Modelling of Mineral Mixture Reflectance Spectra

Introduction In the forthcoming years, high spectral resolution imaging spectroscopy in the visible/near-infrared range will be used to obtain compositional information for planetary surfaces (especially for Mars surface with the space missions Mars Express and Mars Reconnaissance Orbiter). Deconvolving a reflectance spectrum to mineral abundance in an umabiguous way is difficult, because the spectra are complex nonlinear functions of grain size, abundance, material opacity, and type of mixtures (dust, sand, areal). The common approach is to extract information from the measured spectrum itself, using e.g. the Modified Gaussian Model [1] or automated band detection [10]. The strength of these methods is to aid in determining appropriate endmembers to model, but no information can be derived on abundance, grain size, and type of mixtures of the end-members. Multiple scattering models can provide approximate solutions to the radiative transfer in a particulate medium. One such model is the semiempirical Hapke model [2] which has been used by most modelers of natural and laboratory common geologic mixtures. Applications of the Hapke model to quantitative analysis of laboratory mineral intimate mixtures give abundances estimates better than 10% if an estimate of the particle sizes of the respective mixture components is known [3,4]. A geometrical optics model for albedo spectral dependence of regolith-like surfaces was presented by Skhuratov et al. [5]. The purpose of the paper is to use the Shkuratov scattering theory to determine the type of mixture, the relative proportions, and the grain sizes of components (minerals) of laboratory common geologic mineral mixtures given reflectance spectra of the endmembers only. Modelling method The formulation of the Shkuratov model can be summarized into three steps: 1) to derive the albedo of a particle of given composition (defined by the optical constants n k ) and size d; 2) to derive the reflectance of an homogeneous and single-member particulate surface; 3) to derive the reflectance of a certain type of mixture. Here, two kinds of mixtures are considered: intimate (or “salt-and-pepper”) mixture of coarse particles of size and areal mixture (or “checkerboard”). Note that mixture of particles of size (dust mixture) can be in principle constructed from the Shkuratov theory. In terms of formulation, the main difference between the Hapke and the Shkuratov theory is the role of the phase function of individual particles, which is automatically calculated (and generally forward directed) in the case of the Shkuratov model instead being a free parameter (but effectively dominated by isotropic scattering) as formulated in the Hapke model [6]. Shkuratov theory incorporates realistic grain forward scattering at all orders of scattering, while most applications of Hapke theory include it only to first order. One of strengths of the Shkuratov theory is that the model is analytically invertible: the imaginary index fraction k can be found if the spectral albedo is known and estimates for n and d are available. Our procedure of deconvolution is first to calculate k from each endmember spectrum; we then use the scheme described above to reproduce the mixture spectra by optimizing the type of mixture (intimate or areal), the grain sizes and the abundances of endmembers.