Radiative Transfer in Half Spaces of Arbitrary Dimension

Abstract

We solve the classic albedo and Milne problems of plane-parallel illumination of an isotropically-scattering half-space when generalized to a Euclidean domain ${\mathbb{R}}^d$ for arbitrary $d\ge 1.$ A continuous family of pseudo-problems and related H functions arises and includes the classical 3D solutions, as well as 2D “Flatland” and rod-model solutions, as special cases. The Case-style eigenmode method is applied to the general problem and the internal scalar densities, emerging distributions, and their respective moments are expressed in closed-form. Universal properties invariant to dimension d are highlighted and we find that a discrete diffusion mode is not universal for d > 3 in absorbing media. We also find unexpected correspondences between differing dimensions and between anisotropic 3D scattering and isotropic scattering in high dimension.

KEYWORDS:

• Albedo problem
• Flatland
• MacDonald kernel
• hypergeometric
• Case’s method
• Wiener-Hopf

Acknowledgments

Thanks to B. D. Ganapol for pointing out the numerical errors in (McCormick 2015), for the benchmark values in Table 1, and for tracking down a twice-scattered BRDF solution for 3D (Sears 1975). We also appreciate the erudite suggestions of reviewer J. A. Grzesik.

Notes

1 112 years later, the computer rendering of a glass of milk with multiple scattering was one of the iconic images in a seminal paper (Jensen et al. 2001) that sparked the subsurface revolution in film rendering and earned the authors an Academy award.

2 It was incorrectly reported in (d’Eon 2013) that the number of discrete eigenvalues increases past d = 4.

3 Tables 1, 3, and 4 of reference (McCormick 2015) have numerical errors. The values for the column labeled $j_{\mathit{ratio},1}({\mu }_0)$ in Table 1 are all incorrect. The correct values for Table 3 with c = 0.7 are $<x^2({\mu }_0)> =3.36647$ and $<x^3({\mu }_0)> =12.45400$ for ${\mu }_0=0.9$ and $<x^2({\mu }_0)> =3.83320 \text{and} <x^3({\mu }_0)> =14.86410 \text{for} {\mu }_0=1.0.$ The top right value of Table 4 should be $1.91257.$

4 Equation (XX) from (McCormick 2015) will be cited as Equation (IXX)