated compaction anomalies are typically around 20 cm over 2000–2017 and thus come with an uncertainty of the order of ±4 cm. Since pore close-off age here is around 250 years, a reasonable uncertainty range on this value is ±19 years. In contrast, on the drier AIS, pore close-off age is about 1000 years; thus this range increases to ±75 years. Compaction anomalies hover around 0 cm on most of the dry zone of the AIS because it has not experienced the strong recent surface warming of the GrIS. Absolute uncertainty is thus reduced but still critical given the large area of the AIS over which uncertainties are aggregated when mass balance trends are evaluated. The uncertainty ranges calculated from the CV values provide an order of magnitude of errors in firn model outputs that must be accounted for in altimetry-based mass balance assessments and in ice core studies.

We further investigate how using different models and different parameterizations leads to discrepancies in the modelled compaction. We compute monthly values of compaction anomalies over the 2000–2017 period with the original and MAP models of HL, Ar and LZ (Fig. 7). Ar shows the strongest sensitivity to climatic conditions diverging from these of the reference period; compaction responds strongly to the general increases in GrIS in temperature and accumulation rate, especially in late summer. Due to its lower values for α, β and Eg, MAPAr exhibits fewer extreme compaction anomalies than the original Ar and thus less seasonal variability. In sharp contrast to Ar, HL-computed compaction rates remain relatively stable, due to low activation energy values that smooth out the seasonal variability. Firn core observations provide little information and constraints on seasonal patterns of densification. However, it is noteworthy that MAPAr and MAPLZ tend to show comparable short-timescale sensitivities (insets in Fig. 7), despite structural differences in the models' governing equations. This might indicate that these models fare relatively well in capturing seasonal fluctuations of densification rates and their sensitivity to climate shifts.

We have implemented a Bayesian calibration method to estimate optimal parameter combinations applicable to GrIS and AIS firn for three benchmark firn densification models (HL, Ar, LZ). An extensive dataset of 91 firn cores was separated into calibration and independent evaluation data. Two optimized models (MAPHL, MAPAr) showed significant improvement against the evaluation data, while MAPLZ reached results close to, but slightly worse, than its original version and inferior to MAPHL and MAPAr. When compared to other models of greater complexity, the MAP models showed comparable or even improved performances. Furthermore, the Bayesian approach provides a robust way to evaluate the uncertainty related to parameter value choice, which is a major deficiency of current models. By introducing realistic climatic perturbations in the calibration process, the uncertainty intervals obtained account for the effects of an uncertain climatic forcing. However, at most sites where we evaluated, all three models' uncertainty intervals do not cover observed DIP values. As such, although model results can be improved by re-calibration methods, model tuning alone is insufficient to reach exact fidelity of firn densification models. The formulation of models' governing equations impacts the remaining errors with respect to observations, which highlights deficiencies in our understanding of dry firn densification. Developing a well-constrained physically detailed model is challenging given the number of mechanisms affecting densification rates and their dependency on microstructural properties of firn, which are difficult to observe. Our study demonstrates that, despite these observational limitations, thorough calibration methods relying only on climatic variables can substantially improve firn model accuracy, and constrain uncertainties.