Abstract
We address the problem of unpolarized light spectroscopy of geological materials. Using infrared radiation, the aim of this technique is to learn about the absorbing species, such as hydroxyl. The use of unoriented samples leads to the need to perform a rigorous statistical analysis, so that the three principal absorbances of the crystal can be retrieved. We present here such an analysis based on a derivation of the probability density function for a single random measurement. Previous methods for retrieval of the absorbances are shown to be suboptimal, producing biased results that are sometimes even unphysical (e.g., negative estimates for an inherently positive quantity). The mathematical structure of the problem is developed to use the maximum likelihood estimation method, and we show how to optimize for the three absorbance parameters. This leads to good parameter retrieval on both synthetic and real data sets.
Acknowledgments
This work was performed while A.J. was a visitor at RSES, Australian National University, and IGPP, Scripps Institution of Oceanography. He thanks both institutions for their hospitality. We thank István Kovács for supplying the olivine data and for helpful discussions and editorial comments that improved the manuscript.
References cited
Abramowitz, M., and Stegun, I.A. (1970) Handbook of Mathematical Functions. Dover Publications.Search in Google Scholar
Bell, D.R., Rossman, G.R., Maldener, J., Endisch, D., and Rauch, F. (2003) Hydroxide in olivine: A quantitative determination of the absolute amount and calibration of the IR spectrum. Journal of Geophysical Research: Solid Earth, 108(B2). https://doi.org/10.1029/2001JB00067910.1029/2001JB000679Search in Google Scholar
Hao, Y.-T., Xia, Q.-K., Tian, Z.-Z., and Liu, J. (201) Mantle metasomatism did not modify the initial H2O content in peridotite xenoliths from the Tianchang basalts of eastern China. Lithos, 260, 315–327.10.1016/j.lithos.2016.06.003Search in Google Scholar
Hillier, G. (2001) The density of a quadratic form in a vector uniformly distributed on the n-sphere. Econometric Theory, 17(1), 1–28.10.1017/S026646660117101XSearch in Google Scholar
Kovács, I., Hermann, J., O’Neill, H.St.C., Fitz Gerald, J., Sambridge, M., and Horvath, G. (2008) Quantitative absorbance spectroscopy with unpolarized light: Part II. Experimental evaluation and development of a protocol for quantitative analysis of mineral IR spectra. American Mineralogist, 93, 765–778.10.2138/am.2008.2656Search in Google Scholar
Libowitzky, E., and Rossman, G.R. (1996) Principles of quantitative absorbance measurements in anisotropic crystals. Physics and Chemistry of Minerals, 23, 319–327.10.1007/BF00199497Search in Google Scholar
Rice, J.A. (2007) Mathematical Statistics and Data Analysis, 3rd ed. Duxbury Press.Search in Google Scholar
Sambridge, M., Fitz Gerald, J., Kovács, I., and O’Neill, H.St.C. (2008) Quantitative absorbance spectroscopy with unpolarized light: Part I. Physical and mathematical development. American Mineralogist, 93, 751–764.10.2138/am.2008.2657Search in Google Scholar
Withers, A.C. (2013) On the use of unpolarized infrared spectroscopy for quantitative analysis of absorbing species in birefringent crystals. American Mineralogist, 98, 689–697.10.2138/am.2013.4316Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston