Introduction
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
().
In the last decades, there has been rapid shrinkage of Arctic sea ice cover
, particularly in summer. In addition, the Arctic sea ice
is also experiencing dramatic thinning in recent years , 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 which bears great
deviation from climatology , 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 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)
, AMSR-E and AMSR2
. 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 , 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
. 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 for multi-year sea ice (MYI) and halved for the
first-year sea ice (FYI). For laser altimetry as in ICESat
, 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
and , the uncertainty
in snow depth is the most important contributor to that of the sea ice
thickness and volume.
Sea ice parameters in the active and passive remote sensing of the
sea ice cover, including sea ice thickness (hi), snow depth
(hs) and snow freeboard (FBs).
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
are derived from reanalysis data and satellite retrieved sea
ice motion, while the climatological snow depth data in as
used by CryoSat-2 contain large uncertainty due to
interpolation and interannual variability and may not be adequate for the
present day under the context of climate change
. The retrieval of snow depth with passive
microwave satellite remote sensing has been carried out in various studies.
In , multi-band data from AMSR-E are
utilized, but only for snow cover over FYI. 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
and . 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 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 .
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. 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. . By integrating the covariability
information, we propose the retrieval algorithm and carry out evaluation and
analysis in Sect. . Section summarizes the
article and provides discussion of related topics and future work.
Data and models
Data
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.
Data match between OIB and SMOS data. SMOS TB product is provided on
the 12.5 km EASE grid (shown by blue rectangular cells). However, the
inherent resolution of SMOS TB is of about 40 km. The red and black line
represents the OIB track. Therefore, in order to accommodate the resolution
differences, OIB samples that reside within the nine cells (red) are considered
to be of equal contribution to the TB value at the central EASE grid cell
(outlined by the thick blue line).
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 .
Sea ice thickness is retrieved from snow freeboard (denoted FBs)
and snow depth (denoted hs), which is measured by the University
of Kansas' snow radar . Surface temperature is determined
from the IceBridge KT-19 infrared radiation pyrometer dataset .
There is also accompanying sea ice type information, which is from EUMETSAT
OSI-SAF system . Therein, the OIB Level-4 product IDCSI4
is adopted 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.
Data usage protocols
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 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. ) of
equal contribution to the SMOS TB at the central cell (the one bounded by
thick blue lines in Fig. ). 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 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
or sea ice concentration lower than 1 according
to . 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 (h‾s) and mean
sea ice thickness (h‾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.
L-band radiation model. Panel (a) shows sea ice salinity
profile for FYI (dotted lines) and MYI (solid line). The vertical axis (z)
is normalized with respect to the sea ice thickness. The comparison of the
simulated TB based on OIB data and the observed SMOS TB is presented in
panel (b). Blue triangles represent FYI, while red circles are MYI. The
dashed (dotted) line is the least squares fit (least squares fit under the
constraint that slope is 1). The root mean square error of TB is 3.1 K.
Panels (c) and (d) show the modeled TB under typical sea
ice parameters (hi and hs), assuming Arctic winter
conditions (surface temperature of -30 ∘C). The green lines
represent constant snow freeboard lines.
L-band radiation model
The L-band (1.4 GHz) radiative property of the sea ice cover is
characterized through numerical modeling based on . The
model was originally designed for the modeling of radiative transfer of the
X- and L-band soil moisture. In , 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 . 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 a 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
and . The convergence of the modeled TB with respect to the
layer count is witnessed, which is consistent with the study in
. In , 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 c 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 FBs lines. With the thickening
of sea ice cover, the value of TB increases and saturates when hi
is large enough (larger than 2.5 m). The value of TB is not monotonic with
respect to FBs, 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. .
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 b
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
R2 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. , 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 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.
Isostatic equilibrium model
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 ρice), seawater density (denoted ρwater), snow density (denoted
ρsnow) and the equilibrium state, the sea ice thickness, snow
depth and snow freeboard FBs are constrained according to
Eq. (). The sea ice thickness can be derived given
the snow depth, according to Eq. (). This model is widely
applied for both radar and laser altimetry for the retrieval of sea ice
thickness.
ρice⋅hi+ρsnow⋅hs=ρwater⋅hi+hs-FBshi=ρwaterρwater-ρice⋅FBs-ρwater-ρsnowρwater-ρice⋅hs
In this study, ρwater and ρice are taken to be
1024 and 915 kg m-3, which are derived from field measurements
discussed by , and ρsnow is
320 kg m-3, derived from .
Statistics of snow depth from OIB at the local scale of retrieval.
Panel (a) shows the mean and the ±1 standard deviation of the
snow depth within each snow freeboard bin (from 0 to 1.5 m by an interval
of 5 cm), shown by lines and shaded areas for four realistic cases of OIB.
Panel (b) shows the nonlinear fitting of snow depth over snow
freeboard (Eq. ) under representative s values (0.71
for FYI and 0.95 for MYI) and various values of α. Solid color lines
are for MYI and dashed ones for FYI. The solid black line is y=x.
Retrievability analysis
Under the observational constraints of TB and FBs, both sea ice
thickness and snow depth over sea ice can be retrieved .
Figure c and d show TB as simulated by the radiation model
(Sect. 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
FBs, 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. ) is studied in , 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. , 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
hs and FBs 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 FBs
and hs, which can be better characterized by a nonlinear fitting.
Furthermore, the effect of the covariability on retrievability is analyzed in
Sect. .
Covariability analysis based on OIB data
For the covariability between FBs and hs, 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 FBs and hs. We
divide these samples into FBs 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 hs.
Figure a shows the mean hs and the ±1
standard deviation range and their relationship with FBs, for
four
representative TB points. Furthermore, we carry out least squares linear
fitting (weighted according to sample count in each bin) between
FBs of the bins and the corresponding mean hs in each
bin. Among all available data, there is statistically significant positive
correlation between hs and FBs for over 90 % of
all points. The values of R2 are in the range of 0.06 and 0.89 (95 %
percentile), with the mean value of R2 as 0.53. This indicates that there
is consistent covariability between snow depth and snow freeboard across
Arctic sea ice cover.
Typical distributions of FBs and the range of mean
hs for 0<α<1. Panel (a) shows the four
distributions (two for FYI and two for MYI) and the corresponding mean value of
FBs. Global values of s for FYI and MYI are adopted. For these
four distributions, panel (b) shows the mean hs for the
range of α between 0 and 1. Mean hs increases
monotonically with α and saturates when α is
large.
Typical scenarios for retrievability studies. The mean sea ice
thickness (hi‾), mean snow depth
(hs‾), mean snow freeboard
(FBs‾), observed TB from SMOS and the simulated
TB from forward radiation model are shown. Scenarios I and II are FYI, and
scenarios III, IV and V are MYI.
Ice type
Scenario
hi‾ (m)
hs‾ (m)
FBs‾ (m)
TB (K)
Simulated
Observed
FYI
I
1.28
0.12
0.2212
245.84
246.38
II
2.25
0.20
0.3790
242.92
243.14
MYI
III
2.46
0.17
0.3807
245.29
245.46
IV
3.01
0.32
0.5419
246.66
246.61
V
4.13
0.31
0.6509
245.84
246.38
However, for both FYI and MYI ice, there is saturation of the mean
hs with respect to FBs. Besides, in the Arctic
inundation is generally uncommon (i.e.,
hs<FBs). In order to accommodate these
characteristics, we propose a nonlinear fitting, as shown by
Eq. (). The parameters α and β are fitted
according to observations. According to the equation, the value of
hs saturates to α⋅π/2 when FBs is
large, and the value of α⋅β (denoted s), which is the
slope of the function at FBs=0, should be lower than 1 in order to
avoid any inundation.
hs(FBs)=α⋅arctanβ⋅FBs
Using Eq. , the overall quality of the fitting for all
available local OIB segments is improved, with mean value of R2 rising
from 0.53 to 0.67, and the 95 % percentile of R2 rises to 0.23 and
0.92, respectively. Detailed distribution of the fitted parameters for all OIB
data is shown in Appendix B (Fig. for FYI and
Fig. 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. c
and c). 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 b shows fitting
function of snow depth over snow freeboard based on these representative
values of s under various values for α.
Effects of covariability on retrievability
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 , 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
(FBs|m, where m is the index of the samples, and
1≤m≤M), we carry out the scanning of the (uniform) snow depth
hs from 0 m (snow free) to 1 m (sufficiently deep). Under a
certain value of hs, we retrieve the sea ice thickness
hi|m for each FBs|m with
Eq. (), based on the current value of hs.
Then the TB value for this sample (TB|m) can be calculated according to
the L-band radiation model, with hi|m, hs and
surface temperature Tsfc|m. The mean TB value is then computed
as the arithmetic mean of all TB|m's, for the current value of
hs. For any OIB sample, if the value of freeboard is smaller than
the current value of hs, in order to avoid inundation, the snow
depth for this sample is assumed to be the same as FBs. If the
number of samples that witness potential of inundation over 50 % of M,
we stop the scanning even if hs has not reached 1 m.
Retrievability study with different retrieval scenarios. The
horizontal solid (dotted-dashed) lines are the SMOS (modeled) TB. The
vertical solid lines represent the values of the mean snow depth from OIB
observation. The black dashed curves denote the values of TB generated by
scanning of hs under the flat snow cover assumption, and the
vertical dashed lines denote the values of hs that result in
50 % OIB samples to be inundated. The red (blue) dashed curves (with the
corresponding mean snow depth) are the values of TB generated by scanning of
α with the local (global) values of s as in
Eq. ().
Flow chart for retrieval algorithm. Two phases are marked out. The
red box includes the scanning process for the potential solutions to the
retrieval problem, and the blue box shows the iterative binary search for the
solving process.
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 a shows four typical distribution of FBs, and
Fig. b shows a range of values for α (0 to 1) and the
resulting mean value of hs for the four typical distributions. For
the range of 0 to 1, the resulting mean hs covers a continuous
range for each distribution. For each distribution, when α is very
small, the corresponding hs is very small for whole range
FBs, resulting in a very small value of mean hs.
Furthermore, the value of mean hs 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 hs; and
when α is large enough, the mean hs saturates. For all four FBs distributions, we consider that the resulting mean
hs 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 hs. With each
scanned value of α, a corresponding value for β, can be computed
as s/α, and the snow
depth hs|m for each sample can be computed with
Eq. (). Then the hi|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 shows the results of
scanning for the five scenarios in Table . Note that for
the lines that represent scanning of α (i.e., involving
covariability), the x axis is the resulting values of mean hs,
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. ) 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.
Retrieval algorithm and evaluation
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 . The incorporation of covariability
is further integrated, based on the nonlinear fitting in
Eq. () 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. , 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.
Retrieved results (hi‾ and
hs‾, in units of meters) for five scenarios under
different retrieval algorithms. In scenarios II, IV and V, the retrieval with
flat snow cover assumption is unsuccessful. The values in the brackets for
scenario V denote the other (possible) solution for sea ice
parameters.
Scenario
hi‾ (m)
hs‾ (m)
Observed
Retrieval
Retrieval
Retrieval
Observed
Retrieval
Retrieval
Retrieval
w/flat
w/local s
w/global s
w/flat
w/local s
w/global s
snow cover
snow cover
I
1.28
–
0.95
0.93
0.124
–
0.167
0.171
II
2.25
2.00
2.23
2.30
0.202
0.263
0.207
0.195
III
2.46
–
2.50 (1.69)
2.45 (1.88)
0.172
–
0.168 (0.293)
0.175 (0.263)
IV
3.01
–
3.25
2.38
0.321
–
0.285
0.419
V
4.13
3.88
4.09
4.11
0.308
0.350
0.313
0.310
For the typical scenarios in Table , we carry out the
retrieval for the mean sea ice thickness (hi‾) and the
mean snow depth (hs‾) using the standard algorithm with
either the global or the local values of s, as well as the reference
algorithm. Table 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 hi‾ and
hs‾ 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. .
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
hi‾ and hs‾ against OIB
measurements. Section covers the retrieval and
analysis. Furthermore, we construct several representative retrieval
scenarios in Sect. and analyze the uncertainty in the
retrieved parameters and carry out attribution of the uncertainty to input
parameters of the retrieval.
Large-scale retrieval of mean sea ice thickness (a, c) and
mean snow depth (b, d) and verification with OIB observations. In
each panel, blue triangles (red rectangles) denote FYI (MYI), the solid line
is the 1:1 line and the dashed (dashed dotted) line represents the linear
fitting (linear fitting line with the constraint that the slope be 1). The
quality of fittings in terms of R2 are also shown. Panels (a, b)
represent the comparison results for the retrieval with modeled TB and the
local values of s. Panels (c, d) represent the results with SMOS
TB and the global values of s as derived from OIB
data.
Large-scale retrieval
For the systematic verification of the proposed algorithm, we carry out the
retrieval with all the available OIB data (as mentioned in
Sect. ) 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 shows the
comparison of retrieved mean sea ice thickness and snow depth with
observations. Figure a and b shows the results for
hi‾ and hs‾, 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. a, the LSQ fit for hi‾
(dash line) features a R2 value of 0.966, while the LSQ fit under the
extra constraint on slope (dotted dash line) features a R2 value of 0.964.
For snow depth (Fig. b), the R2 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. c and d show the evaluation for
hi‾ and hs‾, respectively. The
fitting quality (in terms of R2) 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 R2 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
R2 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. . The areas
with more extensive OIB scans are shown of lower TB error (see
Fig. ), 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 FBs. This retrieval problem ignores the
resolution difference between altimetry scans and L-band radiometry and
generally corresponds to the theoretical retrieval problem analyzed in
. Specifically, for the use of OIB data, the mean value of
M samples of FBs is computed and further combined with TB for
the retrieval of a single value for both hi and hs.
Since only the mean FBs 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. , the
retrieval yields R2 of 0.78 and 0.50 for hi and hs
(fitting between the retrieved and the observed parameter). For comparison,
under the realistic retrieval results (Fig. c
and d), the quality of retrieval is much improved for both hi
(R2 from 0.78 to 0.89) and hs (R2 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.
Uncertainty analysis
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 a. Due to the nonlinear relationship between sea ice
parameters and TB, we cannot directly compute the uncertainty in
hi or hs. Instead, Monte Carlo (MC) simulation is
adopted. For each scenario in Table a, four sets of MC
simulations are constructed, each containing: (1) random perturbations to TB
only, (2) random perturbations to FBs only, (3) random
perturbations to s only, and (4) random perturbations to TB,
FBs and s altogether. Each set contains 1000 random sampling to
these parameters.
Uncertainty estimation for typical retrieval
scenarios.
(a) Typical scenarios for uncertainty estimation
Scenario
Ice type
hi‾ (m)
hs‾ (m)
I
FYI
1.307
0.127
II
FYI
2.549
0.171
III
MYI
3.009
0.265
IV
MYI
4.736
0.348
(b) Results
Scenario
Relative
Perturbations
uncertainty
TB
FBs
s
All
I
σhi/hi
9.44 %
15.75 %
8.19 %
15.30 %
σhs/hs
14.38 %
25.08 %
13.06 %
24.21 %
II
σhi/hi
4.90 %
4.36 %
3.42 %
5.60 %
σhs/hs
11.33 %
9.99 %
7.89 %
12.93 %
III
σhi/hi
11.15 %
12.23 %
5.02 %
10.24 %
σhs/hs
19.49 %
21.41 %
8.83 %
17.88 %
IV
σhi/hi
6.62 %
5.19 %
4.71 %
6.37 %
σhs/hs
13.92 %
10.94 %
9.92 %
13.28 %
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 FBs 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. (), where a, b and const are fitted parameters by
using OIB data at 40 m resolution (see Fig. ). 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.
f(x|a,b,constant)=constB(a,b)x(a-1)(1-x)(b-1)
The perturbations to s follow the fitted beta distribution. Furthermore,
the perturbations to TB, FBs 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 hi and
hs are computed by biased standard deviation estimation with
respect to the original retrieval which involves no perturbation.
Table b shows the relative uncertainty of hi and
hs for each experiments for all scenarios. First, the relative
uncertainty for hi or hs is at most about 25 %.
Also, all scenarios show that s plays a minor role in terms of uncertainty,
as compared with TB or FBs. TB and FBs play a
comparable role in the uncertainty of the retrieved parameters. Moreover, for
both FYI and MYI, the uncertainty in the retrieved hi and
hs is relatively lower for thicker ice and deeper snow cover. The
uncertainty of TB (or FBs) is not correlated spatially and that
of s is based on basin-scale statistics from OIB. Therefore, the
uncertainty of the retrieved hi (or hs) is not
spatially correlated, resulting in effective reduction of the uncertainty in
the sea ice volume (or snow volume).
Summary and discussion
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
.
Difference with existing retrieval algorithms
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
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.
Covariability analysis
In , statistical analyses are carried out between snow depth
and snow freeboard, which also show covariability between the two. As also
noted in , 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
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 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.
Uncertainty estimation related to model parameters
Besides the input parameters to the retrieval (TB, FBs and s),
model parameters also play an important role in modulating the uncertainty
for retrieval . 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 . MC simulations can be adopted
for the quantification of the uncertainty through perturbations to both
input and model parameters.
Outlook of satellite-based retrieval
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 , 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 , a MODIS-based product
. 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 , 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
effectively reduces the overestimation of TB, as
studied by . 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
. 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 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 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. . 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
.
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 , 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.