Reflection of Solar Light from Surface Snow Loaded with Light-Absorbing Impurities: A Case Study of Black Carbon, Mineral Dust, and Ash.

Using a hemispherical directional reflectance factor instrument, spectral data of dirty snow containing black carbon (BC), mineral dust (MD), and ash was collected from multiple locations to investigate the impact of these light-absorbing impurities (LAIs) on snow reflectance characteristics. The findings revealed that the perturbation of snow reflectance caused by LAIs is characterized by nonlinear deceleration, indicating that the reduction in snow reflectance per unit ppm of LAIs declines as snow contamination increases. The reduction in snow reflectance caused by BC may reach saturation at elevated particle concentrations (thousands of ppm) on snow. Snowpacks loaded with MD or ash initially exhibit a significant reduction in spectral slope around 600 and 700 nm. The deposition of numerous MD or ash particles can increase snow reflectance beyond the wavelength of 1400 nm, with an increase of 0.1 for MD and 0.2 for ash. BC can darken the entire measurement range (350-2500 nm), while MD and ash can only affect up to 1200 nm (350-1200 nm). This study enhances our understanding of the multi-angle reflection characteristics of various dirty snow, which can guide future snow albedo simulations and improve the accuracy of LAIs' remote sensing retrieval algorithms.

Comparative analysis of radiative transfer approaches for calculation of plane transmittance and diffuse attenuation coefficient of plane-parallel light scattering layers.

We present an analysis of a number of different approximations for the plane transmittance Tp and diffuse attenuation coefficient Kd of a semi-infinite, unbounded, plane-parallel, and optically homogeneous layer. The maximally wide optical conditions (from the full absorption to the full scattering and from the fully forward to the fully backward scattering) were considered. The approximations were analyzed from the point of view of their physical limitations and closeness to the numerical solution of the radiative transfer equation for the plane transmittance. The main criterion for inclusion of the models for analysis was the possibility of practical use, i.e., approximations were well parameterized and included only easily measured or estimated parameters. A detailed analysis of errors for different Tp and Kd models showed that the two-stream radiative transfer Ben-David model yields the best results over all optical conditions and depths. However, the quasi-single-scattering and polynomial Gordon's approximations proved to be the best for the depths close to zero.

Comparative analysis of radiative transfer approaches for calculation of diffuse reflectance of plane-parallel light-scattering layers.

We present an analysis of a number of different approximations for the diffuse reflectance (spherical and plane albedo) of a semi-infinite, unbounded, plane-parallel, and optically homogeneous layer. The maximally wide optical conditions (from full absorption to full scattering and from fully forward to fully backward scattering) at collimated, diffuse, and combined illumination conditions were considered. The approximations were analyzed from the point of view of their physical limitations and compared to the numerical radiative transfer solutions, whenever it was possible. The main factors impacting the spherical and plane albedo were revealed for the known and unknown scattering phase functions. The main criterion for inclusion of the models in analysis was the possibility of practical use, i.e., approximations were well parameterized and only included easily measured or estimated parameters. We give a detailed analysis of errors for different models. An algorithm for recalculation of results under combined (direct and diffuse) illumination also has been developed.

Radiative properties of optically thick fluorescent turbid media.

In this paper simple analytical equations for the reflection and transmission coefficients of fluorescent turbid media are given. The case of weakly absorbing optically thick media is considered (e.g., paper, textiles, tissues). The calculations are performed in the framework of the two-flux approximation for finite layers under monochromatic illumination conditions. The relationships of Kubelka-Munk parameters to the true absorption and transport extinction coefficients of fluorescent turbid media are derived. The results can be used for the development of various optimization procedures in the paper and textile industries and also in the area of fluorescence spectroscopy of turbid media.

Scattered light corrections to Sun photometry: analytical results for single and multiple scattering regimes.

Analytical equations for the diffused scattered light correction factor of Sun photometers are derived and analyzed. It is shown that corrections are weakly dependent on the atmospheric optical thickness. They are influenced mostly by the size of aerosol particles encountered by sunlight on its way to a Sun photometer. In addition, the accuracy of the small-angle approximation used in the work is studied with numerical calculations based on the exact radiative transfer equation.

Scaling constant and its determination from simultaneous measurements of light reflection and methane adsorption by snow samples.

We describe a fast and accurate method for the determination of the specific surface area of snow samples based on the measurements of the snow reflection function at a single wavelength and geometry. The method is less sensitive to the assumed shape of particles as compared with other techniques. The concept of the snow scaling constant is introduced, and its value is derived from simultaneous measurements of light reflectance and methane adsorption.

Scattering optics of snow.

Permanent snow and ice cover great portions of the Arctic and the Antarctic. It appears in winter months in northern parts of America, Asia, and Europe. Therefore snow is an important component of the hydrological cycle. Also, it is a main regulator of the seasonal variation of the planetary albedo. This seasonal change in albedo is determined largely by the snow cover. However, the presence of pollutants and the microstructure of snow (e.g., the size and shape of grains, which depend also on temperature and on the age of the snow) are also of importance in the variation of the snow's spectral albedo. The snow's spectral albedo and its bidirectional reflectance are studied theoretically. The albedo also determines the spectral absorptance of snow, which is of importance, e.g., in studies of the heating regime in snow. We investigate the influence of the nonspherical shape of grains and of close-packed effects on snow's reflectance in the visible and the near-infrared regions of the electromagnetic spectrum. The rate of the spectral transition from highly reflective snow in the visible to almost totally absorbing black snow in the infrared is governed largely by the snow's grain sizes and by the load of pollutants. Therefore both the characteristics of snow and its concentration of impurities can be monitored on a global scale by use of spectrometers and radiometers placed on orbiting satellites.

Statistical properties of a photon gas in random media.

This paper is devoted to a derivation of the probability distribution of photon escape from a semi-infinite random medium, depending on the number of its interactions with macroscopic particles inside the medium. The consideration is limited to the case of highly developed multiple light scattering. The distribution function found facilitates the solution of both direct and inverse problems in light scattering media optics.

Determination of the droplet effective size and optical depth of cloudy media from polarimetric measurements: theory.

We present the development of a semi-analytical algorithm for optical particle sizing in disperse media. The algorithm is applied to the specific case of water clouds. However, it can be extended with minor modifications to other types of light-scattering medium. It is assumed that the optical thickness tau of the medium is large and the probability of photon absorption beta is small. Thus the optical particle-sizing problem is studied in the regime of highly developed multiple light scattering. It was found that the degree of polarization in visible and near-infrared channels provides us with information both on the effective size of droplets and on the optical thickness tau.

Simple approximate formula for the reflection function of a homogeneous, semi-infinite turbid medium.

A simple, approximate analytical formula is proposed for the reflection function of a semi-infinite, homogeneous particulate layer. It is assumed that the zenith angle of the viewing direction is equal to zero (thus corresponding to the case of nadir observations), whereas the light incidence direction is arbitrary. The formula yields accurate results for incidence-zenith angles less than approximately 85 degrees and can be useful in analyzing satellite nadir observations of optically thick clouds.