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**Research article**
18 Apr 2019

**Research article** | 18 Apr 2019

Estimating sea ice parameters

^{1}Sorbonne Université, Observatoire de Paris, Université PSL, CNRS, LERMA, Paris, France^{2}Danish Meteorological Institute, Copenhagen, Denmark^{3}Institute of Environmental Physics, University of Bremen, Bremen, Germany

^{1}Sorbonne Université, Observatoire de Paris, Université PSL, CNRS, LERMA, Paris, France^{2}Danish Meteorological Institute, Copenhagen, Denmark^{3}Institute of Environmental Physics, University of Bremen, Bremen, Germany

Abstract

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Mapping sea ice concentration (SIC) and understanding sea ice properties and
variability is important, especially today with the recent Arctic sea ice
decline. Moreover, accurate estimation of the sea ice effective temperature
(*T*_{eff}) at 50 GHz is needed for atmospheric sounding applications
over sea ice and for noise reduction in SIC estimates. At low microwave
frequencies, the sensitivity to the atmosphere is low, and it is possible to
derive sea ice parameters due to the penetration of microwaves in the snow
and ice layers. In this study, we propose simple algorithms to derive the
snow depth, the snow–ice interface temperature (*T*_{Snow−Ice}) and
the *T*_{eff} of Arctic sea ice from microwave brightness temperatures
(TBs). This is achieved using the Round Robin Data Package of the ESA sea ice
CCI project, which contains TBs from the Advanced Microwave Scanning
Radiometer 2 (AMSR2) collocated with measurements from ice mass balance buoys
(IMBs) and the NASA Operation Ice Bridge (OIB) airborne campaigns over the
Arctic sea ice. The snow depth over sea ice is estimated with an error of
5.1 cm, using a multilinear regression with the TBs at 6, 18, and 36 V. The
*T*_{Snow−Ice} is retrieved using a linear regression as a function
of the snow depth and the TBs at 10 or 6 V. The root mean square errors
(RMSEs) obtained are 2.87 and 2.90 K respectively, with 10 and 6 V TBs.
The *T*_{eff} at microwave frequencies between 6 and 89 GHz is
expressed as a function of *T*_{Snow−Ice} using data from a
thermodynamical model combined with the Microwave Emission Model of Layered Snowpacks. *T*_{eff} is estimated from the *T*_{Snow−Ice}
with a RMSE of less than 1 K.

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How to cite.

Kilic, L., Tonboe, R. T., Prigent, C., and Heygster, G.: Estimating the snow depth, the snow–ice interface temperature, and the effective temperature of Arctic sea ice using Advanced Microwave Scanning Radiometer 2 and ice mass balance buoy data, The Cryosphere, 13, 1283-1296, https://doi.org/10.5194/tc-13-1283-2019, 2019.

1 Introduction

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In situ observations of the variables controlling the sea ice energy and momentum balance in polar regions are scarce. One way to overcome this observational gap is to use satellites for measuring sea ice properties. The objective of this study is to estimate key sea ice variables from satellite remote sensing to improve current sea ice models and prediction, sea ice concentration (SIC) mapping in the EUMETSAT Ocean and Sea Ice Satellite Application Facility (OSISAF) project, and polar atmospheric sounding applications.

Sea ice thermodynamics is controlled by the regional heat budget
(Maykut and Untersteiner, 1971). In general, sea ice is covered by snow, which can reach a
mean thickness of up to ∼50 cm in the Arctic (Sato and Inoue, 2018). Snow on
sea ice strongly affects the sea ice energy and radiation balance, with its
high insulation of heat and reflectivity of solar radiation. Snow is a poor
conductor of heat: it insulates the sea ice and reduces the winter ice growth
(Fichefet and Maqueda, 1999). In summer, its high albedo reduces the sea ice melting
rate. The high albedo of snow on sea ice compared to open-water albedo plays
an important role in the sea ice albedo feedback mechanism and Arctic
amplification (Hall, 2004). Sato and Inoue (2018) suggest that the recent sea
ice growth has been effectively limited by the increase in snow depth on thin
ice during winter. Current sea ice models include snow schemes (e.g.
Lecomte et al., 2011), with the snow depth and temperature gradient in the
snow pack modulating the sea ice growth and melt. Improved estimates of snow
depth (*D*_{s}), as well as snow–ice interface temperature
(*T*_{Snow−Ice}) from satellite observations would provide valuable
information on the vertical thermodynamics in the snow and ice to improve
current sea ice models and therefore the prediction of sea ice growth.

Here we propose using a simple algorithm to retrieve *D*_{s} and
*T*_{Snow−Ice} from passive microwave observations from the Advanced
Microwave Scanning Radiometer 2 (AMSR2), based on a large data set of
collocated in situ and satellite observations. An extensive Round
Robin Data Package (RRDP) (Pedersen et al., 2018,
https://figshare.com/articles/Reference_dataset_for_sea_ice_concentration/6626549,
last access: 15 January 2019) has been developed
during the European Space Agency (ESA) sea ice Climate Change Initiative
(CCI) project and the SPICES (Space-borne observations for detecting and
forecasting sea ice cover extremes) project
(http://www.seaice.dk/ecv2/rrdb-v1.1/, last access: 15 June 2017). It contains in situ
data from the ice mass balance buoys (IMBs), and the Operation Ice Bridge
(OIB) airborne campaigns collocated with AMSR2 brightness temperature
measurements between 6 and 89 GHz.

Algorithms already exist to retrieve the snow depth from microwave observations. Markus and Cavalieri (1998) and Comiso et al. (2003) use the spectral gradient ratio of the 19 and 37 GHz (GR37/19) in vertical polarization to deduce the snow depth over sea ice. This method has been developed for dry snow on first-year ice (FYI) in Antarctica, and it is applicable only to this ice type. Sea ice emissivity depends on the ice type. At frequencies ≥18 GHz, the ice emissivity is higher for FYI than for multi-year ice (MYI) (Comiso, 1983; Spreen et al., 2008). The difference of emissivity between the 19 and 37 GHz can be used to retrieve the snow depth or the sea ice type. Therefore, the snow depth algorithms which use this gradient ratio (GR37/19) are strongly dependent on the ice type. Improvements by Markus and Cavalieri (1998) have been suggested by Markus et al. (2011) and Kern and Ozsoy-Çiçek (2016). More recently, Rostosky et al. (2018) revisit the methodology for the Arctic region, using a new gradient ratio between 7 and 19 GHz (GR19/7) to derive snow depths over both FYI and MYI. For their study, they use the snow depths of OIB campaigns obtained in March and April. With the help of the RRDP, we will extend the methodology to the full winter (from 1 December to 1 April) for the Arctic region using the IMB snow depth data.

Tonboe et al. (2011) showed from radiative transfer simulations that there is a
high linear correlation between the *T*_{Snow−Ice} and the passive
microwave observations at 6 GHz. Preliminary results from
Grönfeldt (2015) evidenced the possibility of deriving the temperature of
sea ice from passive microwave observations using simple regression models.
This work will be extended here to estimate *T*_{Snow−Ice} over
Arctic sea ice.

Passive microwave satellite observations between 50 and 60 GHz are
extensively used to provide the atmospheric temperature profiles in Numerical
Weather Prediction (NWP) centres, with instruments such as the Advanced
Microwave Sounding Unit-A (AMSU-A) or the Advanced Technology Microwave
Sounder (ATMS). For an accurate estimation of the temperature profile in the
lower atmosphere, quantifying the surface contribution is required. The
surface contribution, i.e. the surface brightness temperature (TB), depends on
the frequency, and it is the product of a surface effective emissivity
(e_{eff}) and a surface effective temperature (*T*_{eff}):

$$\begin{array}{}\text{(1)}& {\displaystyle}\text{TB}={e}_{\mathrm{eff}}\cdot {T}_{\mathrm{eff}}.\end{array}$$

*T*_{eff} is defined as the integrated temperature over a layer
corresponding to the penetration depth at the given frequency: the larger the
wavelength, the deeper the penetration into the medium. In the same way,
*e*_{eff} represents the integrated emissivity over a layer
corresponding to the penetration depth. It depends on the frequency, the
incidence angle, and the sub-surface extinction and reflections between snow
and sea ice layers (Tonboe, 2010). Therefore, estimating the surface
contribution is particularly complicated over sea ice due to the layering
and the vertical structure of the snowpack, which affect the microwave
emission processes
(Mathew et al., 2008; Rosenkranz and Mätzler, 2008; Harlow, 2009, 2011; Tonboe, 2010; Tonboe et al., 2011),
and to the large spatial and temporal variability of sea ice and snow cover
(English, 2008; Tonboe et al., 2013; Wang et al., 2017). The understanding of the
relationship between *T*_{eff} and the physical temperature profile is
complicated, especially at microwave frequencies ≥18 GHz, when
scattering occurs, but it has been shown that from 6 to 50 GHz there is a
high correlation between the *T*_{eff} and the *T*_{Snow−Ice}
(Tonboe et al., 2011). With *T*_{Snow−Ice} estimated from the AMSR2
observations, we will deduce the sea ice *T*_{eff} at AMSR2
frequencies between 6 and 89 GHz, using linear regression.

Section 2 describes the data set and the methodology
used in this study. The snow depth retrieval is presented in
Sect. 3. Section 4 reports on the
*T*_{Snow−Ice} retrieval. Finally, microwave sea ice
*T*_{eff} at 50 GHz is derived for application to temperature
atmospheric sounding (Sect. 5).
Section 6 discusses the snow depth and the
*T*_{Snow−Ice} retrieval results over a winter in Arctic.
Section 7 concludes this study.

2 Material and methods

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The RRDP from the ESA sea ice CCI project is an openly available data set (Pedersen et al., 2018, https://figshare.com/articles/Reference_dataset_for_sea_ice_concentration/6626549, last access: 15 January 2019). It contains an extensive collection of collocated satellite microwave radiometer data with in situ buoy or airborne campaign measurements and other geophysical parameters, with relevance for computing and understanding the variability of the microwave observations over sea ice. It covers areas with 0 % and 100 % of SIC and different sea ice types (thin ice, first-year ice, multiyear ice), for all seasons including summer melt. In our study, we will focus on Arctic sea ice during winter in regions with 100 % sea ice cover. Two different data sets from the RRDP are used: AMSR2 brightness temperatures (TBs) collocated with IMB measurements and AMSR2 TBs collocated with OIB airborne campaign measurements.

AMSR2 is a passive microwave radiometer on board the JAXA GCOM-W1 satellite
(launched on 18 May 2012). AMSR2 has 14 channels at 6.9, 7.3, 10.65, 18.7,
23.8, 36.5, and 89 GHz for both vertical and horizontal polarizations and it
observes at 55^{∘} of incidence angle. In the RRDP, the spatial
resolution of each channel is resampled by JAXA to the 6.9 GHz resolution
(32×62 km) (see AMSR2 L1R products, Maeda et al., 2011, 2016)
before collocation with buoy or airborne campaign measurements (RRDP report,
Pedersen and Saldo, 2016; Pedersen et al., 2018).

IMBs are installed by the Cold Regions Research and Engineering Laboratory (CRREL) to measure the ice mass balance of the Arctic sea ice cover (Richter-Menge et al., 2006; Perovich and Richter-Menge, 2006). Buoy components include acoustic sounders and a string of thermistors. The thermistor string extends from the air, through the snow cover and sea ice, into the water and has temperature sensors located every 10 cm along the string. It measures the physical temperature with an accuracy of 0.1 K. There are two acoustic sounders located above the snow surface and below the sea ice. The acoustic sounders measure the position of snow and ice surfaces (top and bottom) with a precision of 5 mm, from which the snow depth is computed. The buoys also include instruments that measure air temperature, barometric air pressure, and GPS geographical position (Perovich et al., 2019). Several IMBs are deployed by the CRREL at different locations and times during the year. We only use Arctic buoy data recorded during winter (1 December to 1 April) to avoid cases where ice starts to melt. The IMBs available for this study are all located on MYI, with an ice thickness ≥1 m. A summary of buoy information corresponding to these criteria is given in Table 1 and the IMB locations are shown in Fig. 1. IMB measurements collocated with AMSR2 TBs used in this study totalize 2845 observations.

For snow depth retrieval, we also used data from the OIB airborne campaign. The NASA OIB project has collected ice and snow depth data in the Arctic during annual flight campaigns (March–May) since 2009. The data are especially valuable in this context, since they contain snow depth information from the snow radar on board the aircraft, not only from single points but continuously along the flight path. The vertical resolution of the OIB snow radar is 3 cm, and the uncertainty on the snow depth is around 6 cm compared with in situ measurements (Kurtz et al., 2013). Recent studies evidence larger errors on OIB snow depth (Kwok and Maksym, 2014) with issues to detect snow depth under 8 cm (Kwok and Maksym, 2014; Holt et al., 2015). These different limitations are summarized in Kwok et al. (2017). In the RRDP, the snow depth data from OIB snow radar are averaged into 50 km sections to be collocated with AMSR2 observations. For our study we use the OIB data from the 2013 campaign. It totalizes 408 observations over 8 d in March and April and covers FYI and MYI areas. Figure 1 summarizes the locations of IMBs and OIB campaigns over the Arctic ocean.

It is important to note that there are discrepancies due to the scale when comparing point measurements from buoys with the spatially averaged data from satellites or aircrafts (Dybkjær et al., 2012).

For the estimation of *T*_{eff}, we use a microwave emission model
coupled with a thermodynamic model. The emission model uses the temperature,
density, snow crystal and brine inclusion size, salinity, and snow or ice
type to estimate the microwave emissivity, the *T*_{eff}, and the TB
of sea ice. It is coupled with a thermodynamic model in order to provide
realistic microphysical inputs. The thermodynamic model for snow and sea ice
is forced with ECMWF ERA40 meteorological data input: surface air pressure,
2 m air temperature, wind speed, incoming shortwave and longwave radiation,
relative humidity, and accumulated precipitation. It computes a centimetre-scale profile of the parameters used as inputs to the emission model. The
emission model used here is a sea ice version of the Microwave Emission Model of Layered Snowpacks (MEMLS) (Wiesmann and Mätzler, 1999) described in
Mätzler (2006). The simulations were part of an earlier version of the
RRDP and the simulation methodology is described in Tonboe (2010). This
MEMLS simulation uses, among its inputs, the snow depth and the
*T*_{Snow−Ice} and computes *T*_{effs} and TBs at different
frequencies (from 1.4 to 183 GHz). The data set contains 1100 cases and is
called the MEMLS-simulated data set in the following.

In this study, we propose simple algorithms, using multilinear regressions,
to derive the snow depth, the *T*_{Snow−Ice}, and the
*T*_{eff} of sea ice from AMSR2 TBs.

The measurements from the IMB 2012G, 2012H, 2012J, and 2012L, collocated with
AMSR2 TBs, are used as the training data set for the different regressions to
retrieve snow depth and *T*_{Snow−Ice}. These buoys have been
selected because they are located in different regions across the Arctic and
show a large range of snow depths. The measurements from IMB 2013F, 2013G,
2014F, and 2014I, which are all located in the Beaufort Sea, are used as the
testing data set.

First, the IMB snow depth is expressed as a function of the AMSR2 TBs using a
multilinear regression (see Sect. 3.1). The OIB data
are used for the forward selection and the IMB training data set is used to
perform the regression. Second, the *T*_{Snow−Ice} is expressed as a
function of TBs and snow depth, using linear regressions. An automated method is developed
that detects the position of the snow–ice interface on the vertical temperature
profile measured by the IMB thermistor string (see
Sect. 4.1). Then, the IMB training data set is
used to perform the regressions (see Sect. 4.3). For
this part there are two consecutive regressions: the first one is done
between the centred (the average was subtracted) *T*_{Snow−Ice} and
TBs; the second one is done between the *T*_{Snow−Ice} corrected for
the TB dependence and the snow depth. Third, the sea ice *T*_{eff} at
different microwave frequencies is expressed as a function of the
*T*_{Snow−Ice} (see Sect. 5.2). This final
step uses the simulations from a thermodynamical model and MEMLS to
derive linear regression equations for the *T*_{eff} at frequencies
between 6 and 89 GHz. The *T*_{eff} at 50 GHz is of special interest
for atmospheric sounding applications.

3 Snow depth estimation

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A forward selection method is used to choose the best AMSR2 channels to retrieve snow depth. It is a statistical method to determine the best-predictor combinations (here, AMSR2 TBs) to retrieve a variable (here, snow depth). We use the stepwise regression (Draper and Smith, 1998). It is a sequential predictor selection technique: at each step statistic tests are computed, and the predictors included in the model are adjusted. Our training data set for this forward selection is the OIB snow depth from the 2013 campaign included in the RRDP. OIB data are chosen for forward selection because the data cover a large area with a wide range of snow depths. In addition, the scale of the averaged OIB data is closer to satellite footprint than buoy measurements, increasing the consistency with the satellite observations. Forward selection tests have also been done with the IMB training data set, but the results were not satisfactory. We find that the best channel combination for snow depth retrieval is the combination of the three channels at 6.9, 18.7, and 36.5 GHz in vertical polarization (6, 18, and 36 V).

Then, a multilinear regression is conducted using the IMB training data set (buoys G, H, J, L in 2012 collocated with AMSR2 TBs). The snow depth is given as a linear combination of the TBs at 6, 18, and 36 V:

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{D}_{\mathrm{s}}=\mathrm{1.7701}+\mathrm{0.0175}\cdot {\text{TB}}_{\mathrm{6}\phantom{\rule{0.125em}{0ex}}\mathrm{V}}-\mathrm{0.0280}\cdot {\text{TB}}_{\mathrm{18}\phantom{\rule{0.125em}{0ex}}\mathrm{V}}\\ \text{(2)}& {\displaystyle}& {\displaystyle}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}+\mathrm{0.0041}\cdot {\text{TB}}_{\mathrm{36}\phantom{\rule{0.125em}{0ex}}\mathrm{V}},\end{array}$$

with *D*_{s} the snow depth expressed in metres and TB in kelvin. This model was
trained with snow depths between 5 and 40 cm.

The forward selection has also been tested by constraining the number of predictors to 2 and 4. The combinations obtained are 18 and 36 V for two channels and 6, 18, 36, and 89 V for four channels. Then, the multilinear regression has been performed using these combinations of two or four channels. The results show that the three-channel combination is the best in terms of RMSE and correlation compared to the two- or four-channel combination (see Sect. 3.2).

Figure 2 shows the comparison between the observed snow depth measured by the acoustic sounder of IMB and the regressed snow depth computed from AMSR2 TBs with Eq. (2). The RMSE between the IMB snow depth observations and our snow depth regression is 12.0 cm and the correlation coefficient is 0.66, using the IMBs 2013F, 2013G, 2014F, and 2014I (which are not in the training data set). The buoy 2013F observes a large snow depth (> 40 cm), which is outside the bounds of our snow depth model. Tests are conducted to improve the estimation, including the 2013F buoy in the training data set, with equal numbers of observations for different ranges of snow depths: it does not improve the results. Our model obtained the same snow depth estimation between buoys 2013G and 2013F. It is consistent because these buoys are spatially very close. Therefore, we suspect that the 2013F buoy is located nearby a ridge or hummock, where the local snow depth is large but not detectable at the satellite footprint scale. Without including the buoy 2013F in the computation, the RMSE for our snow depth model is 5.1 cm and the correlation coefficient is 0.61.

We also compare the snow depth retrievals with the measurements of the 2013 OIB campaigns (see Fig. 3) with the ice type computed from the gradient ratio between 19 and 37 GHz (Baordo and Geer, 2015). Our snow depth regression (Eq. 2) RMSE is 6.26 cm and the correlation coefficient with OIB observations is 0.87. Note that the uncertainties on OIB data for the 2013 campaigns are between 2 and 22 cm with a mean standard deviation (SD) of 11 cm (OIB snow depth Dsnow provided in the RRDP). Looking at Fig. 3, our snow depth regression is applicable to both ice types. The RMSEs computed for MYI and FYI are 7.2 and 3.9 cm, and the correlations are 0.71 and 0.03. The RMSE is smaller for FYI because the snow depth variability of FYI is also smaller. The low correlation obtained for FYI can come from the limited number of observations and because the snow depth variability observed is within the signal noise.

Spatial scales are different when comparing satellite measurements or airborne campaign measurements with buoy measurements. Discrepancies can appear due to the spatial variability of the snow depth. It can explain that the correlation is higher when comparing snow depth estimated from AMSR2 TBs with the snow depth observed from OIB radar. It is also important to note that the OIB campaign data are from late winter to beginning of spring (March to April), while IMB measurements are from winter (December to March). With the snow depth regression being developed on IMB measurements, this small change in season can contribute to the larger RMSE observed with OIB data.

4 Snow–ice interface temperature estimation

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During winter, the air temperature is very cold, meaning that the snow surface temperature is cold compared to ice and water temperatures. Through sea ice, the temperature profile is piecewise linear and temperature increases with depth (see Fig. 4). In the air, the temperature gradient is small because of turbulent mixing. In the snow, the temperature gradient is larger due to the thermal properties of snow. Therefore, air–snow and snow–ice interface positions can be detected by changes in the temperature gradient. At the air–snow interface, the second derivative of the temperature profile reaches a maximum. At the snow–ice interface, the temperature gradient being lower in the ice than in the snow, the second derivative of the temperature profile reaches a minimum. Using these properties of the sea ice temperature profile, an automated method is implemented to detect the air–snow and the snow–ice interface positions in the temperature profile measured by the buoy thermistor string.

Figure 4 shows an averaged temperature profile
through sea ice during winter, with the air–snow and snow–ice interface
positions detected with our automated method. This method performs best
during winter when the air is cold. It may not be applicable if the snow
depth is lower than the vertical resolution of the thermistor string
(10 cm) or if sea ice starts to melt and the temperature profile develops
gradually toward an isothermal state. The method selects the thermistor which
is located the closest to the interface. Note that the real interface
position can be located between two thermistors. Therefore, the shift between
the real interface position and the thermistor the closest to the interface
can be up to 5 cm. This can introduce uncertainties in our
*T*_{Snow−Ice} regression.

During winter, the vertical position of the snow–ice interface is fixed with respect to the buoy thermistor string. The thermistor string is frozen into the ice which means that the thermistor at the snow–ice interface will stay at that interface unless there is surface melt or snow ice formation and this rarely happens during winter. For each IMB, the snow–ice interface is detected with our automated method described in Sect. 4.1.

We use a correlation analysis to select the TBs at different frequencies
describing the variability of the *T*_{Snow−Ice}.
Figure 5 shows the correlation coefficient between
*T*_{Snow−Ice} and AMSR2 TBs computed using the data from all IMBs
(Table 1). The 89 GHz TBs are highly correlated with
the air temperature (*R*>0.75). The 18.7, 23.8, and the 36.5 GHz TBs have a
low correlation with *T*_{Snow−Ice} because of microwave scattering
in the snow and/or shallow microwave penetration into the snow. The 7.3 GHz
channel is ignored because it contains practically the same information as
the 6.9 GHz channel. The TBs at 6.9 and 10.65 GHz at vertical polarization
have the highest correlation with *T*_{Snow−Ice} (*R*>0.5).
Therefore, the 10.65 and the 6.9 GHz at vertical polarization (10 and 6 V)
channels are selected as inputs to the linear regression to retrieve the
*T*_{Snow−Ice}.

To express the *T*_{Snow−Ice} as a function of the TB at 6 and
10 V, the linear regressions are calculated on centred data (i.e. the
anomaly). For each buoy, the averaged *T*_{Snow−Ice} is subtracted
from the *T*_{Snow−Ice} measurements and the same is done with the
TB measurements. Thus, the temperature offset between the buoys is removed
and the slope of the linear regression is unchanged:

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}\mathrm{\Delta}{T}_{\mathrm{Snow}-\mathrm{Ice}}={a}_{\mathrm{1}}\cdot \mathrm{\Delta}{\text{TB}}_{\mathrm{6}\phantom{\rule{0.25em}{0ex}}\mathrm{or}\phantom{\rule{0.25em}{0ex}}\mathrm{10}\phantom{\rule{0.125em}{0ex}}\mathrm{V}}\iff {T}_{\mathrm{Snow}-\mathrm{Ice}}\\ \text{(3)}& {\displaystyle}& {\displaystyle}={a}_{\mathrm{1}}\cdot {\text{TB}}_{\mathrm{6}\phantom{\rule{0.25em}{0ex}}\mathrm{or}\phantom{\rule{0.25em}{0ex}}\mathrm{10}\phantom{\rule{0.125em}{0ex}}\mathrm{V}}+{\text{offset}}_{\mathrm{buoy}},\end{array}$$

with Δ*T*_{Snow−Ice} and ΔTB describing the centred
*T*_{Snow−Ice} and TB. Figure 6 shows the
linear regression between the *T*_{Snow−Ice} and the TB at 6 and
10 V, using the measurements from buoys 2012G, 2012H, 2012J, and 2012L. The
slope coefficients (*a*_{1}) estimated between the *T*_{Snow−Ice} and
the TB at 6 and 10 V are 1.086±0.020 and 1.078±0.019.

The offset (offset_{buoy}) in the linear regression equations
between *T*_{Snow−Ice} and the TB is different for each buoy,
because it depends on the snow depth. The *T*_{Snow−Ice} dependence
on snow depth can be explained by the thermal insulation of snow
(Maaß et al., 2013; Untersteiner, 1986). Here, we establish an empirical
relationship between the *T*_{Snow−Ice} corrected for the TB linear
dependence at 10 or 6 V, and the snow depth as follows:

$$\begin{array}{}\text{(4)}& {\displaystyle}{T}_{\mathrm{Snow}-\mathrm{Ice}}-{a}_{\mathrm{1}}\cdot {\text{TB}}_{\mathrm{10}\phantom{\rule{0.25em}{0ex}}\mathrm{or}\phantom{\rule{0.25em}{0ex}}\mathrm{6}\phantom{\rule{0.125em}{0ex}}\mathrm{V}}={a}_{\mathrm{2}}\cdot f\left({D}_{\mathrm{s}}\right)+{a}_{\mathrm{3}},\end{array}$$

with *f*(*D*_{s}) a function of snow depth.

Three different linear regressions have been tested to relate the
*T*_{Snow−Ice} using the snow depth directly, the inverse of the
snow depth, and the logarithm of snow depth. Figure 7
shows the *T*_{Snow−Ice} corrected from TB dependence as a function
of snow depth. The different regressions are tested using the training
data set (IMB G, H, J, and L in 2012). The regression showing the best results
uses the logarithm of the snow depth (solid black line in
Fig. 7). The linear regression using the snow depth
directly (dashed red line in Fig. 7) leads to an
overestimation of the *T*_{Snow−Ice} for large snow depth. The
regression using the inverse of the snow depth (red dotted line in
Fig. 7) leads to an underestimation for small snow
depth. The RMSEs obtained on the *T*_{Snow−Ice} are compared and the
relation using the logarithm of snow depth shows the lowest RMSE. Based on
these results, the final equations to relate the *T*_{Snow−Ice} to
the snow depth and the TB at 10 and at 6 V are as follows:

$$\begin{array}{}\text{(5)}& {\displaystyle}{\displaystyle}{T}_{\mathrm{Snow}-\mathrm{Ice}}=\mathrm{1.078}\cdot {\text{TB}}_{\mathrm{10}\phantom{\rule{0.125em}{0ex}}\mathrm{V}}+\mathrm{5.67}\cdot \mathrm{log}\left({D}_{\mathrm{s}}\right)-\mathrm{5.13}\end{array}$$

$$\begin{array}{}\text{(6)}& {\displaystyle}{\displaystyle}{T}_{\mathrm{Snow}-\mathrm{Ice}}=\mathrm{1.086}\cdot {\text{TB}}_{\mathrm{6}\phantom{\rule{0.125em}{0ex}}\mathrm{V}}+\mathrm{3.98}\cdot \mathrm{log}\left({D}_{\mathrm{s}}\right)-\mathrm{10.70},\end{array}$$

where *T*_{Snow−Ice} and TB are expressed in kelvin, and *D*_{s}
is expressed in metres.

Figure 8 shows the comparisons between
the observed *T*_{Snow−Ice} and the regressed
*T*_{Snow−Ice} using the 10 and 6 V TBs (Eqs. 5
and 6), and the in situ snow depth measured by the
acoustic sounder of IMB. The RMSEs are computed using the IMB 2013F, 2013G,
2014F, and 2014I. The regression of the *T*_{Snow−Ice} using the
in situ snow depth with the 10 V TBs (Eq. 5) is
slightly better (RMSE =1.78 K) than the regression with the 6 V TBs
(Eq. 6) (RMSE =1.98 K). The variability due to the snow
depth is better described with the regression using the 10 V TBs.
Figure 9 is the same as
Fig. 8 but with our snow depth estimation
(Eq. 2). The RMSEs are 2.87 K for the 10 V regression and
2.90 K for the 6 V regression. The results are degraded because of the snow
depth regression, especially for the buoys with thick snow (∼50 cm) or
thin snow (∼5 cm) (e.g. buoy 2013F and buoy 2012L). Note that the
regression is tested with IMBs, which are all located on MYI. However, using our
algorithm to derive the *T*_{Snow−Ice} is also applicable over FYI
areas, as our snow depth algorithm is applicable to both ice types and our
*T*_{Snow−Ice} algorithm uses the channels 10 or 6 V, which have
limited sensitivity to the ice type (Comiso, 1983; Spreen et al., 2008).

5 Sea ice effective temperature estimation

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*T*_{eff} is related to the frequency and the incidence angle of the
satellite observations. It is not a geophysical variable that we can measure
directly as an in situ parameter. A microwave emission model has to
be used to computed the *T*_{effs} from the geophysical parameters.
The *T*_{eff} used here is available from a simulated data set using a
thermodynamical model and the microwave emission model, MEMLS. The model
set-up and the simulations are described in Tonboe (2010). In this
data set, the TBs and the *T*_{effs} are simulated using the
*T*_{Snow−Ice} and the input snow and ice profiles from the
thermodynamical model. Even though the simulated TB data are comparable to
observations in terms of mean and standard deviation, both the
thermodynamical model and the emission model are based on physical equations
and are not tuned to observations. TBs simulated with MEMLS are not fitted to
AMSR2 TBs, meaning that a bias is expected between the *T*_{Snow−Ice}
of the MEMLS-simulated data set (*T*_{Snow-Ice MEMLS}) and the
*T*_{Snow−Ice} estimated with our regression.

The bias obtained is the mean value of the difference between the
*T*_{Snow-Ice MEMLS}, and the *T*_{Snow−Ice} regressed
from Eqs. (5) and (6) using the TBs of the MEMLS-simulated data set as inputs. Biases of 3.97 and 4.01 K are estimated for
the regressions with 10 and 6 V respectively. The RMSEs computed between the
*T*_{Snow-Ice MEMLS} and the *T*_{Snow−Ice} regressed
and corrected for the biases at 10 and 6 V are 2.7 and 2.07 K.

Figure 10 shows the *T*_{Snow−Ice} from the MEMLS-simulated data set as a function of TB at 10 and 6 V, and the
*T*_{Snow−Ice} computed from our regressions (Eqs. 5
and 6), with and without the bias correction. We can see that
the slopes of our linear regressions are consistent with the data simulated
from MEMLS.

The *T*_{eff} near 50 GHz in vertical polarization is correlated with
the *T*_{Snow−Ice} (Tonboe et al., 2011) and it can be expressed as a
linear function of the *T*_{Snow−Ice}:

$$\begin{array}{}\text{(7)}& {\displaystyle}{T}_{\mathrm{eff}(\mathrm{freq},\phantom{\rule{0.125em}{0ex}}\mathrm{pol})}={b}_{\mathrm{1}(\mathrm{freq},\phantom{\rule{0.125em}{0ex}}\mathrm{pol})}\cdot {T}_{\mathrm{Snow}\text{-}\mathrm{Ice}\phantom{\rule{0.25em}{0ex}}\mathrm{MEMLS}}+{b}_{\mathrm{2}(\mathrm{freq},\phantom{\rule{0.125em}{0ex}}\mathrm{pol})},\end{array}$$

with *T*_{eff}, *b*_{1}, and *b*_{2} depending on the frequency (freq)
and on the polarization (pol). We use the MEMLS-simulated data set to
calculate the linear regression between the *T*_{Snow−Ice} and the
*T*_{eff} at 6.9, 10.65, 18.7, 23.8, 36.5, 50, and 89 GHz in vertical
polarization. *T*_{effs} at vertical and horizontal polarizations are
about the same. Only the vertical polarization is considered here, because
TBs measurements are noisier at horizontal polarization due to the
variability of sea ice emissivity at this polarization.

Figure 11 shows the *T*_{eff} at 50 V as a
function of *T*_{Snow−Ice}. The linear regressions between the
*T*_{Snow−Ice} and the *T*_{eff} at different frequencies are
computed. The coefficients *b*_{1} and *b*_{2} of Eq. (7) are
given in Table 2. The slope coefficient of the regression
increases with frequency, meaning that the sensitivity of the
*T*_{eff} to the *T*_{Snow−Ice} is increasing with frequency
between 6 and 89 GHz. A slope coefficient lower than 1 means that the
penetration depth at the given frequency is deeper than snow–ice interface.
At 50 GHz the slope coefficient is near to 1, meaning that the penetration
depth is close to the depth of the snow–ice interface. The RMSEs are below
1 K, with the regression of *T*_{eff} at 50 V showing the lowest
RMSE (0.33 K), and at 89 V showing the highest RMSE (0.92 K).

These linear regressions between the *T*_{eff} and the
*T*_{Snow-Ice MEMLS} (Eq. 7) are the final step in
retrieving the *T*_{eff} of sea ice at microwave frequencies as a
function of TBs, using the work in the previous sections to express the
*T*_{Snow−Ice} as a function of TBs (Eqs. 2,
and 5 or 6). The biases between the AMSR2
observations and the MEMLS-simulated data set are taken into account, replacing
*T*_{Snow-Ice MEMLS} by *T*_{Snow−Ice} estimated from
AMSR2 TBs with a bias correction (see Table 2):

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{T}_{\mathrm{eff}(\mathrm{freq},\phantom{\rule{0.125em}{0ex}}\mathrm{pol})}={b}_{\mathrm{1}(\mathrm{freq},\phantom{\rule{0.125em}{0ex}}\mathrm{pol})}\cdot \left({T}_{\mathrm{Snow}-\mathrm{Ice}}-\mathrm{3.97}\right)+{b}_{\mathrm{2}(\mathrm{freq},\phantom{\rule{0.125em}{0ex}}\mathrm{pol}),}\\ \text{(8)}& {\displaystyle}& {\displaystyle}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\text{for the regression using 10\hspace{0.17em}V TB}\end{array}$$

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{T}_{\mathrm{eff}(\mathrm{freq},\phantom{\rule{0.125em}{0ex}}\mathrm{pol})}={b}_{\mathrm{1}(\mathrm{freq},\phantom{\rule{0.125em}{0ex}}\mathrm{pol})}\cdot \left({T}_{\mathrm{Snow}-\mathrm{Ice}}-\mathrm{4.01}\right)+{b}_{\mathrm{2}(\mathrm{freq},\phantom{\rule{0.125em}{0ex}}\mathrm{pol}),}\\ \text{(9)}& {\displaystyle}& {\displaystyle}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\text{for the regression using 6\hspace{0.17em}V TB}.\end{array}$$

6 Discussion

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For days in November, January, and April, Fig. 12 shows the maps
of the snow depth estimated with our multilinear regression
(Eq. 2), the *T*_{Snow−Ice} estimated with our
multilinear regression (Eq. 5), and the MYI concentration
products from the University of Bremen (https://seaice.uni-bremen.de,
last access: 1 November 2018).
Maps of the MYI concentration from University of Bremen are derived from
AMSR2 and from the Advanced SCATterometer (ASCAT) with the method of
Ye et al. (2016a, b). To perform our regressions, we use the AMSR2 TBs
(Level L1R) provided by JAXA and the SIC from the European Centre for
Medium-Range Weather Forecasts (ECMWF) Reanalysis Interim (ERA-Interim)
data. Only the areas with 100 % SIC are considered to compute the snow
depth on sea ice and the *T*_{Snow−Ice} with our method.

The results show that the snow depth is larger (40 cm) in the north of Greenland (Warren et al., 1999; Shalina and Sandven, 2018) due to the presence of drift snow caused by the numerous pressure ridges present in this area (Hanson, 1980), as anticipated. We can observe that the snow depth is larger in areas with larger MYI concentrations. The variability of the snow cover is low during winter, as the snow depth reaches a maximum by December and remains relatively unchanged until snowmelt (Sturm et al., 2002).

For *T*_{Snow−Ice}, in January and April when the air temperature is
cold (between −20 and −30 ^{∘}C over the whole Arctic, on 5 January
and 5 April 2016 from ERA-Interim air temperature), the areas with large snow
depth show larger *T*_{Snow−Ice} because of the thermal insulation
power of the snow. It is different in November: the air temperature is warmer
(∼ −5 ^{∘}C near Kara Sea, ∼ −15 ^{∘}C near
Laptev Sea, and ∼ −25 ^{∘}C in the central Arctic and Beaufort seas,
on 5 November 2015 from ERA-Interim air temperature) and the areas with
thinner snow show larger *T*_{Snow−Ice} which are close to the air
temperature (Perovich and Elder, 2001). Note that we can observe low
*T*_{Snow−Ice} in some locations near the sea ice margins due to the
presence of open ocean in the satellite footprint. As the brightness
temperature of open water is low, the total brightness temperature measured
is decreased and it impacts our *T*_{Snow−Ice} estimation.

Visually the *T*_{Snow−Ice} shows a high correlation with the
distribution patterns of multiyear ice concentration on the same days: the
highest values are found in the north of Greenland and in the Canada Basin,
with some branches of higher values extending from there towards the Siberian
coast, marking the Beaufort Gyre of the Arctic sea ice drift (see the
animations for the same year at
https://seaice.uni-bremen.de/multiyear-ice-concentration/animations/,
last access: 1 November 2018).
The main differences between FYI and MYI are, on average, the higher
thickness of MYI and its higher snow load. Both effects will influence the
*T*_{Snow−Ice}. Under the same conditions, a higher ice thickness
will lead to a lower *T*_{Snow−Ice}. In contrast, it will be higher
if only the snow depth is increased. The positive correlation between MYI
concentration and *T*_{Snow−Ice} suggests that the influence of the
higher snow depth on MYI outbalances that of the higher ice thickness on the
*T*_{Snow−Ice}, emphasizing the important role of snow on sea ice in
its thermodynamic balance.

The similar patterns observed between the maps of the *T*_{Snow−Ice}
and the MYI concentration on Fig. 12 are encouraging and give
confidence in the methodology developed here, as these MYI concentration
products are from independent work done at the University of Bremen and
distributed daily to users. However it should be noted that the input
channels of both methods overlap in some AMSR2 channels, and even different
channels show some covariance (Scarlat et al., 2017).

7 Conclusions

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We derive simple algorithms to
estimate sea ice parameters such as the snow depth, the
*T*_{Snow−Ice}, and the *T*_{eff} of sea ice at microwave
frequencies, from AMSR2 channels. This is achieved using the ESA RRDP, which
contains AMSR2 data collocated with IMB data and OIB campaign data. In
addition, simulated TB outputs from a sea ice version of MEMLS are used for
the regression of the *T*_{eff}. All the equations used to retrieve these
sea ice parameters are derived using several linear and multilinear
regressions.

Our regression to retrieve the snow depth over winter Arctic sea ice uses the
TBs at 6.9, 18.7, and 36.5 GHz in vertical polarization. A RMSE of 5.1 cm is
obtained between the estimated and the IMB snow depths using an independent
IMB test data set. This snow depth retrieval is applicable to FYI and MYI,
with lower uncertainties for FYI than for MYI (3.9 cm compared to 7.2 cm).
To retrieve the *T*_{Snow−Ice}, two relations are derived using two
different AMSR2 channels (10 or 6 V) and the estimated snow depth. The two
regressions show similar results. The errors are 2.87 and 2.90 K
at 10 and 6 V. This *T*_{Snow−Ice} retrieval has been
tested only for MYI. It can also be applied to FYI, as the 6 and 10 V
channels have limited sensitivity to the ice type
(Comiso, 1983; Spreen et al., 2008). Finally the *T*_{effs} at 6.9, 10.65,
18.7, 23.8, 36.5, 50, and 89 GHz in vertical polarization are retrieved as a
function of *T*_{Snow−Ice} using linear regressions. At the final
step, the RMSEs of the linear regressions between the simulated
*T*_{Snow−Ice} and the *T*_{eff} for all channels are lower
than 1 K, with a minimum value of 0.33 K at 50 GHz, which is a key
frequency for atmosphere temperature retrieval. The methodology used to estimate
snow depth and *T*_{Snow−Ice} has been applied to several days
during winter. It shows consistent results with MYI concentration
estimates obtained independently.

These algorithms can be used to create snow depth and *T*_{Snow−Ice}
products which can improve the study of sea ice variability (e.g. sea ice
growth). Information on the *T*_{Snow−Ice} may help in sea ice
models by constraining the sea ice temperature gradient and the
thermodynamical ice growth. The *T*_{eff} estimations can be used in
atmospheric radiative transfer calculations and to reduce noise in SIC
retrieval algorithms (Tonboe et al., 2013) (e.g. EUMETSAT OSISAF global SIC
product).

Data availability

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Data availability.

The round robin data package used for this study is publicly accessible at https://figshare.com/articles/Reference_dataset_for_sea_ice_concentration/6626549 (Pedersen et al., 2018).

Author contributions

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Author contributions.

This study was conducted by LK and supervised by RTT and CP. GH contributed to the analysis and to the correction of the draft.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

This research was funded by EUMETSAT OSISAF (OSI VS17 03) and the PNTS (Programme national de télédédtection spatiale). The authors acknowledge the support from the EUMETSAT OSISAF visiting scientist programme and the Danish Meteorological Institute for its welcome. We also acknowledge the reviewers for their precious comments, which improved this manuscript a lot.

Review statement

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Review statement.

This paper was edited by John Yackel and reviewed by Leif Toudal Pedersen and one anonymous referee.

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Short summary

In this study, we develop and present simple algorithms to derive the snow depth, the snow–ice interface temperature, and the effective temperature of Arctic sea ice. This is achieved using satellite observations collocated with buoy measurements. The errors of the retrieved parameters are estimated and compared with independent data. These parameters are useful for sea ice concentration mapping, understanding sea ice properties and variability, and for atmospheric sounding applications.

In this study, we develop and present simple algorithms to derive the snow depth, the snow–ice...

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