Topics in Particle and Dispersion Science

Transmission function in the small-angle approximation (SAA) for a medium illuminated at normal incidence: SAA and exact solutions

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Results of calculations according to Eq. 6 in Transmission function in the small-angle approximation .. of the transmission function (Eq. 5 in that topic), are compared to those of exact numerical simulations of the radiative transfer by using the vector code SCIAPOL (Rozanov and Kokhanovsky 2006) in Fig. 1 at an observation angle, γ = 1°. The SCIAPOL code is freely available (see SCIAPOL in RTE numerical solution methods: Software (by name)). As it can be seen in that figure, both methods give the same results up to optical thickness, τ ~ 25, at this observation angle. The accuracy of the analytical solution generally decreases at the larger observation angles, especially at the large optical thickness, τ (Fig. 2).

Such results are difficult to obtain from the numerical solution of the RTE because it requires many hundreds of Legendre polynomial terms to represent the small-angle peak of the phase function correctly. This implies lengthy calculations at each Gaussian quadrature point in the numerical solution. The number of such points and, therefore, the time of calculations increases with the number of polynomials used. The analytical solution of the RTE, i.e. Eq. 11 in Solving RTE in the small-angle scattering approximation .., is computationally much faster and has comparable accuracy. Hence, it is a valuable tool in studies of multiple light scattering (see Single and multiple scattering) by a plane-parallel turbid medium with a highly asymmetric scattering phase function, i.e. a medium containing particles with relative sizes much greater than unity.

CITATION: Kokhanovsky A. A. 2007. Transmission function in the small-angle approximation (SAA) for a medium illuminated at normal incidence: SAA and exact solutions (www.tpdsci.com/Tpc/RTSAANmlTsmCmp.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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