the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Bayesian calibration of firn densification models

### Vincent Verjans

### Amber A. Leeson

### Christopher Nemeth

### C. Max Stevens

### Peter Kuipers Munneke

### Brice Noël

### Jan Melchior van Wessem

Firn densification modelling is key to understanding ice sheet mass balance, ice sheet surface elevation change, and the age difference between ice and the air in enclosed air bubbles. This has resulted in the development of many firn models, all relying to a certain degree on parameter calibration against observed data. We present a novel Bayesian calibration method for these parameters and apply it to three existing firn models. Using an extensive dataset of firn cores from Greenland and Antarctica, we reach optimal parameter estimates applicable to both ice sheets. We then use these to simulate firn density and evaluate against independent observations. Our simulations show a significant decrease (24 % and 56 %) in observation–model discrepancy for two models and a smaller increase (15 %) for the third. As opposed to current methods, the Bayesian framework allows for robust uncertainty analysis related to parameter values. Based on our results, we review some inherent model assumptions and demonstrate how firn model choice and uncertainties in parameter values cause spread in key model outputs.

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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.

## 2.1 Firn densification data

In order to calibrate three firn densification models, we use observations
of firn depth–density profiles from 91 firn cores (see Data Availability and
Supplement) located in different climatic conditions on both
the GrIS (27 cores) and the AIS (64 cores) (Fig. 1). Using cores from both
ice sheets is important since we seek parameter sets that are
generally applicable and not location-specific. We only consider dry
densification since meltwater refreezing is poorly represented in firn
models and wet-firn compaction is absent
(Verjans et al., 2019). As such, we select
cores from areas with low mean annual melt (<0.006 m w.e. yr^{−1}) but spanning a broad range of annual average temperatures (−55 to
−20^{∘}) and accumulation rates (0.02 to 1.06 m w.e. yr^{−1}). For
each core, we use the depth-integrated porosity (DIP), also called firn
air content. We calculate DIP until 15 m depth (DIP15, Eq. 1). For
sufficiently deep measurements, we also calculate DIPpc, Eq. (2), taken
below 15 m and until pore close-off depth (*z*_{pc}, where a density of 830 kg m^{−3} is reached). These are the observed quantitative values used for
the calibration:

where *z* (m) increases downwards, *ρ* is the density of firn (kg m^{−3}) and *ρ*_{i} is the density of ice (917 kg m^{−3}). In Eq. (2), we consider porosity only below 15 m to avoid dependency between
DIP15 and DIPpc. We choose to use both DIP15 and DIPpc in order to
account for first- and second-stage densification. One of the cores has only
a single density measurement above 15 m depth, and thus its DIP15 value is
discarded. We note that 48 cores are too shallow to reach *z*_{pc} and so
cores which do reach this depth provide a stronger constraint to the
Bayesian inference method. This is sensible because these deep cores carry
information about both stages of the densification process.

We use DIP as the evaluation metric for the models because of the crucial role of this variable in both surface mass balance modelling and altimetry-based ice sheet mass balance assessments (Ligtenberg et al., 2014). We note that it is commonly used in firn model intercomparison exercises (Lundin et al., 2017; Stevens et al., 2020) and is a quantity of interest for field measurements (Vandecrux et al., 2019). Due to its formulation (Eqs. 1 and 2), DIP represents the mean depth–density profile and thus is robust to the presence of individual errors and outliers in density measurements.

Observed firn density can be prone to measurement uncertainty, which previous studies point out is about 10 %, though it is variable in depth and between measurement techniques employed (Hawley et al., 2008; Conger and McClung, 2009; Proksch et al., 2016). We outline our procedure to account for measurement uncertainty in Sect. 2.4.

We separate the dataset into calibration data (69 cores) and independent
evaluation data (22 cores). The latter are selected semi-randomly; we ensure
that they include a representative ratio of GrIS–AIS cores and that they cover
all climatic conditions, including an outlier of the dataset with high
accumulation and temperature (see Supplement). The resulting
evaluation data have 8 GrIS and 14 AIS cores; 11 of the 22 cores extend to
*z*_{pc}.

## 2.2 Climate model forcing

At the location of each core, we simulate firn densification under climatic
forcing provided by the RACMO2.3p2 regional climate model (RACMO2 hereafter)
at 5.5 km horizontal resolution for the GrIS
(Noël et al., 2019) and 27 km for the AIS
(van Wessem et al., 2018).
Each firn model simulation consists of a spin-up by repeating a reference
climate until reaching a firn column in equilibrium, which is followed by a
transient period until the core-specific date of drilling. The reference
climate is taken as the first 20-year period of RACMO2 forcing data
(1960–1979 and 1979–1998 for the GrIS and AIS respectively). The number of
iterations over the reference period depends on the site-specific
accumulation rate and mass of the firn column (mass from surface down to
*z*_{pc}). We ensure that the entire firn column is refreshed during the
spin-up but fix the minimum and maximum number of iterations to 10 (200 years spin-up) and 50 (1000 years spin-up). We note that at 33 sites, the
core was drilled before the last year of the reference climate and so the
transient period is effectively a partial iteration of the spin-up period.

Results of the calibration would depend on the particular climate model used for forcing. We thus propagate uncertainty in modelled climatic conditions into our calibration of firn model parameters by perturbing the temperature and accumulation rates of RACMO2 with normally distributed random noise. Standard deviations of the random perturbations are based on reported errors of RACMO2 (Noël et al., 2019; van Wessem et al., 2018 – see more details in the Supplement). By introducing these perturbations, uncertainty intervals on our parameter values encompass the range of values that would result from using other model-based or observational climatic input.

In addition to the climatic forcing, another surface boundary condition is
the fresh snow density, *ρ*_{0}. At each site, the *ρ*_{0} value is
taken in agreement with the shallow densities measured in the corresponding
core of the dataset. However, measurements of fresh snow density are highly
variable (e.g. Fausto et al., 2018). We account for uncertainty in this
parameter by adding normally distributed random noise with standard
deviation 25 kg m^{−3} to *ρ*_{0} at every model time step (see
Supplement). We prefer this approach to the use of available
parameterizations of *ρ*_{0}
(Helsen
et al., 2008; Kuipers Munneke et al., 2015) to avoid any error in the fresh
snow parameterization to affect the calibration process.

## 2.3 Firn densification models

We use the Community Firn Model (Stevens et al., 2020) as the
framework of our study because it incorporates the formulations of all three
densification models investigated: HL (Herron and Langway, 1980),
Ar (Arthern et al., 2010) and LZ
(Li and Zwally, 2011). The Robin hypothesis
(Robin, 1958) constitutes the fundamental assumption of HL, Ar
and LZ. It states that any fractional decrease in the firn porosity,
$\frac{{\mathit{\rho}}_{i}-\mathit{\rho}}{{\mathit{\rho}}_{i}}$, is proportional to an increment in
overburden stress. This translates into densification rates depending on a
rate coefficient *c*, assumed different for stage-1 and stage-2
densification.

The formulations of the rate coefficients rely on calibration and thus differ between the three models investigated.

HL

Ar

LZ

with

Here $\dot{b}$ is the accumulation rate (m w.e. yr^{−1}), *T* the
temperature (K), *T*_{av} the annual mean temperature, *R* the gas constant,
*g* gravity and *ρ*_{w} the water density (1000 kg m^{−3}). All
remaining terms are model-specific tuning parameters. For $\dot{b}$, we use
the mean accumulation rate over the lifetime of each specific firn layer
because it better approximates the overburden stress than the annual mean
(Li and Zwally, 2011). HL and Ar use Arrhenius
relationships with activation energies (*E* terms) capturing temperature
sensitivity and exponents characterizing the exponential proportionality of
the rate coefficients to the accumulation rate. Originally,
Herron and Langway (1980) inferred all values from calibration
based on 17 firn cores, from which they inferred the values for the six free
parameters (Table 1) of HL. In contrast, Arthern et al. (2010) fixed the
accumulation exponents in advance ($\mathit{\alpha}=\mathit{\beta}=\mathrm{1}$) and took activation
energies (*E*_{c}, *E*_{g}) from measurements of microscale mechanisms:
Nabarro–Herring creep for *E*_{c} and grain growth for *E*_{g}. Still, they
noted a mismatch with the activation energy fitting their data best. The
${k}_{\mathrm{0}}^{\mathrm{Ar}}$ and ${k}_{\mathrm{1}}^{\mathrm{Ar}}$ parameters were tuned to three measured time
series of strain rates collected in relatively warm and high-accumulation
locations of the AIS. Here, we consider all five *α*, *β*, ${k}_{\mathrm{0}}^{\mathrm{Ar}}$, ${k}_{\mathrm{1}}^{\mathrm{Ar}}$ and *E*_{g} as free parameters (Table 1) but keep *E*_{c}
fixed because of its strong correlation with *E*_{g}; our use of monthly
model time steps and depth–density profiles as calibration data is not
suitable for differentiating effects of $\frac{{E}_{\mathrm{g}}}{R{T}_{\mathrm{av}}}$ and
$\frac{{E}_{\mathrm{c}}}{RT}$. Equation (6) shows that LZ has eight free parameters
(Table 1), all denoted by lz in this paper. In contrast to our approach to
Ar, we do not add additional accumulation rate exponents to $\dot{b}$ in Eq. (6) because the dependence of *c*_{0} and *c*_{1} on $\dot{b}$ also involves
the coefficients lz_{12} and lz_{22} in the definition of *β*_{0}
and *β*_{1}. Li and Zwally (2011) performed their
calibration of Eq. (6) against firn cores only from the GrIS. Later,
Li
and Zwally (2015) developed a densification model calibrated for Antarctic
firn. The latter model uses the same governing equations as LZ for *c*_{0}
and *c*_{1} but different formulations for *β*_{0} and *β*_{1}
(Eq. 6). Since one of the goals of this study is to find a densification
formulation applicable to firn in both the GrIS and AIS, we choose to apply
our calibration method only to the formulations of *β*_{0} and *β*_{1} specified in Li and Zwally (2011) (Eq. 6).
However, in our results' analysis (Sect. 3), we also consider the
performance of the
Li
and Zwally (2015) model on the AIS cores of our dataset.

## 2.4 Bayesian calibration

In our approach, the free parameters of the firn models are identified as
the quantities of interest and we define this parameter set as *θ*.
Hereafter, “original model values” refers to the values originally
attributed by Herron
and Langway (1980), Arthern et al. (2010), and Li and Zwally (2011) to their
respective sets of free parameters *θ*. The calibration process relies
on Bayes' theorem (Eq. 7), which allows the update of a prior probability
distribution *P*(θ) for *θ* based on observed data
*Y*.

We use normal and weakly informative priors centred about the original model
values so that the constraint of the prior on *P*(*θ*|*Y*) is minor (Table 1). As indicated by
Morris and Wingham (2014), in HL and Ar, the values of the Arrhenius
pre-exponential factors (${k}_{\mathrm{0}}^{\ast}$, ${k}_{\mathrm{1}}^{\ast}$, ${k}_{\mathrm{0}}^{\mathrm{Ar}}$ and
${k}_{\mathrm{1}}^{\mathrm{Ar}}$) are correlated with their corresponding activation energies
(*E*_{0}*E*_{1} and *E*_{g}). At a given temperature, a change of the value in
the pre-exponential factor can be compensated for by adjusting the activation
energy to keep the densification rates constant. We express our a priori knowledge
of these correlations in the prior distributions (see Supplement). No other pair of parameters in HL, Ar or LZ are clearly
correlated a priori, but the calibration process captures a posteriori correlations by
confronting the models with data. The data *Y* consist of the observed
DIP15 and DIPpc values of the calibration data. The marginal likelihood,
*P*(Y), is a constant term independent of *θ* and does
not influence the calibration. We use a normal likelihood function *P*(*Y*|*θ*), which quantifies the match of
the modelled DIP values with the observed:

where *X*_{15} and *Y*_{15} are vectors containing all modelled and observed
values for the calibration data of DIP15 respectively, and similarly for
*X*_{pc} and *Y*_{pc}. We use diagonal covariance matrices Σ_{15}
and Σ_{pc} with site-specific variances. The variances determine the
spread allowed for the model outputs compared to the observed values and are
calculated by taking 10 % and 20 % margins around DIP15 and DIPpc
measurements respectively. Allowing for such spread is necessary because
multiple causes may lead to model–observation discrepancy such as firn model
errors, measurement uncertainties and discrepancies induced by the random
perturbations applied to RACMO2 forcing and to *ρ*_{0}. This particular
form of the likelihood function assumes independence between model errors in
DIP15 and in DIPpc, which is ensured by our calculation of DIPpc only
from 15 m depth to *z*_{pc} (Eq. 2). It also assumes normally distributed
model errors with respect to the observed values. Both these aspects were
verified with preliminary assessments, along with our calculations for the
covariance matrices Σ_{15} and Σ_{pc}, as discussed in the
Supplement. The posterior distribution *P*(*θ*|*Y*) gives a probability distribution over
the parameter space of a given model conditioned on the calibration data. In
our case, with weakly informative priors (Table 1), the distribution
*P*(*θ*|*Y*) is essentially governed
by the likelihood function (Eq. 8). We note here that extreme parameter
combinations in the LZ model can lead to negative densification rates. In
such cases, we set the modelled DIP values to 0, which leads to extremely
low values for the likelihood and for the posterior probability of such
parameter sets.

There is no analytical form of *P*(*θ*|*Y*) and we must investigate the parameter space to generate an
ensemble of *θ*_{i} approximating *P*(*θ*|*Y*). Such an investigation is achieved
efficiently using Markov chain Monte Carlo (MCMC) methods. We apply the well-known
random walk Metropolis (RWM) algorithm (Hastings, 1970) and
summarize it in Fig. 2, on which we base the brief following description. A
given model (HL, Ar or LZ) starts with the original model parameter values
and simulates firn profiles at all the calibration sites. Its DIP15 and
DIPpc results are compared with observations, and the general performance
of the model is quantified by the likelihood. From there and with the prior
distributions assumed, the posterior probability is computed following Eq. (7). At this point, the RWM algorithm starts and the state of the chain,
*θ*_{i} (Fig. 2a), is set to the original model values, and its
posterior probability is saved as *P*(*θ*_{i}|*Y*). It should be noted that the *i*
subscript designates the iteration number, which is equal to 0 at this
initial step. The RWM algorithm then proposes a new ${\mathit{\theta}}_{i}^{\ast}$ from a
proposal distribution (Fig. 2b). For the latter, we use the symmetric
multivariate normal (MVN) distribution which is centred about *θ*_{i}.
This implies that the random choice of ${\mathit{\theta}}_{i}^{\ast}$ depends only on
the current state *θ*_{i} and on the proposal covariance in the MVN distribution,
Σ_{prop}, which is discussed below. Using the parameter combination
${\mathit{\theta}}_{i}^{\ast}$, the model simulates profiles at all calibration sites
again (Fig. 2c) and $P\left({\mathit{\theta}}_{i}^{\ast}|Y\right)$ is computed (Fig. 2d). From there, we either accept or reject the
proposed ${\mathit{\theta}}_{i}^{\ast}$ in the ensemble approximating *P*(*θ*|*Y*). By using the previously computed
*P*(*θ*_{i}|*Y*), the probability of
accepting ${\mathit{\theta}}_{i}^{\ast}$ depends on the ratio $P\left({\mathit{\theta}}_{i}^{\ast}|Y\right)/P\left({\mathit{\theta}}_{i}|Y\right)$ (Fig. 2e). The set saved in the
ensemble (Fig. 2g) is ${\mathit{\theta}}_{i}^{\ast}$ if accepted or *θ*_{i} if
${\mathit{\theta}}_{i}^{\ast}$ was rejected. The saved set becomes the updated
current status for the next iteration *θ*_{i+1} (Fig. 2a) with its
associated posterior probability, $P\left({\mathit{\theta}}_{i+\mathrm{1}}|Y\right)$. The algorithm iterates this
process and reaches a final posterior distribution over *θ*. The RWM algorithm
has the property that the chain will ultimately converge to a stationary
distribution that represents the posterior *P*(*θ*|*Y*). Thus, after a sufficiently high number
of iterations of the algorithm, the ensemble of parameter sets is
representative of *P*(*θ*|*Y*). We
verify adequate convergence using a number of tests, which are shown in the
Supplement. The proposal covariance Σ_{prop} must
account for dependence between the different components of *θ*; i.e.
the value of one free parameter can influence the value of another free
parameter for the model to reach a good match with the observed data.
Σ_{prop} can capture this dependence between parameters and, for
optimality, it is updated every given number of iterations (100 in our
study) using Eq. (9) (Rosenthal, 2011):

where Σ_{cov} is the covariance matrix between the free parameters
of the model at this stage of the iterative chain, and *p* is the number of
free parameters.

From the posterior probability distributions, we can infer the maximum a
posteriori (MAP) estimates of each model (MAP_{HL}, MAP_{Ar},
MAP_{LZ}). These are the modes of the multi-dimensional distributions over
the space of free parameters and have been identified as the most likely
sets by the RWM algorithm. The MAP estimates can be compared to the corresponding
original model values of the parameters. The posterior distributions
additionally incorporate the uncertainty in the parameter values. By
performing posterior predictive simulations on the evaluation data, we can
assess this remaining uncertainty (Gelman et al., 2013). More
specifically, we can assume that a large (500) random sample of the ensemble
of accepted *θ* is representative of the posterior distribution. As
such, model results computed with all sets of this sample inform about model
performance accounting for uncertainty. Intuitively, a large spread in
results from 500 random samples would indicate a large range of possible
sets for the free parameters and thus a high uncertainty in parameter
values.

Since there is no analytical form of our posterior distributions, and to facilitate future firn model uncertainty assessments, we can approximate the posterior distributions with MVN distributions whose means and covariances are set to the posterior means and posterior covariance matrices of the calibration. This allows straightforward sampling of random parameter sets instead of relying on posterior samples of the MCMC. We provide information about the normal approximations and assess their validity in the Supplement. Such normal approximations are asymptotically exact and are commonly applied to analytically intractable Bayesian posterior distributions (Gelman et al., 2013).

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 (*E*_{g}, *α*, *β*) lie in the tails of the posterior distributions (Fig. 3a, b) and even
outside these intervals in the case of *b*, *E*_{g}, *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 MAP_{HL} (Fig. 4a) and even more for MAP_{Ar} (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 MAP_{LZ} 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 MAP_{LZ} 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 MAP_{LZ} 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, MAP_{HL} reaches improvements in DIP15 for
12 of the 22 evaluation cores and in DIPpc for 5 of the 11 evaluation
cores (Fig. 6a). Generally, MAP_{HL} performs better at AIS sites and
worse at GrIS sites. An analysis of the improvement of MAP_{HL} as a
function of climatic variables (Fig. 6a) shows that the original HL gives
better results in a narrow range of *T*_{av}: 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, MAP_{HL} seems more appropriate for covering a wider range of
climatic conditions. For Ar, the original model shows better performance
than MAP_{Ar} 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 MAP_{LZ} 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, MAP_{LZ}
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 MAP_{LZ} 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 MAP_{Ar} and
MAP_{LZ} but worse than MAP_{HL}, 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, MAP_{HL} performs best in DIP15
and MAP_{Ar} in DIPpc (Table 2). MAP_{LZ} 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, lz_{12} and lz_{22},
is significantly positive (0.74, Fig. S5 in the Supplement). High values of lz_{12} make
*β*_{0} more sensitive to $\dot{b}$ (Eq. 6). However, *β*_{0}
appears in the numerator of the *β*_{1} calculation (Eq. 6), and
higher values of lz_{22} thus moderate the sensitivity of stage-2
densification to $\dot{b}$. As such, positively correlated lz_{12} and
lz_{22} 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 $\dot{b}$ can be computed from the derivative of the rate coefficient:

Similarly, the derivative $\frac{\partial {c}_{\mathrm{1}}}{\partial \dot{b}}$ is
obtained by replacing ${k}_{\mathrm{0}}^{\mathrm{Ar}}$ and *α* with ${k}_{\mathrm{1}}^{\mathrm{Ar}}$ and
*β*. Our calibration process strongly favours smaller values of *α*, *β* and *E*_{g} 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 *T*_{av} and $\dot{b}$. 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 $\dot{b}$.

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 MAP_{LZ},
which relies on eight free parameters, performs worse on the evaluation data
than MAP_{HL} and MAP_{Ar} 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 *z*_{pc} depth, age_{pc} (years).
At every site *i* of our dataset, we compute the 2000–2017 total compaction
anomaly, cmp_{an, i} (m) and the age_{pc,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
(cmp_{an}) – 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

where ${\mathrm{cmp}}_{\mathrm{tot}}^{\mathrm{00}\text{\u2013}\mathrm{17}}$ (m) is the total firn compaction over 2000–2017,
and ${\mathrm{cmp}}_{\mathrm{ref}}^{\mathrm{yr}}$ (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 cmp_{an} and age_{pc} 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 cmp_{an} can inflate its CV, we consider only half of the sites at
which the mean computed cmp_{an} is highest. For all three models, the
CV values for both cmp_{an} and age_{pc} 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 cmp_{an} and 7.5 % for
age_{pc}, demonstrating larger inter-model disagreement on cmp_{an}
calculations (Table 3). By using the CV values, we can calculate reasonable
uncertainty estimates for cmp_{an} and age_{pc}. 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 *E*_{g}, MAP_{Ar} 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 MAP_{Ar} and MAP_{LZ} 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 (MAP_{HL}, MAP_{Ar}) showed significant improvement
against the evaluation data, while MAP_{LZ} reached results close to, but
slightly worse, than its original version and inferior to MAP_{HL} and
MAP_{Ar}. 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.

In total 41 of the 91 firn cores are from the SUMup dataset (2019 release), which is publicly available from the Arctic Data Center (https://doi.org/10.18739/A26D5PB2S, Koenig and Montgomery, 2019). A total of 41 of the 91 firn cores are from the dataset compiled by Matt Spencer (Spencer et al., 2001), which is available upon request. Five of the 91 firn cores were provided by Joe McConnell and Ellen Mosley-Thompson and are available on request through PKM. Two of the 91 cores are available via the PANGAEA website (https://doi.org/10.1594/PANGAEA.227732, Gerland and Wilhelms, 1999; https://doi.org/10.1594/PANGAEA.615238, Wilhelms, 2007). One of the 91 cores is available via the NOAA website (https://www.ncdc.noaa.gov/paleo-search/study/2427, Mayewski et al., 1995, 2020). One of the 91 cores is available via the USAP website (https://doi.org/10.7265/N5CR5R88, Cole-Dai, 2004). All Antarctic RACMO2.3p2 climate data used are available on request through JMVW. All Greenland RACMO2.3p2 climate data used are available on request through BN, and yearly SMB and components are free to download (https://doi.org/10.1594/PANGAEA.904428, Noël, 2019). The Community Firn Model is available for download on GitHub (https://github.com/UWGlaciology/CommunityFirnModel; https://doi.org/10.5281/zenodo.3585884, Stevens, 2019).

The supplement related to this article is available online at: https://doi.org/10.5194/tc-14-3017-2020-supplement.

VV, AAL and CN conceived this study. VV performed the development of the calibration method, performed the model experiments and led writing of the manuscript. AAL and CN supervised the work. CMS developed the Community Firn Model. PKM provided firn core data. BN and JMvW provided the RACMO2 forcing data. All authors provided comments and suggested edits to the manuscript.

The authors declare that they have no conflict of interest.

We thank Lora Koenig and Lynn Montgomery for making the SUMup dataset of firn cores available and easily accessible (Koenig and Montgomery, 2019). Matt Spencer is also acknowledged for publishing a separate dataset of firn cores (Spencer et al., 2001). We thank Joe McConnell and Ellen Mosley-Thompson, supported by the NSF–NASA PARCA project, for providing additional firn core data (Bales et al., 2001; Banta and McConnell, 2007; McConnell et al., 2000; McConnell, 2002; Mosley-Thompson et al., 2001). We thank Malcolm McMillan for his interest in the study and for providing insight into the subject of ice sheet mass balance assessments. Vincent Verjans thanks Elizabeth Morris for pointing out errors in geographical coordinates of some of the firn cores and for her endless interest in firn densification. We thank all contributors to the development of the Community Firn Model (CFM) who are not authors of this study. All authors thank the two anonymous referees for their time and effort in reviewing the manuscript.

This research has been supported by the Centre for Polar Observation and Modelling, EPSRC (A Data Science for the Natural Environment, grant no. EP/R01860X/1), NESSC (Netherlands Earth System Science Centre), and NWO (Netherlands Organisation for Scientific Research, grant no. VI.Veni.192.019).

This paper was edited by Pippa Whitehouse and reviewed by two anonymous referees.

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