A method of simultaneously estimating snow depth and sea
ice thickness using satellite-based freeboard measurements over the Arctic
Ocean during winter was proposed. The ratio of snow depth to ice thickness
(referred to as α) was defined and used in constraining the
conversion from the freeboard to ice thickness in satellite altimetry
without prior knowledge of snow depth. Then α was empirically
determined using the ratio of temperature difference of the snow layer to
the difference of the ice layer to allow the determination of α
from satellite-derived snow surface temperature and snow–ice interface
temperature. The proposed method was evaluated against NASA's Operation
IceBridge measurements, and results indicated that the algorithm adequately
retrieves snow depth and ice thickness simultaneously; retrieved ice
thickness was found to be better than the methods relying on the use of snow
depth climatology as input in terms of mean bias. The application of the
proposed method to CryoSat-2 radar freeboard measurements yields similar
results. In conclusion, the developed α-based method has the
capacity to derive ice thickness and snow depth without relying on the snow
depth information as input for the buoyancy equation or the radar penetration
correction for converting freeboard to ice thickness.
Introduction
Satellite altimeters have been used to estimate sea ice thickness for nearly 2 decades (Laxon et al., 2003, 2013; Kwok et al., 2009). The
altimeters do not measure sea ice thickness directly but measure the sea ice
freeboard which is then converted to sea ice thickness with assumptions, for
example, regarding the snow depth, snow–ice densities, and radar penetration
(Ricker et al., 2014). We hereafter refer to this procedure as “freeboard to
thickness conversion”.
Generally, there are two types of satellite altimeters measuring different
sea ice freeboards. (1) Lidar altimeters such as NASA's ICESat (Zwally et
al., 2002) and ICESat-2 (Markus et al., 2017) missions measure the total
freeboard (Ft), which is the height from the sea surface in leads to the snow
surface. (2) Radar altimeters such as ESA's CryoSat-2 (CS2) (Wingham et al.,
2006) measure the radar freeboard (Fr), which is the difference in the radar ranging
between the sea surface and the radar scattering horizon. By applying two
correction terms regarding the wave propagation speed change in the snow
layer (Fc) and displacement of the scattering horizon from the ice
surface (Fp), the radar freeboard is converted to the ice freeboard
(Fi), which is the height from the sea surface to the snow–ice interface
(Fi). Several studies indicate that the radar scattering horizon is at
or above the snow–ice interface depending on ice type and snow–ice
conditions (Nandan et al., 2017; Armitage and Ridout, 2015; Willatt et al.,
2011; Tonboe et al., 2010). However, the radar scattering horizon is often
treated as the snow–ice interface (Kurtz et al., 2014; Kwok and Cunningham,
2015; Hendricks et al., 2016; Guerreiro et al., 2017, Tilling et al., 2018).
The three different freeboards are indicated in Fig. 1.
Schematic diagram of a typical snow–ice system during the winter.
Snow depth (hs), ice thickness (Hi), total freeboard (Ft), radar
freeboard (Fr), and ice freeboard (Fi) are indicated. Correction
terms regarding the wave propagation speed change in the snow layer (Fc) and
the displacement of the scattering horizon from the ice surface (Fp) are
indicated by blue arrows. The red line denotes a typical temperature profile
with air–snow interface temperature (Tas), snow–ice interface
temperature (Tsi), and ice–water interface temperature (Tiw).
For both lidar and radar altimeters, snow depth (hs) is required as an
input to constrain the freeboard to thickness conversion; thus, the
conversion results are highly dependent on snow depth (Ricker et al., 2014;
Zygmuntowska et al., 2014; Kern et al., 2015). The buoyancy equation used in
the freeboard to thickness conversion describes the balance between buoyancy
and the weight of snow and ice. For a given freeboard, snow–ice densities,
and assumptions on radar penetration of the snow layer, sea ice thickness
(Hi) is a function of hs. According to Zygmuntowska et al. (2014),
up to 70 % of uncertainty in the freeboard to thickness conversion stems
from the poorly constrained snow depth. However, mapping the Arctic-scale
snow depth distribution is challenging. The most commonly used snow depth
information necessary for the freeboard to thickness conversion is the
modified version of the snow depth climatology by Warren et al. (1999)
(hereafter W99). W99 is based on in situ measurements from Soviet drifting
stations (1954–1991) mostly of multiyear ice (MYI). Kurtz and Farrell
(2011) compared W99 with Operation IceBridge (OIB) snow depth measurements
in 2009 and claimed that W99 was still valid in the MYI region and
significantly different from OIB snow depth on first-year ice (FYI). Based on
that study, modified W99 (hereafter MW99) was developed, which halves W99
snow depth in regions covered by FYI. MW99 is often used in ESA's CryoSat-2 (CS2) ice
thickness products available from the Center for Polar Observation and Modeling Data Portal (CPOM-UCL; Laxon et al., 2013), the Alfred Wegener Institute (AWI; Ricker
et al., 2014), and the National Snow and Ice Data Center (NSIDC; Kurtz and Harbeck, 2017).
However, the use of MW99 for the freeboard to thickness conversion
understandably yields a substantial error, considering that W99 is
climatology and not actual snow depth. This is because the actual snow depth
distribution is subject to the year-to-year variation of the snow–ice
system; thus, the climatology based on the 37-year measurements of snow depth
would deviate significantly from the actual distribution (Webster et al.,
2014). Accordingly, such deviation causes errors in the estimation of ice
thickness. Thus, additional snow observations covering both MYI and FYI at
the Arctic basin scale would be ideal as a replacement of MW99.
There have been various approaches aimed at obtaining the snow depth
distribution over the Arctic scale using satellite observations. Markus and
Cavalieri (1998) developed an algorithm based on the brightness temperatures
(TBs) of the Special Sensor Microwave/Imager (SSM/I) based on the negative
correlation of the snow depth with the spectral gradient ratio between 18
and 37 GHz of vertically polarized TBs on the Antarctic FYI. Comiso et al. (2003) have updated the coefficients of this algorithm for the Advanced
Microwave Scanning Radiometer for the Earth Observing System (AMSR-E). However, snow depth
retrieval using this algorithm is relatively less accurate when the MYI
fraction within the grid cell is significant (Brucker and Markus, 2013).
Recently, Rostosky et al. (2018) suggested a new method: using the lower
frequency pair of 7 and 19 GHz to overcome this limitation. Nonetheless,
estimating the basin-scale snow depth distribution seems to be a difficult
task.
There are other approaches involving the use of the lower frequency
measurements at L-band. Using Soil Moisture Ocean Salinity (SMOS)
measurements, Maaß et al. (2013) found that 1.4 GHz TB depends on the
snow depth through the insulation effect of the snow layer, and they determined
snow depth by matching TBs simulated with a radiative transfer model (RTM) with SMOS-measured TBs. Zhou et al. (2018) simultaneously estimated the sea ice
thickness and snow depth by adding additional laser altimeter freeboard
information, improving the Maaß et al. (2013) approach. However, both of
these RTM-based approaches require a priori information on ice properties
(e.g., temperature and salinity profiles).
Other satellite remote sensing approaches include the snow depth retrieval
using dual-frequency altimetry (Guerreiro et al., 2016; Lawrence et al.,
2018; Kwok and Markus, 2018), multilinear regression (Kilic et al., 2019),
and a neural network approach (Braakmann-Folgmann and Donlon, 2019). In
spite of promising results, the dual frequency altimetry method is available
only for regions where two altimeters overlap with each other, reducing a
great deal of spatial coverage. On the other hand, the regression/neural
network methods based on AMSR-2 TBs are prone to the overfitting problem,
limiting their applications to other microwave sensors.
Here, let us switch our point of view to solving the buoyancy equation
instead of retrieving snow depth directly. Remember that there are two
unknowns (snow depth and ice thickness) in the buoyancy equation for given
snow–ice densities, freeboard, and assumptions on radar penetration of the
snow layer. The attempt so far has been to add one constraint (snow depth
information) to the buoyancy equation for solving ice thickness. However, if
a particular relationship between two unknowns is available, it can be used
to constrain the equation yielding both ice thickness and snow depth
simultaneously.
To identify such a relationship, this study examines the vertical thermal
structure within the snow–ice layers observed by drifting buoys. The
vertical thermal structure of a snow–ice system in winter is rather simple;
the temperature profile of the snow–ice system can be assumed to be
piecewise linear, as illustrated in Fig. 1. Therefore, the temperatures at
three interfaces can represent the thermal state of the snow–ice system
fairly well: they are (1) air–snow interface temperature (Tas), (2) snow–ice interface temperature (Tsi), and (3) ice–water interface
temperature (Tiw). Tiw is assumed to be nearly constant at the
freezing temperature of seawater (Maaß et al., 2013), implying that the two
other interface temperatures (Tas and Tsi) are sufficient to
describe the thermal structure of the system.
Based on this thermal structure, there is a constraint relating the snow
depth and ice thickness. In identifying this constraint, conductive heat
flux is assumed to be continuous through the snow–ice interface (Maykut and
Untersteiner, 1971), implying that conductive heat fluxes within the snow
and ice layers are the same under the steady state assumed in the given thermal
structure. As the conductive heat flux is proportional to the bulk
temperature difference of the layer divided by its thickness, it is possible
to deduce the relationship between snow depth and ice thickness from the
given thermal structure.
Once the relationship is obtained, then it is possible to apply it at the
Arctic Ocean basin scale because the thermal structure can be resolved from
satellites, as shown in the recently available basin-scale and long-term
satellite-derived interface temperatures (Dybkjær et al., 2020; Lee et
al., 2018). In determining the snow depth along with the ice thickness,
instead of using the snow depth as an input to solve for the ice thickness,
we intend to (1) examine the relationship between the vertical thermal
structure of a snow–ice system (Tas and Tsi) and the thicknesses of
the snow and ice layer (hs and Hi) using buoy measurements and (2) retrieve the sea ice thickness and the snow depth simultaneously by applying
their relationship to the freeboard to thickness conversion as a constraint,
thus replacing the snow depth information. The result may reduce uncertainty
in the freeboard to ice thickness conversion by replacing the currently used
snow depth climatology.
Method
Here we provide the theoretical background of how the snow–ice thickness
ratio (α=hs/Hi) can be related to Tas and
Tsi. Then, after empirically determining the relationship of α
to Tas and Tsi from buoy-measured temperature profiles, α
obtained from satellite observed Tas and Tsi is then used to
constrain the conversion from freeboard to ice thickness over the Arctic
Ocean during winter.
Theoretical background
We intend to find a relationship between snow depth and ice thickness in
terms of the vertical thermal structure of the snow–ice system. Because the
temperature gradients within the snow and ice layers are linked to both
temperature and thickness, we focus on the temperature gradient. Owing to
the physical reasoning that the conductive heat flux is continuous across
the snow–ice interface (Maykut and Untersteiner, 1971), the following
relationship is valid at the snow–ice interface:
ksnow∂Tsnow∂zz=0=kice∂Tice∂zz=0.
In Eq. (1), the subscripts snow and ice denote their respective layers, while T, k,
and z denote temperature, thermal conductivity, and depth, respectively. The
snow–ice interface is defined as z=0. Assuming a piecewise linear
temperature profile within the snow–ice layer, Eq. (1) can be rewritten as
follows:
ksnowTas-Tsihs=kiceTsi-TiwHi,
where subscripts as, si, and iw denote the air–snow, snow–ice, and ice–water
interface, respectively, and Hi and hs denote the sea ice thickness
and snow depth as in Fig. 1. Introducing a variable α, which is the
snow–ice thickness ratio, Eq. (2) becomes the following:
α=hsHi=ksnowkiceΔTsnowΔTice.
Here, ΔT denotes the temperature difference between the top and
bottom of each of the snow and ice layers (i.e., ΔTsnow=Tas-Tsi, ΔTice=Tsi-Tiw). As explained in
detail in Sect. 2.3, α can be used to constrain the freeboard to
thickness conversion. Thus, once α is known, both snow depth and ice
thickness can be simultaneously estimated from altimeter-measured freeboard
instead of using snow depth data for ice thickness retrieval.
Empirical determination of “α-prediction equation” from buoy
measurements
To obtain α, the conductivity ratio (ksnow/kice) should be
known even if the temperature difference ratio (ΔTsnow/ΔTice) is given. In this study, instead of using the
conventional conductivity ratio found in the literature, it is empirically
determined using buoy-measured α and ΔTsnow/ΔTice. Thus, the interface should be defined and determined from
buoy-measured temperature profiles, which show a piecewise linear
temperature profile as shown in Fig. 1.
The buoy-measured temperature profiles at the vertical resolution of 10 cm
are used in this study (Sect. 3.1). Although the instrument initially sets
the zero-depth reference position to be approximately at the snow–ice
interface, the reference position can deviate from the initial location if
the ice deforms or if the snow refreezes after the temporary melt into
snow ice. In addition, the interfaces (air–snow, snow–ice, and ice–water)
may be located in between measurement levels in a 10 cm spacing. Therefore,
an interface searching algorithm is developed to determine three interfaces
(yas, ysi, yiw) and their respective temperatures (Tas,
Tsi, Tiw) by extrapolating each piecewise linear temperature profile
iteratively.
The flow chart of the interface searching algorithm: yi and
Ti denote the position and temperature of a data point in the
temperature profile, yas,ysi, and yiw denote the position of the
interfaces, and Tlayer denotes a set of temperature data points.
The interface searching algorithm iterates three processes to find the
location and temperature of each interface: it (1) divides the temperature
profile into four layers using the most recently available locations of the
three interfaces, (2) finds a linear regression line of the temperature
profile at each layer, and (3) updates the location and temperature of each
interface by finding an intersection between two adjacent regression lines.
The algorithm fails if the temperature profile is far from linear or if the
thickness of a certain layer is too thin to have fewer than two data points.
More detailed procedures for determining the interface are provided in Fig. 2 as a flow chart. The outputs are Tas, Tsi, Tiw, Hi (Hi=yas-ysi), and hs (hs=ysi-yiw). Examples of the
interface searching results for 15 d averaged temperature profiles are
shown in Fig. 3. The algorithm works adequately for both Cold Regions Research and
Engineering Laboratory Ice Mass Balance (CRREL-IMB; Fig. 3a–c) and Surface Heat Energy Budget of the Arctic (SHEBA; Fig. 3d–f) buoy data.
Examples of interface searching results with an averaging period
of 15 d: (a) 2012G period: 2, (b) 2013F period: 8, (c) 2014G period: 1, (d) Q2 period: 6, (e) R4 period: 6, and (f) SEA period: 10. The period number
indicates the sequential 15 d period from 1 November (e.g., period: 2
denotes a time-averaging period of 16 to 30 November). Blue
dots are buoy-measured temperature profiles, and red lines are regression
lines. Dashed black lines indicate the intersections between adjacent
regression lines.
Since Tas, Tsi, Tiw, Hi, and hs can be obtained from the
previous interface determination with buoy data, the calculation of
ΔTsnow/ΔTice and α is straightforward. An
empirical relationship can then be obtained by relating α to ΔTsnow/ΔTice by running a regression model, and details are
given in Sect. 4. However, for the time being, we assume that the regression
equation (referred to as an “α-prediction equation”, which will be
discussed in Sect. 4) is used to predict α from ΔTsnow/ΔTice.
Simultaneous estimation of ice thickness and snow depth from
satellite-based freeboard using α
In this section, we describe how α can be used to constrain the
freeboard to thickness conversion. Based on the assumed hydrostatic balance,
ice thickness can be obtained from satellite-borne total freeboard or ice
freeboard as follows:
4Hi=ρwρw-ρiFt-ρw-ρsρw-ρihs,5Hi=ρwρw-ρiFi+ρsρw-ρihs.
Here, ρw, ρi, and ρs denote the bulk densities
of the water, ice, and snow layers, respectively. Ice freeboard is obtained from
radar freeboard by applying two correction terms regarding the change in the
wave propagation speed in snow layer (Fc) and the displacement of the
scattering horizon from the ice surface (Fp) (Kwok and Cunningham,
2015):
Fi=Fr+Fc-Fp.
The correction terms are expressed in the following equations (Armitage and
Ridout, 2015; Kwok and Markus, 2018):
7Fc=ηs-1fhs,8Fp=1-fhs.
Here, ηs denotes the refractive index of the snow layer and f
denotes the radar penetration factor (Armitage and Ridout, 2015), which is
the depth of the radar scattering horizon relative to the snow depth (e.g.,
f=1 if the radar scattering horizon is at the snow–ice interface and f=0
if the radar scattering horizon is at the air–snow interface). A combination of
Eqs. (6)–(8) yields the following relationship:
Fi=Fr+fηs-1hs.
Ice freeboard in Eq. (5) can be substituted by radar freeboard and snow
depth using Eq. (9), i.e.,
Hi=ρwρw-ρiFr+fηs-1ρw+ρsρw-ρihs.
According to Eq. (10), the ice thickness can be estimated from the radar
freeboard and the snow depth. Note that Eq. (10) becomes equivalent to the
equation for the total freeboard (Eq. 4 if f=0, i.e., if there is no
radar penetration into the snow layer). With the use of α, defined in
Eq. (3), Eqs. (4) and (10) become the following:
11Hi=ρwρw-ρi+αρw-ρsFt,12Hi=ρwρw-ρi-αfηs-1ρw+ρsFr.
From Eqs. (3), (11), and (12), it is evident that the snow depth and ice
thickness can be simultaneously estimated from the freeboards once α, ρ, f, and ηs are known.
In order to obtain α from satellite measurements of Tas and
Tsi, we need to calculate the temperature difference ratio (ΔTsnow/ΔTice). For the calculation, Tiw is set to be
-1.5∘C. The freezing temperature of seawater is often assumed to
be -1.8∘C; however, the value of -1.5∘C has been chosen
based on the buoy observations. A sensitivity test indicated that the
influence of a 0.3 ∘C difference in the freezing temperature on
α was negligible (e.g., approximately 1.2 % difference for typical
interface temperatures of Tas=-30∘C and Tsi=-20∘C). The α values are calculated only at the pixel whose
monthly sea ice concentration (SIC) is greater than 95 % and rejected if
Tas is warmer than Tsi. The densities are prescribed with those used
for OIB data processing: ρs, ρi, and ρw are
0.320, 0.915, and 1.024 g cm-3, respectively
(Kurtz et al., 2013). Although ρs varies seasonally (Warren et
al., 1999) and ρi is greater for MYI than FYI (Alexandrov et al.,
2010), we use the same densities as those of OIB data because we intend to
compare outputs against OIB data. In solving Eq. (12), cases showing
negative ice thickness (α≥αcrit=0.291 for
the given densities and radar penetration factor) are rejected. Radar
penetration factor f is set to be 0.84 for CS2 (Armitage and Ridout, 2015),
and ηs is parameterized as a function of the snow density, i.e.,
ηs=(1+0.51ρs)1.5 (Ulaby et al., 1986).
Before the Arctic-basin-scale retrieval, ice thickness is estimated from OIB
total freeboard measurements using Eq. (11) and from OIB-derived radar
freeboards (Sect. 3.3) using Eq. (12) and using satellite-derived α as
a constraint. At the same time, the corresponding snow depth is derived by
multiplying the obtained sea ice thickness and α (Eq. 3). Sea ice
thicknesses are also calculated from Eqs. (4) and (10) using MW99 as the snow
depth to examine how simultaneous retrievals compare with ice thickness
estimation using MW99. To differentiate various outputs, obtained snow depth
and ice thickness are expressed with nomenclature such as (constraint,
freeboard source). For example, the snow depth estimated from
satellite-derived α and OIB total freeboard is referred to as
hs (αsat, FtOIB), and sea ice thickness from the
MW99 and OIB radar freeboard is referred to as Hi (hsMW99,
FrOIB). Finally, ice thickness and snow depth are estimated from
CS2 radar freeboard (Sect. 3.4) over the Arctic Ocean.
Data
Here we provide detailed information on the datasets used for the
development of the retrieval algorithm, evaluation, and application at the
Arctic Ocean basin scale.
CRREL and SHEBA buoy data
To determine the empirical relationship between α and ΔTsnow/ΔTice using Eq. (3), we need information regarding
hs, Hi, Tas, Tsi, and Tiw (as depicted in Fig. 1). These
are sourced from temperature profiles observed by buoys deployed over the
Arctic as part of the Surface Heat Energy Budget of the Arctic (SHEBA)
campaign (Perovich et al., 2007) and the Cold Regions Research and
Engineering Laboratory Ice Mass Balance (CRREL-IMB) buoy program (Perovich
et al., 2019). Those buoy observations are stored for further analysis if
there are no missing records over the entire period ranging from November to
March of the following year. Detailed information regarding ice type and
initial snow–ice thickness at deployment locations is given in Table 1.
Information on the measurement sites of buoys whose observations
were used in this study.
∗ The initial snow depth and ice thickness at the SHEBA sites are average
values of all thickness gauge measurements in the corresponding site because
there was one thermistor string but several thickness gauges at each
measurement site.
Time averages of temperature profiles are used as input for the interface
searching algorithm (described in Sect. 2.2) to meet the required
near-equilibrium states (e.g., linear temperature profile). However, because
of the possibility that the results are dependent on the averaging period,
we examine the results using various averaging periods from 1 to 30 d.
Satellite-derived skin and interface temperatures
For applying the buoy-based α-prediction equation in retrieving the
snow–ice thicknesses over the Arctic Ocean, satellite-derived Tas and
Tsi data are necessary. In this study, Tas is obtained from the Arctic
and Antarctic ice Surface Temperatures from thermal Infrared satellite
sensors version 2 (AASTI-v2) data (Dybkjær et al., 2020), and the
monthly mean for the 1982–2015 period is obtained from daily products.
AASTI-v2 Tas is derived from the CM SAF cLouds, Albedo and surface RAdiation
dataset from AVHRR (Advanced Very High Resolution Radiometer) data edition 2 (CLARA-A2) dataset (Karlsson et al.,
2017), which is based on the algorithm described in Dybkjær et al. (2018).
Information on the validation of this product is found in Dybkjær and
Eastwood (2016). It is available in a 0.25∘ grid format; however,
because other satellite datasets such as SIC are available in a 25 km Polar
Stereographic SSM/I grid, AASTI-v2 data are re-gridded in the same 25 km
grid format. This reformatted AASTI-v2 dataset is called “satellite skin
temperature”.
Tsi is obtained from snow–ice interface temperature (SIIT) produced by
Lee et al. (2018) over 30 years (1988–2017) of winters (December to
February) using SSM/I and the Special Sensor Microwave Imager/Sounder (SSMIS)
homogenized TBs (Berg et al., 2018). The daily data are in the 25 km grid
format. Lee et al. (2018) reported that the satellite-derived Tsi is
consistent with snow–ice interface temperatures observed by CRREL-IMB
buoys, with a correlation coefficient, bias, and RMSE of 0.95, 0.15, and
1.48 K, respectively. In this study, we also produced Tsi for March
using the same algorithm as Lee et al. (2018) for evaluating results against
OIB data, which are mostly collected during spring. Monthly composites are
constructed by averaging daily data for grid cells where the data frequency
is over 20 d. This product is called “satellite interface temperature”.
OIB data
In this study, OIB snow depth (hsOIB) and total freeboard
(FtOIB) are used as a reference in the evaluation of snow depth and
ice thickness retrieved from the developed algorithm. NASA's OIB is an
aircraft mission, and it measures snow depth and total freeboard over the
Arctic using snow radar, Digital Mapping System (DMS), and Airborne
Topographic Mapper (ATM) (Kurtz et al., 2013). OIB ice thickness is derived
from measured snow depth and total freeboard for the given snow and ice
densities using Eq. (4). In this study, the OIB radar freeboard
(FrOIB) is derived from FtOIB and hsOIB using the
combined relationship of Fi=Ft-hs and Eq. (9) as follows:
FrOIB=FtOIB-hsOIB-fηs-1hsOIB.
Because the main objective of using OIB data is to evaluate the relative
performance of the simultaneous retrieval method when the method is applied
to CS2 data, the radar penetration factor (f) for OIB data processing is also
set to be 0.84. In the data processing chain, hsOIB is removed if
it is smaller than the given uncertainty level of the dataset
(∼5.7 cm) or it is larger than the total freeboard
FtOIB.
The 5 years of OIB data during the 2011–2015 period are utilized in this study.
The level 4 dataset (Kurtz et al., 2015) during the 2011–2013 period and Quick Look dataset during the 2014–2015 period are obtained from the NSIDC website
(see the Data Availability section). Because we use the November–March
period for the buoy analysis, only March OIB data are considered for the
evaluation. The OIB data are also reformatted into the 25 km grid format by
averaging pixel-level OIB observations on the 25 km grid.
CS2 data
For examining the Arctic Ocean basin distribution of ice thickness and snow
depth, CS2 freeboard measurement summary data are used (Kurtz and Harbeck,
2017). They are monthly mean composites of CS2 ice freeboard data in the 25 km Polar Stereographic SSM/I grid format covering the entire Arctic and
available from September 2010. Detailed descriptions of the retracker
algorithm used in this dataset are found in the study by Kurtz et al. (2014). The dataset also includes MW99 (hsMW99) and W99 snow
density climatology used for producing the ice freeboard.
The CS2 ice freeboard data (FiCS2) distributed by NSIDC (Kurtz and
Harbeck, 2017) assumed that the radar scattering horizon is at the snow–ice
interface and applied a wave propagation speed correction. However, the
correction was made using hsMW99 and W99 snow density climatology
with an erroneous form of hc=(1-ηs-1) hs instead of the proper form of hc=(ηs-1) hs (Mallett
et al., 2020). Thus, at this point, it is straightforward to derive the CS2
radar freeboard by removing the correction term, as in the following
equation:
FrCS2=FiCS2-1-ηs-1hsMW99.
Here, ηs was parameterized as a function of the snow density,
i.e., ηs=(1+1.7ρs+0.7ρs2)0.5 (Tiuri et al., 1984), and ρs is taken from
the W99 climatology based on Kurtz and Harbeck (2017). Then CS2 ice thickness
is reproduced from FrCS2 and hsMW99 by using Eq. (10)
with the constant densities and the radar penetration factor described in
Sect. 2.3. Those hsMW99 and Hi(hsMW99,
FrCS2) values are used for comparison with results from our
simultaneous method.
Sea ice concentration
Calculation of α is done for those pixels whose monthly SIC is
greater than 95 % (as described in Sect. 2.3). To determine pixels that
meet this SIC criterion, “bootstrap sea ice concentrations from Nimbus-7 SMMR [Scanning Multichannel Microwave Radiometer] and DMSP [Defense Meteorological Satellite Program] SSM/I-SSMIS version 3” produced by Comiso (2017) are used.
This SIC dataset is provided in the 25 km Polar Stereographic SSM/I grid
format.
(a) Scatterplots of the temperature difference ratio of the snow
and ice layer (ΔTsnow/ΔTice) and the snow–ice
thickness ratio (α). Color denotes the collected year of buoy data.
The red lines are the regression lines (defined in Eq. 15). (b) The
scatter plot of observed and regressed α.
ResultsThe empirical relationship between α and ΔTsnow/ΔTice
We examine variables (i.e., Tas, Tsi, Tiw, Hi, and hs)
obtained from buoy observations by applying the interface searching
algorithm. In the scatter plot of weekly averaged ΔTsnow/ΔTice vs. α (Fig. 4a), it appears that
α linearly increases with ΔTsnow/ΔTice when
the ratio is smaller than 1.8, but the linear slope becomes smaller when
ΔTsnow/ΔTice is larger than 1.8. This pattern of the
slopes is found to be nearly invariant from year to year, as is observed in the
different colors appearing in the entire range of ΔTsnow/ΔTice in Fig. 4a. We also found that this slope
pattern is of a consistent nature even for different datasets; two different
datasets (red points for SHEBA and other points for CRREL) covering various
ranges of ΔTsnow/ΔTice show similar distributions
along the two different slopes. Thus, the slope pattern is not due to
different data sources or different data periods. Further analysis of the
two slopes is found in Appendix A.
Taking such a two-slope pattern with ΔTsnow/ΔTice
into account, we introduce a piecewise linear function that may express the
slope pattern, i.e.,
y=a1x+b1x≤x0a2x+b2x>x0,x0=b1-b2a2-a1.
In Eq. (15), x and y correspond to ΔTsnow/ΔTice and
α, respectively, and x0 is the point where the slope transition
takes place. Applying Eq. (15) to data points from buoy-based variables, the
regression coefficients (a1, b1, a2, b2) and transition
point (x0) are determined by minimizing the total variance – the obtained
regression line is plotted in Fig. 4a. The α value is predicted using the
determined regression equation (hereafter referred to as the α-prediction equation) and compared to the original α values to see
how well the regression was performed. The comparison of α with
predicted values in Fig. 4b shows that the regression equation is well
fitted because of the zero bias and 91.9 % of explained variance.
(a) The regression coefficients (a1, b1, a2,
b2) in Eq. (15) and (b) the error statistics of the regression with
averaging periods from 1 to 30 d.
Coefficients of the regression equation for averaging periods of 1,
7, 15, and 30 d; a1, b1, a2, b2, and x0 are given in
Eq. (15).
Although the slope pattern discussed with Eq. (15) and Fig. 4 is based on
the weekly averages, the slope pattern seems to be consistent among the data
averaging periods except for an averaging period shorter than 5 d.
Regressions in the form of Eq. (15) are performed with buoy data averaged
with different averaging periods to understand the slope pattern. The regression
coefficients and transition points for the chosen averaging periods are
examined, and results for four averaging periods are given in Table 2.
Detailed information on the coefficients and associated statistics varying
with the averaging period is given in Fig. 5. The positions of slope change
(x0) are located at approximately 1.8, delineating a nearly invariant
slope pattern regardless of different data averaging periods. Figure 5a shows
that coefficients do not vary much with different averaging periods, while
coefficients of the first part of the regression line (a1 and b1,
x≤x0) vary less than those of the second part (a2 and
b2, x>x0). The regression equations show that the
explained variance (R2) rises quickly when the averaging period is
longer but levels off when data are averaged over a period that is longer
than 7 d. The bias appears to be near zero over the various averaging
periods. Thus, the regression performance is found to be comparable if data are
averaged over a period that is longer than 1 week.
Evaluation against OIB estimates
According to the regression results, it is possible to estimate α
from ΔTsnow/ΔTice. Since ΔTsnow/ΔTice can be calculated from the satellite skin and
interface temperature (as described in Sect. 3.2), the corresponding
α can be estimated from satellite measurements. Thus, we are able to
simultaneously retrieve sea ice thickness and snow depth from
altimeter-based freeboard measurements following Eqs. (11) and (12). We
test and evaluate this simultaneous retrieval approach using OIB data.
Accordingly, ice thickness and snow depth are simultaneously estimated from
OIB freeboard measurements and evaluated against the OIB snow depth
(hsOIB) and ice thickness (HiOIB).
To calculate α, a data averaging period must be selected.
Considering that the monthly composite of satellite freeboard measurements
is needed to retrieve snow–ice thickness at the Arctic basin scale, it seems
appropriate to use the monthly averaging period to calculate the monthly
α distribution. Thus, we use the monthly averaged satellite
temperatures and the coefficients for the 30 d averaging period (Table 2)
to calculate α.
We simultaneously retrieved Hi and hs for March of each year during the
2011–2015 period from the reformatted OIB freeboard measurements (Sect. 3.3), together with satellite-derived α (αsat). As
expressed in Eqs. (11) and (12), two different ice thickness retrievals are
possible depending on the use of the freeboard type (i.e., total freeboard
Ft vs. radar freeboard Fr). Two accordingly associated retrievals of
snow depth are available. Retrieved results of ice thickness (Hi) and
snow depth (hs) from the use of OIB total freeboard and radar freeboard
are given in the first and second row of Fig. 6, respectively. Corresponding
OIB measurements are given at the bottom of Fig. 6. The comparison between
any snow–ice retrievals and OIB measurements appear to be consistent with
each other for both snow depth and ice thickness in terms of magnitudes and
distribution.
Simultaneously retrieved ice thickness and snow depth from OIB
total and radar freeboards in March of the 2011–2015 period. Corresponding OIB
products are at the bottom.
To compare the results quantitatively, scatterplots comparing retrievals
against OIB measurements are made, along with statistics for the snow depth
and ice thickness retrievals, in the top four panels of Fig. 7. The two top-left panels are derived from the use of OIB total freeboard
(FtOIB), while the two top-right panels are derived from the OIB
radar freeboard (FrOIB). The comparison is done only for pixels
for which all four products (i.e., snow–ice thicknesses from two different
freeboards) are available. This indicates that the snow depth from the total
freeboard (top left) is fairly consistent with the OIB snow depth with a
correlation coefficient of 0.73 and with a near-zero bias. The retrieved ice
thickness from the total freeboard (middle left) appears to be consistent
with OIB ice thickness with a correlation coefficient of 0.93 and a bias
around 8.5 cm. The RMSEs for snow depth and ice thickness are 6.8 cm and
44.3 cm, respectively. Based on the comparison results, Eq. (15) obtained
from buoy measurements can be successfully implemented with space-borne
total freeboard measurements for the simultaneous retrieval of snow depth
and ice thickness.
Scatter plots between OIB products and the simultaneously
retrieved snow depth and ice thickness from OIB total and radar freeboards
during the March 2011–2015 period. Corresponding ice thicknesses estimated
from MW99 are in the third row. The red lines are linear regression lines.
Following Eq. (12), snow depth and ice thickness retrievals are made from
the use of radar freeboard measurements, and results are presented in the
two top-right panels in Fig. 7. On the one hand, the comparison of obtained
ice thickness against the OIB ice thickness indicates that the retrieved ice
thickness shows a similar quality as that retrieved from the total freeboard
measurements. On the other hand, snow retrievals from the radar freeboard
show more scattered features compared to snow retrieval results from the
total freeboard. The more scattered features found in the snow depth from the
radar freeboard are likely due to the greater sensitivity of the retrieved
α and the prescribed densities, as noted in Eq. (12). Note that Eq. (12) has a smaller denominator than that of Eq. (11). Results of the associated
sensitivity analysis can be found in Appendix B.
We now examine how the use of MW99 for retrieving sea ice thickness from
ICESat and CS2 measurements compares with results from our simultaneous
method. To do so, OIB-measured total freeboard and radar freeboard are
converted into ice thickness using MW99 as the input to solve Eqs. (4) and (10).
In this study, these two ice thickness retrievals with the use of MW99 are
referred to as “ICESat-like” thickness and “CS2-like” thickness,
respectively, and their comparisons are now observed in the two panels at the
bottom of Fig. 7. According to our analysis, ICESat-like thickness tends to
underestimate the ice thickness by about 47.9 cm when MW99 is used in
comparison to OIB thickness, and CS2-like ice thickness shows an overestimate
of about 25.5 cm. Nevertheless, their correlation coefficients and RMSEs are
similar to the results obtained from the α method.
The better agreement of Hi from the simultaneous method with
HiOIB may be due to the fact that the simultaneously estimated
hs is more consistent with hsOIB (hsMW99 is likely
larger than hsOIB, as shown in Fig. S1). Note that all inputs are
the same except the snow depth. The negative bias of ICESat-like thickness
and positive bias of CS2-like thickness reflect expected responses in
different signs to the same snow depth error, as shown in different signs in
the last terms of Eqs. (4) and (10) (also note Eq. B2 in Appendix B).
Because of this reasoning, if there are decreasing trends not only in ice
thickness but also in snow depth, the decreasing trend of ice thickness
estimated from the constant snow depth will be diminished in radar while
being amplified in lidar. Because of this, the construction of the ice
thickness (or volume) trend from the two different satellite altimeters
would be problematic if MW99 is used for the freeboard to thickness
conversion. For example, it would be hard to compare the sea ice thickness
records estimated from ICESat and CS2 observations and to extend the current
ice thickness record from CS2 with NASA's recently launched ICESat-2, which
carries a lidar altimeter, for the same reason.
Simultaneous retrieval of ice thickness and snow depth from CS2
measurements
We have demonstrated that the method of simultaneously retrieving the sea
ice thickness and snow depth was successfully implemented with OIB
measurements. Now we extend the proposed approach to satellite freeboard
measurements. Here the method is tested with CS2 freeboard measurements,
solving for Hi in Eq. (12), and α is obtained from the
collocated satellite skin and interface temperature data.
Geographical distributions of observed CS2 radar freeboard
(Fr) and estimated snow–ice thickness ratio (α), ice thickness
(Hi), and snow depth (hs) from December 2013 to March 2014. Gray
areas in the second row denote where α retrievals failed because
Tas was warmer than Tsi.
Monthly means of CS2-estimated freeboard (Fr), retrieved α, ice
thickness (Hi), and snow depth (hs) for December 2013 to March 2014
are given in Fig. 8. The geographical distribution of α indicates
that α is largest in January and becomes smaller during the
following months. Geographically, there seems to be no particular
distribution of α between months, although interestingly the lowest
α values are always found over the north of the Canadian Archipelago,
and the western part of the Arctic Ocean shows α values that are
generally larger than those over the eastern part.
Retrieved ice thickness from the CS2 freeboard (Fr) using obtained
α is presented in the third row of Fig. 8. As expected and as noted in
Eq. (12), Hi shows a similar geographical distribution to radar
freeboard (the first row). The thickest area is located north of the
Canadian Archipelago where the ice appears thicker than 4 m. On the other
hand, most of the FYI thickness appears to range from 1.0 to 2.0 m. The snow
depth hs is obtained by multiplying α by Hi (in 2nd and 3rd
rows), following Eq. (3), and results are shown in the bottom row. The
obtained snow distribution indicates that thicker (thinner) snow areas are
generally coincident with thicker MYI (thinner FYI) areas. Such a similarity
should be consistent with the notion that MYI should accumulate more
precipitation than FYI because of its longer existence.
To assess the accuracy of CS2 retrievals, reference snow depth and ice
thickness collocated with CS2 freeboard in space and time are necessary.
However, different from simultaneous retrievals from OIB freeboards in Sect. 4.2, the evaluation with the required matching data may not be possible from the
monthly composite of CS2 data used in this study. Here, instead of using
monthly collocated match-up data, an indirect way is used to examine the
accuracy of CS2 retrievals. We do so by examining whether the relationship
between the simultaneous method and the MW99 method, based on retrievals
from the OIB freeboard, can be reproduced by CS2-based retrievals. If
similar results are obtained, respective accuracies can be deduced against
those noted from the evaluation against OIB measurements.
Comparison of simultaneously retrieved snow depth and ice thickness
to those from the MW99 method. (a) Snow depth from OIB radar freeboard, (b) snow depth from CS2 radar freeboard, (c) ice thickness from OIB radar
freeboard, and (d) ice thickness from CS2 radar freeboard.
The relationships which can be obtained from the analysis in Sect. 4.2 – i.e.,
hs (αsat, FrOIB) vs. hsMW99 and Hi(αsat, FrOIB) vs. Hi (hsMW99, FrOIB) –
are compared with the relationships found in the current results in Fig. 8 – i.e., hs (αsat, FrCS2) vs. hsMW99 and
Hi(αsat, FrCS2) vs. Hi (hsMW99,
FrCS2); the results are presented in Fig. 9. Observably, the
relationships from CS2 freeboard data (Fig. 9b, d) are very similar to the
relationship obtained from the comparison results from OIB measurements
(Fig. 9a, c). This similarity of the slope strongly indicates that the
CS2-based sea ice thickness from the current α method has similar
accuracy to that found in the evaluation against OIB measurements (Sect. 4.2). Further uncertainty estimates for CS2-derived products can be found in
Appendix C.
Conclusions and discussion
A new approach towards simultaneously estimating snow depth and ice
thickness from space-borne freeboard measurements was proposed and tested
using OIB data and CS2 freeboard measurements. In developing the algorithm,
the vertical temperature slopes were assumed to be linear within the snow
and ice layers so that a continuous heat flux could be maintained in both
layers. This assumption allowed for the description of the snow–ice
vertical thermal structure with snow skin temperature, snow–ice interface
temperature, the water temperature at the ice–water interface, snow depth,
and ice thickness. Based on the continuous heat transfer assumption, the
snow–ice thickness ratio (α=hs/Hi) was introduced and
could then be embedded into the freeboard to ice thickness conversion
equations. Thus, information on both ice thickness and snow depth can be
derived once α is known in the case of the availability of a freeboard
without relying on the snow depth information as an input for the conversion
from freeboard to ice thickness. From the drifting buoy measurements of the
temperature profile, snow depth, and ice thickness over the Arctic Ocean, we
demonstrated that α can be reliably determined using the ratio of the
vertical difference of the snow layer temperature to the vertical difference
of ice layer temperature (ΔTsnow/ΔTice). An
empirical regression equation was obtained for predicting α from
three interface temperatures.
Before applying the α-prediction equation to simultaneously retrieve the
ice thickness and snow depth from satellite-borne freeboard measurements,
the algorithm was evaluated using OIB measurements in conjunction with
satellite-derived snow skin temperature and snow–ice interface temperature.
The evaluation of results demonstrated that our proposed algorithm adequately
retrieved both parameters simultaneously. As a matter of fact, the ice
thickness results were more accurate than they were from the current
retrieval methods relying on the input of snow depth (this time MW99 snow
climatology) in terms of mean bias. It should be noted that in this case,
snow depth is a retrieval product instead of being input for the freeboard
to ice thickness conversion adopted by CS2 or ICESat retrieval. The
application was finally made for the retrieval of the snow depth and ice
thickness from CS2 radar freeboard measurements from December 2013 to March
2014 using α as a constraint. Results showed that the quality of the
obtained ice thickness was similar to that obtained from evaluation results
against OIB measurements. Retrieved snow depth distributions were also found
to be consistent with expectations.
In the retrieval process, we may be concerned about the applicability of the
algorithm developed with buoy observations representing the point
measurements to the larger spatial and temporal scales of satellite
measurements. This concern may be relevant upon observing the range of
α values. The α value in the satellite's monthly and 25 km × 25 km
spatial scales was found to be generally smaller than 0.2. The smaller range
of α compared to that shown in the buoy analysis results is likely
due to the scale differences, indicating that extreme α values often
shown in buoy measurements (due to very thick snow and/or very thin ice) may
never be observed in satellite measurements. However, the range may not be a
problem because the relationship (Eq. 3) expresses the thermal equilibrium
condition described by the temperature at three interfaces, the ratio of
snow and ice thickness, and the ratio of thermal conductivity between snow
and ice. Considering that the algorithm is based on the equilibrium
conditions, results should be valid regardless of spatial and temporal
scales if the prerequisite equilibrium conditions are met. Apparently, buoy
observations contain so many different cases that equilibrium conditions are
met with different thermal and physical conditions of the snow–ice system. The sound evaluation results and the consistency between OIB and CS2 ice
thickness retrieval results, which are subject to different scales, all
suggest that the point-measured α-prediction equation can apply to
satellite measurements.
Overall, the developed α-based method yields ice thickness and snow
depth without relying on a priori “uncertain” snow depth information
(MW99), which results in uncertainty in the ice thickness retrieval. The
proposed method applies to both lidar and radar altimeter data, although
lidar-based altimeter data tend to offer relatively more suitable snow depth
information with a smaller RMSE. We expect to continuously monitor the Arctic-scale snow depth and ice thickness by applying the proposed α method
to total freeboard observations from the recently launched ICESat-2 and using
temperature observations from the upcoming MetOp-SG meteorological imager
(MetImage), the microwave imager (MWI), and the proposed Copernicus Imaging
Microwave Radiometer (CIMR).
Physical interpretation of the piecewise linearity between
α and ΔTsnow/ΔTice
The relationship found between α and ΔTsnow/ΔTice showed a piecewise linearity which is almost invariant to the
data averaging period. Because the slope change is attributable neither to
different data sources nor to different data periods, it is likely caused by
the physical properties of the snow and ice, as shown in Fig. A1. If the
slope change is caused by the snow–ice condition, there will be a
significant difference in snow–ice properties between the two parts showing
different slopes. Here we examine the possibility of different physical
properties causing the difference in slopes. Through this comparison using
buoy data, we may identify important properties that might be responsible
for the piecewise linearity.
Distribution of physical variables on scatterplots of the
temperature difference ratio of snow and ice layer (ΔTsnow/ΔTice) and the snow–ice thickness ratio (α).
Color denotes the value of physical variables: (a) ice thickness
(Hi), (b) snow depth (hs), (c) air–snow interface temperature
(Tas), (d) snow–ice interface temperature (Tsi), (e) temperature
difference within the snow layer (|ΔTsnow|), and (f) temperature difference within the ice layer (|ΔTice|).
Histogram of estimated (a, b)kice and (c, d)ksnow. The top and bottom rows denote the first and the second parts,
respectively. The size of the bins is 0.05 W K-1 m-1.
First, the averages of basic properties available from buoy measurements are
compared. They include ice thickness, snow depth, snow–ice interface
temperature, ice temperature – Tice=(Tas+Tsi)/2 – and
so on. The comparison revealed that snow–ice system within the first part
(x≤x0) is found to consist of relatively thicker ice (mean value:
1.84 m), thinner snow (0.29 m), and colder ice (-9.13∘C), while
the second part (x >x0) is found to consist of relatively
thinner ice (1.10 m), thicker snow (0.46 m), and warmer ice (-5.00∘C). In general, a thicker snow or ice layer exhibits a greater
temperature difference from the top to the bottom of the layer. There is no
significant difference between the air–snow interface temperature
(Tas) in the two slope parts.
The thermal conductivities, ksnow and kice, are also compared
because what connects α and ΔTsnow/ΔTice is
the ratio of thermal conductivities. Before showing the results, we describe
how to calculate ksnow and kice. First, the thermal conductivity
ratio is calculated from buoy-measured variables (i.e., Tas, Tsi,
Tiw, hs, and Hi) using Eq. (3). Because the underlying physics
in ksnow are significantly more complex, kice is estimated first, and
then ksnow is obtained by multiplying the calculated kice by
ksnow/kice. To calculate kice, the parameterization of Maykut
and Untersteiner (1971), which describes kice as a function of salinity
and temperature, is used:
kice=2.03+0.117SiceTice.
Here, Sice and Tice are the salinity (in parts per thousand, ppt) and temperature (in
Celsius) of sea ice, respectively. For the calculation, Sice is
estimated according to the empirical relationship between sea ice thickness
and mean salinity from Cox and Weeks (1974) as follows:
Sice=14.24-19.39Hi,Hi≤0.4m7.88-1.59Hi,Hi>0.4m.
Although Trodahl et al. (2001) reported that kice depends on depth and
temperature, here we do not estimate accurate thermal conductivities but
attempt to examine the physical consequences of the total ice layer.
The calculated thermal conductivities are presented in Fig. A2. The
calculated kice ranges from 1.8 to 2.0 W K-1 m-1 (two left panels in Fig. A2). These values are consistent with
the in situ measurements by Pringle et al. (2006). The mean values of
kice of the first part (1.96 W K-1 m-1) and the second part
(1.88 W K-1 m-1) show almost no difference. The calculated
ksnow ranges from 0.2 to 1.05 W K-1 m-1 (two right panels in Fig. A2). This range is consistent with reported
values in Sturm et al. (1997). The first part shows the greater spread in
the distribution of ksnow compared to the second part. The mean
ksnow values are 0.44 and 0.27 for the first part and second part,
respectively.
As a significant difference is observed in ksnow, we would like to find
a possible reason for this difference. To do so, we should first review the
factors determining ksnow; they are density, temperature, and crystal
structure (Sturm et al., 1997). Snow is a mixture of ice particles and air,
and air has lower thermal conductivity than ice. Thus, snow with a
relatively lower density including a greater portion of air should have
relatively lower thermal conductivity. Besides, the thermal conductivity of
ice particles depends on the temperature, and the path of heat transfer
depends on the crystal structure which describes how the particles are
connected. The heat transfer occurs not only by conduction but also by water
vapor latent heat transportation and convection through the pore spaces
(Sturm et al., 2002), which are hard to quantify explicitly. These two
factors are closely related to the temperature gradient (or difference)
imposed within the snow layer.
Based on this knowledge, we can infer the condition of the snow layer of the
two parts. The relatively higher and varying ksnow of the first part
would be related to the compaction process resulting in high density and
metamorphic diversity, which changes the crystal structure. According to
Sturm et al. (2002), the value of ksnow of a hard wind slab is up to
0.5 W m-1 K-1, while that of ksnow of depth hoar is below
0.1 W m-1 K-1. On the other hand, the lower and nearly constant
ksnow of the second part implies that the snow layer of the second part
would consist of fresh and dry snow having relatively lower density and a
relatively lower likelihood of experiencing particular metamorphism.
In summary, it is concluded that the physical properties of snow and ice can
account for the piecewise linearity based on the differences in the
physical properties between the first and second parts. Especially the
thermal conductivity of the snow, ksnow, seems to play an important
role. Nevertheless, further analysis is required to fully understand this
phenomenon.
Sensitivity test for the proposed method
Here we present results of a sensitivity test for showing how the snow depth
and ice thickness retrieval results are dependent on the uncertainties in
α. To do so, the uncertainty in the snow depth (Δhs) due
to the α error (i.e., Δα) and associated ice
thickness error (ΔHi) are estimated. From this sensitivity test,
we expect to understand why the simultaneous method for the radar freeboard
shows more scattered features than those from the lidar total freeboard.
First, Δhs is defined by the difference of retrieved hs
with error (α+Δα) and without error
(α):
Δhs=hsα+Δα,Ft-hsα,Ft(usingFt)hsα+Δα,Fr-hsα,Fr(usingFr).
Then Δhs can be converted to the error in the ice thickness
(ΔHi) using the following equation derived from Eq. (10):
ΔHi=fηs-1ρw+ρsρw-ρiΔhs=-6.46Δhs(usingFt)3.44Δhs(usingFr).
Because Hi and hs are the combination of freeboard and α, as
in Eqs. (3), (11), and (12), we only examine the uncertainty with some
typical sea ice types. Here, physical states for thicker ice (type A),
moderate ice (type B), and thinner ice (type C) are chosen, which are
summarized in Table B1. Typical values for those three types are shown in
the scatterplots of OIB-based αOIB vs. FtOIB and of
satellite-based αsat vs. FrCS2 – Fig. B1. It is
shown that the majority of data points are located around type B, followed
by type A. There seems to be a very small portion of total samples showing values
around type C.
With Δα=±0.05, which is the root mean square
difference (RMSD) value between αOIB and αsat,
Δhs and ΔHi are estimated for three ice types. Table B2 summarizes the results and shows that |Δhs| is within
8 cm and that it tends to decrease as the ice becomes thinner when the current
method is applied to the total freeboard. On the other hand, the use of
radar freeboard shows that |Δhs| tends to be more
sensitive for the same Δα. Especially the sensitivity of
type C is the greatest. This is because the denominator of Eq. (12) becomes
smaller when α approaches αcrit, resulting in an
unstable solution. For the ice thickness, |ΔHi| is smaller when the total freeboard is used since ΔHi is proportional to Δhs. However, the gap between the
results from the two freeboards has narrowed because Hi from the total
freeboard is more sensitive than the radar freeboard to Δhs,
according to Eq. (B2). The sensitivity characteristics shown here are
consistent with the analysis results given in Sect. 4.2. Because there is a
much smaller number of data points belonging to type C, at least in the data
used for this study, the overall sensitivity would likely be in between types B and A.
It is also of importance to ask what degree of retrievals was yielded successfully. In this study, cases showing Tas>Tsi or
retrieved α≥αcrit are considered to be failures.
Statistics on success/fail ratio of α retrieval for December–March
of the 2011–2015 period are provided in Table B3. Overall, the success ratio
was over 82 % in December-February, while it was reduced to
∼74 % in March. Most of the failures appear to be associated with
cases showing the temperature inversion (i.e., Tas>Tsi), whose areas are shaded with gray in the α distributions of
Fig. 8. Those failure areas are generally found around the marginal ice
zones and in the east of Greenland. On the other hand, there was a near-zero
failure (0.02 % of total pixels) for retrieved α≥αcrit. This near-zero failure implies that almost all calculated
α values meet the satisfactory condition after the removal of cases showing
the temperature inversion. It may be concluded that the calculated α
appears to be physically reasonable (i.e., α<αcrit) as long as presumed thermodynamic conditions are met.
The physical state of typical cases of points A, B, and C.
∗ Retrieval fail occurs if α+Δα>αcrit (αcrit=0.291 for ρs=320 kg m-3, ρI=915 kg m-3, ρw=1024 kg m-3, and f=0.84).
Locations of physical states for typical types (A, B, C) on the
freeboard-thickness ratio space. Blue dots are from (a) OIB data and
(b) retrieved thickness ratio and CS2 radar freeboard.
Statistics of success/fail ratios of α retrieval for winter 2011–2015.
αcrit=0.291 for ρs=320 kg m-3, ρi=915 kg m-3, ρw=1024 kg m-3, and f=0.84.
Uncertainty estimation for CS2 retrievals
Although the sensitivity test regarding uncertainty of satellite-derived
α has been conducted in Appendix B, the uncertainty of CS2 freeboard
measurements and prescribed parameters should be considered as well for the
satellite snow depth and ice thickness estimates. To do so, a simple
propagation analysis of errors is performed, regarding the uncertainty of
satellite products (αsat and FrCS2) and prescribed
parameters (ρi, ρs, and f). Uncertainty due to the
variability of ρw is neglected (Kurtz and Harbeck, 2017;
Hendricks et al., 2016; Ricker et al., 2014). Here we assume that αsat and FrCS2 are not correlated and have no systematic bias.
Such an assumption may not be true in the real world. However, it allows us to
estimate the retrieval uncertainty from satellite-derived products with a
certain limit. Uncertainty of ice thickness can be estimated by the following
Gaussian error propagation equation:
ϵy,total2=∑xϵyx2.
Here, εy,total denotes the total uncertainty of retrieved
variable y (hs or Hi) and εy(x) denotes the
uncertainty of y related to input variable x (α,Fr, ρi,
ρs, or f). The uncertainties on the right-hand side are obtained
by the following equation:
ϵyx=∂y∂xσx=yx+δ-yxδσx.
Here, σx denotes the uncertainty of x, and δ is set to be
10-6 for the numerical calculation of the partial derivative using Eqs. (3)
and (12); σα is estimated to be an RMSD value between
αOIB and αsat, σFr is given by Kurtz
and Harbeck (2017), and σf is adopted from Armitage and Ridout
(2015). Uncertainties of snow–ice densities are from the relevant literature
(Alexandrov et al., 2020; Hendricks et al., 2016; Kern and Spreen, 2015;
Ricker et al., 2014; Warren et al., 1999). Those values are summarized in
Table C1.
Using Eqs. (C1) and (C2), uncertainties of snow depth and ice thickness
retrievals can be estimated. Ice thickness uncertainty estimates are
presented in Fig. C1. Total uncertainty of ice thickness estimates ranges
from 0.8 to 2.0 m. Generally, Fr-related uncertainty in the third row
is greater than α-related uncertainty in the second row. Snow depth
uncertainty estimates are presented in Fig. C2. Total uncertainty of snow
depths ranges from 0.04 to 0.4 m. In the case of the snow depth, α-related uncertainty is greater than Fr-related uncertainty. Both
uncertainties of ice thickness and snow depth are greater for MYI regions
than FYI regions. It is thought that the improvement of accuracy in satellite-derived temperatures can reduce the snow depth uncertainty, while the
improvement of freeboard accuracy can reduce the ice thickness uncertainty.
Uncertainties induced from densities and radar penetration factors are found
to be relatively smaller than uncertainties related to α and
Fr (shown in Figs. S2 and S3).
Values and uncertainties of input variables for uncertainty
estimation.
Geographical distributions of sea ice thickness uncertainty:
(first row) total uncertainty, (second row) α-related uncertainty,
and (third row) Fr-related uncertainty for the period from December 2013
to March 2014.
Geographical distributions of snow depth uncertainty: (first row)
total uncertainty, (second row) α-related uncertainty, and (third
row) Fr-related uncertainty for the period from December 2013 to March
2014.
Data availability
The SHEBA buoy data were obtained from NCAR/EOL
(https://doi.org/10.5065/D6KS6PZ7, last access: 14 September 2020, Perovich et al., 2007), and CRREL-IMB buoy data were obtained from the CRREL-Dartmouth Mass Balance Buoy
Program (http://imb-crrel-dartmouth.org, last access: 14 September 2019, Perovich et al., 2019).
AASTI-v2 and SIIT data are available upon request to the authors. Other datasets were obtained from NSIDC. They are OIB data
(https://doi.org/10.5067/G519SHCKWQV6, last access: 10 September 2019, Kurtz et al., 2015), OIB
Quick Look data (https://doi.org/10.5067/GRIXZ91DE0L9, last access: 28 July 2020, Kurtz, 2019), CS2 data (https://doi.org/10.5067/96JO0KIFDAS8, last access: 10 September 2019, Kurtz and Harbeck, 2017), and SIC data (https://doi.org/ 10.5067/7Q8HCCWS4I0R, last access: 12 September 2019, Comiso, 2017).
The supplement related to this article is available online at: https://doi.org/10.5194/tc-14-3761-2020-supplement.
Author contributions
HS and BJS conceptualized and developed the methodology, and HS conducted
data analysis and visualization. GD, RTT, and SML gave important feedback for the algorithm development and interpretation of results. GD provided AASTI data. All
of the authors participated in writing the paper; HS prepared the
original draft under the supervision of BJS and GD, and BJS critically
revised the paper.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
We appreciate NSIDC for
producing and providing the OIB, CS2, and SIC datasets. We also give thanks
to CRREL and NCAR/EOL under the sponsorship of the National Science
Foundation for providing IMB and SHEBA buoy data. The authors express their
sincere thanks to an anonymous reviewer and to Isobel R. Lawrence for
their valuable comments that led to the improvement of the paper.
Financial support
This study has been supported by the Space Core Technology Development Program (NRF-25 2018M1A3A3A02065661) of the National Research Foundation of Korea and by Korea Meteorological Administration Research and Development Program under grant KMIPA KMI2018-06910. This study has also been supported by the International Network Program of the Ministry of Higher Education and Science, Denmark (grant ref. no. 8073-00079B).
Review statement
This paper was edited by Yevgeny Aksenov and reviewed by Isobel Lawrence and one anonymous referee.
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