2.1. Theory
Snow is composed of transformed ice crystals that precipitate from the atmosphere. The crystals accumulate on the ground and metamorphose in place. The transformed snow may ultimately melt, slide and sublimate. The volumetric concentration of ice in snow is about 30%, although this number varies depending on the snow type and age. The effective grain size (diameter) d of crystals in snow is much larger as compared to the wavelength λ of incident radiation in the visible and near infrared regions of electromagnetic spectrum. Therefore, the geometrical optics approach is suitable for the description of light scattering, extinction and absorption in a snowpack. The processes of light reflection and refraction occur at the surfaces of crystals. Additionally, light absorption by bulk ice occurs after penetration of photons to the interior of ice crystals. Through a semi-infinite snow layer approximation, we assume that snowpack is so optically thick that the influence of the underlying surface can be neglected. This is a valid assumption in the visible part of the electromagnetic spectrum for snow layers having a thickness larger than about 10–30 cm, depending on snow type, pollution level and underlying surface type. The penetration depth is much smaller for the near-infrared radiation (below 2–5 cm depending on the wavelength) mostly used for the retrievals discussed in this paper. This means that we retrieved snow surface parameters (e.g., grain size at the top of snow layer). We also ignored vertical and horizontal inhomogeneities of snow variables and assumed that the snow is a plane-parallel medium. The retrieval of properties of snowpack covering the tilted surfaces is considered in the framework of the simplified theory.
In our theory, the reflection function of snow depends on just two inherent snow properties: single scattering albedo (SSA)
(or, alternatively, the probability of photon absorption (PPA)
) and the probability of photon scattering (phase function
p (θ)) in a given direction (specified by the scattering angle θ) by a unit volume of snowpack [
19,
20]. The phase function
p(θ) depends on the scattering angle being larger in the forward scattering region, where the diffraction of light on ice crystals also takes place. In the visible spectral domain and in the absence of pollutants in snow, the single scattering albedo is spectrally neutral and close to one due to extremely weak absorption of radiation by bulk ice [
21,
22]. This explains the brightness and white color of freshly accumulated dry snow. The theoretical value of fresh clean snow albedo (planar and spherical) is close to one in the visible wavelengths. Its dependence on the size and shape of ice crystals can be neglected in the visible. For instance, it has been found from spectral radiance measurements [
23] that the snow albedo in Antarctica is in the range 0.96–0.98 in the visible range of the electromagnetic spectrum almost independently of the snow grain size, shape and illumination conditions. The small difference of albedo from one (0.02–0.04) can be attributed to the snow contamination and measurement errors. Indeed, in the absence of absorption processes, all photons penetrating through the snowpack surface must return back toward the atmosphere and outer space, resulting in albedo equal to 1.0. This is true independently on the type of local scattering law as long as the medium can be treated as semi-infinite. The fresh snow bidirectional reflection distribution function (BRDF) does not depend on the grain size in the visible wavelengths because the light scattering takes place in the geometrical optics domain and the absorption processes by bulk ice in the visible are weak. Although it depends on the shape of particles. It is known that the single scattering pattern (phase function) reaches its asymptotic shape as d/λ→∞. This asymptotic pattern is different for absorbing and non-absorbing particles. Furthermore, the pattern is influenced by the shape of grains. Therefore, for instance, the assumption of spherical particles cannot be used in snow optics because this assumption leads to BRDFs never observed for snow [
24]. Experimental measurements demonstrate that the snow BRDF can be quite accurately modeled by assuming that the snow phase function is flat in the backward hemisphere. The characterization of semi-infinite non-absorbing snow BRDF using theoretical calculations is a complicated matter because the snow is composed of large and irregularly shaped close-packed ice grains. Internally and externally mixed pollutants (such as mineral dust and black carbon) may be present in a snowpack, strongly affecting its optical properties [
25,
26].
Let us introduce the snow reflectance
, where
is the reflected light intensity,
is the cosine of the solar zenith angle and
is the solar flux incident on the area perpendicular to the solar beam.
R is equal to 1.0 for the ideal Lambertian surface with albedo 1.0 at any observation/illumination geometry. Although visible snow is close to the Lambertian surface with albedo 1.0, the deviations from the Lambertian surface model are significant and cannot be ignored [
27,
28,
29].
An important result of the asymptotic radiative transfer theory (see
Appendix A) is the fact that the snow reflectance at no absorption
(e.g., in the visible in absence of pollutants) can be related to the same snow (with the same phase function) reflectance with single scattering albedo not equal to 1.0. In particular, it follows in the framework of the exponential approximation [
30,
31,
32] that
where
where
g is the average cosine of scattering angle in snow, which is defined as
The angular function
f satisfies to the reciprocity principle and depends on the viewing/observation geometry via the reflectance of non-absorbing snow layer
and via the escape function, which can be approximated by the linear function of the cosine of the viewing zenith angle μ [
33]
The angular function f also depends on the type of snow/shape of ice crystals (via reflectance/phase function), but not on the snow grain size.
The snow plane albedo
can be derived from Equation (1). In particular, it follows by definition that
where
where φ is the relative azimuthal angle. The double integral (6) can be evaluated numerically, but we will use the accurate analytical approximation described below. As
y → 0, it follows from Equation (1) that
Because
is referred to as a reflecatance for a non-absorbing snow, the double integral (6) with
is equal to the plane albedo of non-absorbing snow. Therefore, this integral is equal to 1.0 by definition. Then, it follows from Equation (8) (see also Equations (5)–(7)) that
We will use the exponential approximation similar to Equation (1)
Equation (10) coincides with Equation (9) as y → 0. However, it can be used in a wider range of the parameter y as compared to Equation (9).
The spherical albedo is defined as
It follows from Equations (5) and (9) that
Therefore, one derives in the framework of the exponential approximation(see Equations (9) and (10))
Equation (12) can be used to make an interpretation of the similarity parameter
y. Namely, it follows for the absorptance
A of the snow layer under the diffuse illumination conditions and small values of the probability of photon absorption
This means that the parameter y coincides with the value of the snow layer absorptance at small values of the probability of photon absorption in snow. The value of A increases as g (see Equations (3) and (14)) increases as one may expect (stronger forward scattering and, therefore, light penetration to a layer, where light can be effectively absorbed).
It follows from the above equations that the reflectance and plane albedo of snow can be expressed via spherical albedo in the approximation under study
and
It follows from Equations (5) and (16) that the plane and spherical albedo coincide at the cosine of the solar zenith angle equal to 2/3. Equation (15) allows the estimation of spherical (and, therefore, plane albedo (see Equation (16)) from the reflectance measurements at fixed observation geometry avoiding integration procedure as shown in Equations (6), (7), (11). It should be stressed that most of optical instruments do not sample all observation directions; therefore, the theory presented here gives an important hint on how to avoid poor sampling of angular measurements in the procedures for the determination of snow albedo.
2.2. The Accuracy of Exponential Approximation
In this section, we study the accuracy of the presented approximations at various values of the single scattering albedo using the exact radiative transfer calculations. The Henyey-Greenstein phase function
where
η is the cosine of the scattering angle, with the average cosine of scattering angle g = 0.75 similar to that for snow and crystalline clouds, is assumed in all calculations. As a matter of fact, it is not the specific phase function, but rather the average cosine of scattering angle that determines the relationship between
R and
to be used in this work (see Equation (1)). We assumed an optical thickness of 5000 in our numerical solution of the integro-differential radiative transfer equation for determination of snow reflectance and in the evaluation of the approximations presented in this work. Such an assumption is needed to have the case of a semi-infinite medium considered in the previous section. In this paper, we made use of Intensity and POLarization calculation radiative transfer code (IPOL [
34]). IPOL is a Fortran 90/95, BLAS/LAPACK based, discrete-ordinates matrix-operator vector radiative transfer code that computes intensity and polarization (including ellipticity) of the solar radiation reflected from various turbid media. The code also computes planar and spherical albedos by numerical integration of the azimuthally averaged intensity. Recent intercomparison of radiative transfer codes [
35] revealed high accuracy of IPOL, including the case of cloud layers, which are in many respects analogous to snowpack.
The results of inter-comparison of exact and approximate results for the escape function are illustrated in
Figure 1. It follows that the approximation can be used for solar zenith angles under 80 degrees. The accuracy for the reflectance is illustrated in
Figure 2 at the nadir observation and for various values of single scattering albedo. It follows that the reflectance decreases for the cases of high solar zenith angles. The reflectance increases for the overhead Sun position. It follows that the approximation is highly accurate even for low (0.2) reflectance values (and PPA equal to 0.05), when the dependence of reflectance on the solar zenith angle is low. The errors are also small (see
Figure 3) for the plane albedo in the range 0.3–1.0 typical for clean and polluted snow. In
Figure 3, the accuracy of two approximations for the plane albedo is shown (via the similarity parameter
y as shown in Equation (10) and also using Equation (16) with spherical albedo derived from Equation (15)). Both approximations produce acceptable results. The accuracy of approximation for the spherical albedo is illustrated in
Figure 4. It follows that the approximation provides accurate results even at comparatively large values of similarity parameter (
y = 2.5). The summary of relative errors for various parameters is illustrated in
Figure 5 for the nadir observation case. The spread along the vertical axis is due to solar zenith angle. We conclude that errors are smaller than 5% for the majority of cases. Therefore, the derived approximations are highly accurate and can be used for the solution of the inverse problem: determination of snow properties using optical measurements. Similar conclusions have been reached using a ray tracing model at the grain scale for a wavelength of 1300 nm [
36]. A slight discrepancy appeared only at 1550 nm, where the ice absorption is extremely strong [
37].
The indirect confirmation of the applicability of the approximate theory also follows from
Figure 6, where we present the results of plane and spherical albedo calculations for the various values of single scattering albedo (as in
Figure 2). The spherical albedo and planar albedo curves indeed intersect at
, as suggested by the theory presented above. In this case, the solar zenith angle (SZA) is equal to 48 degrees.