We propose a system of analytical equations to retrieve snow grain size and absorption coefficient of pollutants from snow reflectance or snow albedo measurements in the visible and near-infrared regions of the electromagnetic spectrum, where snow single-scattering albedo is close to 1.0. It is assumed that ice grains and impurities (e.g., dust, black and brown carbon) are externally mixed, and that the snow layer is semi-infinite and vertically and horizontally homogeneous. The influence of close-packing effects on reflected light intensity are assumed to be small and ignored. The system of nonlinear equations is solved analytically under the assumption that impurities have the spectral absorption coefficient, which obey the Ångström power law, and the impurities influence the registered spectra only in the visible and not in the near infrared (and vice versa for ice grains). The theory is validated using spectral reflectance measurements and albedo of clean and polluted snow at various locations (Antarctica Dome C, European Alps). A technique to derive the snow albedo (plane and spherical) from reflectance measurements at a fixed observation geometry is proposed. The technique also enables the simulation of hyperspectral snow reflectance measurements in the broad spectral range from ultraviolet to the near infrared for a given snow surface if the actual measurements are performed at a restricted number of wavelengths (two to four, depending on the type of snow and the measurement system).
The reflective properties of clean and polluted snow are of importance for various applications, including climate (Hansen and Nazarenko, 2007) and environmental pollution (Nazarenko et al., 2017) studies. The spectral snow reflectance is usually studied in the framework of the radiative transfer theory. The application of the numerical methods for the solution of the radiative transfer equation for snow layers has been performed by, among others, Mishchenko et al. (1999), Stamnes et al. (2011) and He et al. (2018). The approximate solutions of the radiative transfer equation useful for snow optics and spectroscopy applications have been developed by Warren and Wiscombe (1980), Wiscombe and Warren (1980) and Kokhanovsky and Zege (2004). In this work, we propose an analytical snow albedo and reflectance model which can be used to derive near-surface snow optical and microphysical properties using measurements at just two to four wavelengths in the visible and near infrared depending on the measurement system and type of snow. In particular, we present the method for the determination of snow grain size, absorption Ångström coefficient and spectral absorption coefficient of impurities embedded in the snow matrix assuming an external mixture of snow grains and impurities. The technique to derive the snow albedo from reflectance measurements is also presented. The absorption and extinction of light by snow grains is treated in the framework of a geometrical optical approximation. The absorption coefficient of impurities is modeled using the Ångström power law. All derivations are performed in the framework of the asymptotic radiative transfer theory (see, e.g., Kokhanovsky and Zege, 2004; Zege et al., 2011). It is assumed that the snow layer is vertically and horizontally homogeneous and semi-infinite. Therefore, the effects of the finite layer thickness are ignored.
The snow reflectance
The product of the effective diameter
We present the absorption coefficient of pollutants in snow as
It follows from Eqs. (4)–(8) that
The parameter
Using the EAL, the equations for the snow reflectance and spherical albedo may be simplified.
Namely, it follows that
Equations (25) and (26) can be used to find the pair
One may also derive the impurity absorption coefficient at the wavelength
To determine the concentration of pollutants
The value of
In particular, it follows for soot impurities that
If the plane albedo is the measured physical quantity, one needs to find only
three constants:
The respective analytical equations can be presented as
In the case of unpolluted snow, one derives
We have applied the technique developed above to the measured spectral plane albedo both for polluted and pure snow. Therefore, in situ spectral albedo measurements were obtained from two different field sites located in the French Alps (polluted snow) and in Antarctica (clean snow).
The spectral albedo of a spring alpine snowpack was measured at the Col du
Lautaret field site (
The intercomparison of theory (symbols) with experimental measurements
of plane albedo (line, no noise removed) performed in the French Alps
(
The results of comparison of measurements and the theory presented above are
illustrated in Fig. 1 at the Col du Lautaret field site. The parameters
The derived spectral probability of photon absorption for the case presented in Fig. 1.
The derived snow parameters for the five samples. The value of
The spectral albedo of pure snow (very low amount of impurities) was measured
at Dome C (
The results of the application of the proposed technique to the pure snow (no
pollution) albedo measured in Antarctica are illustrated in Fig. 3.
Application of our technique results in excellent agreement with measured
albedo over pure snow (no pollution) in Antarctica. Because the snow at Dome
C is clean/pristine, the value of
The intercomparison of theory (symbols) with experimental measurements
of plane albedo (line) performed in Antarctica (Dome C;
The application of the developed theory to the measurements of the spectral
reflectance is presented in Fig. 4 for two locations with different dust
loads (39.6 and 107.4 ppm). The spectral reflectance of snow was measured in
the European Alps (Artavaggio plains; 1650 m a.s.l.;
The intercomparison of theory (symbols) with experimental measurements
(line) in the European Alps (
One can see that the theory works well not only for the albedo measurements
(see the previous section) but also for the reflectance measurements for
polluted snow layers. In particular, our results are closer to the
measurements than the theoretical model described by Flanner et
al. (2007) (see Fig. 4b in Di Mauro et al., 2015). The derived parameters are
given in Table 1 (lines 6–7). The value of
The mass absorption coefficient (MAC) can be estimated using
In this work, we have presented a sequence of analytical equations, which can be used to determine the snow grain size, the absorption coefficient of impurities and the absorption Ångström coefficient of surface snow impurities from the snow reflectance measured at four wavelengths: two in the visible and two in the near infrared, as suggested by Warren (2013). In the case of albedo measurements just three wavelengths can be used to find main snow properties. For unpolluted snow, it is enough to perform the measurements at two wavelengths (for reflectance measurements) or just at a single wavelength (for albedo measurements) in the near infrared to determine the snow grain size.
In principle, the refractive index of dust and dust size distribution can also be determined using derived the spectral absorption coefficient of dust and assuming the shape of dust particles. However, we did not make an attempt at such retrievals in this work. A method for the retrieval of the complex refractive index and single-scattering optical properties of dust deposited in mountain snow based on exact radiative transfer calculations was proposed by McKenzie Siles et al. (2017) under the assumption that local optical properties of dust grains can be simulated assuming the spherical shape of particles. Their method is based on the extraction of dust grains from snowpack. Our technique does not require such a complicated procedure.
We have demonstrated how snow albedo can be derived from spectral reflectance
measurements avoiding complicated integration with respect to the observation
geometry (azimuth, viewing angle). The last point is useful for the
determination of the snow
The determination of the EAL
The data are available from the first author upon request.
Nomenclature.
Let us consider the error budget for the retrieved snow parameters. To simplify, we assume that the snow parameters are derived using albedo measurements.
The value of
The uncertainty in the parameter
Let us consider the error budget for the retrieved spectral absorption
coefficient of pollutants. The absolute error of the retrieved parameter
It follows from Eqs. (46) and (B3) that
The absorption coefficient of pollutants is given by the following equation
(see Eqs. 8, 13, 30):
One derives for this coefficient
In particular, one finds that the positive bias in the measured albedo in the visible will lead to the underestimation of the concentration of pollutants (assuming that the grain size is exactly known). It should be pointed out that in most cases the concentration of pollutants is so small that it can not be assessed using optical instruments (change in reflectance is inside experimental measurement error). This issue has been discussed by Zege et al. (2011) and Warren (2013). Similar conclusions hold also if the reflectance (and not albedo) is the measured quantity.
AK prepared the first draft of the paper and also derived all equations presented in the paper. The measurements were performed by ML, BDM, GP, LA, MD, and FT. CB and JEB contributed to the discussion of the results and preparation of the final version of the paper.
The authors declare that they have no conflict of interest.
This work was mainly supported by the European Space Agency in the framework of ESRIN contract no. 4000118926/16/I-NB, “Scientific Exploitation of Operational Missions (SEOM) Sentinel-3 Snow (Sentinel-3 for Science, Land Study 1: Snow)”. CNRM/CEN and IGE are part of LabEx OSUG@2020. Measurements in the French Alps were funded by the ANRJCJ grant EBONI 16-CE01-0006 and at Dome C by ANR JCJC MONISNOW 1-JS56-005-01. Edited by: Benjamin Smith Reviewed by: two anonymous referees