#### 2.1. Theory

We perform the retrievals using separate retrieval chains for clean snow (Case 1 snow) and for polluted snow (Case 2 snow). Here, we use the analogy with the classification of Case 1 and Case 2 water as proposed in [

8] (see also [

9]). Case 1 water corresponds to relatively clean water where most of the absorption is due to phytoplankton, and Case 2 water contains other impurities including mineral particles. In our application, we define Case 1 as the situation where snow properties are determined just by snow grains without significant interference from impurities or living matter (cells, algae, etc.). The snow Case 1 is often met in Antarctica—far from any significant aerosol sources and limited algal populations. The areal extent of the clean dry snow areas on Greenland and Antarctica ice sheets makes the Case 1 snow dominant on a global scale. Additionally, a simplified atmospheric correction is possible in this case [

6]. The selection of clean snow pixels is performed as follows. First, we check the reflectance in OLCI band 1. If it is larger than the dynamic threshold value (THV), it is assumed that the ground scene is covered by unpolluted snow (the majority of pixels in the terrestrial cryosphere). The THV is derived from the synthetic radiative transfer calculations for the assumed (default: 0.1) aerosol optical thickness at 550 nm (see

Appendix A).

Case 1 snow

The simplified atmospheric correction for Case 1 snow is described in [

6] and summarized below. It is based on the fact that the pure snow spherical albedo can be accurately parameterized using the following equation:

where

$\alpha \left(\lambda \right)=4\pi \chi /\lambda $ is the bulk ice absorption coefficient for a given wavelength λ,

$\chi $ (see e.g.

https://atmos.washington.edu/ice_optical_constants/, last access: 07/01/2020) is the imaginary part of ice refractive index, and

$l$ is the effective absorption length. The snow spectral reflectance

${R}_{s}$ is related to the snow spherical albedo, which is a three dimensional integral of

${R}_{s}$ with respect to solar and viewing zenith angles and relative azimuthal angle (RAA) [

6] via the following approximate equation [

6]:

where

x is a geometrical correction coefficient depending on

R_{0} and on the angular function

u [

6] evaluated at the cosine of the solar zenith angle (SZA)

${\mu}_{0}$ or at the cosine of the viewing zenith angle (VZA)

$\mu $:

and we use the following approximation for the angular function [

6]:

The value of

${R}_{0}$ gives the non-absorbing underlying surface reflectance (

${r}_{s}=1$). One can use OLCI measurements at 865 and 1020 nm to determine both effective absorption length and

${R}_{0}$ from Equation (2) under the assumption that the atmosphere does not affect the satellite signal at these channels [

6].

The determined value of effective absorption length makes it possible to derive the snow spherical albedo at any wavelength using Equation (1). The plane albedo

${r}_{p}$is defined via the integral of the azimuthally averaged reflection function with respect to the viewing zenith angle [

6]. As a matter of fact,

${r}_{p}$ can be also derived from the spherical albedo using the following simple approximation [

6]:

or with account for Equation (1):

Also, one can derive the underlying snow spectral reflectance function using Equations (1) and (2). Therefore, the procedure for the determination of Case 1 spectral albedo from space is straightforward. It was validated in [

6]. Generally, the errors in the retrieved albedo are below 1–3% depending on the wavelength

$\lambda $.

Case 2 snow

The retrievals for the Case 2 snow are more complicated. In this case, the satellite measurements of snow spectral reflectance in the visible are influenced by various pollutants or living matter (cells, algae, etc.). Therefore, there is no way to estimate snow spectral reflectance/albedo in the visible using measurements in the near infrared as it is done for the Case 1 snow (see above). Then, we use yet another approach described below.

The top-of-atmosphere reflectance for the atmosphere-underlying snow system can be presented in the following way [

10,

11]:

where

${R}_{ag}$ is the atmospheric contribution to the measured signal,

${r}_{ag}$ is the spherical albedo of the atmosphere,

${r}_{s}$ is the bottom-of-atmosphere snow spherical albedo, and

${T}_{ag}$ is atmospheric transmittance from the top-of-atmosphere to the underlying surface and back to the satellite position. In the case of Lambertian underlying surfaces, the underlying surface reflectance does not depend on solar and viewing observation directions, and Equation (7) is valid with

${r}_{s}={R}_{s},$ where

${R}_{s}$ is the underlying Lambertian surface reflectance. The snow is not exactly the Lambertian reflector; therefore, we replace

${r}_{s}$ in the numerator of Equation (7) by the snow reflectance [see Equation (2)]. Such an approximation makes it possible to have the correct limit for the top-of-atmosphere reflectance [see Equation (2)] in the case of absence of atmosphere. The term in the dominator of Equation (7) accounts for multiple reflections between snow and atmosphere, and the account for the snow reflectance directional nature in the dominator of this equation is of secondary importance. Then, it follows:

The reflectance of non-absorbing snow

${R}_{0}$ in Equation (8) is calculated using simple analytical approximation, as discussed in [

6]. We do not derive the value of

${R}_{0}$ from OLCI measurements themselves because such a derivation for the polluted snow can be influenced by the type and the load of pollutants.

We use channels that are not influenced by water vapor and oxygen absorption effects, although we account for the ozone absorption effects. Equation (8) is very general and valid outside and inside molecular absorption bands. We account for the ozone absorption in a simplified way. Namely, we derive free of ozone absorption top-of-atmosphere reflectance

${R}_{c}$ using the following equation:

${R}_{c}=\frac{{R}_{meas}}{{T}_{O3}},$ where

${T}_{O3}$ is the atmospheric transmittance with account for the ozone absorption (see

Appendix A). Then, Equation (8) is transformed to a simplified approximation:

where the functions

${R}_{a},$ ${r}_{a},{T}_{a}$ (see

Appendix A) have the same meaning as

${R}_{ag}$ ${r}_{ag},{T}_{ag}$, respectively, except for atmosphere not influenced by gaseous absorption processes (e.g., ozone absorption). The spherical albedo of underlying snow surface can be found from Equation (9) provided that the aerosol model is known. In this case, the snow spherical albedo

${r}_{s}$ is the only unknown parameter in Equation (9) and can be readily calculated, solving the transcendent Equation (9) with respect to

${r}_{s}$. For the wavelengths where the aerosol contribution is low and can be neglected,

${R}_{a}~0$,

${r}_{a}~0$,

${T}_{a}~1$, and an analytical solution of Equation (9) is possible:

where the analytical expression for

${R}_{0}$ is given in [

6]. The functions

${R}_{a},$T, and

${r}_{a}$ depend on aerosol and molecular scattering parameters and can be stored in look-up-tables for various aerosol models. Because aerosol load is weak in the Arctic and Antarctica, various approximations for the functions mentioned above can be used. In particular, we calculate these functions in the framework of approximations described in the

Appendix A. We solve the transcendent Equation (9) with respect to

${r}_{s}$ for all OLCI wavelengths free of water vapor and oxygen absorption in the Case 2 snow.

The broadband albedo (BBA), either plane or spherical, is calculated from the spectral plane or the spherical albedo using the integration between the wavelengths

${\lambda}_{a}$ and

${\lambda}_{b}$ as shown below [

7]:

where

$F\left(\lambda \right)$ is the incident solar flux at the snow surface, and

${r}_{p,s}\left(\lambda \right)$ is plane (

p) or spherical (

s) albedo depending on whether plane or spherical BBA

${\overline{r}}_{p,s}\left({\lambda}_{a},{\lambda}_{b}\right)$ is to be calculated. The indices

a and

b signify the wavelengths

$\lambda $ used. We assume that the incident solar flux can be approximated by the following analytical function:

where we ignore rapid oscillations of

$F\left(\lambda \right)$, which are due to gaseous absorbers. This is possible because

${r}_{p,s}\left(\lambda \right)$ is a continuous function, which acts as a filter of high frequencies. The coefficients in Equation (12) are derived from the fit of

$F\left(\lambda \right)$ calculated using the Santa Barbara DISORT Radiave Transfer (SBDART) code [

12] to Equation (12) in the spectral range 0.3–2.4 μm. They are given in

Table 1. The calculations of

$F\left(\lambda \right)$are performed at the parameters listed in

Table 2 for the rural aerosol model [

13]. Clearly,

$F\left(\lambda \right)$ depends on the location and the time. We find that the choice of aerosol model in the calculation of

$F\left(\lambda \right)$ only weakly influences the calculations of BBA (see Equation (11)). The spectral snow albedo needed as input for SBDART is calculated assuming clean snow with the effective diameter of spherical ice grains equal to 0.25 mm. Generally, the results are only weakly sensitive to the variation of the function

$F\left(\lambda \right)$ [

7]. We therefore assume solar flux independent from the location of the retrieval and from solar zenith angles.

For the Case 1 snow, the broadband albedo is calculated numerically using Equations (1), (5), (11), and (12) in the spectral range 0.3–2.4 micrometers. Also, other limits of integration can be used (say, to derive visible or near-infrared BBA).

For the Case 2 snow, the spherical albedo is known only for selected OLCI channels as derived from Equation (9). Therefore, we use interpolation to get the spherical albedo between the measurement points needed for the evaluation of integral (11). For the spectral range below 865 nm, we use:

While, for wavelengths larger than 865 nm, we use:

We use the dependencies as shown in Equations (13) and (14) because we find that the measurements can be approximated by the second order polynomial for the spectral range below 865 nm and the exponential function for the wavelengths above 865 nm. The coefficients (a, b, c) are found separately for the intervals 400–709 nm and 709–865 nm using the following wavelength triplets: (400, 560, 709 nm) and (709, 753, and 865 nm), respectively.

The coefficients $\left(\u03f5,\sigma \right)$ are derived from OLCI measurements at 865 and 1020 nm at the values of ${R}_{meas}$(1020 nm) equal to or smaller than 0.5. Otherwise, Equation (1) [and not Equation (14)] is used at $\lambda >865\mathrm{nm}$with the effective absorption length derived from the value of spherical albedo at 1020 nm. We use different approaches for the pixels with small and large values of ${R}_{meas}$(1020 nm) because the case of comparatively large values of ${R}_{meas}$(1020 nm) corresponds to snow. Otherwise, ice or extremely dirty snow is present. Then, Equation (1) is not valid.

Integral (11) for the spherical broadband albedo with account for Equations (12)–(14) can be evaluated analytically. The answer is:

where

Here, the coefficients

${a}_{j},{b}_{j},{c}_{j}$ are the same as presented in Equation (13) with

j = 1 for the first spectral interval (0.3–0.709 microns) and

j = 2 for the second spectral interval (0.709–0.865 microns).

${\lambda}_{a}=0.3\mathsf{\mu}\mathrm{m},{\lambda}_{1}=0.709$ $\mathsf{\mu}\mathrm{m},{\lambda}_{2}=0.865\mathsf{\mu}\mathrm{m},{\lambda}_{b}=2.4$ $\mathsf{\mu}\mathrm{m},$ $J\left({\lambda}_{a},{\lambda}_{b}\right)$is the integral given in the dominator in Equation (11) [evaluated analytically with account for Equation (12)] and:

At the R (1020 nm) equal to or above 0.5, the analytical expression for the BBA cannot be derived (because one accounts for Equation (1) and not Equation (13) in Equation (11)). Then, the numerical integration procedure is followed.

The broadband plane albedo is calculated in a similar way as a broadband spherical albedo using Equation (5) for the transformation of spherical to plane albedo.

This concludes the description of this new fast radiative transfer Snow and ICE surface albedo retrieval (SICE) that accounts for atmospheric scattering and absorption effects. The SICE algorithm can be considered as an update of the previous version of the algorithm (called S3Snow [

6]) that appeared in the Snow Properties module of SNAP.