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The integro-differential radiative transfer equation (RTE, Eq. 1 in Radiative transfer equation) is used to solve multiple light scattering problems, for example, reflectance and transmittance of turbid media. Consider the RTE that is re-written in the following form:
| κ dL(κ, τ) / dτ = -L(κ, τ) + (ω0 / 2) ∫-11 L(κ', τ) pa(κ', κ) dκ' | (1) |
where L is the radiance, κ is the cosine of the observation angle, κ' is the cosine of the incidence angle (Fig. 1), τ is the optical thickness of the scattering medium, ω0 is the single-scattering albedo, and pa is the azimuthally-averaged phase function, defined as follows:
| pa(κ', κ) = [1 / ( 2π) ] ∫02π p(κ', κ, φ - φ' ) d(φ - φ' ) | (2) |
where p is the phase function defined in Eq. 3 of Phase function. The solution of the RTE is generally obtained by using numerical techniques because exact analytical solution of this equation are possible only in certain asymptotic cases (Ishimaru 1999, Kokhanovsky 2006, 2004, 2003a).
One such analytical solution is derived in Solving RTE in the small-angle scattering approximation .., after Kokhanovsky (2007f), for a widely-useful case of a turbid medium composed of plane-parallel layers and having a scattering phase function that strongly peaks in the forward direction. The high asymmetry of the phase function makes it possible to neglect the effect of light scattering at scattering angles much greater than 0°. For the normal incidence of the unidirectional illumination of the medium, considered here, this implies limiting the observation angle range so that κ ~ 1, i.e. centering a narrow range of observation angles about a direction normal to the scattering medium layer. Thus, κ dL(κ, τ) / dτ ~ dL(κ, τ) / dτ. This approximation is referred to as the small-angle scattering approximation, or small-angle approximation for short (SAA).
Hence, in the small-angle approximation, Eq. 1 becomes:
| dL(κ, τ) / dτ = -L(κ, τ) + (ω0 / 2) ∫1-1 L(κ', τ) pa(κ', κ ) dκ' | (3) |
See next Solving RTE in the small-angle scattering approximation ...
| CITATION: Kokhanovsky A. A. 2007. Radiative transfer equation (RTE) in the small-angle approximation: Medium illuminated at normal incidence (www.tpdsci.com/Tpc/RTESAANml.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 14-Feb-2007 Modified: 05-Feb-2007 Peer-reviewed: PENDING |
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