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Bi-directional reflection spectroscopy based on multiple scattering of particulate surfaces is employed in identifying the optical properties of titanium-iron oxide mineral from laboratory reflection measurements.However, the approach suffers from issues including: i) both n and k are to be extracted from a single spectroscopy spectrum, ii) imposing constraints of the K-K correlation relating spectral n and k is weakened by its fundamental insensitivity, and iii) incapability in addressing intrinsic strong absorption features of absorbing materials. We resolve these issues by employing additional optical information of directional-hemispherical reflection and emission/absorption for a slab and a stratified multi-layer medium of the material, respectively. The accompanied analyses consist of radiation transfer in a slab medium investigated using the two-flux approximation method and electromagnetic radiation propagation in a stratified multi-layer medium investigated using the electric filed transfer matrix. We further find that an understanding of the internal scattering coefficient of grains in the multiple scattering model paves the way in successfully predicting absorption features of materials. A wavelength-dependent internal scattering coefficient of the material is then found to be 50 1/micrometer and 1/10000000 1/micrometer in regions of strong absorption and high transmission (between 7 and 13 micrometer), respectively. The value of the refractive index n varies weakly on the wavelength. A pronounced change in the determined absorptive index k with the wavelength is observed. Low values of the absorptive index k on the magnitude of 0.01 are obtained in the transmission window spectral range. In the strong absorption spectral range starting from 13 micrometer, values of the absorptive index k are higher than 0.1., Comment: 26 pages, 14 figures

We study the asymptotic error between the finite element solutions of nonlocal models with a bounded interaction neighborhood and the exact solution of the limiting local model. The limit corresponds to the case when the horizon parameter, the radius of the spherical nonlocal interaction neighborhood of the nonlocal model, and the mesh size simultaneously approach zero. Two important cases are discussed: one involving the original nonlocal models and the other for nonlocal models with polygonal approximations of the nonlocal interaction neighborhood. Results of numerical experiments are also reported to substantiate the theoretical studies., Comment: 33 pages, 7 figures and 10 tables