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**Research article**
22 Mar 2018

**Research article** | 22 Mar 2018

Sea ice parameter retrieval

^{1}Ministry of Education Key Laboratory for Earth System Modeling, Department of Earth System Science, Tsinghua University, Beijing, China^{2}Department of Atmospheric and Environmental Sciences, University at Albany, State University of New York, Albany, NY, USA^{3}State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics (LASG), Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China

Abstract

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The accurate knowledge of sea ice parameters, including sea ice thickness and snow depth over the sea ice cover, is key to both climate studies and data assimilation in operational forecasts. Large-scale active and passive remote sensing is the basis for the estimation of these parameters. In traditional altimetry or the retrieval of snow depth with passive microwave remote sensing, although the sea ice thickness and the snow depth are closely related, the retrieval of one parameter is usually carried out under assumptions over the other. For example, climatological snow depth data or as derived from reanalyses contain large or unconstrained uncertainty, which result in large uncertainty in the derived sea ice thickness and volume. In this study, we explore the potential of combined retrieval of both sea ice thickness and snow depth using the concurrent active altimetry and passive microwave remote sensing of the sea ice cover. Specifically, laser altimetry and L-band passive remote sensing data are combined using two forward models: the L-band radiation model and the isostatic relationship based on buoyancy model. Since the laser altimetry usually features much higher spatial resolution than L-band data from the Soil Moisture Ocean Salinity (SMOS) satellite, there is potentially covariability between the observed snow freeboard by altimetry and the retrieval target of snow depth on the spatial scale of altimetry samples. Statistically significant correlation is discovered based on high-resolution observations from Operation IceBridge (OIB), and with a nonlinear fitting the covariability is incorporated in the retrieval algorithm. By using fitting parameters derived from large-scale surveys, the retrievability is greatly improved compared with the retrieval that assumes flat snow cover (i.e., no covariability). Verifications with OIB data show good match between the observed and the retrieved parameters, including both sea ice thickness and snow depth. With detailed analysis, we show that the error of the retrieval mainly arises from the difference between the modeled and the observed (SMOS) L-band brightness temperature (TB). The narrow swath and the limited coverage of the sea ice cover by altimetry is the potential source of error associated with the modeling of L-band TB and retrieval. The proposed retrieval methodology can be applied to the basin-scale retrieval of sea ice thickness and snow depth, using concurrent passive remote sensing and active laser altimetry based on satellites such as ICESat-2 and WCOM.

How to cite

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How to cite.

Zhou, L., Xu, S., Liu, J., and Wang, B.: On the retrieval of sea ice thickness and snow depth using concurrent laser altimetry and L-band remote sensing data, The Cryosphere, 12, 993-1012, https://doi.org/10.5194/tc-12-993-2018, 2018.

1 Introduction

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Sea ice is an important factor in the global climate system, playing key roles in modulating atmosphere and ocean interaction in the polar regions, the radiation budget through albedo effects, the ocean circulation through salinity and freshwater distribution (Kurtz et al., 2011; McPhee et al., 2009; Perovich et al., 2011; Screen and Simmonds, 2010). In the last decades, there has been rapid shrinkage of Arctic sea ice cover (Comiso et al., 2008; Laxon et al., 2013; Rothrock et al., 1999; Stocker et al., 2013; Stroeve et al., 2012), particularly in summer. In addition, the Arctic sea ice is also experiencing dramatic thinning in recent years (Kwok et al., 2009; Laxon et al., 2013), with the transition to overall younger sea ice age. Besides, the snow as accumulated over the sea ice cover is important as thermal insulation, which further hinders atmosphere–ocean interaction, and due to its higher albedo as compared with sea ice. With respect to changes in the sea ice cover, there is also significant decrease of the snow depth over the sea ice cover in the Arctic (Webster et al., 2014) which bears great deviation from climatology (Warren et al., 1999), indicating changes in the hydrological cycles such as late accumulation due to late freeze onset. The accurate knowledge of the sea ice cover and the snow over the sea ice is key to the understanding of related scientific questions in climate change as well as operational usage such as seasonal forecast.

Basin-scale observation of the sea ice cover mainly relies on satellite-based remote sensing. Among the various sea ice parameters retrieved from satellite data, the most established is the sea ice concentration (or coverage). Figure 1 shows the various parameters related to satellite-based laser altimetry and (L-band) passive radiometry for the sea ice cover. Passive microwave remote sensing of both the Arctic and Antarctic is the basis of the retrieval of sea ice extent, with near-real-time coverage since about 1979 based on satellite campaigns such as Scanning Multichannel Microwave Radiometer (SMMR), the Special Sensor Microwave/Imager (SSM/I) (Cavalieri et al., 1999), AMSR-E (Comiso et al., 2003) and AMSR2 (Toudal Pedersen et al., 2017). However, the sea ice thickness is generally not retrievable through passive remote sensing techniques due to the saturation of radiative properties especially for high-frequency ranges such as SMMR or SSM/I. In situ measurements of ice thickness through moored upward-looking sonar instruments and electromagnetic induction sounders mounted on sledges, ships or helicopters/airplanes can provide sea ice thickness at specific locations or cross sections (Stroeve et al., 2014), so they are limited in terms of spatial coverage. Active remote sensing of satellite altimetry measures the overall height of the sea surface, serving as the major approach for the thickness retrieval of the sea ice. For radar altimetry (RA), it is usually assumed that the radar signals penetrate the snow cover, and the main reflectance plane is the sea ice–snow interface (Laxon et al., 2003, 2013). Therefore in RA, the sea ice freeboard is measured. The sea ice thickness can be retrieved under certain assumption of the snow loading, such as climatological snow depth data in Warren et al. (1999) for multi-year sea ice (MYI) and halved for the first-year sea ice (FYI). For laser altimetry as in ICESat (Kwok and Cunningham, 2008; Kwok et al., 2009), the main reflectance surface is the snow–air interface, and the directly retrieved value is actually the snow (or total) freeboard. The snow loading is also required for the conversion of the snow freeboard to the sea ice thickness. As analyzed in Tilling et al. (2015) and Zygmuntowska et al. (2014), the uncertainty in snow depth is the most important contributor to that of the sea ice thickness and volume.

The major reason for the uncertainty in snow depth and the loading on the sea ice cover is the lack of stable product for snow depth over the sea ice with good temporal and spatial coverage. The snow data as used in ICESat (Kwok and Cunningham, 2008) are derived from reanalysis data and satellite retrieved sea ice motion, while the climatological snow depth data in Warren et al. (1999) as used by CryoSat-2 (Laxon et al., 2013) contain large uncertainty due to interpolation and interannual variability and may not be adequate for the present day under the context of climate change (Kwok et al., 2011; Webster et al., 2014). The retrieval of snow depth with passive microwave satellite remote sensing has been carried out in various studies. In Comiso et al. (2003), multi-band data from AMSR-E are utilized, but only for snow cover over FYI. Maaß et al. (2013b) explored the retrieval of snow depth over thick sea ice with L-band data from Soil Moisture Ocean Salinity (SMOS). SMOS provides full coverage of polar regions on a near-real-time (daily) basis. It has great advantage over satellite altimetry, which can only achieve basin coverage on the scale of about 1 month. However, the sea ice thickness is required for the retrieval. Besides, with the better penetration of L-band signal in the sea ice cover, it is also demonstrated that there is retrievability of thin sea ice thickness with L-band data, as in Kaleschke et al. (2010) and Tian-Kunze et al. (2014). Although airborne remote sensing methods have limited spatial and temporal coverage, campaigns such as NASA's Operation IceBridge (OIB) carry out high-resolution scanning of the sea ice cover (Brucker and Markus, 2013; Kurtz and Farrell, 2011; Kurtz et al., 2013; Kwok et al., 2011) and provide invaluable data that are organized into flight-track-based segments of the sea ice cover. The data can be adopted for the analysis of the status and variability of the sea ice cover at fine scale, as well as basin-scale studies as in Webster et al. (2014).

In this article, we propose a new algorithm that achieves simultaneous retrieval of both sea ice thickness and snow depth, based on two observations: the L-band passive microwave remote sensing and the laser altimetry that measures the total freeboard of sea ice. The potential of retrieval of these parameters lies in that both observations (freeboard and L-band radiative properties) are determined by these sea ice parameters. Specifically, we use OIB data (sea ice thickness, snow depth and snow freeboard) and concurrent SMOS L-band brightness temperature (TB) to simulate the simultaneous retrieval. It is found that the covariability of snow depth and freeboard at the local scale can greatly affect the well-posedness of the retrieval problem, and it is crucially important to include such covariability in the retrieval algorithm. Based on both realistic retrieval scenarios and large-scale retrieval with OIB and SMOS data, we demonstrate that the proposed algorithm can simultaneously retrieve both sea ice thickness and snow depth, and the error in the retrieved parameters mainly arises from the discrepancy between the sea ice area that corresponds to the SMOS measurement and that scanned by OIB. In Sect. 2 we first introduce the data, the models and the protocol of the combined retrieval. Detailed statistics of snow depth and the effects of covariability is covered in Sect. 3. By integrating the covariability information, we propose the retrieval algorithm and carry out evaluation and analysis in Sect. 4. Section 5 summarizes the article and provides discussion of related topics and future work.

2 Data and models

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In order to construct and evaluate the retrieval algorithm, we mainly utilize
two datasets, SMOS and OIB. SMOS measures the microwave radiation emitted
from the Earth's surface in L-band (1.4 GHz). In this article, we adopt the
L3B TB product from SMOS. The daily gridded SMOS TB data field is generated
from multiple snapshots within a day, with each snapshot involving multiple
incident angles (ranging from 0 to 40^{∘}) and spatially varying gain.
The data are provided on the Equal-Area Scalable Earth (EASE) grid with a grid
resolution of 12.5 km. However, due to the limitation of the satellite's
antenna size, the effective resolution of L-band radiometer onboard SMOS is
about 40 km.

High-resolution airborne remote sensing of sea ice parameters is available
from OIB missions, starting in 2009 and covering the western Arctic during winter
months (mainly around March). This paper utilizes OIB measurements from 2012
to 2015, during which the measurements include surface temperature of the sea
ice cover. The product is organized into tracks and includes along-track
measurements of total (or snow) freeboard, surface temperature and snow depth.
Due to the nature of the airborne measurements, the observations are limited
to a narrow swath on the order of 100 m. Snow freeboard products are
produced from Airborne Topographic Mapper (ATM) laser altimeter (Studinger, 2010).
Sea ice thickness is retrieved from snow freeboard (denoted FB_{s})
and snow depth (denoted *h*_{s}), which is measured by the University
of Kansas' snow radar (Leuschen, 2014). Surface temperature is determined
from the IceBridge KT-19 infrared radiation pyrometer dataset (Shetter et al., 2010).
There is also accompanying sea ice type information, which is from EUMETSAT
OSI-SAF system (Aaboe et al., 2016). Therein, the OIB Level-4 product IDCSI4
is adopted (Kurtz et al., 2013) for 2012–2013 and the remaining OIB data for
2014–2015 are from IDCSI2 Quicklook product, which is also available at NSIDC
DAAC. Both of these datasets are 40 m in resolution in the along-track
direction.

Due to the difference between OIB and SMOS data in both temporal and spatial coverage, we outline the following protocols of using the two datasets. OIB and SMOS data are taken from the same day. Spatially, for each OIB flight track, we locate all the EASE grids that contain OIB measurements. Figure 2 shows a typical case. Since OIB measurements are of a small swath, we consider the OIB data (of 40 m resolution) as samples of the underlying sea ice cover that contributes to a single SMOS TB measurement. However, because the inherent resolution of SMOS is about 40 km and the daily gridded field is used in this study, we approximate the correspondence of OIB and SMOS TB by considering OIB measurements in the adjacent 3 × 3 cells (the red segment in Fig. 2) of equal contribution to the SMOS TB at the central cell (the one bounded by thick blue lines in Fig. 2). In total, the nine cells cover an area of about 37.5 km × 37.5 km, which is coherent with the physical resolution of SMOS data.

It is worth noting that the area as covered by a single scan of the OIB track
consists of less than 5 % of the total area that contributes to the SMOS
TB. Therefore, we only treat the OIB data as samples of the underlying sea
ice cover. The OIB sample count (denoted *M*) ranges from several hundreds to
over 1000. The mean value of *M* is about 700, but there exist certain areas
that are scanned more extensively, which correspond to large values of *M*.
Figure A1 shows the distribution of *M* for all
available OIB data.

In order to exclude the potential effect of insufficient sampling or the
inhomogeneity of the sea ice cover, we further exclude the following data for
the analysis and evaluation. First, if an area is under-sampled by OIB
(*M* < 100), it is not considered for further analysis. Second, we exclude
the cases in which a single SMOS TB corresponds to OIB samples with different
sea ice types (i.e., mixed MYI and FYI). Third, we also exclude the cases
involving sea ice leads as detected by the sea ice lead map in
Willmes and Heinemann (2015a) or sea ice concentration lower than 1 according
to Cavalieri et al. (1996). The purpose of these treatments is to rule out the
factors that may compromise the quality of the OIB samples and allow focus on
the discussion of the retrieval algorithm.

The snow freeboard as measured by OIB and the SMOS TB is used as the input
to the retrieval. The mean snow depth (${\stackrel{\mathrm{\u203e}}{h}}_{\mathrm{s}}$) and mean
sea ice thickness (${\stackrel{\mathrm{\u203e}}{h}}_{\mathrm{i}}$) among OIB samples are used for
verification of the retrieval. Additionally, since we assume the underlying sea
ice cover as homogeneous within the retrieval scale (within nine cells) and
treat OIB measurements as samples to it, we also use the *M* measurements of
snow depth to study the statistics of the snow depth and its covariability
with snow freeboard.

The L-band (1.4 GHz) radiative property of the sea ice cover is
characterized through numerical modeling based on Burke et al. (1979). The
model was originally designed for the modeling of radiative transfer of the
X- and L-band soil moisture. In Maaß et al. (2013b), this model is applied to
sea ice and further used for the retrieval of snow depth over thick sea ice.
In these works, a simple one-layer formulation is used for both the sea ice and
the snow cover over it. In order to better characterize the radiative
properties of the sea ice, in this article we use a multi-layer formulation
of the model with sea ice type-dependent vertical salinity and temperature
profile (Zhou et al., 2017). The temperature profile in the vertical
direction is linear in either the snow cover or the sea ice, assuming
homogeneous thermal conductivity within the snow or the sea ice. Therefore
the temperature in each sea ice or snow layer can be fully decided given the
parameters of thermal conductivity, the ice bottom temperature (assumed to be
−1.8 ^{∘}C) and the snow surface temperature. The salinity profile
of FYI differs from that of MYI. For FYI, the salinity of all layers of the
sea ice all equals the bulk salinity, which decreases with the sea ice
thickness. For MYI, a surface-drained profile is adopted to reflect the
effect of summer melt and flushing. Figure 3a shows the sea
ice salinity profiles under the different sea ice types or thickness. The
dielectric properties, the emissivity of the layers and the overall radiative
properties of the sea ice cover are modeled, following Kaleschke et al. (2010)
and Maaß et al. (2013b). The convergence of the modeled TB with respect to the
layer count is witnessed, which is consistent with the study in
Maaß et al. (2013b). In Zhou et al. (2017), it is demonstrated that the
multi-layer treatment and the salinity profile MYI yield good fit between
the simulated TB and SMOS TB. Appendix A covers details of the model and the
verification with OIB and SMOS data. Figure 3c and d show
the modeled TB under typical sea ice parameters for FYI and MYI under typical
winter Arctic conditions (surface temperature of −30 ^{∘}C). The
green contour lines are constant FB_{s} lines. With the thickening
of sea ice cover, the value of TB increases and saturates when *h*_{i}
is large enough (larger than 2.5 m). The value of TB is not monotonic with
respect to FB_{s}, and two solutions are possible for certain value combinations of snow
freeboard and TB. This results in the potential
problem of ill-posedness of the retrieval with realistic observational data,
as is discussed in Sect. 3.2.

In order to match the protocol of the SMOS TB data product, we also simulate
the mean of horizontal and vertical polarization TB from 0 to 40^{∘}. We
consider the correspondence between a single TB value from SMOS and the
arithmetic mean of all the *M* TB values simulated by the radiation model
using the *M* corresponding OIB samples (each with sea ice thickness, snow
depth, surface temperature and sea ice type). Figure 3b
shows the comparison of modeled TB and SMOS TB, by using all available data.
The least squares (LSQ) fit line (dashed line) and the LSQ fit line with the
constraint that the slope be 1 (dotted line) are shown. The root mean square
error (RMSE) in modeled TB as compared with SMOS data is about 3.1 K. The
*R*^{2} value for the second fit is 0.54 with an intercept of −1.637 K,
which is treated as a model bias and canceled in further studies. As noted in
Sect. 2.2, there is potentially insufficient sampling of OIB
data, so we further consider areas with more extensive OIB sampling.
Specifically, cells with large values of sample count *M* (over 95th
percentile) are considered to be more thoroughly scanned spatially, and the
RMSE of TB for these cells drops to 1.41 K. Figure A2 shows
the relationship between RMSE of TB to the value of *M*, which demonstrates
that the lack of sufficient spatial coverage is an important source for the
difference between the modeled TB and the SMOS observation. Based on the
aforementioned RMSE of 1.41 K for well-surveyed regions, we only consider
the retrieval for cells with an TB error within 1.5 K for further studies.
In all 412 TB cells are available, containing 35 OIB tracks and 321 168 OIB
measurements. They account for about 50 % of all available TB cells. We
consider this a limitation of combined usage of OIB and SMOS data, and the
retrieval with actual satellite laser altimetry and L-band TB can be free
from this limitation through better altimetric scanning and wider swath as
compared with OIB.

Apart from the L-band radiation model, the other model as used by the
retrieval is the equilibrium model based on the buoyancy relationship. Under
certain assumptions of the sea ice density (denoted *ρ*_{ice}), seawater density (denoted *ρ*_{water}), snow density (denoted
*ρ*_{snow}) and the equilibrium state, the sea ice thickness, snow
depth and snow freeboard FB_{s} are constrained according to
Eq. (1). The sea ice thickness can be derived given
the snow depth, according to Eq. (2). This model is widely
applied for both radar and laser altimetry for the retrieval of sea ice
thickness.

$$\begin{array}{}\text{(1)}& {\displaystyle}& {\displaystyle}{\mathit{\rho}}_{\mathrm{ice}}\cdot {h}_{\mathrm{i}}+{\mathit{\rho}}_{\mathrm{snow}}\cdot {h}_{\mathrm{s}}={\mathit{\rho}}_{\mathrm{water}}\cdot \left({h}_{\mathrm{i}}+{h}_{\mathrm{s}}-{\text{FB}}_{\mathrm{s}}\right)\text{(2)}& {\displaystyle}& {\displaystyle}{h}_{\mathrm{i}}={\displaystyle \frac{{\mathit{\rho}}_{\mathrm{water}}}{{\mathit{\rho}}_{\mathrm{water}}-{\mathit{\rho}}_{\mathrm{ice}}}}\cdot {\text{FB}}_{\mathrm{s}}-{\displaystyle \frac{{\mathit{\rho}}_{\mathrm{water}}-{\mathit{\rho}}_{\mathrm{snow}}}{{\mathit{\rho}}_{\mathrm{water}}-{\mathit{\rho}}_{\mathrm{ice}}}}\cdot {h}_{\mathrm{s}}\end{array}$$

In this study, *ρ*_{water} and *ρ*_{ice} are taken to be
1024 and 915 kg m^{−3}, which are derived from field measurements
discussed by Wadhams et al. (1992), and *ρ*_{snow} is
320 kg m^{−3}, derived from Warren et al. (1999).

3 Retrievability analysis

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Under the observational constraints of TB and FB_{s}, both sea ice
thickness and snow depth over sea ice can be retrieved (Xu et al., 2017).
Figure 3c and d show TB as simulated by the radiation model
(Sect. 2.3 and Appendix A) under a range of sea ice
parameters. Specifically, the constant snow freeboard lines (with freeboard
values) are shown. With the observed TB and the corresponding observation of
FB_{s}, the values of sea ice thickness and snow depth can be
attained through a solving process that involves the two aforementioned
forward models. The theoretical retrieval problem (shown in
Fig. 3) is studied in Xu et al. (2017), with treatment of
ill-posed cases which involve two potential solutions.

For the retrieval with actual observational data, the resolution difference
between the two types of observations should be accounted for. Previously in
Sect. 2.2, we used a high-resolution altimetry scans as samples
for L-band passive radiometry, which is of relatively coarser resolution. In
this section we further analyze the statistical covariability between
*h*_{s} and FB_{s} on the scale of retrieval. Under the
context of retrieval, we base the analysis with the freeboard measurements as
a priori and focus on how the snow depth changes with freeboard in a
statistical sense. For each TB measurement, the multiple (*M*) OIB samples
are subjected to statistical analysis, which shows that among these samples
there exists statistically significant correlation between FB_{s}
and *h*_{s}, which can be better characterized by a nonlinear fitting.
Furthermore, the effect of the covariability on retrievability is analyzed in
Sect. 3.2.

For the covariability between FB_{s} and *h*_{s}, we choose
the native resolution of the OIB product (40 m) as the spatial scale for
analysis. Each TB corresponds to multiple (*M*) OIB samples, with each sample
containing the measurement for both FB_{s} and *h*_{s}. We
divide these samples into FB_{s} bins, with each bin covering 5 cm.
In total there are 30 bins, covering the range of 0 to 1.5 m. For samples in
each bin, we compute the percentiles and the mean value of *h*_{s}.
Figure 4a shows the mean *h*_{s} and the ±1
standard deviation range and their relationship with FB_{s}, for
four
representative TB points. Furthermore, we carry out least squares linear
fitting (weighted according to sample count in each bin) between
FB_{s} of the bins and the corresponding mean *h*_{s} in each
bin. Among all available data, there is statistically significant positive
correlation between *h*_{s} and FB_{s} for over 90 % of
all points. The values of *R*^{2} are in the range of 0.06 and 0.89 (95 %
percentile), with the mean value of *R*^{2} as 0.53. This indicates that there
is consistent covariability between snow depth and snow freeboard across
Arctic sea ice cover.

However, for both FYI and MYI ice, there is saturation of the mean
*h*_{s} with respect to FB_{s}. Besides, in the Arctic
inundation is generally uncommon (i.e.,
*h*_{s} < FB_{s}). In order to accommodate these
characteristics, we propose a nonlinear fitting, as shown by
Eq. (3). The parameters *α* and *β* are fitted
according to observations. According to the equation, the value of
*h*_{s} saturates to $\mathit{\alpha}\phantom{\rule{0.125em}{0ex}}\cdot \phantom{\rule{0.125em}{0ex}}\mathit{\pi}/\mathrm{2}$ when FB_{s} is
large, and the value of *α* ⋅ *β* (denoted *s*), which is the
slope of the function at FB_{s}=0, should be lower than 1 in order to
avoid any inundation.

$$\begin{array}{}\text{(3)}& {\displaystyle}{h}_{\mathrm{s}}\left({\text{FB}}_{\mathrm{s}}\right)=\mathit{\alpha}\cdot \mathrm{arctan}\left(\mathit{\beta}\cdot {\text{FB}}_{\mathrm{s}}\right)\end{array}$$

Using Eq. 3, the overall quality of the fitting for all
available local OIB segments is improved, with mean value of *R*^{2} rising
from 0.53 to 0.67, and the 95 % percentile of *R*^{2} rises to 0.23 and
0.92, respectively. Detailed distribution of the fitted parameters for all OIB
data is shown in Appendix B (Fig. B1 for FYI and
Fig. B2 for MYI). Based on statistics of all the
available OIB data, the value of *s* for the local OIB segment is in the
range of 0.49 and 0.96 (95 % percentile) with a single mode distribution
for both MYI and FYI (Fig. B1c
and B2c). For FYI, the mean value of *s* is 0.71
and for MYI 0.95, which implies a generally thicker snow cover over MYI.
Among all the local OIB segments, 80 % of them witnessed a value of *s*
lower than 1.

Furthermore, we consider the value of *s* to be stable across either FYI or
MYI sea ice and choose these values as universal parameters for the design
of the retrieval algorithm. Figure 4b shows fitting
function of snow depth over snow freeboard based on these representative
values of *s* under various values for *α*.

We evaluate the covariability and its effect on retrieval from several
aspects. We choose five realistic retrieval scenarios among all the OIB and SMOS
data, with two of them representing FYI retrieval and three of them for MYI.
As shown in Table 1, they represent typical retrieval
problems for Arctic sea ice. Besides, the simulated TB values by the
radiation model is close to the corresponding SMOS TB values (within 1.5 K).
Based on these scenarios, we examine whether it is possible to retrieve the
actual sea ice thickness and snow depth, with or without the covariability.
Firstly we ignore the covariability and assume a flat snow cover: for the
*M* OIB samples, we assume that the snow depth is uniform. For the retrieval
problem, since the directly observed values are freeboard samples
(FB_{s}|_{m}, where *m* is the index of the samples, and
$\mathrm{1}\phantom{\rule{0.125em}{0ex}}\le \phantom{\rule{0.125em}{0ex}}m\phantom{\rule{0.125em}{0ex}}\le \phantom{\rule{0.125em}{0ex}}M$), we carry out the scanning of the (uniform) snow depth
*h*_{s} from 0 m (snow free) to 1 m (sufficiently deep). Under a
certain value of *h*_{s}, we retrieve the sea ice thickness
*h*_{i}|_{m} for each FB_{s}|_{m} with
Eq. (2), based on the current value of *h*_{s}.
Then the TB value for this sample (TB|_{m}) can be calculated according to
the L-band radiation model, with *h*_{i}|_{m}, *h*_{s} and
surface temperature *T*_{sfc}|_{m}. The mean TB value is then computed
as the arithmetic mean of all TB|_{m}'s, for the current value of
*h*_{s}. For any OIB sample, if the value of freeboard is smaller than
the current value of *h*_{s}, in order to avoid inundation, the snow
depth for this sample is assumed to be the same as FB_{s}. If the
number of samples that witness potential of inundation over 50 % of *M*,
we stop the scanning even if *h*_{s} has not reached 1 m.

In order to incorporate the effect of covariability, we adopt either the
global value of *s* (0.71 for FYI and 0.95 for MYI) or the locally fitted
value of *s* (specific to each scenario) and carry out the retrieval.
Figure 5a shows four typical distribution of FB_{s}, and
Fig. 5b shows a range of values for *α* (0 to 1) and the
resulting mean value of *h*_{s} for the four typical distributions. For
the range of 0 to 1, the resulting mean *h*_{s} covers a continuous
range for each distribution. For each distribution, when *α* is very
small, the corresponding *h*_{s} is very small for whole range
FB_{s}, resulting in a very small value of mean *h*_{s}.
Furthermore, the value of mean *h*_{s} approaches 0 when *α*
approaches 0, which in effect corresponds to “bare ice”. With the grow of
*α*, there is a monotonous increase in the mean *h*_{s}; and
when *α* is large enough, the mean *h*_{s} saturates. For all four FB_{s} distributions, we consider that the resulting mean
*h*_{s} is reasonable for the range of *α*. Therefore, the
retrieval of snow depth is attained by locating the proper value of *α*.
Due to the potential of double solution in the retrieval, the solving of
*α* is attained by a scanning process that covers the reasonable range
for *α*. The scan starts from 0.001 and steps by 0.01, and it is limited to
a large value that yields saturation for mean *h*_{s}. With each
scanned value of *α*, a corresponding value for *β*, can be computed
as *s* ∕ *α*, and the snow
depth *h*_{s}|_{m} for each sample can be computed with
Eq. (3). Then the *h*_{i}|_{m}, the TB values for
each sample can be computed, as well as the mean snow depth and mean TB.

We record the (mean) snow depth and the corresponding mean TB across the
scanning process. Figure 6 shows the results of
scanning for the five scenarios in Table 1. Note that for
the lines that represent scanning of *α* (i.e., involving
covariability), the *x* axis is the resulting values of mean *h*_{s},
not *α*. The observed TB and the simulated TB (with OIB data) are shown
by solid and dot-dashed horizontal lines, respectively. Besides, the observed
mean snow depth and the 50 % inundation with flat snow cover are shown by
solid and dashed vertical lines, respectively. The simulated TB with flat
snow cover (black dashed curve in each subfigure) is always lower than that
with covariability information (blue dashed curves for results with global
*s* and red ones for those with local *s*). For all the scenarios, the TB
values that are attained through scanning can reach the observed TB with the
incorporation of covariability, while the values of TB in two scenarios (III and IV) fail to reach the observation with the flat snow cover assumption.
This implies that with the flat snow cover assumption, there is no solution
to the retrieval problem. We further examine the other three scenarios; the
solutions of the retrieval problem reside at the cross point of the scanned TB
curves and the horizontal bars that represent observational TB values. The
solutions of mean snow depth under the flat snow cover assumption are always
larger than the observed mean snow depth by over 5 cm.

For the comparison between the covariability incorporated scanning with local
*s* and global *s*, we show that for scenarios I, II, III and IV the
solutions of the two scannings are close to each other (within 2 cm). For
scenarios II, III and IV, the solutions as produced by the scanning is close
to the observed snow depth. The differences between the solutions produced by
scanning and the observed snow depth are 5 cm or larger for scenarios I and
V, while the scanning with local *s* produces smaller errors. It is worth
noting that for the actual retrieval process, the local value of *s* is not
available, and only the global value of *s* is usable. Lastly, for
scenario III, two potential solutions exist (two crossing points between the
TB scanning curve and the observational TB). Without extra observational data
during retrieval, it is not possible to judge which solution is the true (or
better) one. Therefore the retrieval algorithm should be able to locate both
possible solutions.

The covariability as observed with OIB data plays an important role in the
retrievability of the sea ice parameters. Also with OIB data, we extract the
statistical relationship (Eq. 3) that characterizes the
covariability which can be incorporated in the retrieval. However, during
retrieval, the parameter *s* is generally not available for the local area,
and the global values of *s* (for FYI and MYI) as computed from
high-resolution OIB data can be adopted.

4 Retrieval algorithm and evaluation

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With the statistically significant covariability, we design the retrieval
algorithm for sea ice thickness and snow depth that includes two distinctive
phases. The overall structure of the algorithm is similar to the theoretical
retrieval algorithm in Xu et al. (2017). The incorporation of covariability
is further integrated, based on the nonlinear fitting in
Eq. (3) and the fixed value of *s* for both FYI and MYI
derived from OIB data. The first phase involves the scanning of possible snow
depth configurations. This phase is in effect carried out by the scanning of
the value of *α* from 0.001 to 3 (or sufficiently large). A possible
solution is detected between two adjacent values of *α*, when the TB
values as generated with these two values of *α* are on different sides
of the observed TB. During the second phase, all the possible solutions are
then attained with an iterative binary search of *α*. All possible
solutions are reported by the retrieval algorithm. The outline of the
algorithm is presented in Fig. 7, with the two phases marked
out by red and blue boxes, respectively. We also construct a reference
retrieval algorithm based on the flat snow cover assumption, for which the
scanning is over the snow depth instead of *α*. The details of this
reference algorithm is omitted for brevity.

For the typical scenarios in Table 1, we carry out the
retrieval for the mean sea ice thickness ($\stackrel{\mathrm{\u203e}}{{h}_{\mathrm{i}}}$) and the
mean snow depth ($\stackrel{\mathrm{\u203e}}{{h}_{\mathrm{s}}}$) using the standard algorithm with
either the global or the local values of *s*, as well as the reference
algorithm. Table 2 shows the comparison of the
retrieval results and observations. The reference algorithm (with flat snow
cover assumption) consistently performed worse than the standard algorithm.
For scenarios I and IV, it failed to attain any solution. For the standard
algorithm, small error in both $\stackrel{\mathrm{\u203e}}{{h}_{\mathrm{i}}}$ and
$\stackrel{\mathrm{\u203e}}{{h}_{\mathrm{s}}}$ is attained with the local values of *s* specific
to each scenario, as compared the retrieval with the global values of *s*.
Besides, for scenario III for which two solutions are possible, the retrieval
algorithm addresses both of them. The retrieval results are consistent with
the retrievability analysis in Sect. 3.2.

We further carry out verification of the retrieval algorithm in two aspects. First, by using all available OIB data, we simulate the retrieval problem with laser altimetry measurements and verify the retrieved $\stackrel{\mathrm{\u203e}}{{h}_{\mathrm{i}}}$ and $\stackrel{\mathrm{\u203e}}{{h}_{\mathrm{s}}}$ against OIB measurements. Section 4.1 covers the retrieval and analysis. Furthermore, we construct several representative retrieval scenarios in Sect. 4.2 and analyze the uncertainty in the retrieved parameters and carry out attribution of the uncertainty to input parameters of the retrieval.

For the systematic verification of the proposed algorithm, we carry out the
retrieval with all the available OIB data (as mentioned in
Sect. 2.3) from 35 OIB tracks and 412 SMOS TB
measurements, which correspond to 412 retrieval cases. For each SMOS TB, the
corresponding samples (snow freeboard, surface temperature and sea ice type)
from OIB dataset are used as the input for the retrieval. The
retrieval with the flat snow cover assumption (the reference algorithm) is
only successful for 50 cases, which accounts for about 12 % of available
cases. For comparison, the (standard) algorithm achieves retrieval for 391
cases (95 %) with the global *s* values and for all the TB values with
the locally fit *s* values. Figure 8 shows the
comparison of retrieved mean sea ice thickness and snow depth with
observations. Figure 8a and b shows the results for
$\stackrel{\mathrm{\u203e}}{{h}_{\mathrm{i}}}$ and $\stackrel{\mathrm{\u203e}}{{h}_{\mathrm{s}}}$, based on
(1) simulated TB (as computed from the radiation model) and (2) the local
value of *s*. This represent the “idealized” retrieval problem in which
there exists no extra uncertainty. As shown in
Fig. 8a, the LSQ fit for $\stackrel{\mathrm{\u203e}}{{h}_{\mathrm{i}}}$
(dash line) features a *R*^{2} value of 0.966, while the LSQ fit under the
extra constraint on slope (dotted dash line) features a *R*^{2} value of 0.964.
For snow depth (Fig. 8b), the *R*^{2} values for the
two fittings are both 0.844. This indicates that the retrieval is in good
agreement with the observations.

For the actual retrieval problem for which the local value of *s* is unknown,
and the observational TB values from SMOS are used,
Fig. 8c and d show the evaluation for
$\stackrel{\mathrm{\u203e}}{{h}_{\mathrm{i}}}$ and $\stackrel{\mathrm{\u203e}}{{h}_{\mathrm{s}}}$, respectively. The
fitting quality (in terms of *R*^{2}) for sea ice thickness is as high as 0.89
and that for snow depth is 0.637. It is worth noting that these results are
achieved with only statistical data derived from large-scale OIB surveys.
Furthermore, if the retrieval is based on (1) observed TB from SMOS and
(2) the locally fitted value of *s*, the *R*^{2} values for the fitting are
0.91 and 0.65 for sea ice thickness and snow depth, respectively, with
virtually no change in the fitting lines (not shown). There is minor increase
in quality (0.91 versus 0.89 and 0.65 versus 0.637) and a relatively large
gap to the “idealized” retrieval. As a comparison, we also carry out
retrieval with the TB with forward model and the local values of *s*, and the
*R*^{2} for fittings between the retrieved and the observed parameters for sea
ice thickness and snow depth are 0.96 and 0.84, respectively. This indicates
that the difference (or error) of the modeled and the observed TB plays an
important role in affecting the quality of the retrieval. The discrepancy
between the observed TB and the modeled TB may arise from
(1) the imperfect radiation model, including its formulation as well as the
model parameters, or (2) the mismatch between the altimetry scans and L-band
passive observations, as introduced in Sect. 2.2. The areas
with more extensive OIB scans are shown of lower TB error (see
Fig. A2), indicating that the error in the retrieved
parameters can be potentially reduced with better altimetry coverage.

For comparison, we also carry out the retrieval which only involves TB and
the mean value of FB_{s}. This retrieval problem ignores the
resolution difference between altimetry scans and L-band radiometry and
generally corresponds to the theoretical retrieval problem analyzed in
Xu et al. (2017). Specifically, for the use of OIB data, the mean value of
*M* samples of FB_{s} is computed and further combined with TB for
the retrieval of a single value for both *h*_{i} and *h*_{s}.
Since only the mean FB_{s} is involved in the retrieval,
covariability does not play a role in the retrieval. By using the same SMOS
and OIB data as the evaluation in Fig. 8, the
retrieval yields *R*^{2} of 0.78 and 0.50 for *h*_{i} and *h*_{s}
(fitting between the retrieved and the observed parameter). For comparison,
under the realistic retrieval results (Fig. 8c
and d), the quality of retrieval is much improved for both *h*_{i}
(*R*^{2} from 0.78 to 0.89) and *h*_{s} (*R*^{2} from 0.50 to 0.64). This
demonstrates that the high-resolution altimetry samples and the accompanying
covariability information play an important role in improving the quality of
the retrieval.

Based on the retrieval with large-scale observational data, the proposed algorithm achieves effective retrieval of both sea ice thickness and snow depth, by using simultaneous remote sensing of the sea ice cover, i.e., laser altimetry and L-band passive microwave sensing. The statistics of snow depth and its covariability with snow freeboard on the spatial scale of retrieval play an important role in improving the well-posedness of the retrieval problem as well as the quality of the retrieved parameters.

In order to assess the uncertainty of the retrieved parameters, we further
design four realistic retrieval scenarios from OIB and SMOS data listed in
Table 3a. Due to the nonlinear relationship between sea ice
parameters and TB, we cannot directly compute the uncertainty in
*h*_{i} or *h*_{s}. Instead, Monte Carlo (MC) simulation is
adopted. For each scenario in Table 3a, four sets of MC
simulations are constructed, each containing: (1) random perturbations to TB
only, (2) random perturbations to FB_{s} only, (3) random
perturbations to *s* only, and (4) random perturbations to TB,
FB_{s} and *s* altogether. Each set contains 1000 random sampling to
these parameters.

The perturbations to TB follow normal distribution and SMOS dataset (in terms
of standard deviation of the uncertainty). The perturbations to the *M*
values of FB_{s} are based on OIB data specification and follow
log-normal distribution. The perturbations to *s* are specific to sea ice
type (FYI or MYI) and based on the statistics of *s* as derived from all OIB
data. As shown in Appendix B, the distribution of *s* can be well
characterized by beta distribution for both FYI and MYI. The fitting to beta
distribution is then carried out for both FYI and MYI according to
Eq. (4), where *a*, *b* and const are fitted parameters by
using OIB data at 40 m resolution (see Fig. B3). For FYI,
*a*, *b* and const are 4.31, 2.00 and 1.00, respectively, and for MYI are 4.25,
2.06 and 1.2.

$$\begin{array}{}\text{(4)}& {\displaystyle}f\left(x\right|a,b,\text{constant})={\displaystyle \frac{\text{const}}{B(a,b)}}{x}^{(a-\mathrm{1})}(\mathrm{1}-x{)}^{(b-\mathrm{1})}\end{array}$$

The perturbations to *s* follow the fitted beta distribution. Furthermore,
the perturbations to TB, FB_{s} and *s* are treated as independent.
Each MC simulation (of 1000 simulations) contains a set of perturbed input
parameters and corresponds to a retrieval problem. Based on the results from
the 1000 simulations, the uncertainty of the retrieved *h*_{i} and
*h*_{s} are computed by biased standard deviation estimation with
respect to the original retrieval which involves no perturbation.
Table 3b shows the relative uncertainty of *h*_{i} and
*h*_{s} for each experiments for all scenarios. First, the relative
uncertainty for *h*_{i} or *h*_{s} is at most about 25 %.
Also, all scenarios show that *s* plays a minor role in terms of uncertainty,
as compared with TB or FB_{s}. TB and FB_{s} play a
comparable role in the uncertainty of the retrieved parameters. Moreover, for
both FYI and MYI, the uncertainty in the retrieved *h*_{i} and
*h*_{s} is relatively lower for thicker ice and deeper snow cover. The
uncertainty of TB (or FB_{s}) is not correlated spatially and that
of *s* is based on basin-scale statistics from OIB. Therefore, the
uncertainty of the retrieved *h*_{i} (or *h*_{s}) is not
spatially correlated, resulting in effective reduction of the uncertainty in
the sea ice volume (or snow volume).

5 Summary and discussion

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In this study, we introduce a new algorithm for retrieving multiple Arctic sea ice parameters based on a combination of L-band passive microwave remote sensing and active laser altimetry. Two physical models, the L-band radiation model and the buoyancy relationship, are adopted to constrain the sea ice thickness and snow depth. They are used as forward models during an iterative retrieval process that solves the sea ice parameters that satisfy the observed L-band TB and snow freeboard values. Specifically, according to high-resolution observations, there is statistically significant covariability between the snow depth and the snow freeboard. This information of covariability is further incorporated in the retrieval algorithm, and it is demonstrated that the covariability plays a key role in the retrievability. Specifically, a nonlinear fitting that characterizes the covariability is derived from OIB data, and a parameter (initial slope of the fitting function) is considered invariant for FYI and MYI and further adopted by the retrieval algorithm. Verification with available OIB data shows that both sea ice thickness and snow depth are retrieved, with the error in both parameters mainly arising from the mismatch between modeled and observed TB values. This algorithm can be applied to the large-scale retrieval of sea ice thickness and snow depth using concurrent L-band satellite remote sensing and satellite altimetry of the sea ice cover such as Abdalati et al. (2010).

In traditional (laser) satellite altimetry, the retrieval of sea ice thickness mainly relies on (adapted) climatological snow depth or data as derived from reanalyses, which may contain unconstrained uncertainty due to model biases and missing physical processes. Besides, these snow depth data usually lack fine-scale details that match the resolution of satellite altimetry, such as the covariability characteristics. In contrast, the retrieval of snow depth using L-band SMOS data as in Maaß et al. (2013b) relies on the a priori knowledge of the thickness of the (thick) sea ice. Contrary to these existing retrieval algorithms, the proposed algorithm carries out retrieval of both sea ice thickness and snow depth, with the concurrent active and passive remote sensing of the sea ice cover. Since no climatological snow depth or any other derived snow data are used in the algorithm, the retrieved sea ice thickness does not suffer from the potential lack of efficacy of these data.

In Kwok et al. (2011), statistical analyses are carried out between snow depth and snow freeboard, which also show covariability between the two. As also noted in Kwok et al. (2011), the derivation of these two parameters is measured with independent instruments by OIB. The statistically significant relationship as represented by covariability is due to physical processes relating to the snow loading and its effects on the total freeboard. However, it is worth noting that the scale and the resolution as adopted in Kwok et al. (2011) are about 400 and 4 km, respectively. They are both much larger than those used in this study (about 40 km and 40 m). While the analysis in Kwok et al. (2011) is carried out on coarser spatial scales, our work focuses on the spatial scale that is relevant to the retrieval of sea ice parameters. We demonstrate that on this relatively small spatial scale, there still exists covariability between snow depth and snow freeboard.

Besides the input parameters to the retrieval (TB, FB_{s} and *s*),
model parameters also play an important role in modulating the uncertainty
for retrieval (Xu et al., 2017; Zygmuntowska et al., 2014). For this study, we adopt
constant parameters for density values following protocols of OIB, mainly for
the direct comparison with OIB dataset. However, their effect on the
uncertainty of retrieved parameters should be accounted for in a systematic
approach, similar to Xu et al. (2017). MC simulations can be adopted
for the quantification of the uncertainty through perturbations to both
input and model parameters.

The proposed retrieval method is the basis for the retrieval of sea ice
parameters with data from concurrent satellite campaigns. Although there was
no concurrent L-band satellite observation with the ICESat campaign, there
are candidate satellite campaigns such as WCOM (Shi et al., 2016), which
provides concurrent L-band observation with the planned ICESat-2 campaign.
For the study with satellite data, there are several practical issues.
First, the snow surface temperature is provided by airborne sensors in OIB
but is not generally available with laser altimetry. Several data sources serve
as candidate data for the concurrent surface temperature field, such as
reanalysis data (Dee et al., 2011), a MODIS-based product
(Hall et al., 2004). Second, there is small-scale variability of the
sea ice cover such as leads, which were not considered for the analysis and
verification in this study. As shown in Zhou et al. (2017), the presence of
sea ice leads has a profound effect in lowering the overall TB on the scale of
SMOS observations. Leads can be treated as small-scale heterogeneity of the
sea ice cover, and the incorporation of lead maps such as
Willmes and Heinemann (2015b) effectively reduces the overestimation of TB, as
studied by Zhou et al. (2017). Specifically, the lead map can be adopted by
the retrieval through the integration with the forward radiation model. Other
types of small-scale variability such as mixture of FYI and MYI, should be
also accounted for using sea ice type maps. Third, the covariability explored
in this study is on the spatial scale of the original OIB data (i.e., 40 m).
For each specific satellite altimetry, we consider the freeboard measurement
the mean freeboard value within a certain spatial range. For ICESat-2, each
laser scan dot covers a circular region of about 70 m in diameter
(Abdalati et al., 2010). The scaling of the covariability should be studied
for the specific resolution of the satellite altimetry. By using 70 m as the
typical resolution of ICESat-2, we deduce the value of *s* at this resolution
by manual coarsening OIB's data by averaging adjacent points. In effect, the
value of *s* at 80 m is computed, which shows a slight decrease of *s* for
both FYI and MYI. Figure B3 shows the general scaling of *s*
for the resolution range from 40 to 240 m. Fourth, in order to estimate the
uncertainty of the retrieved parameters, the effects of surface temperature,
as well as other data sources (including TB, freeboard measurements and the
value of *s*), should be evaluated in a systematic way. Due to the nonlinear
relationship between TB and the sea ice parameters, MC simulations
can be carried out for the quantification of the uncertainty. Besides, for
the historical data from ICESat (Kwok and Cunningham, 2008) during the first decade of
the 21st century, due to the lack of basin-scale L-band observation for the
Arctic, other passive remote sensing data such as C-band data from AMSR-E can
be exploited in a similar manner for the retrieval of these historical data.

The native spatial resolution of AMSR-E based C-band remote sensing product is over 60 km, which is coarser than that of SMOS L-band data but provides similar, daily coverage for the Arctic. Therefore, the resolution difference between AMSR-E based C-band data products and ICESat data should be accounted for in a similar approach as in Sect. 2.2. Besides, due to the relatively shorter wavelength of C-band as compared with L-band, the penetration depth of C-band signal in sea ice cover is potentially shallower, resulting in more premature saturation of C-band signal to sea ice thickness. Under the assumption of a uniform and dry snow cover, the relatively long wavelength of C-band and L-band ensures that the snow cover is “transparent” to the L- or C-band signal. For L-band and C-band, there is good potential for retrieval through the thermodynamical modulation of the sea ice thickness by the snow cover, as indicated by Maaß et al. (2013a).

The L-band radiation model as adopted by this article can also be used for the concurrent retrieval of sea ice parameters with L-band passive radiometry and RA. While laser altimetry with ICESat covered the historical era in the 2000s, CryoSat-2 based RA is an ongoing campaign which started in early 2010s and overlaps with existing L-band and C-band passive campaigns, including SMOS, SMAP and AMSR2. According to the theoretical study by Xu et al. (2017), the retrieval that combines RA with L-band data is potentially free of the ambiguous solutions present in this study. Besides, there also exists resolution differences between RA (e.g., 300 m for CryoSat-2) and L-band data such as SMOS. Measurements from RA can be treated as high-resolution sampling of the sea ice area that corresponds to a single L-band TB. Furthermore, based on the analysis and methods proposed in this article, the covariability between snow depth and sea ice freeboard can be further incorporated in the combined retrieval with RA and L-band passive remote sensing data.

Data availability

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Data availability.

SMOS data are provided by the Integrated Climate Data Center (ICDC), University of Hamburg, Germany, http://icdc.cen.uni-hamburg.de/1/daten/cryosphere/l3b-smos-tb.html (last accessed: 25 October 2017). OIB and SSM/I sea ice concentration data are provided by NASA National Snow and Ice Data Center Distributed Active Archive Center, Boulder, Colorado, USA, https://doi.org/10.5067/7XJ9HRV50O57 (last access: 25 October 2017). Daily pan-Arctic sea ice lead maps for 2003–2015 are provided by Sascha Willmes and Guenther Heinemann, with links to maps in NetCDF format. https://doi.org/10.1594/PANGAEA.854411 (last access: 14 March 2018).

Appendix A: L-band radiation model

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The L-band (1.4 GHz) radiation model as used for retrieval describes the radiation emitted from snow-covered sea ice that floats over seawater. The model was originally developed for soil moisture applications in Burke et al. (1979) and further adopted for sea ice in Maaß et al. (2013b). As introduced in Zhou et al. (2017), improvements to the model are made to better characterize the vertical structure of the sea ice that is specific to each sea ice type. Details are provided below.

The modeling of the radiative properties of the sea ice cover includes four types
of media in the vertical direction: seawater beneath the sea ice,
sea ice, snow cover over the sea ice and air. The seawater (air)
is considered to be semi-infinite beneath (above) the sea ice cover. The sea
ice is further divided into *N* layers in the vertical direction, with each
layer of the same height. For the snow cover, a homogeneous structure is
assumed, with prescribed parameters such as thermal conductivity and
permittivity. Also, a dry snow cover is assumed, and snow morphology
features (such as differentiation between wind slab and depth hoar) and other
vertical structures are not considered. The (SMOS) observed brightness
temperature (TB) is assumed to be the multi-angle mean (0–40^{∘}) TB as
radiated from the aforementioned multi-layer media.

The radiation model characterizes the vertical structure of the sea ice
cover by specifying the temperature and salinity of each layer based on the
sea ice type and the snow surface temperature (*T*_{surf}). The
vertical temperature profile is determined by the overall thermal condition
as defined by *T*_{surf} and the thermal conductivity for sea ice
(*k*_{ice}) and that of snow (*k*_{snow}). The bottom of the
sea ice is assumed to be at freezing temperature of −1.8 ^{∘}C
(denoted *T*_{water}). Based on observation-based fittings in
Untersteiner (1964) and Yu and Rothrock (1996), *k*_{ice}
and *k*_{snow} are defined as follows.

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{k}_{\mathrm{ice}}=\mathrm{2.034}\phantom{\rule{0.125em}{0ex}}\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}+\mathrm{0.13}\phantom{\rule{0.125em}{0ex}}\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{kg}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}{\displaystyle \frac{{S}_{\mathrm{ice}}}{{T}_{\mathrm{ice}}-\mathrm{273.15}}}\\ {\displaystyle}& {\displaystyle}{k}_{\mathrm{snow}}=\mathrm{0.31}\phantom{\rule{0.125em}{0ex}}\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}{\mathrm{K}}^{-\mathrm{1}}\end{array}$$

In this study, we consider the change of *k*_{ice} within the sea ice
of minor effects and use a bulk value for *k*_{ice}, resulting in a
linear temperature profile within the sea ice. This bulk value is determined
by the bulk value of *S*_{ice}. The temperature profile is assumed to
be continuous through the media interfaces, and ice temperature is assumed to
equal the snow temperature at the snow–ice interface. Given *T*_{surf}
based on other observations (such as MODIS), the bulk ice and snow
temperatures *T*_{ice} and *T*_{snow} can be written as follows
($K=({k}_{\mathrm{snow}}{h}_{\mathrm{i}}+{k}_{\mathrm{ice}}{h}_{\mathrm{s}}{)}^{-\mathrm{1}}$).

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{T}_{\mathrm{ice}}={T}_{\mathrm{water}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}K\left({T}_{\mathrm{surf}}-{T}_{\mathrm{water}}\right){k}_{\mathrm{snow}}{h}_{\mathrm{i}}\\ {\displaystyle}& {\displaystyle}{T}_{\mathrm{snow}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left({T}_{\mathrm{water}}+{T}_{\mathrm{surf}}+K\left({T}_{\mathrm{surf}}-{T}_{\mathrm{water}}\right){k}_{\mathrm{ice}}{h}_{\mathrm{s}}\right)\end{array}$$

Since a bulk value is adopted for both *k*_{ice} and
*k*_{snow}, given any *T*_{surf}, the temperature profile is
linear within the snow cover, as well as the sea ice. Then, the temperature
of each layer of the sea ice cover can be computed.

For the salinity, sea ice type is considered with differentiation between MYI
and FYI. For FYI, the salinity is assumed to be homogeneous in the vertical
direction and equals the bulk salinity as prescribed by the sea ice
thickness. The bulk salinity for FYI is in turn adapted from the multi-linear
structure in Cox and Weeks (1974) and defined as follows (where the ice
salinity, denoted *S*_{ice}, is in ppt).

$$}{S}_{\mathrm{ice}}=\mathrm{6.08}\cdot {e}^{(-\mathrm{5.81}\cdot {h}_{\mathrm{i}})}+\mathrm{7.409}\cdot {e}^{(-\mathrm{0.5228}\cdot {h}_{\mathrm{i}})$$

With the deepening of the FYI sea ice cover, the bulk salinity decreases, and
its minimum value is kept above 1.5 ppt. In contrast, for MYI, in
order to reflect the effect of brine drainage and flushing during the melt
season, a vertical salinity profile is adopted following
Schwarzacher (1959). For the *k*th sea ice layer (*k*=0 for the
surface layer of the sea ice), the mean salinity (*S*_{i,k}) is prescribed
as

$$}{S}_{i,k}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{S}_{\mathrm{max}}\left[\mathrm{1}-\mathrm{cos}\left(\mathit{\pi}{z}^{a/(z+b)}\right)\right].$$

A normalized vertical coordinate (*z*) is adopted with respect to sea ice
thickness, starting from *z*=0 for the ice surface to *z*=1 for the bottom of
the ice. For layer *k*, $z=(k-\mathrm{1}/\mathrm{2})/N$ and the corresponding salinity of the
layer can be computed. *N* is the total number of ice layers, and
*S*_{max}=3.2 ppt, *a*=0.407 and *b*=0.573, which are the fitted
parameters from in situ observations of MYI salinity. Therefore, for MYI, the
sea ice salinity ranges from 0 at the top of the surface (*z*=0) to
*S*_{max} at the bottom (*z*=1). The seawater salinity is fixed at
constant 33 g kg^{−1}.

The radiation model describes the radiation emitted from snow cover, sea ice
and seawater; the brightness temperature at the top of atmospheric
(TB_{TOA}) can be described as (Maaß et al., 2013b)

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{\text{TB}}_{\mathrm{TOA}}=(\mathrm{1}-c)\cdot \left({\text{TB}}_{\mathrm{water}}+\left(\mathrm{1}-{e}_{\mathrm{water}}\right)\cdot {\text{TB}}_{\mathrm{cosm}}\right)\\ \text{(A1)}& {\displaystyle}& {\displaystyle}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}+c\cdot \left({\text{TB}}_{\mathrm{ice}}+\left(\mathrm{1}-{e}_{\mathrm{ice}}\right)\cdot {\text{TB}}_{\mathrm{cosm}}\right)+\mathrm{\Delta}{\text{TB}}_{\mathrm{atm}}.\end{array}$$

In Eq. (A1), *c* is sea ice concentration, *e*_{ice} and
TB_{ice} are the emissivity and TB of sea ice, *e*_{water} and
TB_{water} are the emissivity and TB of seawater, and
TB_{cosm} is cosmic microwave background radiation, which can be
considered as uniform and constant (2.7 K). ΔTB_{atm}
is TB from atmospheric contribution ranging from −0.36 to +5.67 K.
Emissivity parameters are computed as follows: *e*_{water} is based on
the Fresnel equations in different directions of polarization
(Ulaby et al., 1986) and *e*_{ice} is a function of parameters such
as polarization, incidence angle, sea ice thickness, temperature, density,
salinity, surface roughness, snow depth and temperature. Based on
Maaß et al. (2013b), the permittivity of snow (*ϵ*_{snow}) is
determined by a polynomial fit obtained from measurements at microwave
frequencies ranging between 840 MHz and 12.6 GHz (Tiuri et al., 1984)
as follows.

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{\mathit{\u03f5}}_{\mathrm{snow}}=\left(\mathrm{1}+\mathrm{0.7}{\mathit{\rho}}_{\mathrm{snow}}+\mathrm{0.7}{\mathit{\rho}}_{\mathrm{snow}}^{\mathrm{2}}\right)\\ {\displaystyle}& {\displaystyle}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}+i\cdot \left(\mathrm{1.59}\times {\mathrm{10}}^{\mathrm{6}}\times \left(\mathrm{0.52}{\mathit{\rho}}_{\mathrm{snow}}+\mathrm{0.62}{\mathit{\rho}}_{\mathrm{snow}}^{\mathrm{2}}\right)\right.\\ {\displaystyle}& {\displaystyle}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left.\cdot \left({f}^{-\mathrm{1}}+\mathrm{1.23}\times {\mathrm{10}}^{-\mathrm{14}}\sqrt{f}\right){e}^{\mathrm{0.036}T}\right),\end{array}$$

where *ρ*_{snow} is the relative density of snow (compared to
water), *T* the temperature of snow in degrees Celsius and *f* the microwave
frequency. It should be noted that *ϵ*_{snow} depends on the
snow wetness, which is not considered by the current model. Permittivity of
sea ice (*ϵ*_{ice}) is confirmed by brine volume fraction
(*V*_{b}) using empirical relationship in Vant et al. (1978).

$${\mathit{\u03f5}}_{\mathrm{ice}}={a}_{\mathrm{1}}+{a}_{\mathrm{2}}{V}_{\mathrm{b}}+i\cdot \left({a}_{\mathrm{3}}+{a}_{\mathrm{4}}{V}_{\mathrm{b}}\right)$$

*V*_{b} is given in ‰, and the values of *a*_{1}, *a*_{2},
*a*_{3} and *a*_{4} follow Kaleschke et al. (2010). Similar to
Maaß et al. (2013b), for the permittivity of seawater
(*ϵ*_{water}), the empirical relationship from
Klein and Swift (1977) is adopted, and the permittivity of air
(*ϵ*_{air}) is assumed to be 1. The brine volume fraction
*V*_{b} can be expressed in the following (Cox and Weeks, 1983).

$${V}_{\mathrm{b}}={\displaystyle \frac{{\mathit{\rho}}_{\mathrm{ice}}{S}_{\mathrm{ice}}}{{\mathit{\rho}}_{\mathrm{brine}}{S}_{\mathrm{brine}}(\mathrm{1}+k)}}$$

*S*_{ice} is the ice salinity, *ρ*_{ice} the ice density,
*S*_{brine} the brine salinity and *ρ*_{brine} the brine
density. *ρ*_{brine} can be fitted with *S*_{brine} (in
‰) according to Cox and Weeks (1983).

$${\mathit{\rho}}_{\mathrm{brine}}=\mathrm{1}+\mathrm{0.0008}\cdot {S}_{\mathrm{brine}}$$

Then the following equation is adopted to relate *S*_{brine} with
*T*_{ice} (Vant et al., 1978):

$$}{S}_{\mathrm{brine}}=a+b\cdot {T}_{\mathrm{ice}}+c\cdot {T}_{\mathrm{ice}}^{\mathrm{2}}+d\cdot {T}_{\mathrm{ice}}^{\mathrm{3}},$$

where *T*_{ice} is in ^{∘}C and *a*, *b*, *c* and *d* are fitted
parameters in Vant et al. (1978). These polynomial approximations agreed
well with the experimental data in Zubov (1963). Also,
*ρ*_{ice} can be expressed by ice temperature (*T*_{ice}:
^{∘}C) in Pounder (1966):

$$}{\mathit{\rho}}_{\mathrm{ice}}=\mathrm{0.917}-\mathrm{1.403}\times {\mathrm{10}}^{-\mathrm{4}}{T}_{\mathrm{ice}$$

Therefore, *V*_{b} can be expressed as a function of
*ρ*_{ice}, *S*_{ice} and *T*_{ice}.

As derived model from Burke et al. (1979), the radiation model is a
non-coherent model. However, the effect of non-coherency is considered to be
mitigated by several factors. First, with the SMOS observations, there
is large variability of both sea ice thickness and snow depth within the
typical resolution of 40 km. There usually exists large variability of the
sea ice cover within the spatial scale of 40 km (variability of
*h*_{i} larger than one-quarter of the L-band wavelength), which effectively
mitigates the effect of non-coherency, as indicated in
Kaleschke et al. (2010). Furthermore, multi-angle mean of SMOS TB further
introduces a range of integration path of radiations. The multi-layer
treatment of the sea ice is also explored in Maaß et al. (2013b). According to
the study in Zhou et al. (2017), with treatment of the salinity profile in
MYI (i.e., salinity drainage in the top layers), the modeled TB is more
consistent with the SMOS TB.

Under typical winter Arctic conditions
(*T*_{surf} = −30 ^{∘}C), simulated TB over different sea ice type from reformulated radiation model is shown in
Fig. 3c and d, with constant snow freeboard lines shown.

Verification is carried out between OIB data and SMOS, with specific
attention to the effect of better OIB sampling. Due to the resolution
difference between SMOS and OIB (or any type of satellite altimetry), we
consider the case involving multiple (*M*) OIB samples and a single SMOS TB.
Section 2.2 and Fig. 2 provide details of
the correspondence between the two types of observations. Using all OIB data,
Fig. A1 shows the distribution of *M* for all the
retrieval problems (each corresponding to an area of
37.5 km × 37.5 km). The mean value of *M* is about 700. The RMSE
of TB (modeled vs. observed by SMOS) is 3.1 K for all available OIB data
(see also Fig. 3c). If we further limit the computation of
RMSE to the points with large values of *M* (95th percentile for *M*,
corresponding to areas with good OIB coverage), the RMSE drops to 1.41 K.
Figure A2 shows the relationship of TB error and *M*. As
shown, there is a drop in both RMSE and the maximum error of TB with better
spatial coverage of OIB. The lead information can be further incorporated in
the radiation model (Zhou et al., 2017), which effectively reduces the
overestimation of TB as caused by refrozen leads or open water.

Appendix B: Statistical analysis for covariability

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We summarize the statistics of the fitted parameters of *α*, *β* and
*s* ($s=\mathit{\alpha}\cdot \mathit{\beta}$) as in Eq. (3), based on all
available OIB data. Figures B1
and B2 show the distribution of these parameters
for FYI and MYI, respectively. The original resolution of OIB dataset (i.e.,
40 m) is adopted. Further, the scaling of the distribution of *s* is
analyzed by manually coarsening the OIB measurements by averaging adjacent
samples. There is a statistically significant positive correlation between
*h*_{s} and FB_{s} across the scales from 40 to 240 m, and
Fig. B3 shows that the distribution of *s* is stable across
these scales, with a slight shift of its mode to a larger value at coarser
scales. By cutting the distribution of *s* at its 99th percentile for FYI and
MYI, we find that it can be well captured by beta distributions.
Section 4.2 contains the quantified results of the fitted
parameters for beta distributions at 40 m resolution for both FYI and MYI.

Competing interests

Back to toptop
Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

This work is partially supported by the National Key R & D Program of China
under the grant number 2017YFA0603902 and the General Program of National
Science Foundation of China under the grant number 41575076. The authors
would like to thank the editors and referees for their invaluable efforts in
improving the manuscript. Additionally, the authors are grateful to Sascha
Willmes and Guenther Heinemann for the provision of Arctic sea ice lead map.

Edited by: Julienne Stroeve

Reviewed by: four anonymous referees

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