Ice is a highly transparent material in the visible. According to the most
widely used database

The spectral absorption coefficient of ice is one of the primary variables
controlling the reflectance and transmittance of snow-covered surfaces, sea
ice, and ice clouds. It is responsible for the main spectral features of snow
albedo: in the visible, the albedo is close to 1 and it smoothly decreases in
the near-infrared with significant absorption bands around 1030, 1300, and
1550 nm

Knowledge of the ice absorption coefficient is essential but some significant
uncertainties remain in the lower range of the microwaves

Estimation of the absorption coefficient from light transmission measurements
directly in ice slabs proved to be difficult for several reasons. First, to
measure significant decrease due to the absorption, the path length in the
slab needs to be large compared to the inverse of the absorption coefficient,
i.e., of the order of kilometers. Second, the residual scattering and
reflections in the slab reduce the transmission and can be misinterpreted as
absorption

In this paper, we aim at refining the ice absorption coefficient using a
method similar to

Scheme and picture of SOLEXS. The rod (black) is vertically guided and inserted in the snow with the rope manipulated by the operator. The quadripod (gray) is oriented to minimize the shadows. At the bottom end of the rod, the light first traverses the semi-transparent 0.5 mm thin Teflon cylinder and is reflected by a Teflon block towards the fiber tip, from which it is transferred to the spectrometer.

The paper is organized as follows: Sect.

Solar Extinction in Snow (SOLEXS) is a device to measure the rate of radiance
decrease in snow

Similar radiance profilers were used by

SOLEXS is accompanied by a dedicated software with graphical user interface
to control, visualize, and annotate the acquisitions directly in the field
and a post-processing library to produce the profiles. The following
processing steps are applied: (1) subtraction of the dark current and
normalization by the integration time as in

A total of 56 profiles have been acquired during the summer seasons 2012–2013
and 2013–2014 between 3 and 25 km from Concordia station and in
different directions. The prevailing winds come from the south-southwest
sector around Dome C

According to radiative transfer theory, the decrease of intensity (or diffuse
radiance, actinic flux, etc.)

Combining Eqs. (

To derive the ice absorption, the first method used here follows

The second method follows the same physical principles but accounts in
addition for the uncertainties in the observations and the variable thickness
of the zones. It uses Bayesian inference (BAY) to deduce not only the most
likely

Samples (gray) of the prior distribution of the ice absorption to be used by the BAY method. Ice absorption coefficients IA1984 and IA2008 are presented for reference.

The statistical problem so stated is solved for the 70 homogeneous zones and
29 wavelengths from 320 to 880 nm (20 nm step).
Some deep zones have no data for the longer wavelengths because the
measurements are under the noise level. The problem is huge with a total of
1417 unknowns and 40 261 observations. The computation of the posterior
distributions is performed using a Markov chain Monte Carlo sampler called
No-U-Turn Sampler (NUTS;

Both estimation methods (WBG and BAY), whatever their degree of
sophistication, consider that the errors are independent random noise with
zero mean, i.e., that the observations are non-biased measures of the
profile of radiance (at least proportional to the radiance). To test this
assumption, we investigate the impact of the presence of the rod in the snow.
To this end, the 3-D radiative transfer model “Monte Carlo modeling of light
transport in multi-layered tissues” (MCML;

The rod housing the optical fiber is modeled by a cylinder (radius

Principle of the 3-D radiative transfer model MCML adapted to snow.
The rod
(gray) is a cylinder of diameter 2

The model records the total absorption as a function of depth and radius (the
calculation is done in three-dimension, but the recording assumes cylindrical
symmetry). In addition, it records rays escaping the medium by the upper
interface (i.e., reflected rays) and lower interface (i.e., transmitted rays).
We extended
the original code, which only considers a narrow beam source, to
support two modes of operations. First mode aims to simulate infinitely large
source (like a diffuse sky). For this, rays are launched from a circular area
of radius

Ice absorption estimated by the WBG method (black) on the SOLEXS profile at 25kmE_1 (25 km east of Concordia). The 95 % confidence interval is shown in gray shade.

The optical single scattering properties of snow (

This section presents the estimation of ice absorption on our dataset using WBG and BAY methods. The 3-D radiative transfer model is then used to study the influence of the insertion of the rod in the snow.

The WBG method is first applied to one of the homogeneous zones for
illustration. We selected the profile 25kmE_1 between 20 and 30 cm
depth which is described and used in

Seventy ice absorption spectra estimated by the WBG method

We applied the WBG method individually to all the homogeneous zones, which
yields 70 ice absorption spectra (Fig.

While the analysis and selection of this set of spectra could be refined, this is not the route we have chosen. The Bayesian inference method is more powerful to deal with the heterogeneity of the data quality, requires less assumption on the error distributions, and is inherently able to provide the best estimate weighted by the quality of each individual members of this set.

Samples (gray) of the posterior distribution of the ice absorption estimated by the BAY method on the 70 homogeneous zones selected in the 56 radiance profiles measured around Concordia.

The BAY method is applied simultaneously to all the homogeneous zones and
wavelengths. It yields a set of 800 absorption spectra drawn from the
posterior. A subset of 100 spectra is shown in Fig.

Ice absorption estimated by the BAY method at different locations around around Concordia.

Posterior of ice absorption at 440 nm independently estimated on different sites by the BAY method.

The influence of site where the profile has been measured is investigated by
applying the BAY method on groups of profiles based on the distance and
direction to the station (Figs.

Snow MCML is used to evaluate the impact of the rod on SOLEXS measurements.
We first consider a homogeneous semi-infinite snowpack with typical values
for Dome C snow (SSA

In the presence of the rod (square symbols in Fig.

The influence of the rod can be explained as follows: with an albedo of 0.90,
the rod absorbs more than snow whose single scattering albedo is larger than 0.99
at any wavelength in the range considered here. Hence, as the rod is
inserted in the snow, the probability that a ray hits the rod before being
captured by the optical fiber increases. Inserting the rod can be compared in
a first approximation to adding LAI. The radiance
decreases more sharply than it does in the ideal case because of the
increasing additional absorption. However, while impurities are usually
dispersed in horizontal layers in the snow, the rod is a highly localized and
concentrated absorber. This difference has strong implications because rays
propagating from the surface to the tip of the fiber do not necessarily
interact with the rod. In other words, there are many ray trajectories from
the surface to the fiber tip that do not touch the rod. Furthermore,
considering only the rays that reach the fiber tip – and that are ultimately
measured by SOLEXS – it is evident that hitting the lower part of the rod is
more likely than the upper one. This is due to dilution in the 3-D space
yielding a probability of interaction with the rod decreasing as the square
power of the distance from the fiber tip. It implies in practice that only
the “lower” part of the rod significantly contributes to the rod absorption.
Conversely, once the “lower” part is completely inserted in the snow, the
absorption due to the rod becomes nearly constant while the signal continues
to decrease as a result of the “normal” extinction caused by snow scattering
and absorption. This explains why (i) the log-radiance is lower when the rod
is present compared to the ideal case (i.e., the ratio in
Fig.

Influence of the

To further investigate the impact of the rod, we run simulations with varying
rod characteristics. Figure

Influence of the air gap between the 10 mm diameter rod and snow
simulated with MCML at 400 nm for a homogeneous snowpack with SSA of
30 m

The rod itself is not the only component perturbing the profiles. To perform
the hole before the insertion of the rod, we use a metal stick similar to the
rod and that uses the same guide as the fiber. Despite this precaution, the
alignment is rarely perfect which enlarges the hole, letting a small void
between the snow and the rod. The light can propagate without loss in this
void, which is illustrated in Fig.

Ratio at 50 cm depth (black contour lines) between the radiances calculated by MCML simulations with rod and without rod for a homogeneous snowpack with varying SSA and density. Low values of the ratio indicate high rod absorptions. Green symbols represent the couples (SSA, density) at several depths measured in the 25kmE_1 snow pit.

The rod absorption also depends on the snow properties.
Figure

Vertical profiles of SSA (in orange) and density (in blue) measured in the 25kmE_1 snow pit.

Log-radiance profiles at 400, 500, 600, and 700 nm measured by
SOLEXS at 25kmE_1 (25 km east of Concordia). Simulations with the 1-D model
TARTES (without rod) and the 3-D model MCML using density and SSA profiles
measured at the same point

Same as Fig.

The first series of simulations shows that the measured radiance is
significantly lower than in the ideal case and that the decrease rate is
negatively biased with respect to the theoretical AFEC, but under the
transition zone this bias becomes small compared to the snow extinction. It
implies that to retrieve ice absorption, the top

The snowpack 25kmE_1 is now investigated. The MCML simulations are run with
layers every 2.5 cm using measured SSA and density
(Fig.

The simulations at 700 and 600 nm with MCML show a very good
agreement with SOLEXS observations up to 20 cm depth and only
slightly degrade below. It means that SSA and density measurements as well as
the values of the constants

Figures

A practical consequence of this issue is that even in homogeneous snow
layers, the gradient over a few centimeters in the upper part of this layer
is affected by the rod absorption in the overlying layer. This upper part
must be excluded from the estimation of the AFEC. Fortunately, this is what
happens in practice because the selection of the homogeneous zones is done
directly on the radiance profiles. We can expect that selecting linear pieces
of profiles remove the effect of sharp transitions. Conversely, if the snow
properties vary gradually with depth, the variations of the rod absorption
might be smooth and be misinterpreted as linear piece. For instance, if the
rod absorption is decreasing (e.g., because SSA is smoothly increasing), the
gradient underestimates the AFEC of the ideal case, resulting in an
underestimation of the ice absorption. Considering that at Dome C the
density is highly variable while the SSA tends to decrease in the first upper
meter of the snowpack

Ice absorption estimated with the WBG method applied to simulated irradiance profiles with MCML in the presence of the rod for the homogeneous snowpack (triangles up) and the 25kmE_1 snow pit (triangles down) by using IA2008 (green) and BAY (black) ice absorption spectra, respectively, as input.

The uncertainty range caused by the rod–snow interactions is evaluated here
by considering the two snowpacks investigated earlier: the homogeneous
snowpack, which leads to an overestimation of the AFEC, and the 25kmE_1
snow pit, which results in the opposite. To perform this evaluation, we ran
MCML for each snowpack and for each ice absorption spectrum (IA2008 and BAY),
which yields simulated radiance profiles as in Figs.

Figure

The ice absorption obtained with our new observations is 1 order of
magnitude greater than the most recent and widely used compilation proposed
by

First, the large difference has been obtained whatever the estimation method
and filtering of the data, which demonstrates that the difference is
statistically significant and robust against the methodology. The simulations
with MCML showed that the instrument is not neutral, resulting in differences
between the vertical gradient of the measured log-radiance and the AFEC

Second, despite many similarities between our experimental protocol and that
of

Estimates of ice absorption from this paper (black, BAY) from AMANDA
experiment

Third, even if the absorption coefficient is accurately estimated, the
question of the contamination of the snow remains.
Figure

Despite no explanation of the discrepancy can be given, a convergence of
facts tends to suggest that the ice absorption IA2008 is too low: (i) this
value was associated with a large uncertainty (non-monotonicity of the slope
in Fig. 7 of

To briefly illustrate the potential impact of our new estimate, we computed
the change of albedo between IA2008 and BAY for a homogeneous snowpack with
SSA

This study reproduced the method developed by

Considering the lack of reproducibility between

The ice absorption data
are available from

Laurent Arnaud and Ghislain Picard developed SOLEXS. Ghislain Picard developed the 3-D Monte Carlo radiative code. Ghislain Picard and Quentin Libois developed the Bayesian inference method. All the authors contributed to the measurements, data analysis, and preparation of the manuscript.

This study was supported by the ANR program 1-JS56-005-01 MONISNOW. The authors acknowledge the French Polar Institute (IPEV) for the financial and logistic support at Concordia station in Antarctica through the NIVO program. The computations have been done on the CIMENT cluster. The authors thank S. G. Warren and R. E. Brandt for discussions about their experimental setup and their comments on an earlier version of the manuscript. The editor and two anonymous reviewers are acknowledge for their suggestions to improve the manuscript, especially regarding the differences between the various ice absorption estimates. M. Dumont is also acknowledged for her comments on the initial manuscript. Edited by: F. Dominé Reviewed by: two anonymous referees