Topics in Particle and Dispersion Science

Derivation of the RTE in the small-angle approximation

Parent topic Fig. 1

The radiative transfer equation (RTE), which is written for a general case of radiative transfer in a turbid medium as Eq. 1 in Radiative transfer equation, is repeated here for convenience by specifying relevant directions in the polar coordinates (i.e. by using angles instead of vectors):

dL(γ, φ, r) / dr	= -cL(γ, φ, r) + [b / 4π ]
	∫0 2π ∫0 π L(γ', φ', r) p(γ', φ'; γ, φ) sin γ' dγ' dφ'	(1)

where L is the radiance, γ and φ are respectively the elevation and azimuth angles that define the observation direction, angles γ' and φ' define the incidence direction (see also Fig. 1), r denotes the position vector, p is the phase function defined in Eq. 3 of Phase function [hence a factor of 1 / (4π) here], c is the attenuation coefficient of the turbid medium, and b is its scattering coefficient.

Consider a plane-parallel isotropic turbid medium with vertical illumination (i.e. normal to the boundary of the medium, γ' = 0). In this case dr = dz / cosγ = dz / κ, where z is the vertical axis oriented downward. Further, by the symmetry of the case, the radiance, L, in the path function (the second term of the right side in Eq. 1) is only a function of the elevation angle, γ', and depth, z, in the turbid medium. Hence Eq. 1 can be simplified as follows:

κ dL(κ, z) / dz	= -cL(κ, z) +
	(b / 2) ∫0 π L(γ', z) pa(γ', γ) sin γ' dγ'	(2)

where p_a is the azimuthally-averaged phase function:

pa(γ', γ) = [1 / (2π) ] ∫02π p(γ', γ) dφ

(3)

with p(γ', γ) being the phase function. If the latter is axially symmetrical, i.e. depends only on the scattering angle, θ, then p_a is identical to the phase function itself because of the lack of dependency on the azimuth angle. Let us now rewrite Eq. 2 by using the single-scattering albedo, ω₀ = b / c, where b is the scattering coefficient, and optical thickness, τ = cz (hence dz = dτ / c), as well as by noting that sinγ dγ = - dκ. Notation p_a(γ', γ) is equivalent to p_a(κ', κ). We thus obtain:

κ dL(κ, z) / dτ = -L(κ, z) + (ω0 / 2 ) ∫-11 L(κ', z) pa(κ',κ) dκ'

(4)

which is in the form of the RTE quoted in Eq. 3 of Radiative transfer equation (RTE) in the small-angle approximation: Turbid medium illuminated at normal incidence.

CITATION: Kokhanovsky A. A. 2007. Derivation of the RTE in the small-angle approximation (www.tpdsci.com/Tpc/RTESAANmlDv.php). In: Top. Part. Disp. Sci. (www.tpdsci.com).

HISTORY: Published: 14-Feb-2007 Modified: 02-Feb-2007 Peer-reviewed: PENDING

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