1. Introduction

2. The Analytical Representation of the Top-of-Atmosphere Reflectance over Snow

2.1. The Radiative Transfer in a Snow Layer

The top-of-atmosphere (TOA) reflectance over snow $R_{TOA}$as used in version 1.1 of the SNOWTRAN software, is given by the following analytical expression [16]:

$$R_{TOA}=R{T}_g,$$

(1)

where $T_g$ is the gaseous transmittance, R is the TOA reflectance of the gaseous absorption free atmosphere - underlying snow surface derived as

$$R{=R}_a+\gamma T_a{R}_s,$$

(2)

$R_a$ is atmospheric reflectance for the case of a black underlying surface, $T_a$is the two-way atmospheric transmittance, ${\mathrm{a}\mathrm{n}\mathrm{d}R}_s$ is the snow reflectance (bottom of atmosphere (BOA) reflectance). The parameter $\gamma$ accounts for the atmosphere–snow retro-reflections. The value of $\gamma$ can be calculated using the following approximation strictly valid for the Lambertian underlying surfaces [16]:

$$\gamma ={(1-r_a{r}_s)}^{-1},$$

(3)

where $r_a$ is the atmospheric albedo, $r_s$ is the snow spherical albedo.

It is assumed that the snow can be represented as a semi-infinite layer with irregularly shaped ice crystals. The spherical albedo of such a layer with the single scattering albedo ${\omega }_0$ and the asymmetry parameter $g$ can be estimated using the following approximation [14]:

$$r_s=\frac{(1-as)(1-s)}{(1+bs)},$$

(4)

where a=0.139, b=1.17 and

$$s=\sqrt{\frac{(1-{\omega }_0)}{(1-g{\omega }_0)}}$$

(5)

is the similarity parameter. It follows from Eq. (4) that various turbid media with close values of the similarity parameter s have similar values of spherical albedo. The approximation given by Eq. (4) is very accurate and can be used at any level of light absorption in snow, which is relevant for the modelling of visible-near infrared (VNIR) and SWIR snow spectra.

In this paper we derive the relevant approximation for the value of the nadir snow reflectance $R_s,\mathrm{w}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{s}\mathrm{a}\mathrm{c}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{i}\mathrm{n}\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{V}\mathrm{N}\mathrm{I}\mathrm{R}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{S}\mathrm{W}\mathrm{I}\mathrm{R}\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}.$In principle, $R_s$ can be found solving the respective integro-differential radiative transfer equation [13,15]. It is assumed in the framework of SNOWTRAN that the nadir snow reflectance can be calculated using the following approximation derived from the parameterization of the simultaneous radiative transfer calculations of the snow reflectance $R_s$ and spherical albedo $r_s$as presented in [15]:

$$R_s=a_0+a_1{r}_s+a_2{r}_s^2,$$

(6)

where the parameters $a_0,a_1,a_2$ depend on the cosine of the solar zenith angle${\mu }_0$ (see Appendix A). The accuracy of Eqs. (4), (6) is demonstrated in Figure 1 for the case of the Henyey - Greenstein phase function [13,15] with the asymmetry parameters g equal to 0.75 and 0.875 (typical for snow [17]) and several values of the single scattering albedo ${\omega }_0$.

Eqs. (6) and (4) make it possible to calculate the global radiative transfer characteristics of snow layers for a given value of the similarity parameter s. It follows from Eq. (5) that the similarity parameter s depends on just two local optical characteristics of snow (${\omega }_0$ and $g$). Therefore, an important point is to find analytical relationships of these parameters with the size of ice grains. This also makes it possible to reduce the number of input parameters for the model proposed in this paper. Under assumption of clean snow, we shall use the following approximate equations for the relevant parameters [17,18]:

$${\omega }_0=1-{\beta }_i,{\beta }_i=\frac{1}{2}(1-\rho )(1-e^{-\sigma z}),g=g_{\infty }-(g_{\infty }-g_0)e^{-\epsilon z}$$

(7)

derived using the geometrical optics approximation for the fractal ice grains [17,19]. Here, ${\beta }_i$ is the probability of photon absorption (PPA) by pure snow at the wavelength $\lambda$, $\alpha =4\pi \chi /\lambda ,$ $z=\alpha d_{ef}$ is the bulk ice absorption coefficient, $\chi$ is the imaginary part of the ice refractive index, $d_{ef}$ is the effective diameter of ice grains and parameters $\rho ,\sigma ,\epsilon ,g_0,g_{\infty }$depend on the assumed type/shape of ice crystals in snow. The effective grain diameter is defined as[17,20,21]:

$$d_{ef}=\frac{3V}{2\Sigma },$$

(8)

where V is the average volume of the ice grains and $\Sigma$ is their average cross section perpendicular to the incident beam. It is assumed that particles are randomly oriented.

We have found that the volume of fractal grains used in simulations (Koch fractals of second generation [19]) is related to the side length $\mathcal{l}$ by the following formula:

$$V=b{\mathcal{l}}^3,$$

(9)

where $b=27/52\sqrt{2}\approx \mathrm{0.367.}$ Then it follows: $d_{ef}=\varsigma {\mathcal{l}}^3/\Sigma$, where $\varsigma =1.5b$. The numerical calculations show that the value of $d_{ef}$ is close to $\mathcal{l}$/2.

We use the following equations for the parameters$\rho ,g_0,g_{\infty }$ derived for the real part of the refractive index n in the range 1.25-1.35, which is the case for ice in the spectral region 0.3-2.5 μm [17,22]:

$$\mathsf{\rho }=0.0123+0.1622(n-1),g_0=0.9919-0.769(n-1),g_{\infty }=1.008-0.11(n-1).$$

(10)

In addition, when fitting the numerical Monte–Carlo geometrical optical calculations with the use of Eqs. (7), we found that σ=0.9045, ε=0.8571. These parameters depend on the assumed shapes of ice crystals in snow.

The extinction cross section of large nonspherical randomly oriented particles can be calculated as follows [20,21]:

$$C_{ext}=2\Sigma .$$

(11)

Therefore, it follows for the extinction coefficient [17]:

$$k_{ext}=\frac{3c}{d_{ef}},$$

(12)

where c is the volumetric concentration of particles. Taking into account that c is close to 1/3 for natural snow$\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}$ [17], it is concluded that the extinction length in snow $L_{ext}=k_{ext}^{-1}$ is close to the effective diameter of ice grains.

The intercomparison of results derived using simple Eqs. (7) and time consuming geometrical optical Monte - Carlo calculations [18] is shown in Figure 2 a,b for the case of large nonspherical particles (Koch fractals of the second generation [18] with the side length of the initial tetrahedron $\mathcal{l}$ equal to 50, 100 and 300 μm ( or $d_{ef}$=0.027, 0.054, and 0.162 mm, respectively). The ice spectral refractive index proposed in [22] has been used. The high accuracy of Eqs. (7) is clearly demonstrated. Some deviations could be actually due to the statistical error of Monte – Carlo calculations and not due to the errors of Eqs. (7). Figure 2c shows the results of calculation of the similarity parameter s using Eqs. (5), (7) and Monte – Carlo geometrical optics calculations. Both results almost coincide, which makes it possible to derive analytical relationship between the value of the nadir reflectance (see Eqs. (6), (4)) and the effective diameter of ice grains, which is of importance for the solution of the inverse snow optics problems.

Figure 2. a. The spectral dependence of the probability of photon absorption derived using the geometrical optics Monte Carlo code (lines) and simple Eq. (7) (symbols) for the length $\mathcal{l}$ of the side of the initial ice tetrahedron for the Koch fractal of the second generation [17] equal to 50, 100 and 300µm.

Figure 2. a. The spectral dependence of the probability of photon absorption derived using the geometrical optics Monte Carlo code (lines) and simple Eq. (7) (symbols) for the length $\mathcal{l}$ of the side of the initial ice tetrahedron for the Koch fractal of the second generation [17] equal to 50, 100 and 300µm.

Figure 2. b. The same as in Figure 2a except the asymmetry parameter is considered. The lower and upper lines give the spectral dependencies of the parameters $g_0\left(\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\right)$and $g_{\infty }$ (upper line).

Figure 2. b. The same as in Figure 2a except the asymmetry parameter is considered. The lower and upper lines give the spectral dependencies of the parameters $g_0\left(\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\right)$and $g_{\infty }$ (upper line).

Figure 2. c. The same as in Figure 2a except for the similarity parameter.

Figure 2. c. The same as in Figure 2a except for the similarity parameter.

We have studied the case of type 1 snow (clean snow) in the discussion given above. Let us consider now the case of type 2 snow (snow with impurities). The level of snow pollution is usually low and, therefore, in a good approximation, we can assume that the light scattering and extinction processes in snow are dominated by ice grains. The absorption coefficient of snow with impurities can be presented as:

$$k_{abs}=c_i{\kappa }_i+c_p{\kappa }_p,$$

(13)

where $c_i$ is the volumetric concentration of ice grains and $c_p$is the volumetric concentration of impurities. We assume that just one type of impurity is present in the snow, although the technique can easily take into account multiple types of impurities by adding additional terms in Eq. (13). The values of ${\kappa }_i\mathrm{and}{\kappa }_p$ are volumetric absorption coefficients of ice grains and impurities, respectively. They can be calculated as follows:

$${\kappa }_i=\frac{〈C_{abs,i}〉}{〈V_i〉},{\kappa }_p=\frac{〈C_{abs,p}〉}{〈V_p〉},$$

(14)

where $〈C_{abs,i}〉$ is the average absorption cross section of ice grains, $〈V_i〉$is the average volume of ice grains, $〈C_{abs,p}〉$ is the average absorption cross section of impurities, $〈V_p〉$is the average volume of impurities. It follows for the probability of photon absorption:

$$\beta \equiv \frac{k_{abs}}{k_{ext}}={\beta }_i+c_p\frac{{\kappa }_p}{k_{ext}},$$

(15)

where ${\beta }_i$is the PPA for snow without impurities (see Eq. (7)), and thus (see Eqs. (15), (12)):

$$\beta ={\beta }_i+\frac{c{d}_{ef}}{3}{\kappa }_p,$$

(16)

where $c$=$c_p/c_i$. One can see that the probability of photon absorption for the snow with impurities depend not only on the value of ${\beta }_i$ but also on the relative volumetric concentration of impurities c and their spectral volumetric absorption coefficient ${\kappa }_p$. We shall parameterize the spectrum ${\kappa }_p$ in the following way:

$${\kappa }_p(\lambda )={\kappa }_p({\lambda }_0){(\lambda /{\lambda }_0)}^{-m},$$

(17)

where ${\kappa }_0={\kappa }_p({\lambda }_0)$ and m are the input parameters of the discussed analytical model. We shall assume that ${\lambda }_0$= 550 nm.

3. The Comparison of Theoretical Calculations with Satellite Measurements

Figure 4. a. The intercomparison of SNOWTRAN spectra with the EnMAP measured spectra (75.119S,123.902E, February 15, 2023, 00:36UTC). The EnMAP data in the spectral range with strong water vapor absorption (e.g., around 1800nm) are not provided by the satellite ground system.

Figure 4. a. The intercomparison of SNOWTRAN spectra with the EnMAP measured spectra (75.119S,123.902E, February 15, 2023, 00:36UTC). The EnMAP data in the spectral range with strong water vapor absorption (e.g., around 1800nm) are not provided by the satellite ground system.

Figure 4. b. The relative difference (in percent) between theoretical calculations and experimental data in the VNIR region.

Figure 4. b. The relative difference (in percent) between theoretical calculations and experimental data in the VNIR region.

Figure 4. c. The absolute difference between theoretical calculations and experimental data in the VNIR (upper panel) and SWIR (lower panel).

Figure 4. c. The absolute difference between theoretical calculations and experimental data in the VNIR (upper panel) and SWIR (lower panel).

Figure 5. a. The intercomparison of SNOWTRAN spectra with the PRISMA measured spectra (76.14S,129.93E, December 21, 2022, 00:14UTC).

Figure 5. a. The intercomparison of SNOWTRAN spectra with the PRISMA measured spectra (76.14S,129.93E, December 21, 2022, 00:14UTC).

Figure 5. b. The spectral relative difference (in percent) between theoretical calculations and PRISMA data in the VNIR region.

Figure 5. b. The spectral relative difference (in percent) between theoretical calculations and PRISMA data in the VNIR region.

Figure 5. c. The spectral absolute difference between theoretical calculations and PRISMA data.

Figure 5. c. The spectral absolute difference between theoretical calculations and PRISMA data.

Table 2. The parameters of the model used in simulations for the case of nadir observations shown in in Figs.4a, 5a. The ground-observed TOC and PWV at Dome C (75S, 123.3E) are given in the brackets. It has been assumed that snow is free of impurities. The small difference of satellite measurements ( up to 5 degrees) from the exact nadir direction is ignored.

4. The Determination of the Snow Grain Size

5. Conclusions

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