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The new developments concerning the optical tomography and the propagation of infrared radiation in a biological tissue can be found in review papers by Arridge , Gibson et al. For systematic study of the radiation transfer theory including the theoretical basis and the computational methods one can recommend the excellent textbooks by Sparrow and Cess , Siegel and Howell and Modest In a set of articles following from this introduction, a reader can find a basic knowledge with the references to the above mentioned textbooks and also to some interesting archive papers.
The most important results of the present-day studies reported recently in journal papers are also included. The latter is expected to be important for potential readers who should know the state-of-the-art in the computational radiative transfer in participating media. Moreover, we are going to update the contents of this topic regularly to follow the new methods and findings in the computational radiative transfer field. Adzerikho, K. Arridge, S. Case, K. Chandrasekhar, S. Press also Dover Publ. Davison, B.
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Dombrovsky, L. Gibson, A. Goody, R. Kokhanovsky, A. Liou, K. Optical properties of snow change because of changes in the microphysical properties of snow [e. It is not possible to account for all of the properties of snow and sea ice from satellite measurements. In order to characterize anisotropy of radiation field over snow, however, such detailed information may not always be necessary.
Generally, the anisotropic factors of bright objects tend to be less dependent on viewing geometry than those of darker objects [ Loeb et al. Separating bright snow and sea ice scenes from the darker scenes, therefore, provides a useful way of classification that is readily attainable from satellite measurements. The average reflectance is derived separately for permanent snow, fresh snow, and sea ice Figure 1.
Therefore, even though M is only a function of the solar zenith angle explicitly, snow and sea ice properties also vary with solar zenith angle. The increasing in reflectance at large solar zenith angles is caused by the spherical geometry of the Earth, which is addressed in detail in section 6. The index is determined by comparing all MODIS reflectance in a region with M and evaluating the following expression, In this expression, N is the total number of samples in the grid box, and n s is the number of samples with a reflectance that exceeds M at the corresponding solar zenith angle.
Values of P are determined monthly for each grid box. Note that dark or bright is determined relative to the mean nadir view reflectance for a given surface type. A darker surface over Greenland in April is not identified relative to the surrounding sea ice, but it is relative to the mean permanent snow reflectance. While some differences in atmospheric and surface conditions such as the column amount of water vapor and size of snow grain might be responsible for the Greenland versus Antarctica snow brightness difference, addressing the reason is out of scope of this paper.
Figure 2 also shows that Arctic and Antarctic sea ice in spring are classified as bright sea ice, while Arctic sea ice in August is classified as dark sea ice. Figure 2 shows, therefore, reasonable seasonal variations of snow and sea ice surface brightness at most of polar regions. However, some features, such as a bright surface in southern part of Greenland in June and the sinuous dark streak along the East Antarctic ridge in February, indicate that results by this simple classification algorithm are not always plausible.
Because CERES instruments sample viewing azimuth angle independent of the viewing zenith angle, measurements are obtained in almost all viewing angles. However, viewing angles in the forward observing direction the relative azimuth angle close to 0 are not sampled when the solar zenith angle is large because CERES instruments avoid scanning the Sun.
We use radiative transfer models to account for radiances in angular bins that are not sampled.
Bohren and Barkstrom  and Wiscombe and Warren  demonstrated that the albedo of snow can be modeled with the incoherent scattering approximation; all snow grains are treated independently so that the volume averaged scattering coefficient is the individual scattering coefficient integrated over size distributions. In addition, these models are based on the assumption that the snowpack albedo is independent of the density. The modeling results and observations of Bohren and Beschta  suggest that these approximations are reasonable even for packed snow.
Snow grains assumed to be spherical [ Leroux et al. Other studies [e. An overview of recent measurements of the bidirectional reflectance of snow is given by Nolin and Liang . Because the average value of these properties within an angular distribution model scene type is unknown, we use DISORT [ Stamnes et al.
The optical properties of nonspherical particles are those of bullet rosettes, aggregates and hollow columns given by Yang et al. Note that only the shape of the modeled radiance curve as a function of angle affects values used to fill angular bins by this method. We use the snow and ice fraction derived by Minnis et al. As a consequence, the snow and sea ice fraction must be less than or equal to the clear fraction. Similar to the permanent snow scenes, we use DISORT to estimate the radiance at angles where measurements are unavailable.
We use snow models described in section 4. Bidirectional reflectance models using packed snow grains do not provide an accurate reflectance of an ice surface. Because a few centimeter snow depth is enough to obscure the surface underneath it, we assume that sea ice surfaces are covered by snow. Even though a significant part of snow on sea ice surfaces in Antarctic melts and forms superimposed ice throughout the year [ Massom et al.
Therefore, we treat sea ice as snow in modeling the radiance.
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To model a partial snow cover and melting sea ice, we linearly combine the radiance reflected by either a land or ocean surface with that reflected by snow. We do not use an interpolation between scene types but an estimate of the error caused by this is given in section 8.
The anisotropic factor is selected from the angular distribution models based on the scene identification i. The model that gives a smaller radiance difference, i. We then use the average reflectance o and ADM mean albedo A adm , o from the bright dark surface overcast angular distribution model to derive the anisotropic factor by where In these expressions, p and A adm , p correspond to the average reflectance and ADM mean albedo, respectively, from the partly cloudy angular distribution model at cloud fraction f , and o bd and A adm , o bd are the average reflectance and ADM mean albedo of the overcast dark and bright surface models combined.
Clouds make a notable difference in the angle dependence of the radiance; on average, clear permanent snow scenes appear more isotropic than cloudy scenes. The maximum anisotropic factor in the forward observing direction is 3. The average anisotropic factor of thick clouds over three snow types is similar, indicating that overcast scenes are identified consistently over the three surface types. If we select a uniform scene and footprints fall nearly the same region, the irradiance should be constant with viewing angle [ Loeb et al.
This variation is mainly due to surface and cloud property variations. As a consequence, irradiances for both the Greenland and Antarctica cases are nearly constant with angle.
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All cases are over bright permanent snow surfaces, which occur mostly over Antarctica Figure 2. The snow surface is modeled by placing a snowpack at the 1 km altitude. We also use optical properties of nonspherical particles given by Yang et al. Therefore, if the mode radius of nonspherical particles is increased, we expect the reflectance of the snowpack to decrease. The reflectance close to nadir decreases more than the reflectance at larger viewing zenith angles in the forward observing direction because the path length in the snowpack increases more in the nadir direction than oblique directions when particles are replaced by more forward scattering particles.
As a consequence, the reflectance of a snow surface with larger particles shows a larger viewing zenith dependence than that with smaller particles. Observed particle size by Grenfell et al. Grenfell et al. The soot content in snow collected at remote site was 0.
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A sensitivity study by Jin and Simpson  indicates that mixing soot in snow by 0. Increasing the water vapor amount in the atmosphere reduces the reflectance; an increase in water vapor amount from 8. The column water vapor amount in the subarctic winter standard atmosphere [ McClatchey et al.
The reduction is larger in the forward observing direction than the backward observing direction because a larger area of shadows is seen in the forward observing directions. Warren et al. While the ADM mean albedo of dark permanent snow is nearly constant, the ADM mean albedo of bright permanent snow and overcast with thick and thin clouds increases with solar zenith angles. While part of the albedo change with the solar zenith angle is caused by regional surface and cloud optical property variations correlated with the solar zenith angle, we expect that the albedo depends on the solar zenith angle because measured radiances over permanent snow indicate that permanent snow surfaces are not Lambertian.
As the solar zenith angle increases, photons scattered in the forward direction need less scattering events to escape upward from the snow surface [ Warren , ]. Therefore, the albedo increases with solar zenith angle and the increase is larger for larger particles. Unlike the radiance, however, surface observations indicate that surface roughness does not affect the albedo very much. According to Warren et al. The atmosphere is assumed to be a spherical shell with 6 layers and composed by nonabsorbing molecules. A cloud layer with the optical thickness of 10 is placed between 5 to 1 km over a black surface.
However, if a particular scene type occurs at a particular viewing geometry and the occurrence of scene types varies among viewing angles, the integration of radiances does not necessarily equal to the average irradiance of all sky conditions. Because a degree of the correlation between scene type and viewing geometry in data taken from Terra is unknown, we use an alternative method to estimate the error in the irradiance derived from angular distribution models in the following section.
We then compare the albedo derived from permanent snow angular distribution models described in earlier sections with the albedo derived from Loeb et al. Figure 8 shows the distribution of the relative albedo difference for one month December.