ANALYSIS OF VISIBLE AND NEAR INFRARED REFLECTANCE SPECTRA OF PLANETARY
MATERIALS USING A BAYESIAN ESTIMATION APPROACH. 1Seiji Sugita, 1 Kenji Nagata,
1,2Nobuko Tsuboi, 3 Takahiro Hiroi, 1Masato Okada, 1 Dept. of Complexity Sci. and Eng, Univ. of Tokyo,
2Hewlett Packard Japan, Ltd., 3Dept. of Geological Sci., Brown University
Introduction: Optical reflectance spectroscopy
is a very powerful tool to observe the planetary
surfaces remotely from both ground-based telescopes
and spacecrafts. Because many silicate minerals have
many absorption bands diagnostic to both crystalline
structure and elemental composition in thiswavelengths range, VNIR spectroscopic observations
can reveal a wide variety of information on planetary
surfaces [e.g., 1].
The reflectance spectra observed on theplanetary surfaces are compared with those ofminerals observed in laboratories to infer themineralogical and elemental compositions of thesurfaces. Because silicate minerals often exhibit
complex absorption bands in the UV to NIR
wavelengths range, such a complex absorption band isoften deconvoluted into multiple simple (i.e.,mathematically well-defined forms, such as Gaussianand Lorentzian) absorption bands is often used forquantitative comparisons and analyses. One simple
approach would be to search for the optimum
combination of simple absorption bands thatminimized the difference with an observed spectrumby trying every possible parameter set. Such direct
search is rather effective when only a small number offree parameters are involved. However, a direct search
is not practical for reflectance spectroscopy ofplanetary surfaces because many simple absorption
bands are required to describe their complex
absorption band patterns. When Gaussian orLorentzian is used, each simple band requires threemodel parameters: strength, central wavelength, andbreadth. Thus, the number of parameter sets to try insearching for the best-fit synthetic spectrum becomes
extremely large. For four Gaussians, for example, weneed to try 1012 parameter sets if we choose ten
different values for each parameter. It would require
1024 sets if we examine 100 different values for eachparameters to attain a high accuracy.
Consequently, a more efficient mathematicalmethod is necessary. In previous studies, the steepest
descent method has been used frequently to resolve
this problem. In particular, Sunshine et al. [2] propose
an analytical approach using the steepest descentmethod to deconvoluted reflectance spectra of silicate
minerals with a number of Gaussians and a continuum
as a function of wavelength. Their approach, called
modified Gaussian model (MGM), has been shown
capable of decomvolving olivines [e.g., 3] and Mg-Fe
pyroxenes [e.g., 4,5] very successfully and obtainclear correlations between the central wavelengths ofdeconvoluted Gaussian bands and chemicalcomposition of minerals.

Although the steepest descent method is a very
powerful mathematical method to find a minimum of
a continuous function but is not necessarily good at
finding the global minimum of a complex function
with many local minima. Depending on the initialchoice of parameters, an obtained result may be just alocal minimum of the function. Furthermore, thismathematical procedure needs to be given how manymodel parameters (e.g., the number of Gaussians inthis study) should be used before analysis. Generally,a model with more Gaussians will lead to a smaller E,
but the complexity of the model also increases. Our
recent analysis indicates that the use of different
numbers of Gaussians for MGM analysis may
dramatically change the result of the MGM analysis[6], underscoring the importance of the choice of thenumber of Gaussians.
In reflectance spectroscopy of silicate minerals,
the choice of the number of Gaussians is very
important for accurate description of their complex
absorption bands. A numerical validation discussed inour companion paper by [7] indicates that a Bayesianestimation method using the exchange Monte Carlo(EMC) algorithm can estimate the optimum number ofGaussians much more accurately than otheralgorithms, such as cross validation, WAIC, andRJMCMC. Thus, our spectroscopic analysis ofplanetary surfaces, the newly developed Bayesian
MGM algorithm is better suited for reflectancespectroscopy of silicate materials on planetary
surfaces.
In this study, we derive a new MGM algorithm
using a Bayesian estimation approaches namely
exchange Monte Carlo (EMC) method and theannealing method. Both pure olivine samples withdifferent Fe/Mg ratios and olivine-pyroxene mixture
samples are used for analysis to examine theeffectiveness of our new algorithm. Based on theresults, we discuss the applicability of our new MGM
method.
Spectral Analysis: The mathematical details
and numerical validation of the algorithm used in thisstudy are given in our companion papers by [7,8].
In order to examine the validity of theBayesian MGM algorithm proposed in this study, we
conduct actual spectral deconvolution analyses using aseries of reflectance spectra, olivine powder samples
with different Mg/(Mg+Fe) ratios (mg#) ranging fromzero to one.
The reflectance spectra of olivine samples
were obtained the web site of NASA’s RELAB.
Here, it is noted that the olivine samples whosereflectance spectra were used in this study aredifferent from those used by [4]. As shown in thefollowing, nevertheless, the result of the analyses ofthis study and [4] exhibit very good agreement. Thisstrongly suggests the robustness of the results ofMGM results.
Analysis Results: First of all, the results of
our Bayesian MGM analysis indicate that theoptimum number of Gaussian bands to reproduce thecomplex absorption band around 1 mm predicted isthree for the reflectance spectra of all the olivine
samples (Fig. 1). This optimum number is the same as
the number estimated empirically by [4]. This
coincidence is very important because the number ofGaussians is very difficult to determine objectively
but highly influential to the model parameters tocharacterize individual Gaussian bands. Thisagreement between this study and previous fitting
results by [4] strongly supports the validity of ourBayesian MGM algorithm
Furthermore, all the model parameters ofindividual Gaussian bands obtained in this study werepractically the same as those obtained by [4]. The
parameters include the central wavelengths, breadths,
and strengths of the individual Gaussian absorption
bands.
The central wavelengths of individual bandsof 1-mm olivine increases gradually as mg# decreases;
more iron leads to longer wavelengths (Fig. 2). Theband widths and the relative intensities of the three
individual bands are relatively constant regardless ofmg# (Fig. 2). Here, note that the data sets of olivine
spectra used in this study and those used by [4] aredifferent. This supports that the results of these MGM
analyses are not specific to some particular data sets
but reflect the intrinsic properties of olivine.
Here, it is noted that the results of thepreliminary analysis by [6] suggests that four
Gaussians is more appropriate to fit the 1-mm
absorption band of olivine than three as used byprevious studies [2,4]. However, a detailed
comparison of different algorithms using artificiallygenerated data sets by [7] indicates that MGM
analysis using Bayesian approach is significantly moreaccurate than that with cross-validation method, which
is used by [7]. Thus, we conclude in this study that theoptimum number of Gaussians for the 1-mm
absorption band of olivine is three.
References: [1] Pieters, C.M. and P.A.J. Englert(1993), Remote geochemical analysis: Elemental andmineralogical composition; [2] Sunshine, J. M., et al.
(1990), J. Geophys. Res., 95, 6955; [3] Sunshine, J. M.
and C. M. Pieters (1993), J. Geophys. Res., 98, 9075;[4] Sunshine, J. M. and C. M. Pieters (1998), J.
Geophys. Res., 103, 13,675; [5] Klima, R. L. et al.,(2007), Meteoritics and Planet. Sci., 42, 235. [6]Tsuboi, N. et al. (2009), Proc. of 42th Lunar andPlanet. Symp., 118; [7] Nagata, K. et al. (2010) NeuralNetworks, to be submitted; [8] Sugita, S. et al., (2010)
J. Geophys. Res. to be submitted.

Fig. 1. MGM deconvolution result of an olivine
spectrum using our Bayesian algorithm.

Fig. 2. The trend of central wavelength, width, andnormalized intensities of deconvolved absorption
bands of olivine with a variety of mg#.