Derivation of an Explicit Expression for the Fournier-Forand Phase Function in Terms of the Mean Cosine.
DEFENCE RESEARCH AND DEVELOPMENT CANADA VALCARTIER (QUEBEC)
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An analytic expression is derived for the mean cosine of the Fournier-Forand1-2 phase function. This expression and the power law- index of refraction relationship of Mobley3 are used to parameterize the Fournier-Forand phase function by its mean cosine in a similar manner to the Henyey-Greenstein4 function. Radiative transfer theory makes extensive use of the mean cosine of the phase function to compute the evolution of the light field under scattering. In this paper an analytic expression for the mean-cosine of the Fournier-Forand phase function in terms of Hypergeometric functions is derived. This expression reduces to a simple, easy to evaluate, very strongly convergent series that is a function of both mean index of refraction and the Junge size distribution inverse power law exponent. By judicious use of the relationship between the index of refraction and the power law proposed by Mobley it becomes possible to parameterize completely the Fournier-Forand phase function in terms of its mean cosine. This result is more complex but still analogous to the situation that prevails with the Henye-Greenstein5 phase function where the asymmetry factor is identical to the mean cosine. The implications of using this expression for the mean-cosine in the multiple scattering regime of radiative transfer are explored along the lines recently suggested by Piskozub and McKee6.
- Numerical Mathematics