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140 H2O bend and stretch vibrations, and the narrow cation-OH vibration clay absorption bands between 2.2-2.4 µm [Bishop et al., 1995b]. This wavelength region was chosen because these features are commonly used to identify smectitic clay minerals in CRISM and OMEGA spectra of Mars (e.g., Bibring et al. [2004]; Murchie et al. [2007]). Spectra were measured relative to a diffuse Infragold reflectance standard at a viewing geometry of ~10º for both incident and emergent beam angles. A spectrum was acquired at three different spots on the surface of each mixture to account for possible heterogeneity, with each spectrum representing an average of 200 scans. The three spectra were then averaged to produce a single spectrum representing a total of 600 scans for each mixture and each
‘pure’ endmember, respectively.
4.3.2 Analysis of Band Depths and Band Minima
Wavelength positions of true local band minima (maximum absorption strength) were identified for each reflectance spectrum at the ~1.4 µm (OH vibration associated with the clay cation-OH bond), ~1.45 µm (clay and epsomite H2O stretch overtones), ~1.9 and
~1.95 µm (clay and epsomite H2O vibrations), and 2.2-2.4 µm (clay cation-OH vibration) wavelength regions [Bishop et al., 1995b]. Band minima positions were then plotted against measured clay mass fraction (Figure 4.3) to assess changes in band position as a function of clay content. To analyze changes in absorption band strength in each mixture series as a function of the mixture composition (Figure 4.4), a spectral continuum was defined for each reflectance spectrum over the entire 1.25-2.6 µm wavelength range using the ENVI continuum removal routine in which a convex hull fit is defined by straight-line
141 segments connecting local maxima. After continuum removal over the entire wavelength range, the band depths of the true local minima at 1.4, 1.45, 1.9, 1.95, and 2.2-2.4 µm were calculated for each mixture using the method of Clark and Roush [1984]:
!! =1−(!!/!!) (4.13)
where Db is band depth, Rb is the reflectance defined at the band center, and Rc is the reflectance of the defined continuum at the band center.
4.3.3 Linear (Checkerboard) and Nonlinear (Intimate) Spectral Unmixing
To perform nonlinear intimate spectral unmixing, all reflectance mixture spectra were first converted to SSA using Eq. (2). Backscatter was assumed to be negligible (B(g)
= 0), a reasonable assumption given the phase angles of the spectral measurements, and it was assumed that the fine-grained mixtures were composed of isotropic scatterers (p(g) = 1). Though the latter may not be true in the strictest sense, it is likely a small source of uncertainty when comparing results for different mixtures because all spectra were acquired with an identical viewing geometry. In addition, exact phase function values for the clay and sulfate endmembers used here have not been previously reported.
Clay and epsomite endmember spectral weighting coefficients were modeled for each mixture from reflectance (checkerboard mixing model, Eq. (1)) and SSA (intimate mixing model, Eq. (4)) spectra using linear least squares inversion with a constraint of non- negativity implemented in MATLAB using the lsqnonneg function
142 (www.mathworks.com/help/optim/ug/lsqnonneg.html). Linear least squares was performed for each mixture using the measured mixture spectrum and an input matrix containing the pure clay and epsomite endmember spectra of that series and lines of positive and negative slope. The additional sloped lines were included to allow the model to account for phase behavior and wavelength-dependent scattering effects not accounted for by the mineral endmembers [Combe et al., 2008]. The inversions were performed over the full wavelength range (1.25-2.6 µm) as well as a subset wavelength range (2.1 to 2.6 µm) to exclude H2O absorptions that are dependent on sample hydration state (water content) and not uniquely linked to clay mineral abundance. Since the modeled spectral fits rely on the spectral endmembers, no a priori assumptions about particle size, optical constants, porosity, etc.
were required. In theory the models should be able to accurately fit the mixture spectra simply by varying the proportions of the input spectral endmembers (that is, by varying the fractional contribution of each component), especially given that viewing geometry, sample preparation, and other measurement conditions were identical for all samples.
To converge on the set of spectral weighting coefficients that provided the best model fit to the measured spectrum of each mixture (Table 4.1), the lsqnonneg function performed a series of iterations to minimize the sum of the square of the residuals between the measured and modeled spectral reflectance or SSA. The number of iterations in each optimization was determined by the MATLAB default value for tolerance on the coefficients (www.mathworks.com/help/optim/ug/lsqnonneg.html), so that the iterations terminated when the norm of the difference between coefficients calculated during the n and n+1 iterations was smaller than the allowed tolerance.
143 Linear and intimate mixing modeled spectra were then calculated by summing the reflectance or SSA endmember spectra, respectively, weighted by the fractional coefficients determined during linear least squares inversion, including the coefficients for positive and negative sloped lines (see Figures 4.5-4.10 for linear mixing model reflectance spectra). Residuals between modeled and measured spectra were plotted to determine which wavelength regions were best or worst fit by the models (Figures 4.5-4.10).
4.3.4 Modeling Mass Fraction
To assess the accuracy of the checkerboard and intimate mixing models for estimating mineral abundances, clay and epsomite mass fractions were calculated using Eqs. (10) and (11) for each mixture within a series, over both the full and partial spectrum wavelength ranges. Solid densities were defined as 2.3 and 1.7 g/cm3 for clay and sulfate endmembers, respectively, in line with the mineral product information provided by the Clay Mineral Society (clays) and Mallinckrodt Baker, Inc. (epsomite). Spectral weighting coefficients for clay and epsomite were determined by the method described in the previous section, although the clay and epsomite coefficients derived for each mixture were first normalized such that the fractional contributions of mineral endmembers summed to one.
This normalization step corrected for the positive and negative sloped line contributions, which have no meaning in terms of mineral abundances and typically provided very minor contributions (Table 4.1). Normalization also ensured that model results were geologically and physically plausible given that the prepared mixtures were known to be binary.
Three different values for the particle diameter of each component (mean, mode, and optimized) were used in the model runs, resulting in three sets of modeled mass
144 fraction values for each endmember series for a given wavelength range (full or partial spectrum). The measured mean and modal particle diameter values of the clay and epsomite endmembers were estimated from optical microscopy point counting (Figure 4.1, Table 4.2). The optimized particle diameter ratio (Dclay:Depsomite) was calculated for each mixture according to Eq. (10), using non-linear least squares (lsqnonlin function in MATLAB, www.mathworks.com/help/optim/ug/lsqnonlin.html) to minimize the sum of squared residuals between measured and modeled mass fraction values. In these iterations, the derived clay and epsomite weighting coefficients were those calculated by linear least squares inversion and were considered to be constants since these values provide the best possible fit to the measured spectra. Though there is no strong reason to expect the particle size of clay or epsomite endmembers to vary significantly within a mixture suite, these optimized particle diameter ratios provide insight into the relationship between clay or epsomite abundance, measured particle size distributions, and optical path lengths as clay or epsomite content varies within a mixture suite. In addition, the modeled optimized particle diameter ratios can be directly compared to the mean and mode values estimated from photomicrographs of the samples. We also estimated a single diameter ratio that minimized differences between measured and modeled mass fractions for samples within a mixture suite as a whole (Table 4.2). This single optimized particle diameter ratio was calculated for each mixture suite by simultaneously minimizing the differences between all measured and modeled mass fractions within a series, and this value was used to calculate the optimized modeled mass fractions for each mixture series.
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