Interpreting spectral unmixing coefficients: From spectral weights to mass fractions

Abstract

It is well known that many common planetary minerals exhibit prominent absorption features. Consequently, the analysis of spectral reflectance measurements has become a major tool of remote sensing. Quantifying the mineral abundances, however, is not a trivial task. The interaction between the incident light rays and particulate surfaces, e.g., the lunar regolith, leads to a non-linear relationship between the reflectance spectra of the pure minerals, the so-called “endmembers”, and the surface’s reflectance spectrum. It is, however, possible to transform the non-linear reflectance mixture into a linear mixture of single-scattering albedos of the Hapke model.The abundances obtained by inverting the linear single-scattering albedo mixture may be interpreted as volume fractions which are weighted by the endmember’s extinction coefficient. Commonly, identical extinction coefficients are assumed throughout all endmembers and the obtained volume fractions are converted to mass fractions using either measured or assumed densities. In theory, the proposed method may cover different grain sizes if each grain size range of a mineral is treated as a distinct endmember.Here, we present a method to transform the mixing coefficients to mass fractions for arbitrary combinations of extinction coefficients and densities. The required parameters are computed from reflectance measurements of well defined endmember mixtures. Consequently, additional measurements, e.g., the endmember density, are no longer required. We evaluate the method based on laboratory measurements and various results presented in the literature, respectively. It is shown that the procedure transforms the mixing coefficients to mass fractions yielding an accuracy comparable to carefully calibrated laboratory measurements without additional knowledge. For our laboratory measurements, the square root of the mean squared error is less than 4.82 wt%. In addition, the method corrects for systematic effects originating from mixtures of endmembers showing a highly varying albedo, e.g., plagioclase and pyroxene.