Near-infrared optical constants of naturally occurring olivine and synthetic pyroxene as a function of mineral composition



[1] Radiative transfer theory will assist in determining olivine and pyroxene proportions and compositions from the surface of a planetary body composed of intimately mixed minerals. In order to use radiative transfer techniques, the model requires the optical constants of olivine and pyroxene. Optical constants are parameters that describe the degree light absorbed (k) and refracted (n) in a medium. Here we only parameterize k in the near infrared from 0.6 to 2.5 µm of natural olivine as a function of forsterite number and synthetic pyroxene with respect to wollastonite and ferrosilite number. In contrast to previous work, this study is an improvement on previous work because we have a diverse and larger sample size leading to robust optical parameters. Additionally, we characterize each k-spectrum with the modified Gaussian model (MGM). MGM is a physically realistic model of near-infrared absorptions due to electronic transitions. In each spectrum, we model each absorption and continuum with Gaussians and an inverse of a linear function, respectively. We find that our fitting routine characterizes the olivine and pyroxene k-spectra in a robust and consistent manner. Then we use regression analysis to characterize each parameter of the Gaussians and the continuum as a function of mineral composition. The developed optical parameters from this work will allow calculations of mineral proportions and compositions on planetary surfaces with use of data from missions such as Dawn, MESSENGER, SELENE, and Chandrayaan-1.

1 Introduction

[2] Characterization of common mafic mineral proportions/composition of the surface of a planetary surface is often a critical first step for accurate petrologic characterization and analysis of igneous surfaces. Specifically, quantifying the mineral proportions and composition of a surface can determine its lithology, how petrologically and chemically evolved the rock is, the likely compositions of its primary melt, and the conditions in which this melt crystallized [Basaltic Volcanism Study Project, 1981]. From here, petrologic experiments and models developed for igneous processes on Earth can be applied to physically test or simulate the potential crystallization path of a rock and its origin [e.g., Ghiorso and Sack, 1995; Longhi, 1991]. On Earth, samples of igneous rocks are collected from the field for laboratory analysis. However, when celestial body sampling is limited or nonexistent, remotely sensed spectra are the best way to access the global surface composition of a body.

[3] Spectroscopic tools have successfully detected the presence of olivine and pyroxene in the near infrared on bodies such as the Moon [e.g., McCord and Johnson, 1970; Pieters, 1982] and asteroids [e.g., McCord and Gaffey, 1974]. Radiative transfer theory is a common method used to interpret remotely sensed spectra of surfaces composed of intimately mixed minerals. This method allows a planetary spectrum to be modeled on the basis of optical properties determined for suspected constituents and assumptions regarding the physical state of the surface. Initial applications of radiative transfer theory, based upon work from Bruce Hapke [e.g., Hapke, 1981, 1993], to planetary spectra yield encouraging results [e.g., Harker et al., 2002; Birlan et al., 2007; Cahill et al., 2010; Emery et al., 2011]. The accuracy of such models is limited by the quality of the inputs (i.e., optical constants). The presence of the mafic mineral groups (e.g., olivine and pyroxene), as evidence from spectra and samples, is ubiquitous on rocky planetary surfaces (e.g., Moon, Mars, and asteroids). Hence, optical properties of these minerals are required for “Hapke” modeling of planetary spectra. Both iron-bearing olivine and pyroxene group minerals show continuous and strong variation in optical properties with composition and wavelength in the near infrared, so models must include this effect. These models will be valuable to past (e.g., Earth-based telescopic spectra, Clementine, Chandrayaan-1, SELENE, NEAR, and Galileo) and current missions (e.g., MESSENGER and Dawn) to developing mineral maps of the surface of airless bodies. Here we model the near-infrared optical constants of natural olivine and synthetic pyroxene as a function of mineral composition.

[4] Optical constants are physical parameters that describe how electromagnetic radiation interacts with a medium [Hapke, 1993]. Quantitatively, the optical constants are comprised of two components: the real and imaginary part of the index of refraction (n = n − ik), where for transparent material, n, the real index of refraction, describes the degree light refracts in a material and k, the imaginary index of refraction, relates to how efficiently light is absorbed. For silicate minerals, n is nearly constant in the visible and near-infrared portions of the electromagnetic spectrum (Figure 1c) [Hiroi and Pieters, 1994], whereas k varies with wavelength in this region of the spectrum and is manifested in reflectance spectra as absorption features (Figure 1b). Greater values in k indicate that photons are efficiently absorbed at that particular wavelength as a function of its crystallography and composition. On this basis, mineral proportion and composition is encoded within any given reflectance spectrum including intimately mixed materials, which can be estimated using both Hapke modeling and a complete set of characterized optical constants of each individual mineral.

Figure 1.

A comparison of the real and imaginary indices of refraction for olivine (Fo01). Both n and k were calculated using techniques from Lucey [1998]. (a) Reflectance, (b) k-spectrum, and (c) and n-spectrum. When there is a strong absorption in reflectance, the efficiency of absorbing photons is large (k) at the same wavelength.

[5] In this paper, we derive the optical constant parameters of olivine and pyroxene by first converting laboratory reflectance of natural olivines and synthetic pyroxenes to k-spectra using equations ((1))–(8) found in Lucey [1998]. Next, we characterize the resulting k-spectra features with the modified Gaussian model (MGM). Specifically, each Gaussian approximates a spectral absorption feature by defining the band strength, width, and center. Then we use regression analysis to determine how these parameters vary as a function of mineral composition. This project produces models needed to reconstruct the k-spectrum of olivine as a function of Fe2+ and Mg2+ proportion and mafic pyroxene at any relative proportions of Fe2+, Mg2+, and Ca2+.

2 Background

2.1 Olivine

[6] The absorption feature located at the one micron wavelength in olivine is due to the interaction between electromagnetic radiation and the cation constituents housed within a well-defined crystal lattice [Burns, 1993]. The crystal structure of olivine includes two distinct distorted octahedral sites occupied by metal cations denoted as M1 and M2, where M2 is larger than M1 [Burns, 1993]. The cations are usually Mg2+ and Fe2+. The composition of olivine makes up a solid solution series (Fe,Mg)2SiO4, where the end members are Mg2SiO4 (forsterite or Fo) and Fe2SiO4 (fayalite or Fa). The proportions of Mg2+ and Fe2+ based upon molar concentration are quantitatively represented by Fo number, which is defined by Fo = [Mg2+ / (Fe2+ + Mg2+)] × 100.

[7] In reflectance spectra, olivine exhibits a broad absorption centered at one micron [e.g., Burns, 1970; Adams, 1975; Hazen et al., 1977; King and Ridley, 1987; Burns, 1993; Sunshine and Pieters, 1998]. This broad absorption is a superposition of three absorption features centered near 0.9, 1.1, and 1.3 µm [e.g., Burns, 1970; Sunshine and Pieters, 1998]. The presence of the three absorption features is due to the crystallographic sites containing Fe2+, whereas Mg2+ does not produce near-infrared absorptions [Burns, 1993].

[8] As Fo content increases (increasing magnesium and decreasing iron content) in olivine, the 0.9, 1.1, and 1.3 µm absorptions shift toward shorter, higher-energy wavelengths [Burns, 1993; Sunshine and Pieters, 1998]. In general, these absorption shifts are related to the size of the crystallographic sites where Fe2+ enlarges the sites in contrast to Mg2+ [Burns, 1993]. From this concept, increasing the proportion of Fe2+ will shift the absorptions toward longer wavelengths.

[9] Sunshine and Pieters [1998] showed that the 0.9 and 1.3 µm absorption band centers, widths, and strengths are coupled (cf. Figures 4, 6, and 7). This supports Burns [1993] interpretation that the two absorptions are due to Fe2+ in the M1 sites and the 1.1 µm absorption is from Fe2+ in the M2 sites.

2.2 Pyroxene

[10] The crystal structure of pyroxene is more complex than olivine due to the additional interplay of Ca2+ with Fe2+ and Mg2+ among the crystallographic sites. The molar proportion of Mg2+, Fe2+, and Ca2+ in pyroxene is defined by the ferrosilite (Fs) and wollastonite number (Wo), respectively, where Fs = [Fe2+ / (Ca2+ + Fe2+ + Mg2+)] × 100 and Wo = [Ca2+ / (Ca2+ + Fe2+ + Mg2+)] × 100. The pyroxene structure forms a single chain of silica tetrahedral, and cations occupy two crystallographic sites: a relatively symmetrical octahedral site, denoted as M1, and a larger polyhedral site, M2 [Burns, 1993]. The size of the cation is important to their distribution between the M1 and M2 sites. Due to Ca2+ larger size than Fe2+ and Mg2+, Ca2+ will dominantly occupy M2 sites. Any remaining M2 sites unoccupied by Ca2+ are preferentially filled with Fe2+.

[11] Similar to olivine, Fe2+ is the only cation directly responsible for near-infrared absorptions in pyroxene spectra [Burns, 1993], but Mg2+ and Ca2+ have indirect roles in affecting the overall characteristics. Placement of Fe2+ in either the M1 or M2 site will display unique absorptions and intensities at 1.0, 1.2, and 2.0 µm due to contrasting geometries of the sites [e.g., Hazen et al., 1978; Cloutis and Gaffey, 1991; Burns, 1993; Denevi et al., 2007; Klima et al., 2007, 2011]. When Fe2+ is in the relatively symmetric M1 site, weak absorptions are displayed at 1.0 and 1.2 µm. In contrast, Fe2+ in the more distorted M2 sites produce prominent 1.0 and 2.0 µm absorptions. Notably, when Fe2+ is in both the M1 and M2 sites, the 1.0 µm feature is a superposition of two absorptions, one from each site.

[12] In addition, the absorption properties of the 1.0, 1.2, and 2.0 µm change with the proportions of various cation species and their distribution among the crystallographic sites [e.g., Hazen et al., 1978; Cloutis and Gaffey, 1991; Burns, 1993; Klima et al., 2011]. There are two general patterns: (1) Substitution of Mg2+ for Fe2+ or Fe2+ for Ca2+ will enlarge the crystallographic sites causing the absorption bands to shift toward longer wavelengths. (2) Increasing the Fe2+ proportion in either crystallographic site will strengthen the absorptions caused by those respective sites. For instance, substitution of Ca2+ for Fe2+ results in the 1.0 and 1.2 µm absorption centers shifting toward shorter wavelengths and strengthening of the 2.0 µm absorption band because Fe2+ are filling M2 sites that were occupied by Ca2+. Also, substituting Mg2+ for Fe2+ will cause the 1.0 and 1.2 µm absorptions to shift toward longer wavelength.

[13] Klima et al. [2011] observed that the 2.0 µm absorption band is an exception to the general patterns. In pyroxenes with Wo<20, increasing the proportion of Ca2+ will result in a shift of the 2 µm absorption toward longer wavelengths. However, at Wo>20, the 2.0 µm absorption center is constant at ~2.3 µm regardless of increasing portions of Ca2+ or Fe2+. This lack of shift in the 2.0 µm band is related to the geometry of the crystallographic sites in these high-calcium pyroxenes [see Klima et al., 2011].

2.3 Building on Previous Work

[14] In previous work, Lucey [1998] characterized the optical constants of olivine using samples from King and Ridley [1987] and pyroxene from Cloutis [1985] and Cloutis et al. [1986, 1990a, 1990b]. Lucey [1998] used a linear least-squares fit to model compositional variations in k at each measured wavelength from ~0.22 to 2.4 µm as a function of iron content. Consequently, this method resulted in over a hundred parameters that only define the spectrum at particular wavelengths. In this work, we fit a set of olivine and pyroxene spectra with Gaussians and regress the Gaussian parameters with respect to composition. Gaussian analysis has four advantages over the previous method: (1) A Gaussian is a physically realistic model that can accurately represent spectral manifestation of electronic transitions due to Fe2+ contained in the crystallographic sites. (2) Gaussian analysis uses fewer parameters to characterize an entire k-spectrum by an order of magnitude. (3) This method improves the signal to noise of the result. (4) Gaussians enable calculation of k at any arbitrary wavelength without resampling. We build upon Denevi et al.'s [2007] work by characterizing a larger and more diverse database of synthetic pyroxene that better represents the Fe2+-rich portion of the pyroxene quadrilateral. In section 5.2, we contrast our parameters of synthetic pyroxene to the parameters of Denevi et al. [2007].

3 Methods

3.1 Samples

[15] In this study, we use samples from two suites, which contain a large compositional range and several available diffuse bidirectional reflectance spectra. Our olivine spectra were collected at the NASA/Keck Reflectance Experiment Laboratory (RELAB) at Brown University [Pieters, 1983] and the United States Geological Survey (USGS) Library [Clark et al., 2007]. In the USGS olivine samples, the composition is well distributed along the solid solution and covers the entire range. In addition, we model the olivine spectra from RELAB as a comparison to the USGS spectra and also to increase the overall sample size. Also, we choose to model the synthetic pyroxene spectral data, also collected at RELAB because of their larger compositional coverage of the pyroxene quadrilateral relative to the work by Denevi et al. [2007]. An increase in the compositional range will result in improved accuracy of current pyroxene optical parameters.

3.1.1 Naturally Occurring Olivines

[16] The RELAB data set consists of reflectance spectra measured by E. A. Cloutis and J. M. Sunshine with their compositions as well as minimum and maximum sieve fractions reported in Table 1. Sample descriptions can be found in Sunshine and Pieters [1998] and the University of Winnipeg HOSERLab website ( For USGS contributed spectra, sample descriptions are found in Hunt et al. [1973], Salisbury et al. [1987], and King and Ridley [1987]. Major oxide compositions as well as minimum and maximum sieve fractions for the USGS olivine samples are also reported in Table 1.

Table 1. Composition of Olivine Samples
Sunshine and Pieters [1998]Cloutisa 
Wt. %Havard 103267GSBHawaiian Volc. BombHarvard 118652Harvard 113637OLV025OLV102OLV106OLV107OLV201 
  1. aOxide information generously provided by E.A. Cloutis.
  2. bTotal Fe expressed as FeO.
  3. cMean grain size in parentheses.
  4. n.d., not determined.
  5. tr., trace amount (<0.005 wt.%).
Fe2O3as FeOas FeO1.4103.8305.550n.d.n.d.n.d.n.d.n.d. 
FeOb     13.8109.0909.32010.01012.000 
TiO20.030 0.0100.0100.0100.0000.0000.0000.000tr. 
V2O5     0.0000.0000.0000.0300.000 
Grain Size (µm)c<45 (22.5)<45 (22.5)<45 (22.5)<45 (22.5)<45 (22.5)<45 (22.5)<45 (22.5)<45 (22.5)<45 (22.5)<45 (22.5) 
      45–90 (67.5)45–90 (67.5)45–90 (67.5)   
Wt. %OLV002OLV003OLV005OLV007OLV010OLV011OLV012OLV013OLV020OLV021OLV022
FeO12.6209.250 2.710 61.460    28.520
FeOb13.920 9.590 11.11066.4808.0509.27034.97013.36036.370
Grain Size (µm)c<45 (22.5)<45 (22.5)<45 (22.5)<45 (22.5)<45 (22.5)<45 (22.5)<45 (22.5)<45 (22.5)<45 (22.5)<45 (22.5)<45 (22.5)
 45–90 (67.5)45–90 (67.5)45–90 (67.5)    45–90 (67.5)   
King and Ridley [1987]Salisbury et al. [1987]Hunt et al. [1973]
Wt. %KI3005KI3377KI3291KI4143KI13188KI3189KI3054GDS71GDS70NMNH137044HS285.4B
K2O         0.01 
Na2O0.02        0.02 
Grain Size (µm)c<60 (30)<60 (30)<60 (30)<60 (30)<60 (30)<60 (30)<60 (30)<60 (30)<60 (30)<74 (37)74–250 (162)
       60–104 (82)60–104 (82)74–250 (162)250–1200 (725)
        104–150 (127)  
        150–250 (200)  

[17] The RELAB olivine data set includes 31 spectra with 19 compositionally unique samples that span between Fo0.1 to Fo96.9 (Figure 2a). These spectral data were collected at RELAB with 0° emergence and 30° incidence angles. The spectral range of these reflectance spectra is from about 0.3 to 2.6 µm with measurements at 5 nm intervals. Only five of the 31 samples have Fo<80. The sieve fraction of each sample was either <45 or 45–90 µm.

Figure 2.

(a) The Forsterite (Fo) number of the USGS (solid circles) and RELAB (open triangle) olivine samples. (b) The composition of the synthetic pyroxenes on a pyroxene quadrilateral, where En (enstatite) is the magnesium-rich endmember, Fs (ferrosilite) is the iron-rich end member, Di (diopside) is the magnesium-calcium-rich endmember, and Hd (hedenbergite) is the iron-calcium-rich end member. One sample is missing because it is considered a pyroxenoid (En00Fs49Wo51).

[18] The USGS olivine data set contains 17 spectra. The compositions of the samples range between Fo11 to Fo92 (Figure 2a); seven of the samples have Fo<80. This data set covers a wider range of iron content than those of the RELAB samples. The iron-rich olivine samples (e.g., Fo<66) from the Kiglipait intrusion in Labrador, Canada were sieved to <60 µm [King and Ridley, 1987]. Other olivines with Fo>66 are derived from areas including Papakōlea (Green Sand Beach) on Hawai'i; Twin Sister Peak, Washington; Arizona; and Chavira Mine, Kamargo, Chihuahua, Mexico (Table 1). Some samples feature more than one range of sieve sizes. All of these USGS sample spectra were acquired using the Beckman double-beam spectrometer at the United States Geological Survey (USGS) with a spectral range from ~0.22 µm to 2.4 µm and a spectral resolution of ~15 nm. We omit one of the spectra due to an oversaturation in the reflectance spectrum.

3.1.2 Synthetic Pyroxenes

[19] Synthetic pyroxene samples used in this study were synthesized by D. H. Lindsley and others between 1972 and 2007; the experimental procedure used is given by Turnock et al. [1973]. The starting compositions of these pyroxenes were synthesized from reagent-grade chemicals, and five methods were employed to produce the variety of compositions available and also to avoid crystallization of unwanted pyroxenoids and olivines. Additionally, iron is maintained in the ferrous state (Fe2+). To validate the synthetic pyroxene compositions, Turnock et al. [1973] examined the samples with X-ray diffraction. Later, composition and its spatial homogeneity within single grain samples were re-evaluated by Klima et al. [2007, 2011] using the CAMECA SX-100 electron probe microanalyzer at Brown University (Table 2). Klima et al. [2007, 2011] sieved the samples to <45 µm and obtained the spectra using the bidirectional spectrometer at RELAB with incidence and emission angles of 30° and 0°, respectively. The data are composed of pyroxene compositions between the enstatite-diopside, enstatite-ferrosilite, and ferrosilite-hedenbergite solid solutions and also pyroxene with either Fs>30 or Wo>35 (Figure 2b). One of our samples (not shown in Figure 2) is actually classified as a pyroxenoid based upon its composition (Wo51). Reflectance measurements have a spectral range of 0.3 to 2.6 nm and spectral sampling resolution of 5 nm.

Table 2. Composition of Pyroxene Samples
Fe2O30.  0.00 0.00

3.2 Reflectance- to k-Spectra Conversion

[20] Olivine and pyroxene diffuse bidirectional laboratory reflectance spectra are converted to k-spectra using equations ((1))–(8) from Lucey [1998] based upon the work of Hapke [1981]. This conversion requires knowledge of the grain size of the samples. However, the range of grain sizes in these samples is based upon the maximum and minimum sieve diameters. Lucey [1998] found that computing the mean of the largest and smallest sieve diameter adequately represented the sample as a whole. From this, we use 22.5 µm as the grain size for the synthetic pyroxene samples and for olivine the computed grain sizes are found in Table 1.

3.3 Fitting k-Spectra With the Modified Gaussian Model

[21] We fit our k-spectra with the Modified Gaussian Model (MGM), a type of Gaussian analysis [Sunshine et al., 1990]. In contrast to previous Gaussian analyses [e.g., Roush and Singer, 1986], MGM is a physically realistic model because it can account for the shape of the absorption, which is due to thermal effects on the crystallographic sites [Sunshine et al., 1990]. In this analysis, absorptions are superimposed on a background referred to as the continuum in reflectance. Our analysis includes fitting of the continuum background and the Gaussians appropriate to the mineral under analysis. Finally, we regress the derived Gaussian and continuum parameters against mineral composition.

[22] Each absorption is modeled with a Gaussian based upon the MGM method, which is defined by

display math(1)

where s is the Gaussian strength, μ is the center, σ is the width, and υ is in energy. The exponent, n (not to be confused with the real index of refraction), controls the symmetry of the Gaussian or the slopes of the wings of the Gaussian. Sunshine et al. [1990] found that a value of n equal to −1 is an optimal fit for a spectrum in units of energy. The units in our model are wavelength, inversely proportional to energy. Hence, our value of n is equal to 1; mathematically, the MGM function is identical to the traditional Gaussian but with wavelength rather than energy as the variable. In this paper, Gaussian parameters are defined with respect to equation ((2)).

display math(2)

[23] Some authors have hypothesized that the continuum represents specular reflectance (photons that reflect but never penetrated the crystals) [Huguenin and Jones, 1986] or a broad ultraviolet absorption [Clark, 1999; Denevi et al., 2007]. The continuum functional form varies based upon the units used for interpretation (i.e., energy or wavelength). In energy space, the continuum is linear but is the inverse of a linear function in wavelength space [Sunshine et al., 1990; Denevi et al., 2007].

[24] The continua of olivine and pyroxene k-spectra are modeled with a line

display math(3)

[25] The two points (λC1,yC1) and (λC2,yC2), where λ is wavelength and y is k, are used to define the continuum. These points must be adjacent to and outside of any observed absorption features. Specifically, these points are chosen from k minima between 0.6–0.9 µm and 1.3–2.0 µm for olivine and between 0.5–0.7 µm and 1.85–2.0 µm for pyroxene. Instead of tabulating the (λC1,yC1) and (λC2,yC2) used to fit the continuum for each k-spectrum, we consistently measure the continuum based upon two discrete points, 0.6 and 1.85 µm for olivine and 0.6 and 1.3 µm for pyroxene.

[26] We characterize k-spectra of olivine and pyroxene using equation ((2)) for the absorptions and equation ((3)) for the continuum. The fitting routine consists of Gaussians superimposed upon the continuum with initial guesses of the Gaussian parameters. We initially place each absorption at their respective position (e.g., 1.0 µm absorption positioned at 1.0 µm) or near the closest local k maximum. The initial input for the absorption widths and strengths are orders of magnitude smaller than the resulting strengths and widths. The goal of this fitting routine is to minimize the mean and median of the absolute differences between the measured k-spectrum and the MGM model within the spectral range of interest. The spectral region of interest for olivine is between 0.60 and 1.85 µm. As for pyroxene, we isolate and fit the 1.0 µm and 2.0 µm region to simplify and expedite the fitting routine. The two spectral regions of interest are 0.7 µm to the local k minimum and the local k minimum to 2.6 µm, where the local k minimum is located between 1.2 and 2.0 µm. Then an iterative fitting routine freely adjusts the Gaussian parameters based upon a robust manual search through the parameter space until the mean and median absolute differences are minimized.

3.3.1 MGM Modeling of Olivine k-Spectra

[27] In olivine k-spectra, we have three absorptions that are centered near 0.9, 1.1, and 1.3 µm. Per Sunshine and Pieters [1998], the 0.9 and 1.3 µm absorptions are strongly coupled in reflectance spectra and vary proportionally in absorption strength, width, and center. To ensure that the coupling is evident in k-spectra, we examine preliminary results from Trang et al. [2011], in which it was not assumed that the 0.9 and 1.3 µm absorptions were coupled. In the preliminary analyses, we normalize the strengths and widths of all the absorptions to the 1.3 µm absorption. Regardless of Fo number, the strength of the 0.9 µm absorption relative to the strength of the 1.3 µm absorption is 0.46 ± 0.11 and the width ratio between the two absorptions (width of 0.9 µm absorption/width of 1.3 µm absorption) is 0.85 ± 0.07 (Figure 3). Additionally, the separation between the 0.9 and 1.3 µm band centers is constantly 0.30 ± 0.03 µm. These proportionalities (e.g., relative strength, width, and band center separation) indicate that the two absorptions are coupled, similar to the coupling observed by Sunshine and Pieters [1998]. In this work, we couple the 0.9 and 1.3 µm absorptions in our MGM fit of k-spectra until it is optimized. After optimization based upon coupling of the two Gaussians, we refine the model by allowing the two Gaussians to decouple and find a better fit.

Figure 3.

Preliminary fitting results for olivine from Trang et al. [2011] assumed no coupling between the 0.9 and 1.3 µm absorptions. Here the width ratio of the 0.9 to 1.3 µm absorption is ~0.46, and the strength ratio of the same bands is 0.85. In addition, the 0.9 and 1.3 µm absorption centers are always separated by 0.30 µm. This proportionality suggests that a coupling does exist in k-space and is similar to figures shown by Sunshine and Pieters [1998]. The modeled normalized strengths and the widths of the 0.9 and 1.1 µm absorptions appear similar to Sunshine and Pieters [1998]. However, the 0.9 to 1.3 µm displacement and the widths of the 1.1 µm absorption are different.

3.3.2 MGM Modeling of Pyroxene k-Spectra

[28] For pyroxene, we use three to four absorptions to fit a k-spectrum. The three main absorptions of interest are centered at 1.0, 1.2, and 2.0 µm. Occasionally, in pyroxenes with Wo>20, an additional Gaussian centered near 0.8 µm is required in order to prevent the 1.2 µm absorption from becoming too wide and stretching across the 1.0 µm absorption from ~0.8 µm to ~1.2 µm. Physically, the 0.8 µm absorption may actually be part of a larger 1.0 µm absorption. The 1.0 µm absorption is actually the superposition of two absorptions, one absorption from each crystallographic site. Attempting to model both 1.0 µm absorption features will result in a nonunique solution [Klima et al., 2011]. Thus, we only use one Gaussian to represent the 1.0 µm feature and use a second Gaussian centered near the 0.8 µm to prevent the model from generating unrealistic solutions.

3.4 Optical Parameters as a Function of Mineral Composition

[29] We regress each optical parameter to mineral composition in order to produce a model that accurately predicts the k-spectrum at any Fo number for olivine and at any Fs number for pyroxene, but only for Wo≤50. Only three main absorptions of olivine and pyroxene are modeled. Thus, we produce 11 parameters that characterize each k-spectrum (three parameters for each absorption and two parameters for the continuum). For each parameter, we use a least-squares polynomial fit between Fo and each parameter in olivine. In contrast, pyroxene varies by two independent variables, Fs and Wo. Therefore, we apply a multilinear regression between mineral composition and each parameter.

4 Results

4.1 Olivine

[30] Figure 4 shows an example of the high-quality fit typical for our modeling of laboratory-derived k-spectra for olivine. In Table 3 and Figure 5, we report the MGM and continuum parameters as well as the mean and median absolute differences as the error. In some of our spectra, the continuum exhibits a negative slope (Figure 4), which results in negative k-values at wavelengths shorter than the ultraviolet portion of the spectrum. Negative and positive sloped artifacts within this region typically occur when the minimum k-values at the shortest continuum points are less than the longest continuum points; this is unavoidable. Regardless of which direction the continuum is realistically pointing, the slope is steeper toward shorter wavelengths beyond the visible. This region of the spectrum is not critical because most spacecraft spectral data are concentrated in the visible and near infrared, the region of the electromagnetic spectrum that our study models.

Figure 4.

Model k-spectra fits for RELAB olivines with compositions Fo96.9 (top) and Fo0.01 (bottom). The solid black line indicates the measured k-spectrum, the dashed green line is the continuum, the solid blue lines are the Gaussians, and the solid red line is the residual or the difference between the measured k-spectrum and the modeled k-spectrum.

Table 3. Results From the MGM Fits and Errors for Olivine
Fo1.3 µm Center1.3 µm Strength1.3 µm Width1.1 µm Center1.1 µm Strength1.1 µm Width0.9 µm Center0.9 µm Strength0.9 µm Width0.6 µm Continuum1.85 µm ContinuumErr MeanErr Med
Figure 5.

The results of the modeled fits for each optical parameter. Here the parameters are shown relative to Fo number. The centers and widths of the absorptions and the 0.6 and 1.85 µm continuum points were regressed with a linear least-squares regression, but the strengths were regressed with a second-order least-squares fit.

[31] We use a least-square linear and polynomial fit for the USGS and RELAB olivine sample spectra. For the widths and centers of the Gaussian and continuum, we apply a least-squares linear regression. As for the absorption strengths, they exhibit a nonlinear trend with respect to Fo. Instead of a linear regression, we find that a second-order polynomial is appropriate due to the shape of the 0.9 and 1.3 µm absorption strengths with respect to Fo in the RELAB data set. Consequently, this regression increases the quality of the model prediction. Although the absorption widths and centers are similar in both USGS and RELAB data sets (Figure 5), the absorption strengths and continuum are an order of magnitude stronger in the USGS relative to RELAB spectral data set. Due to contrasting absorption strengths and continuum, we produce an optical parameter for each data set instead of one for both data sets. The cause of this difference in absorption strengths is apparently not dependent on the origin of the samples because all the samples show increase k even though the samples were processed by different workers. Thus, we speculate that the difference may be due to the use of different spectrometers.

[32] We do not use all the results of the model fits from olivine k-spectra in producing the optical properties. In the RELAB data set, we use olivine samples that have sieve fractions <45 µm, and for the USGS data set, we use samples where the sieve fraction is <60 µm and <74 µm. These samples are chosen because a well-characterized grain size is needed to convert from reflectance to k-spectra. Recall that our olivine samples widely range in sieve fractions resulting in various calculated grain sizes. A slight deviation between the input grain size during conversion and the actual grain size will result in an overestimate or underestimate of the continuum and absorption strength.

[33] To visualize how important it is to input the correct grain size, we investigate how the k-spectrum changes as a function of grain size in the synthetic orthopyroxenes. Figures 6a and 6b show the potential variation in the continuum and 2.0 µm absorption of Ca2+-free pyroxenes with input grain sizes ranging between 5 and 45 µm at intervals of 5 µm. The resulting absorption strength of Ca2+-free pyroxenes can vary by up to an order of magnitude; meanwhile, the absorption center and width are constant. To minimize these potential errors due to grain size and for consistency, we constrain our regression analysis to the aforementioned grain size ranges. Furthermore, most of our spectra were measured within these sieve fraction ranges.

Figure 6.

(a) The variation in each parameter of the continuum and 2.0 µm absorption in synthetic Ca-free pyroxene as a function of different assumed grain size. Regardless of grain size, the 2.0 µm width and center do not vary, but the strength and continuum can vary by an order of magnitude. (b) The k-spectrum of a Ca-free pyroxene as a function of grain size. Colors match those in Figure 6a. (c and d) Each tick mark is 100 µm. The synthetic pyroxenes vary in average grain size, which we assumed was 22.5 µm. Figures 6c and 6d represent two different Ca-free synthetic pyroxenes, which display two contrasting grain sizes. This grain size variation may attribute to the larger scatter in the absorption strengths and continuum parameters.

[34] The fit coefficients of olivine are displayed in Table 4. We use a least-squares linear regression for each parameter except for absorption strengths, which form the following equation:

display math(4)

and for absorption strengths, the second-order least-squares regression takes the form of,

display math(5)

where p(Fo) represents any of the 11 optical constant parameters; Fo is the Fo number; and A, B, and C are constants (Table 4). In Figure 5, we superimpose the regression along with the model fit results to visually demonstrate the goodness of fit. In Figure 7, the model fit for each olivine spectrum is compared to the predicted parameter value based upon the optical parameters (Table 4). We can see that the absorption centers, widths, strengths, and continuum are well predicted for the RELAB and the USGS data sets, although the USGS data set does show somewhat more scatter in the continuum and strength parameters.

Table 4. Optical Parameters of Olivine
ParameterABCA 1σB 1σC 1σ
0.60 µm Continuum9.744E−05−9.147E−07 8.63E−061.05E−07 
1.85 µm Continuum1.927E−04−1.854E−06 2.11E−051.05E−07 
0.9 µm Strength9.461E−051.617E−07−1.167E−089.84E−063.91E−073.49E−09
0.9 µm Center9.925E−01−8.247E−04 1.50E−021.83E−04 
0.9 µm Width9.425E−02−1.071E−05 7.79E−039.52E−05 
1.1 µm Strength6.642E−051.158E−06−1.684E−081.30E−055.15E−074.60E−09
1.1 µm Center1.091E + 00−4.195E−04 4.14E−035.05E−05 
1.1 µm Width6.027E−021.503E−05 1.89E−032.31E−05 
1.3 µm Strength2.064E−04−2.500E−07−1.767E−081.20E−054.76E−074.25E−09
1.3 µm Center1.294E + 00−8.719E−04 8.91E−031.09E−04 
1.3 µm Width1.753E−01−2.375E−04 5.80E−037.08E−05 
0.60 µm Continuum1.454E−04−1.275E−06 1.55E−052.52E−07 
1.85 µm Continuum4.111E−04−2.612E−06 7.31E−052.52E−07 
0.9 µm Strength4.235E−046.211E−06−1.115E−079.62E−054.19E−063.79E−08
0.9 µm Center9.942E−01−9.461E−05 1.88E−023.04E−04 
0.9 µm Width1.056E−01−2.542E−04 9.18E−031.49E−04 
1.1 µm Strength5.362E−04−4.047E−061.671E−102.63E−041.15E−051.04E−07
1.1 µm Center1.102E + 00−4.343E−04 3.59E−035.81E−05 
1.1 µm Width5.649E−025.786E−05 4.11E−036.66E−05 
1.3 µm Strength1.783E−03−6.059E−06−1.209E−075.00E−042.18E−051.97E−07
1.3 µm Center1.335E + 00−9.925E−04 9.22E−031.49E−04 
1.3 µm Width1.899E−01−5.093E−04 6.45E−031.04E−04 
Figure 7.

We compared the model fitting results to the predicted parameters for olivine, which is based upon our regression. Points that cluster along the 1:1 line indicate that the samples are well predicted by the optical parameters. The RELAB data set showing tighter fits along the 1:1 line than the USGS data set.

4.2 Pyroxene

[35] Fitting of our models to synthetic pyroxene k-spectra is exemplified in Figure 8. We report the fit coefficients to the Gaussian parameters, the continuum, and the mean and median absolute differences as the error in Table 5 and Figures 9 and 10. Of the 62 synthetic pyroxene k-spectra, we determine model fits to 50 sample spectra and omit eight sample spectra due to being featureless. One sample spectrum is excluded from the regression analyses due to its composition (Wo51, making it a pyroxenoid).

Figure 8.

Model k-spectra fits for synthetic pyroxene with compositions En17Fs83Wo0 (top) and clinopyroxene En18Fs35Wo46 (bottom). The solid black line indicates the measured k-spectrum, the dashed green line is the continuum, the solid blue lines are the Gaussians, and the solid red line is the residual or the difference between the measured k-spectrum and the model k-spectrum.

Table 5. Results From the MGM Fits and Errors for Pyroxene
1 µm Region
#EnFsWo0.9 µm Center0.9 µm Strength0.9 µm Width1.2 µm Center1.2 µm Strength1.2 µm Width0.8 µm Center0.8 µm Strength0.8 µm WidthErr MeanErr Median
High-Ca Pyroxene
86075251.0202.789E−030.0721.2653.230E−040.116   1.90E−051.14E−05
561860220.9806.030E−040.0761.2306.900E−050.102   2.19E−061.49E−06
501958230.9902.227E−030.0731.2501.810E−040.107   1.53E−056.30E−06
573639251.0001.542E−030.0681.2509.800E−050.107   1.56E−055.91E−06
513934271.0001.250E−030.0721.2709.000E−050.102   1.23E−054.19E−06
75469451.0304.330E−040.0621.2403.600E−050.103   5.84E−062.51E−06
Low-Ca Pyroxene
88090100.9951.159E−030.0741.2551.710E−040.114   1.21E−056.64E−06
8909370.9751.873E−030.0781.2403.130E−040.113   1.44E−058.87E−06
9109820.9701.584E−030.0981.2605.210E−040.105   2.36E−051.57E−05
61010000.9502.573E−030.0741.2204.230E−040.104   2.03E−051.06E−05
2189200.9451.652E−030.0711.2102.580E−040.107   1.07E−055.65E−06
491170190.9752.070E−030.0761.2302.640E−040.118   1.10E−056.66E−06
481477100.9552.290E−040.0661.1303.600E−050.133   2.42E−061.51E−06
24178300.9451.905E−030.0701.2152.410E−040.095   1.59E−055.68E−06
20178300.9401.846E−030.0701.2002.720E−040.107   1.08E−055.78E−06
532368100.9701.632E−030.0711.2201.490E−040.111   6.87E−063.22E−06
28257500.9352.649E−030.0701.1953.700E−040.105   1.74E−057.86E−06
23307000.9353.079E−030.0701.1953.460E−040.097   1.84E−058.95E−06
25356500.9351.110E−030.0641.1859.800E−050.099   7.53E−063.39E−06
113650140.9651.824E−030.0721.2251.440E−040.092   1.24E−055.78E−06
8385750.9451.874E−030.0701.1851.700E−040.106   5.35E−063.06E−06
14395930.9402.195E−030.0671.1851.530E−040.097   2.20E−059.70E−06
4505000.9309.780E−040.0641.1806.200E−050.085   6.74E−062.03E−06
26703000.9204.110E−040.0601.1451.800E−050.089   1.91E−069.52E−07
22752500.9202.660E−040.0601.1551.000E−050.075   2.14E−066.12E−07
3752500.9203.100E−040.0601.1501.300E−050.078   2.03E−067.54E−07
2802000.9155.640E−040.0601.1202.200E−050.093   2.32E−061.01E−06
27802000.9154.680E−040.0601.1251.800E−050.089   1.96E−068.66E−07
65901000.9101.250E−040.0571.1103.000E−060.088   6.88E−074.15E−07
6497.52.500.9109.800E−050.0571.1302.000E−060.064   1.03E−064.48E−07
2 µm Region
#EnFsWo2.0 µm Center2.0 µm Strength2.0 µm Width0.6 µm Continuum1.3 µm ContinuumErr MeanErr Median    
High-Ca Pyroxene
Low-Ca Pyroxene
Figure 9.

Results for each optical parameter with respect to Fs number of the modeled fits to synthetic pyroxene k-spectra. Based upon the 1.0 and 2.0 µm absorption centers (a and c), we produced two optical parameters because of the two different trends (blue squares versus black circles). We found that the difference in trend was dependent on Wo number and separated at Wo~20. Blue squares represent pyroxene with Wo> 20, and black circles represent pyroxene with Wo<20.

Figure 10.

Results for each optical parameter with respect to Wo number of the modeled fits to synthetic pyroxene k-spectra. Blue squares represent pyroxene with Wo> 20, and black circles represent pyroxene with Wo<20.

[36] The optical constants of synthetic pyroxene are divided into low- and high-calcium pyroxene groups. We base this division upon the existence of two trends found in the 1.0 and 2.0 µm absorption center as a function of Fs (Figures 9a and 9c). As a result, we produce two sets of optical parameters for synthetic pyroxene based upon low-Ca pyroxene and high-Ca pyroxene, where the separation is at Wo20. The Gaussian and continuum parameters are defined by

display math(6)

where Fs is the Fs number; Wo is the Wo number; and A, B, and C are constants (Table 6). Similar to Figure 7, in Figure 11, we compare the model fit of each parameter against the predicted parameter value based upon the multilinear regression (Table 6). We observe the absorption centers; the widths present a tighter fit, but the strength and continuum present some scatter.

Table 6. Optical Parameters of Pyroxene
ParameterA (Fs)B (Wo)CFs 1σWo 1σ
High-Ca Pyroxene
0.6 µm continuum2.9156E−065.9492E−06−2.0426E−041.76E−063.87E−06
1.3 µm continuum3.4447E−067.0868E−06−2.0686E−042.94E−063.87E−06
1.0 µm strength1.5761E−05−3.4041E−051.7915E−036.25E−061.38E−05
1.0 µm center1.7884E−041.8853E−039.4271E−019.29E−052.05E−04
1.0 µm width2.2488E−042.2091E−044.9872E−029.70E−052.14E−04
1.2 µm strength4.7007E−066.3144E−06−2.4487E−041.04E−062.29E−06
1.2 µm center3.7111E−04−2.4335E−031.2807E + 003.24E−047.14E−04
1.2 µm width−4.6011E−059.1738E−049.9332E−022.32E−045.12E−04
2.0 µm strength7.0097E−06−2.9254E−051.4948E−033.84E−068.46E−06
2.0 µm center−2.8731E−044.6057E−042.3116E + 003.13E−046.90E−04
2.0 µm width−3.0332E−04−1.5923E−032.9361E−011.67E−043.69E−04
Low-Ca Pyroxene
0.6 µm continuum2.9742E−06−4.8289E−06−4.7235E−058.05E−074.42E−06
1.3 µm continuum3.1609E−06−1.5064E−06−3.2622E−051.05E−064.42E−06
1.0 µm strength2.1052E−05−2.3501E−061.2191E−044.77E−062.62E−05
1.0 µm center4.8906E−042.4834E−039.0523E−015.36E−052.94E−04
1.0 µm width2.1932E−042.5113E−045.4576E−023.90E−052.14E−04
1.2 µm strength4.3440E−06−4.0099E−06−7.2422E−056.30E−073.46E−06
1.2 µm center1.1393E−031.8446E−031.1128E + 001.72E−049.45E−04
1.2 µm width3.3186E−041.0618E−037.5426E−026.07E−053.34E−04
2.0 µm strength1.8076E−05−1.9937E−051.0283E−044.59E−062.52E−05
2.0 µm center2.7340E−031.2944E−021.8355E + 001.38E−047.59E−04
2.0 µm width3.2461E−043.9075E−031.7368E−017.02E−053.86E−04
Figure 11.

We compared the model fits results to the multilinear regression of the synthetic pyroxenes. For the absorption centers and the widths, they show much tighter fit. However, the absorption strengths exhibit more scatter.

[37] Similarly to the olivine, the assumed grain size of the pyroxene samples may affect the goodness of fit between the fit coefficient and predicted coefficient of the absorption and continuum parameters. Our observations of the binocular microscope images of synthetic pyroxene (Figures 6c and 6d) show that the samples vary in grain size, which support the assumption that scatter in absorption strength and continuum is due to errors in the assumed grain size.

5 Resulting Optical Parameter Implications and Lessons

5.1 Olivine Optical Parameters: RELAB Versus USGS Sample Sets

[38] For the reasons mentioned in section 4.1, we produce two sets of optical parameters for olivine based upon the USGS and RELAB data. Of the two optical parameters, we recommend using the RELAB parameters even though the USGS data set has the advantage of covering a larger range of Fo number. We favor the RELAB parameters for the following reasons: First, RELAB Fe-rich olivine has a maximum k value near 1 µm, which is consistent with the location of the absorption minimum in reflectance spectra. In contrast, the maximum k value in Fe-rich olivines in the USGS data set unusually occurs at longer wavelengths, closer to 1.3 µm (Figure 12a). This result is considered suspect because it is inconsistent with the absorption minimum of olivine in reflectance spectra. Second, the USGS k-spectra absorption strengths for olivine are an order of magnitude stronger than the RELAB olivines, which suggests that the olivines are anomalously dark (Figure 12b). Therefore, using the USGS parameters on unknown k-spectra will result in overestimates of Fo number; an example is shown in section 6.1. For these reasons, we urge the reader to use the RELAB based parameters.

Figure 12.

The olivine k-spectra of the RELAB and USGS data for olivine are quite different even though the compositions are similar. (a) The peak of the absorption on the USGS Fe-rich olivine is unusual because it is near ~1.3 µm, whereas reflectance spectra of olivine tend to show the peak closer to 1.0 µm. (b) The USGS k-spectra are much darker than the RELAB. In this case, the USGS olivine k-spectrum (Fo41) is 4 times darker than the RELAB k-spectrum (Fo42).

5.2 Comparison to Previous Work

[39] We compare our optical parameters to previous work. We did not make comparisons with Lucey's [1998] olivine and pyroxene optical parameters because their parameters are based upon modeling the changes in k with iron content at every wavelength, whereas our parameters are based upon Gaussians. Thus, there is not a direct or indirect method of comparing the two models.

[40] As for pyroxenes, we compare our model fit results of the synthetic pyroxenes to the predicted values for each Gaussian and continuum parameter based upon Denevi et al.'s [2007] optical parameters of mostly natural pyroxenes, which in a sense, is comparing the two optical parameters indirectly. Denevi et al. [2007] produced two sets of optical parameters, including a set for orthopyroxene (Wo≤10) and a set for clinopyroxene (Wo>10), respectively. Roughly speaking, the optical parameters for orthopyroxenes (Wo≤10) derived by Denevi et al. [2007] somewhat match our model fits for synthetic orthopyroxenes with (Wo≤10; Figure 13, black circles). However, when using the optical parameters derived for clinopyroxene (Wo≥10), by Denevi et al. [2007], an inadequate model prediction of synthetic clinopyroxene is determined (Figure 13, blue squares). And conversely, our optical parameters would produce poor model fits to the natural pyroxenes of Denevi et al. [2007].

Figure 13.

We compared the results of the model fits of the synthetic pyroxene to the predicted results based upon the optical parameters from Denevi et al. [2007]. Synthetic pyroxenes with composition of Wo>10 show considerable scatter are not well predicted by Denevi et al.'s [2007] optical parameters. On the other hand, the orthopyroxene optical parameters can somewhat predict the model fits of the synthetic pyroxenes.

[41] This contrast between natural and synthetic pyroxenes k-spectra is not clear, but the difference is important in predicting pyroxene abundances and composition on planetary surfaces. We indicate two possibilities for the observed differences. First, zonation of pyroxenes causes the absorptions to appear wider [Sunshine and Pieters, 1993] and produces erroneous centers [Hazen et al., 1978; Cloutis and Gaffey, 1991]. The natural clinopyroxenes examined here and their absorption widths and centers are plausibly consistent with this interpretation. Another possibility is that these natural pyroxenes are accommodating minor and/or trace elements that are modifying the crystal structure. If the differences between the two-pyroxene suites are due to zonation, then determination of two pyroxenes from spectral data will improve petrological modeling. Future petrographic and spectroscopic studies in examining the difference in natural and synthetic pyroxene will be key to resolving this issue.

5.3 Ordering of Cations Among the Cation Sites

[42] The strengths of olivine and pyroxene absorptions are dependent upon the proportion of Fe2+ present in each crystallographic site. Also, the strength of each absorption is linked to how Fe2+ is ordered among the crystallographic sites. Two studies compiled data on the ordering between Fe2+ and Mg2+ in the M1 and M2 sites in olivine for temperatures between 0 and 800°C [Princivalle, 1990; Rinaldi et al., 2000]. Within these temperatures, Burns [1993] found that the effects of cation ordering on olivine spectra are negligible. Thus, we can safely assume that olivine cation ordering does not affect our optical parameters significantly.

[43] The effects from Mg2+-Fe2+ ordering are more prominent in pyroxene than olivine [Burns, 1993]. As mentioned above, Fe2+ prefers the M2 to M1 sites in pyroxenes. When each Fe2+ ions is in its preferred site, the pyroxene is considered “ordered,” where the reverse is called “disordered.” The degree of ordering in orthopyroxene is dependent upon the rate of cooling. Pyroxenes that cool slowly, such as pyroxenes originating from metamorphic and plutonic rocks, are more ordered than rapidly cooled pyroxenes from volcanic rocks [Ghose and Hafner, 1967; Virgo and Hafner, 1970]. Burns et al. [1991] reheated samples of orthopyroxenes to 500–700°C to enhance disorder and compared the near-infrared spectra before and after disorder. Using Mössbauer spectroscopy, they found that with increasing disorder, where the proportion of Fe2+ in the M1 sites increased, the strengths of the 1.0 and 2.0 µm absorptions weakened and the 1.2 µm absorption strengthened. On this basis, even though a correlation between absorption strengths and pyroxene composition exist, the absorption strength may not be the lone indicator of mineral composition [Denevi et al., 2007]. For example, during pyroxene synthesis, the samples were quenched, which promotes disordering in the samples. Thus, cation ordering may be a contributing factor to the scatter in our model absorption strengths.

[44] Determining cation ordering from a pyroxene spectrum could be used to model cooling rates on asteroids and planetary surfaces. A number of authors have worked on a method using cation ordering to derive cooling rates in terrestrial orthopyroxenes [e.g., Virgo and Hafner, 1970; Besancon, 1981; Stimpfl et al., 1999], pigeonite [Pasqual et al., 2000], and clinopyroxenes [McCallister et al., 1976; Brizi et al., 2000]. Ganguly and Stimpfl [2000] successfully demonstrated that orthopyroxene cation ordering in orthopyroxenes in meteorites could be used to derive mineral cooling rates. If the absorption strength indicates the proportion of Fe2+ in each site (i.e., order/disorder), then correlating pyroxene absorption strength with cation ordering in pyroxene may result in spectroscopy-based derived cooling rates.

5.4 True Proportion of Fe2+

[45] Optical parameters formulated here are based upon Fo for olivine, and Fs and Wo number for pyroxene. These parameters are generally relied upon for characterization of chemical species within these minerals. In the context of remote sensing, spectral variations are generally attributed to changes in bulk Fe2+, Mg2+, and Ca2+ contents of olivine and pyroxene. However, the cation constituents in natural olivine and pyroxene compositions are not restricted to Fe2+, Mg2+, and Ca2+. Minor cations (e.g., Ni2+ and Mn2+ for olivine and Mn2+, Al3+, and Na+ for pyroxene) are often present at percent levels in these minerals. As a result, the relative Fe2+ proportions to all major (e.g., Mg2+ and Ca2+) and minor cations are not accurately reflected in Fs and Fo numbers because they determine the proportions relative to three major elements. For instance, the overall abundance of Fe2+ and Mg2+ can vary, while the proportionality of the two species can remain constant due to an increase in the abundance of minor elements. What is important, however, is whether these accessory elements influence the abundance and/or crystallographic site of Fe2+, Mg2+, or Ca2+ in a way that would influence our interpretation of a reflectance spectrum. If the effects of minor elements on spectra are observable, then the reported absorption strengths, centers, widths, and continuums may not be as adequately related to Fo, Wo, and Fs number.

5.5 The Presence and Potential Influence of Mn2+

[46] The RELAB olivine data set exhibits a near-linear relation between Fo number and the 0.9 and 1.3 µm absorption strength, but a nonlinear correlation between Fo number and the 1.1 µm strength (Figure 14). The relation between Fo number and strength of the 1.1 µm absorption appears to bifurcate into two trends. We hypothesize that this nonlinear behavior in the 1.1 µm absorption and near-linear behavior in the 0.9 and 1.3 µm absorption may be attributed to the presence of the minor element Mn2+. The influence of Mn2+ in mineral spectra has been inferred before. Previous authors have investigated the effects of Mn2+ on olivine spectra [Burns, 1970, 1993] as well as the proportions needed for a spectral effect to be observable [Cloutis, 1997]. These studies found that the presence of Mn2+ has two major effects on near-infrared absorptions of olivine. First, the 1.1 µm absorption weakens relative to the 0.9 and 1.3 µm absorptions as Mn' proportions increase [Burns, 1993] (in atomic proportions, Mn' = [Mn2+ / (Fe2+ + Mg2+ + Mn2+)] × 100). This weakened absorption is due to Mn2+ tendency to occupy M2 sites, forcing Fe2+ into M1 sites. In extreme cases (70–78% mole % of Mn2SiO4), the 1.1 µm absorption of olivine is no longer visually resolvable [Cloutis, 1997]. Second, an increase in Mn' will also shift the 1.1 µm absorption toward longer wavelengths because Mn2+ is a larger cation than Fe2+ [Huggins, 1973; Burns, 1993]. Cloutis [1997] found that at ~10% mole % Mn2SiO4, the 1.1 µm absorption will begin shifting beyond the potential positions for this absorption observed for the forsterite-fayalite solid solution.

Figure 14.

The presence of Mn2+ in Fe-rich olivine is abundant enough to cause the 1.1 µm absorption strength to become nonlinear and also affects the 0.9 and 1.3 µm absorption strengths.

[47] Upon examining the oxides in the RELAB samples, we observe an increase in Mn2+ (and Mn') in olivine as Fo number decreases (Figure 15). Specifically, the proportion of Mn' is >0.5% at ~ Fo<60 with one sample having a Mn' as high as ~ 7%. We also notice that with increasing Mn2+ content, the scatter is greater in the 1.1 µm absorption strength at samples of Fo<80 relative to olivines with Fo>80. At Fo>80, the 1.1 µm absorption strength increases linearly with decreasing Fo number, but at Fo<80, the absorption strength deviates from this linear trend and becomes weaker relative to the 0.9 and 1.3 µm absorption strength (Figure 14), a pattern that is expected due to positioning the Mn2+ into the M2 sites and forcing more Fe2+ into the M1 sites. Additionally, the 0.9 and 1.3 µm absorptions are not perfectly linear due to Mn2+-induced cation ordering.

Figure 15.

In both the RELAB and USGS olivine samples, as the Fe2+ content increases, so does the Mn2+ content.

[48] The optical parameters we derive for olivine are preliminary because we find that the presence of Mn2+ in the crystal structure is significant enough to alter our spectral models. Aside from the presence of Mn2+, we observe that the two points that define the continuum (i.e., 0.6 and 1.85 µm) in the RELAB data are still quite scattered with respect to Fo number, indicating the possible influence from our grain size assumption. Consequently, the absorption strengths may be stronger or weaker than the reported strengths. Future work that constrains grain size as well as modeling with optical parameters through natural and synthetic olivine samples with respect to Fe2+, Mg2+, and Mn2+ content will produce more accurate models of olivine spectra. Furthermore, this future study will result in determination of Mn2+ proportion in olivine studies of planetary surfaces. This determination of abundance and composition of this additional cation will improve future petrologic modeling.

6 Application

6.1 Olivine Example—Lunar Dunite

[49] To demonstrate the practical application of the olivine optical parameters, we determine the mineral composition of olivine separates from a lunar dunite (72415), which was previously studied by Isaacson and Pieters [2010]. Isaacson and Pieters [2010] produced a reflectance spectrum at RELAB with an incidence angles and emission angles of 30° and 0°, respectively. The sieve fraction of this spectrum is from <125 µm olivine separates. We convert the reflectance spectrum to k based upon equations ((1))–(8) from Lucey [1998] and using a grain size of 66.2 µm. Next, we compare the k-spectrum of the sample to modeled k-spectra, based upon the optical parameters with compositions ranging from Fo0 to Fo100 at intervals of 1. The best matching olivine k-spectrum is determined by the lowest mean absolute difference between the model and the sample k-spectrum at wavelengths between 0.6 to 1.55 µm. We limit the spectral range of interest due to the presence of chromite. Chromite produces a broad absorption, which becomes apparent at ~1.5 µm [Isaacson and Pieters, 2010]. The model determines that the composition of the lunar olivine is Fo83 (Figure 16a), which could be considered approximate to the actual (Fo88). We perform the same routine with the USGS optical parameters, which predict the sample is of Fo100 (Figure 16b). This result is unsurprising because the USGS optical parameters predict olivines of a particular Fo number to be darker than the actual. Thus, the USGS optical parameters would over predict the Fo number of the sample.

Figure 16.

The solid black line is the lunar olivine separate k-spectrum for Figures 16a and 16b and the Tatahouine spectrum in Figure 16c. The violet line is the best-fit spectrum. The solid red line represents the residuals between the two spectra. The fitting for the lunar olivine separate was limited to the relevant wavelengths from 0.6 to 1.55 µm. This limitation is due to the additional absorption beyond 1.5 µm caused by chromite. (a) The best model olivine spectrum based upon the RELAB optical parameters has a mineral composition of Fo83, close to the actual composition (Fo88). (b) The best model olivine spectrum based upon the USGS optical parameters is Fo100. However, the fit is not very good based on the residuals. (c) The pyroxene model appears to replicate the Tatahouine k-spectrum well and also predict the mineral composition.

6.2 Pyroxene Example—Tatahouine Meteorite

[50] We apply our pyroxene optical parameters to determine the mineral composition of a diogenite meteorite, Tatahouine. Diogenites are pyroxene-dominated meteorites thought to derive from 4 Vesta. Hiroi et al. [2001] presented the spectrum (0.3 to 2.6 µm) of a sample of Tatahouine ground to <25 µm grain size. For this analysis, we use a grain size of 12.5 µm. We assume that they used an incidence angle of 30° and emission angle of 0°. Next, we convert the spectrum from reflectance to k by applying equations ((1))–(8) from Lucey [1998], the grain size of the sample, and viewing geometry. After the conversion, we compare the Tatahouine k-spectrum to the modeled pyroxene k-spectra based upon the optical parameters. The best matching pyroxene k-spectrum is obtained by comparing every combination of En, Fs, and Wo that summed to 100 at intervals of 1 until the mean absolute difference between the Tatahouine and pyroxene k-spectrum was close to zero. The actual pyroxene composition of Tatahouine is En75Fs23.5Wo1.5 [Barrat et al., 1999], and we predict the mineral composition to be En68Fs31Wo1, which is a good approximation of the sample (Figure 16c).

7 Conclusion

[51] Here we have determined optical parameters for natural olivines and synthetic pyroxenes in the near infrared from 0.6 to 2.5 µm. These optical parameters will be important for determining abundance and proportion of olivine and pyroxene in intimately mixed soils on bodies such as the Moon and asteroids using a variety of available data sets (e.g., Earth-based telescopic spectra, Clementine, Chandrayaan-1, SELENE, NEAR, Galileo, MESSENGER, and Dawn). Natural olivine optical properties are effectively characterized with two sets of model parameters derived from examination of RELAB and USGS spectra. Of these two sets of parameters, the RELAB set is more robust. However, these newly derived optical constants do have some caveats resulting largely to compositional impurities. The olivine optical constants we report are derived from Mn-bearing olivine samples with up to ~5 MnO wt%. The abundance of Mn2+ is found to increase systematically with decreasing Fo number. Due to Mn2+ in these olivine samples, the optical parameters of Fe2+-rich olivine spectra are less than robust and limit accuracy of modeling of Fe-rich olivines, especially for olivine samples that are Mn free/depleted relative to the calibration set of olivine. Mg-rich olivine samples modeled here are better predicted largely because of a larger sample size and lower trace-element abundances.

[52] In this study, pyroxene optical parameters were derived from synthetic samples. Due to the concentration of samples toward the Fe2+-rich region of the pyroxene quadrilateral, the optical parameters describe this region better. Future work is necessary in examining the petrographic and geochemical differences that reveal the discrepancies between the natural and synthetic optical parameters shown here. Such a study would be highly beneficial for more accurate modeling of the pyroxene optical parameters.

Appendix A: Using the Optical Constant Parameters

[53] We provide a simplified version of how to use the optical parameters presented in this study. The value of k at some wavelength λ (in microns) is the superposition of the continuum and all the constituent absorptions (equation ((A1))),

display math(A1)

where kλ is the value of the imaginary index of refraction, Cλ is the continuum, Gλ are the Gaussians, and N is the number of Gaussians (which is three for olivine and pyroxene). The values of Cλ and Gλ can be obtained from the linear and Gaussian functions (equations ((A2)) and ((A3)), respectively).

display math(A2)
display math(A3)

[54] For the continuum (equation ((A2))), values for the ordered pairs, (λC1,yC1) and (λC2,yC2), are dependent on the mineral of interest. For olivine, λC1 and λC2 are 0.6 and 1.85 µm, respectively; as for pyroxene, the values are 0.6 and 1.3 µm, respectively. The k values for λC1 and λC2 (i.e., yC1 and yC2) are derived from functions dependent on mineral chemistry (equation ((A4)) or ((A5))).

display math(A4)
display math(A5)

[55] The constants A, B, and C are obtained from Table 4 for olivine and Table 6 for pyroxene (Note: Ensure that the correct constants A, B, and C are used for the associated parameter). There are three main Gaussians for olivine (i.e., 0.9, 1.1, and 1.3 µm absorptions) and pyroxene (i.e., 1.0, 1.2, and 2.0 µm absorptions). Computing the Gaussians contribution to k (equation ((A3))) requires calculating the values of s, σ, and μ (equations ((A6)) and ((A7)) or (A8)).

display math(A6)
display math(A7)
display math(A8)

[56] In summary, the value of k for either olivine or pyroxene is calculated from combining equations ((A1))–((A8)) and providing the mineral chemistry and the wavelength of interest. Repeating this method for various wavelengths would produce a k-spectrum.


[57] We would like to thank E.A. Cloutis for providing oxide information of the olivines, B.W. Denevi for sharing her pyroxene spectra, and S.T. Crites for ensuring that the “Using the Optical Parameters” section in the Appendix is written clearly. Also, we would like to thank Karly Pitman and Jessica Sunshine for their constructive critical reviews. This work was supported in part by a NASA LRO participating scientist grant (NNX08AM82G) to J. J. Gillis-Davis and a NASA PG&G grant (NNX11AP51G) to J. T. S. Cahill. HIGP #2001 and SOEST #8853.