Retrieval of snow albedo and total ozone column from single-view MSI/S-2 spectral reflectance measurements over Antarctica

We have proposed a simple algorithm to retrieve the total ozone column and snow properties (spectral albedo and effective light absorption path) using the high spatial resolution single – view MSI/S-2 measurements over Antarctica. Therefore, multiple observations of the same ground pixel from different directions as used, e.g., in the MODIS snow albedo retrieval algorithm (Schaaf et al., 2002) are not needed.


Introduction
Sentinel-2 (S-2) is an Earth observation mission from the Copernicus Programme that systematically acquires optical imagery at high spatial resolution (10 m to 60 m) over land and coastal waters. The mission is a constellation with two twin satellites, S-2A and S-2B. The satellites have been launched on June 23, 2015 and March 7, 2017, respectively. It is planned that S-2C and S-2D will be launched during next 10 years. The S-2 mission is equipped with the Multi-Spectral Instrument (MSI), which uses a push-broom concept and its design has been driven by the large (290km) swath and high (10-60m) spatial resolution requirements.
Revisiting is 5 days per month on average under the same viewing angles close to the nadir direction. MSI operates in the spectral range 0.44-2.2 m. The instrument has already been used for numerous applications including studies of lake ecological quality (Free et al., 2020), winter wheat (Zhang et al., 2019) and seasonal agriculture land use (Debella -Gilo, 2021) mapping, estimation of fractional vegetation cover (Wang et al., 2018), etc. (see, e.g., https://www.mdpi.com/journal/remotesensing/special_issues/sentinel2_sa). Gascoin et al. (2019Gascoin et al. ( , 2020 have used MSI/S-2 data for the estimation of the fractional snow cover (FSC) at 20m resolution in open terrain. However, climatic effects of snow cover depend not only on its extent (Peng et al., 2021), but also on its spectral albedo, which plays an important role in the modification of the backscattered solar energy on local and global scales (Flanner et al., 2007;. Snow albedo products are currently available at moderate resolution (i. e., 300m from Ocean and Land Colour Imager (OLCI) (Kokhanovsky et al., 2019(Kokhanovsky et al., , 2020, 500m from MODerate Imaging Spectrometer (MODIS) (Schaaf et al., 2002;Hall et al, 2002) and Sea and Land Surface Temperature Radiometer (SLSTR) (Mei et al., 2021a,b), and 1km from Second Generation Global Imager (SGLI) ( Chen et al., 2021)). However, they do not allow capturing the fine details of spatial variability of the snow surface properties (Schweizer et al., 2008). The technique presented here provides snow albedo product on the scale of 10-20m and makes it possible to derive and validate subpixel snow cover products as obtained, e.g., from MODIS measurements (Kaufman et al., 2002 ;Painter et al., 2009).
The task of this paper is to derive the spectral snow albedo using MSI/S-2A (B) 10-60m spatial resolution reflectance data. The spatial resolution of the product depends on the spectral channels used. In addition, the algorithm to determine the total ozone column (TOC) over snow fields using MSI/S-2 is proposed. The retrieval algorithm is described in the next section. We also present the validation of the algorithm using ground -based snow spectral albedo and 3 TOC measurements performed at Dome C ( 75 05 59 S, 123 19 56 E, 3233m above sea level in Antarctica. 2. The determination of the total ozone and spectral snow albedo using MSI measurements This work is aimed at the retrievals of total ozone column and snow properties from MSI measurements over Antarctica. The atmospheric light scattering effects are rather weak (Six et al., 2005;Tomasi and Petkov, 2015) in Antarctica for the MSI channels (see Table 1). Also the concentration of light absorbing impurities in most Antarctic snow is very low (Grenfell et al., 1994;Kang et al., 2020) and can be neglected in a first iteration. We shall use the following approximation for the spectral top-of-atmosphere reflectance: where g T is the atmospheric gaseous transmittance, R is the underlying surface -atmosphere reflectance under assumption that there are no absorbing gases in the atmosphere. Ignoring weak atmospheric light scattering effects at the MSI channels, we can write for highly reflective clean Antarctic surfaces (Kokhanovsky et al., 2020a): where 4    is the bulk absorption coefficient of ice,  is the imaginary part of ice refractive index and  is the wavelength. Here, 0 R is the snow reflectance in the absence of absorption and L is the effective light absorption path (ELAP). In the case of dirty snow, Eq.
(2) must be modified to account for the snow pollutants as discussed by Kokhanovsky et al. (2021).
The task of this work is to retrieve the spectral snow albedo and total ozone concentration using MSI measurements in the spectral range 443-865nm, where the accuracy of Eq.
(2) is higher as compared to the case of short-wave infrared wavelengths (Kokhanovsky et al., 2019). Also the influence of all atmospheric gases on the value of transmittance g T (except ozone) can be neglected in this spectral interval (Kokhanovsky et al., 2020b). This is especially valid in central Antarctica, where the atmosphere is thin and air is dry. 4 We shall use the following approximation for the atmospheric ozone transmittance (Kokhanovsky et al, 2020a): where K=mN, m is the airmass factor (AMF) depending on several parameters including the cosine of the solar zenith angle (SZA) ξ and the cosine of the viewing zenith angle , N is the total ozone column and ( ) abs C  is the ozone absorption cross section. The dependence of ( ) abs C  on temperature and pressure is weak in the spectral range under study (Gorshelev et al., 2014). Therefore, it is not accounted for.
Summing up, the spectral MSI reflectance over pure snow fields under clear sky is modelled where we have assumed that light reflectance by snow at absence of absorbers 0 R can be substituted by the MSI measurements at band 1 ( a R ). The weak influences of atmospheric scattering effects on the value of MSI spectral reflectance at 442.7nm can be accounted, if needed, using either exact radiative transfer calculations for a given atmospheric model or various approximations (Kokhanovsky, 2020c). These effects are ignored in this work.
Our simplified model of the MSI spectral reflectance in the visible and near infrared given by Eq. (4) makes it possible to determine two unknown constants (K, L) from Eq. (4) analytically.
Namely, it follows: where indices signify the spectral channels ( , , The fit of the MSI spectral reflectance measured in the vicinity of the Dome C in Antarctica using Eqs. (4), (5) is given in Fig.1. We extracted reflectances from MSI/Sentinel-2 level 1C products at the nearest pixel of the measurement site using Google Earth Engine (Gorelick et al., 2017). The solar and sensor geometry parameters for each spectral band were obtained from 6 the product metadata. The spectral range 400-1050nm is shown in Fig.1. The calculations using Eq. (4) have been performed with the spectral resolution 1nm. The spectral imaginary part of ice refractive index as suggested by Picard et al. (2016a) has been used (at the wavelengths below 600nm). For the larger wavelengths, Warren and Brandt (2008) data for the function ( ) has been used. The spectral absorption cross section of ozone ( ) abs C  measured by Gorshelev et al. at 203K (2012) has been used. The corresponding parameters appearing in Eq.
(5) are given in Table 2. The linear interpolation of tabular spectra ( ) abs C  and ( ) on the 1nm grid in the spectral interval 400-1050nm has been applied.  The retrieved values of the parameters given in Eq. (4) for the case shown in Fig. 1  cm molecule and L=2.13mm. They have been derived using Eq. (5) and data given in Table 2.
The total ozone column can be derived from the value of K as follows: N=K/m. Assuming that m is given by the geometrical approximation ( = + ), one derives for the value of TOC for the case shown in Fig.1 (Pommereau and Goutail, 1988;Sarkissian et al., 1997) at the Dome C site give the value of TOC equal to 158 and 180 DU in morning and evening, respectively, which is close to the TOC derived using MSI/S-2 measurements.
The value of L can be used to derive the effective absorption length  , which determines the snow spherical albedo (Kokhanovsky et al., 2019): Namely, it follows (Kokhanovsky et al., 2019): where (Kokhanovsky et al., 2021): The function ( ) u  describes angular distribution of light escaping from semi-infinite nonabsorbing turbid media with light sources placed at infinity. Taking into account that a R  0.92,  =0.41,  =1, and L=2.13mm for the case shown in Fig.1, it follows that 1.79mm   .
This makes it possible to derive both spectral spherical r (see Eq. (6)) and plane ( ( ) u p r r   ) albedo (Kokhanovsky et al., 2019). Moreover, the snow effective diameter can be derived: where  is the shape factor, which depends on the assumed shape of snow grains.
We shall use the value of  =16 as suggested by Kokhanovsky et al. (2019). Therefore, it follows for the case shown in Fig.1: ef d =0.11mm, which is a reasonable estimation for the snow surface at Dome C (Grenfell et al., 1994;Flanner et al., 2021). In particular, Grenfell et al. (1994) has found the values of ef d =0.1mm in the first 5mm of snow at South Pole (January 23, 1986). The value of ef d was two times smaller at the same location immediately after the snowfall (Grenfell et al., 1994).
The total ozone column over Dome C retrieved using the technique described above for November -December, 2020 is shown in Fig.2 (MSI). The cloudy scenes have been removed using MSI measurements at 2.2 m with the assumption that clouds have higher reflectance as compared to snow at this channel due to smaller size of crystals in clouds as compared to snow on the surface. The threshold value of R(2.2 m)=0.2 has been used. The air mass factor has been calculated as (Iqbal, 1983;Savastiouk and McErloy, 2004) where = , ℎ = 0.26 − 0.1 , is the latitude in degrees (the negative number in the southern hemisphere), is the radius of the Earth, ℎ is the height of the ozone layer in .
Eq. (9) coincides with the geometrical AMF at s=0. We also show the TOC derived from other satellite/ground observations (see Table 3 and Kokhanovsky et al. (2021b) for details on various satellite measurements) and European Centre for Medium-Range Weather Forecasts (ECMWF) re-analysis in Fig.2. The temporal (daily product) and spatial 1deg averaging around the site has been performed (except for MSI, where data for a given pixel or the average of two pixels (S-2A, S-2B), if available, is given). It follows that all instruments show the existence of ozone hole in early November and its disappearance in the last week of December, 2020.
Data from all instruments closely follows all temporal oscillations of the TOC (see, e.g., the TOC wave centered at the day 45 (December 15) in Fig.2). There is temporal/ spatial mismatch between MSI/S-2 and other satellite measurements. In particular, the MSI/S-2 measurements provide the total ozone column on the spatial scale of 10m, which is not possible for other instruments (see Table 1). This could explain some differences seen in TOC derived from MSI as compared to other instruments.  Table 3) at DOME C (Antarctica) on November-December 2020.
The data derived from the ECMWF re-analysis is given as well. The intercomparison of ground-based and satellite -retrieved albedo is given in Fig. 3 for December 9, 2016 (22:00:11UTC -ground measurements, 23:57UTC -satellite measurements, SZA= 65degrees). The ground measurements of the plane albedo have been performed in the spectral range 400-1050nm. Further details on the ground measurements are given by Picard et al. (2016b,c) and also by Kokhanovsky et al. (2019), where the same spectral albedo measurements have been used to validate the OLCI/S-3 snow albedo retrievals.
It follows that ground and satellite measurements of spectral albedo give very similar results, which indirectly confirm the validity of our assumptions used in the retrieval procedure. We also note that although we have used the spectral MSI/S-2 reflectance measurements at just two wavelengths (443 and 865nm), the spectral albedo is estimated in much larger spectral range useful also for the shortwave broadband albedo determination. This is due to the fact that the spectral pure plane snow albedo depends largely on two parameters -the effective absorption length and solar zenith angle. The difference between satellite and ground measurements depends on the wavelength being smaller than 2% in the spectral range 400-1100nm. This difference is below of the error of respective optical measurements. The largest differences (2%) occur around 1050nm, where MSI/S-2 does not perform any measurements.
The ELAP estimated from the 864.7nm MSI-S2 channel may differ from that derived from 1020nm due to vertical snow inhomogeneity effects and spectrally dependant light penetration in snow layers. The difference observed in Fig.3a Fig.3b. The same as in Fig. 3a except for the larger spectral range.

Conclusions
We have proposed a simple algorithm to retrieve the total ozone column and snow properties (spectral albedo and effective light absorption path) using the high spatial resolution singleview MSI/S-2 measurements over Antarctica. Therefore, multiple observations of the same ground pixel from different directions as used, e.g., in the MODIS snow albedo retrieval algorithm (Schaaf et al., 2002) are not needed.
In addition, the algorithm allows the retrieval of the snow grain size on the scale 10 -20m although this was not explicitly evaluated here. This algorithm should be useful for understanding of intra-pixel total ozone and snow albedo variability in complement to satellite observations performed on much coarser spatial resolution scale (0.3-1km and even larger spatial scales).
It has been shown that the MSI reflectance in the spectral range 443-865nm over snow in Antarctica can be modelled using just two parameters (K and L). These parameters are proportional to the total ozone column and the effective diameter of ice grains, respectively.
Therefore, we conclude that the main parameters influencing the MSI spectra in the range 443-865nm are TOC and the effective diameter of snow grains with other effects of a secondary importance for a clean Antarctic atmosphere. In particular, the influence of the molecular and aerosol light scattering on the MSI/S-2 reflectance spectra in Antarctica can be neglected in the first approximation. This is especially true for the channel located at 865nm used for the effective grain diameter (see Eqs. (5)- (7)) and spectral albedo determination. We also assumed that the MSI reflectance at channel 1 (8A) is the same on the scale 10 and 60 (20)m, which is true for the most of Antarctic surfaces. Otherwise, the channels (2, 3, 8, see Table 1) having the same spatial resolution (10m) must be used to derive the pair (K, L) from Eq. (4). It follows from Fig.1 that the reflectances at channels 1 and 2 and also 8 and 8A have close values.
Therefore, two different sets of channels used in the retrieval process produce almost the same retrieval results. Alternatively, the parameters (K, L) can be retrieved using all MSI channels in the framework of the optimal estimation approach.
The determination of the effective light absorption path makes it possible to estimate also snow broadband albedo (Kokhanovsky, 2021c) both from ground and satellite measurements.