A simple model that removes the abrupt transition between Stage 1 and Stage 2 densification can be constructed using an activation function for
c centered around transition density
, which is likely to be close to 550 kg m
(activation functions are used in artificial neural networks and are not connected with the activation energy in an Arrhenius equation). Let
where
/M
is a scaled density variable describing the distance away from the transition. Away from the transition, A
>> 1 and
(D −A
) or (D + A
). D and A can be chosen, so that (D − A
)= −
and (D + A
)= −
, so that
c tends toward its Stage 1 and Stage 2 values. Parameter M controls the abruptness of the transition. This particular activation function was chosen because it leads to an analytic solution for the density profile. Other functions that produce a smooth transition from one value to another exist and produce much of the same effect. Note that the assumption is that
<
.
Equation (
20) allows us to calculate the bubble close-off (BCO) depth, important for the determination of past climate from ice-core chemistry. The transition model also leads to an analytic solution for the depth-integrated porosity (DIP), important for the determination of ice-mass balance from altimetry data.
where
is the surface value of
X. The solution for the integral in this equation is given in
Appendix B. The BCO depth and DIP are key indicators used in the FirnMICE intercomparison of densification models [
1], and are thus useful quantities for demonstrating the effect of using the transition model (
Section 3).
2.2. Strain-Rate Profiles
At most, iSTAR and EGIG line sites neutron probe profiles were obtained at around 13 m depth. At this depth at sites along the EGIG line, density is still below 550 kg m
, so the data can only be used to determine Stage 1 densification rates, and hence the combination of parameters (D − A
). Even for iSTAR sites, with relatively high surface density, the transition to Stage 2 is not necessarily completed at 13 m depth, so the strain-rate data cannot be used to derive the parameters A and D separately a priori. However, if Stages 1 and 2 densification rates from an existing model are accepted, then at least some of the neutron-probe data can be used to derive suitable values for M and
. For simplicity, the Herron and Langway expressions for
and
are used (Equations (
9) and (
10)) to calculate A and D.
As an example of the data available from the iSTAR traverse,
Figure 3 shows density and volumetric strain rate
as a function of
q for iSTAR Site 21 where
= 0.75 m w.e. a
and
= −22.3
C. The variations in density about the fitted curve show the annual layering. The transition density of ≈550 kg m
is reached around 6 m below the surface at
−3.5 m w.e., suggesting that the transition between Stage 1 and Stage 2 densification should be observable at this site. The volumetric strain rate decreases in magnitude with depth, with enhanced values near the surface where summer warming increases the densification rate. The other iSTAR sites showed a similar pattern of behavior.
Figure 4 shows the average density-corrected volumetric strain rate
F measured at iSTAR Site 21 as a function of density and the density-corrected volumetric strain rate
c predicted by the transition model using best-fit values M = 2.8 and
= 590 kg m
. The curve for
c tends to the values expected from the Herron and Langway model away from the transition density. The values of
F are very noisy but it is possible to see that the Herron and Langway densification rates for Stages 1 and 2 are reasonable, provided that a transition region is allowed.
Best-fit values for all sites, taken individually, show wide variation in M, but the values of
cluster around 580 kg m
. Fixing this parameter and optimizing for M produces 22 values with a mean of 17 ± 8. Excluding three values more than three standard deviations from this mean reduces it to 7 ± 2.
Figure 5 shows the effect of using M = 7 and
= 580 kg m
at Site 21.
2.3. Density Profiles
Proper calibration of the transition model requires more extensive strain-rate data, ideally from a site with low surface density, so that there is a clear Stage 1 region unaffected by the surface temperature. In order to observe the full transition to Stage 2, data needed to be collected from depths of the order of 20 m. This poses a methodological challenge for neutron-probe measurements. Nevertheless, it was still interesting to investigate the performance of the model as it stands. With the assumption of a constant accumulation rate, this can be done using profiles of ln( with depth, provided that they are sufficiently detailed.
Figure 6 shows that the transition model does indeed produce a better fit to the neutron probe data at Site 21 than the Herron and Langway model, and that the gradient of a straight line fitted to all the data below the Herron and Langway transition point is greater than the true Stage 2 value to which it tends. Morris et al. [
8] defined transitional densification rates,
, for densification below
= 550 kg m
, but using the new transitional model makes these unnecessary. As an aside, we note that in the upper 4 m of the profile the densification rate was enhanced by summer warming. Both models would be able to simulate this if snow temperature
T rather than the mean annual temperature
were used to calculate
in Equation (
9).
In order to quantify the improvement in fit, we define a cost function
where values of
z are calculated at intervals of 5 kg m
from
= 500 kg m
. For the neutron probe density profiles,
, the sum goes to
= 600 kg m
and
N = 20. The observed values are taken from the polynomial fit
. Thus, the cost function describes the goodness of fit of a given model to the smoothed density data in the deeper part of the profile, where the effect of surface-temperature variations is not significant.
Table 1 shows the cost functions calculated using neutron-probe data from 22 iSTAR sites with mean annual accumulation ranging between 0.23 and 0.80 m w.e. a
.
At all sites, the addition of the transition model improves the performance of the Herron and Langway model, i.e., reduces the cost function. At all sites except Site 15, which had the highest accumulation, the Ligtenberg model was also an improvement on the Herron and Langway model. However, it is also the case that at all sites except Sites 10 and 11, which had the lowest accumulation, the transition model performed better than the Ligtenberg model.
At some iSTAR sites, ice cores were collected and gravimetric density measurements made to a depth of 50 m. These may be used to extend the neutron-probe data and test the model over a wider range of density.
Figure 7 shows a better fit to the neutron probe and core data at Site 4, again obtained by using the transition model rather than the Herron and Langway alone. At this site,
= 0.58 m w.e. a
and
= −23.6
C.
Table 2 shows the cost functions calculated for
= 500–800 kg m
using a polynomial fit to the core densities to give observed values of
z. In this case,
N = 60. As before, at all sites, the addition of the transition model improved the performance of the Herron and Langway model, i.e., reduced the cost function. At all sites, including Site 15 this time, the Ligtenberg model was also an improvement on the Herron and Langway model. However, it was again the case that at all sites except Site 10, the transition model performed better than the Ligtenberg model. Although the cost functions in
Table 2 place more emphasis on higher densities and less on the transition region than those in
Table 1, the conclusions we drew were the same.
Independent high-resolution density data are available from the gamma-ray profiling of ice cores, and provide a test of the transition model in regions outside the Pine Island Glacier basin for which it was optimized.
Figure 8 shows data from an ice core (B39) collected in Dronning Maud Land, Antarctica [
19], at a site where
= 0.77 m w.e. a
and
= −17.9
C. The data are available at 1 mm resolution but, in the figure, the data have been smoothed by a 31-point running mean (≈3 cm) for clarity. The model lines terminate at the BCO horizon (the point when density reaches 815 kg m
[
1]) below which they are not expected to be valid. The cost functions for the Herron and Langway and Ligtenberg models for the density range 500–800 kg m
are
= 0.163 and
= 0.156, respectively. Using the transition model with
= 580 kg m
and M = 7 produces a cost function of
= 0.128. That is, both the Ligtenberg and the transition model improve on the Herron and Langway fit, with the transition model being better for this high-accumulation site. If
is reduced to 550 kg m
an improved fit with
= 0.059 is obtained with the transition model. Note that none of the models predicts the slight increase in densification rate just below the BCO horizon associated with an increase in the density variability. For this, a more complex model including the effect of microstructure and impurities on snow strength would be required [
11].
In their analysis of high-resolution (gamma-ray) density profiles, Hörhold et al. [
7] noted a weak transition in the slope of the density–depth profiles at densities between 550 and 580 kg m
for high accumulation sites (such as B39), whereas lower accumulation sites (such as B26) showed this transition at much lower densities, below 500 kg m
.
Figure 9 shows that a slightly improved fit to the data was obtained with the transition model using
= 530 kg m
for core B26 collected in North Greenland [
20] at a site with
= −30.6
C and
= 0.18 m w.e. a
. The Herron and Langway cost function
= 0.087 is reduced to
= 0.085. Using
= 580 kg m
increased the cost function to
= 0.118. However, even with this transition density, the model performed better that the Ligtenberg model with
= 0.170. It is clear that the transition model could be improved by allowing
to vary with
or
. When more strain-rate data are collected, or more high-resolution (gamma-ray) density profiles are released, this can be investigated.