Nonlinear spectral mixing: Quantitative analysis of laboratory mineral mixtures

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109,, doi: /2003je002179, 2004 Nonlinear spectral mixing: Quantitative analysis of laboratory mineral mixtures F. Poulet and S. Erard Institut d Astrophysique Spatiale, Université Paris-Sud, Orsay, France Received 3 September 2003; revised 24 November 2003; accepted 7 January 2004; published 24 February [1] Applications of nonlinear unmixing models based on the Shkuratov theory to fit laboratory spectra of intimate mixtures and to derive imaginary refraction indices of minerals are presented. The tests are performed using mafic minerals (pyroxenes, olivine, plagioclase). Abundance estimates of end-members are accurate to within 5 10% for the analyzed mixtures, while the estimate particle sizes are within the intervals of actual sizes, given the reflectance spectra of the end-members only. The type of mixture (areal versus intimate) can be tested. A quantitative modeling of basalt, a dark rock mainly composed of bright minerals, is also presented. The limitations of our modeling method are discussed. INDEX TERMS: 5460 Planetology: Solid Surface Planets: Physical properties of materials; 5470 Planetology: Solid Surface Planets: Surface materials and properties; 3934 Mineral Physics: Optical, infrared, and Raman spectroscopy; KEYWORDS: mineral, radiative transfer, spectroscopy Citation: Poulet, F., and S. Erard (2004), Nonlinear spectral mixing: Quantitative analysis of laboratory mineral mixtures, J. Geophys. Res., 109,, doi: /2003je Introduction [2] In the forthcoming years, high spatial resolution imaging spectroscopy in the visible/near-infrared range will be used to obtain compositional and physical information for nonicy planetary surfaces and especially for Mars surface from the space missions Mars Express (MEX, ) and Mars Reconnaissance Orbiter (MRO, ). Similar data will also be acquired on the Moon, Mercury and large asteroids e.g., from Smart-1, Messenger, BepiColombo and Dawn missions. Such observations are expected to provide detailed mineralogical composition of the surfaces, and therefore insights to formation and evolution processes. Deconvolving a reflectance spectrum to mineral abundance in an unambiguous way is difficult however, because the spectra are complex nonlinear functions of particle size, abundance, material opacity, and type of surfaces (e.g., dust, sand, and bedrock, where particles are mixed at different scales). [3] The most simple approach to deconvolve mineral rock types and abundances is to coadd the spectra of planetary bodies that represent compositional end-members [Merényi et al., 1996]. This method avoids the problem of dissimilarities between laboratory samples and planetary surfaces. However, for soils which are likely to dominate any natural surface, the mixing systematics are in general nonlinear, while the choice of end-members gives limited information. [4] An alternative and more reliable approach is to extract the end-member information from the data set itself. Different methods have been used, e.g., Principal Component Analysis (PCA) [Johnson et al., 1985], Modified Gaussian Copyright 2004 by the American Geophysical Union /04/2003JE Model (MGM) [Sunshine et al., 1990], or automated band detection based on wavelet filtering [Gendrin and Erard, 2003]. The strength of these methods is to aid in determining appropriate end-members with limited a priori knowledge. The PCA can be also used to identify the type of mixing (macroscopic versus intimate) and to estimate the relative proportion of end-members [e.g., Johnson et al., 1985; Chevrel et al., 1999; Pieters et al., 2001]. However, it requires careful interpretation of the principal axis of variation and the quantitative determination of end-member abundances becomes complex for mixtures containing more than two components. By using the MGM, information on relative abundances has been derived for some types of surfaces [Sunshine and Pieters, 1993, 1998; Mustard et al., 1993, 1997; Cooper and Mustard, 2002; Steutel et al., 2003]. In these cases, parameterizations of the spectral properties of minerals and mineral mixtures with algorithms sensitive to relative abundances or mineral compositions were done. However, these works were focused on very specific mixtures (primarily olivine and pyroxene mixtures), and the particle size effect was only lightly touched upon, while it has long been known to be a first-order contributor to the spectral properties. Another issue is that the parameterization depends on laborious preparations and measurements of suites of samples for inputs to account for all possible mineral assemblages. This limits the exploration of combined effects of particle size and composition, and therefore its field of application. Finally, the MGM method or its derivatives are subject to large errors in the derivation of the quantitative analysis of multicomponent mixtures when part of the components are spectrally featureless [Moroz and Arnold, 1999]. [5] Multiple scattering theories can provide approximate solutions to the radiative transfer in a compact medium. Employing such a theory in concert with optical constants 1of12

2 Figure 1. Spectra of the imaginary index k of a Sunshine clinopyroxene using the average of the upper and lower size limits of each sample (upper figure) or using optimized particle sizes in order to minimize deviation from the mean (lower figure). of components should allow computational exploration of the extensive parameter space. Unlike the methods mentioned above, scattering models take into account the influence of particle sizes and all end-members including spectrally neutral components. One popular radiative transfer model is the Hapke model [Hapke, 1981] which has been used by most modelers of natural and laboratory common geologic mixtures. Applications of the Hapke model to quantitative analysis of laboratory mineral intimate mixtures give mass abundances estimates better than 10% if an estimate of the particle sizes of the respective mixture components is known [Mustard and Pieters, 1987, 1989; Hiroi and Pieters, 1994]. [6] Another geometrical optics model for spectral albedo dependence of regolith-like surfaces was presented by Shkuratov et al. [1999]. An analysis of this scattering theory showed its degree of realism and of efficiency relative to other scattering theories, and in particular to the Hapke model and its derivatives [Poulet et al., 2002]. The first purpose of the present paper is to test the ability of the Shkuratov scattering model to determine the type of mixture, the relative proportions, and the particle sizes of components in mineral powder mixtures, given the spectra of possible end-members only. These tests are performed using laboratory spectra of well defined materials. [7] The sample materials selected for these investigations are minerals composing basaltic rocks, which are known or assumed to exist on the surfaces of many rocky Solar System bodies and especially Mars. The main reason for this choice is to allow direct comparison with other models, that have been most extensively tested on mafic mineral powder mixtures. Modeling mafic materials correctly is also a crucial issue to interpret spectra of differentiated bodies, including the Moon, Mercury and Vesta, for which spectral NIR data will be acquired in the near future. On Mars, this issue is related to dark materials, in particular basaltic sand dunes which may be commonly exposed on the surface. An important issue in this regard is to assess the efficiency of scattering models to describe mixtures that include spectrally neutral components, such as iron oxides and plagioclase feldpsar in the case of basalts. [8] We have also selected bulk materials for these tests. The spectrometers scheduled for the coming Martian missions will benefit from a dramatic increase in spatial resolution, typically 300 meters for the visible and nearinfrared mapping spectrometer OMEGA mounted on board Mars Express [Bibring and the Omega Team, 2003], or 20 m for CRISM on board MRO. Among the improvements expected from this new ability is the opportunity to detect rock outcrops, which are likely of basaltic composition. The Thermal Emission Imaging System on board Mars Odyssey is currently being used to localize such bedrock units, which are very limited in size (some 100 m to a few km) and pretty sparse [Rogers et al., 2003; Christensen et al., 2003]. Such units are expected to be more difficult to detect in the VNIR range, because of lesser penetration depth and higher sensitivity to very thin layers of fine materials. But even if they represent only a small fraction of future data sets, such areas potentially provide direct information about Mars mantle evolution and volcanic history, and proper analysis of these data will be essential to address these issues. The appropriate laboratory analogs for outcropping rocks are bulk rather than powdered samples, which are more characteristic of regolith or soils. We therefore applied our method to both powdered and bulk samples of basalt. Figure 2. Model fits (dashed line) of reflectance of twocomponent mixtures (continuous line) for the small particle size sample. An intimate mixture is considered. 2of12

3 [9] An important limitation of the radiative transfer modeling approach comes from the fact that the optical constants are available only for a few minerals, owing to the difficulty of such measurements in the laboratory. A possibility to estimate the optical constants from pure powdered materials is to use a scattering model [Hapke, 1993; Lucey, 1998]. One of the strength of the Shkuratov theory is that it is analytically invertible: the imaginary index of refraction k(l) can be computed from reflectance spectra if estimates for the refractive index n and the particle size d of the sample are available. This paper also presents an assessment of the radiative transfer model of Shkuratov et al. [1999] to determine complex indices of refraction suitable for mineral mixing computations. [10] There are two major sections to this paper. First of all, we briefly review the major steps of the Shkuratov radiative transfer theory and present our procedure to model spectroscopic data (section 2). Then, we present applications of our method to laboratory spectra of particulate and bulk mineral surfaces, and discuss the accuracy of the results and the procedure (section 3). Figure 4. sample. Same as Figure 2 but for the large particle size 2. Spectroscopic Modeling Method 2.1. Scattering Model [11] Both endogenic (volcanism, tectonism, surface/atmosphere interaction) and exogenic (solar wind, meteoritic bombardment) processes affect the composition and the structure of planetary surfaces, and change their optical properties. These effects take place in all scales of the surface structure from angstroms to hundreds of kilometers. Shkuratov et al. [1999] presented a geometrical optics model for albedo spectral dependence of regolith-like surfaces which can be adapted in order to represent different kinds of surface structures. As described by Poulet et al. [2002], the general scheme of the model can be summarized into three steps: (1) To derive the albedo of a particle of size d with a given composition defined by its optical constants n, k. Collimated light incident on a regolith particle with randomly oriented facets is approximated by diffuse light Figure 3. Same as Figure 2 but for the medium particle size sample. incident on one side of a planar slab with appropriate optical properties. A ray propagating between scatter points is characterized by the absorption coefficient t =4pkd/l. (2) To derive the reflectance of a homogeneous particulate surface with a single end-member. The summation of multiple-scattered rays is done by the principle of invariant embedding [van de Hulst, 1980]. At this point, the model includes a dependence on the surface porosity P in the summation. Shkuratov et al. [1999] showed that reflectivity is only weakly dependent on P in standard laboratory geometry. (3) To derive the reflectance of a multicomponent surface. Three kinds of mixtures were considered in early implementations of the model [Shkuratov et al., 1999; Poulet et al., 2002]: intimate (or salt-and-pepper ) mixtures of coarse particles with size >l; areal (or checkerboard ) mixtures where most photons only cross one component; intramixtures which consist of coarse particles with small inclusions of different absorbing material(s). A fourth kind of mixtures where all particles have size <l (also called micromixtures ) can be also modeled. This type of mixture is computed through effective optical constants derived from the Bruggeman relation [e.g., Bohren and Huffman, 1983], and are introduced in the Shkuratov scheme at step 2. In the frame of NIR spectroscopy, they correspond to mixtures of submicron particles. [12] The main formal difference between the Hapke and the Shkuratov theories is related to the phase function of individual particles, described by its asymmetry parameter. The particle phase function influences the scattering history of a photon within the regolith and thus the angular distribution of emergent photons. In the Shkuratov model the asymmetry parameter is an internal quantity derived from other particle properties, and provides a realistic particle phase function at all orders of scattering; in the Hapke model it is a free parameter that most applications include only in the single scattering term, assuming isotropic scattering at highest orders. Applications of the Hapke model to rocky planetary surfaces mostly provide backscattering particles, while the Shkuratov model generally 3of12

4 provides forward directed phase functions. Results from observations and modeling are also conflicting, and this issue is actually still debated. It is discussed in more details by Poulet et al. [2002]. These differences could explain why a number of recent studies using Hapke theory or its derivatives derive regolith grain sizes which are actually in violation of the assumptions of the models used [e.g., Cruikshank et al., 1998; Barucci et al., 2000]. The Shkuratov model also has the extra ability to investigate the manner in which different constituents are physically mixed, as mentioned above in step 3. This ability increases the degree of realism of the models used to analyze remote sensing spectroscopic data. The theory of Shkuratov has already been used to study the surface composition of some planetary objects [Shkuratov et al., 1999; Poulet et al., 2002], while our technique of deconvolution was applied to model ISM spectra over Syrtis Major [Poulet et al., 2003b] Calculation of Complex Indices of Refraction [13] The Shkuratov model, as well as other scattering models, relies on the knowledge of the optical constants of the possible end-members. However, optical constants of common minerals are not widely available in the UV/NIR range, which could strongly limit the field of application of these models. The Shkuratov model has an important advantage: it is invertible, i.e., the wavelength behavior of the average absorption coefficient k of a given material can be estimated from reflectance data given the size of particles in the sample as well as a priori knowledge on the real part of refractive indice [Shkuratov et al., 1999]. Unlike in the case of mid-infrared vibrational features, the values of n for the minerals are not expected to vary rapidly with wavelength in the near-infrared. Direct laboratory measurements of the optical constants showed that the variations of n are smaller than 0.1 over the UV/NIR wavelengths range for pure silicates and basaltic materials, with values typically comprised between 1.5 and 1.9 depending on the samples [Deer et al., 1966; Egan and Hilgeman, 1975; Egan et al., 1975; Pollack et al., 1994; Dorschner et al., 1995; Scott and Duley, 1996]. In the next section, we will follow the procedure described by Lucey [1998] which gives the expression of n as a function of the composition for clinopyroxene, orthopyroxene, and olivine. For other minerals, we use the values measured in the experiments listed above. An error of 10% on the estimate of n results in a change of a few percent in the computed value of k. Actually, the most significant source of error on the derived value of k comes from the particle size. We tested the accuracy to which k can be determined by applying the Shkuratov model to spectra of several particle sizes (d < 45 mm, 45 < d <75mm, 75 < d < 125 mm) of a sample of clinopyroxene (PP-CMP-21) measured by Sunshine and Pieters [1993]. If the values of the average particle sizes are chosen correctly, the same k spectrum should be obtained for the three separates. However, a variability of ±15% is observed (Figure 1) when using the average particle sizes of the relatively large particle size ranges. We therefore repeat the calculation of k for each separate, adjusting the values of the average particle size so as to minimize the deviation about the mean spectra of k (Figure 1). The scatter between the three spectra of k is significantly reduced when using the optimal calculated Figure 5. Calculated mass fractions (upper figure) and particle sizes (lower figure, clinopyroxene in thick line) of two end-members for the small particle size sample. Calculations are shown with error bars and the actual mass fractions and average sizes are shown with diamonds. particles sizes. Those remain inside the size ranges and close to the means: 20.7 versus 22.5 mm, 69.3 versus 60.0 mm, versus mm. Recognizing that the choice of the particle size is a significant source of error, this experiment shows that the accuracy of the determination of k can be increased significantly by using reflectance spectra of several particle size separates for each sample under study. 3. Applications to Laboratory Mineral Mixtures [14] The derivation of the composition and particle size of a laboratory intimate mixture of pure end-members is a twostep process. Spectra of each end-member are first converted to optical constants as described above. In a second step, we use the Shkuratov scheme to reproduce the mixture spectra by optimizing the type of mixture (intimate, areal, or intramixtures), the particle sizes, and the abundances of individual end-members. Due to the nonlinear formulation of the Shkuratov theory, the inversion problem has to be solved using an iterative approach. A downhill simplex technique is used here to find the minimum Root-Mean- Squared (RMS) residual between measured spectrum and computed spectrum. The minimization is performed under the constraint that the sum of the fractions of various components be equal to 1. In all fits computed here, the 4of12

5 Figure 6. Same as Figure 5 but for the medium particle size sample. Calculations are shown with error bars and the actual mass fractions and average sizes are shown with diamonds. porosity is assumed equal to 0.8, but this parameter has little influence on the results. This section discusses assessments of our method in various situations, using controlled sample mixtures prepared independently for previous experiments, and publicly available on the network Simple Case: Two-Component Mixtures [15] The validity of the Shkuratov model is first tested using reflectance spectra of 8 intimate mixtures of clinopyroxene/orthopyroxene with three different sizes. These materials were prepared by Sunshine and Pieters [1993] to assess another unmixing method (modified Gaussian model), and are used here because the characteristics of these mixtures are well defined in terms of physical and spectroscopic properties. These samples are fit with our procedure, using an intimate mixture. The measured and calculated spectra are shown in Figures 2 4. The actual and optimized mixing ratios are also plotted in Figures 5 7. These figures allow a rapid qualitative assessment of the accuracy of the fits. Modeling of three intimate pyroxene mixtures (0.25/ 0.75,0.5/0.5,0.75/0.25) for three particle sizes (0 45, 45 75, mm) demonstrates that the method is valid for deriving accurate estimates of the abundances of minerals to within 8% and particle sizes within a factor 2 when a unique particle size is used for all end-members. We observe, however, a derived particle size systematically larger than the measurement for the clinopyroxene component. This may be a limitation of the model: the band contrast in clinopyroxene only marginally depends on the grain size; considering the large overlap between clino- and orthopyroxenes bands and the similarity in overall reflectance, the grain determination relies strongly on the absolute reflectances measured, and is therefore sensitive to calibration errors. The model is highly nonlinear so that it is difficult to estimate directly such errors. We therefore evaluate them by running the model on spectra incorporating the radiometric calibration uncertainties. A change of ±10% of the reflectance of the mixtures of the mm particle size sample results in a change of ±50% on the grain size, and ±4% on the fraction. The potential error in grain size and fraction determinations is also estimated using the basaltic samples described in section 3.3, which represent the most sensitive case. A change of ±10% of the reflectance of the basaltic bulk sample results in a change of a factor 2 or less on the grain size, and about ±2% on the fractions. The current method therefore provides a determination of mineral fractions within 5 10%, an accuracy similar to the MGM [Sunshine and Pieters, 1993]. Its main advantage in the case of these simple mixtures is to provide also an estimate of grain size within a factor of 2, accurate enough to discriminate between large categories of grains and therefore between formation processes. [16] On the basis of the good results of the intimate mixture model, the type of mixture is then studied. Table 1 compares the optimized parameters for the particle size mm samples when both an areal and an intimate mixture are considered. The noticeable differences in terms of RMS deviation and abundance between the two types of mixtures allow us to reject the areal mixture as a possible solution, and therefore to show the efficiency of the method to discriminate the type of mixture involved in the case of a two-component mixture with a single size range Soils [17] Further tests are done using a subset of the complex mixtures prepared by Pieters et al. [1993]. These soils were used by Mustard et al. [1993] to assess the accuracy of fractions estimates through the Hapke model. Three minerals (clinopyroxene, orthopyroxene and olivine) were sieved to three particle sizes (<25 (S), (M), (L) mm). Particle size separates were then combined to form mono-mineralic soils in the following proportions: small soil (SS = 65% S 25% M 10% L), medium soil (MS = 25% S 50% M 25% L), and large soil (LS = 10% S 25% M 65% L). Mono-mineralic soils were finally combined with a constant mass fraction ratio of 3:1:1, similar to mafic mineral ratios in gabbroic rocks, with three different soil combinations: SOIL 1 = MS:MS:MS, SOIL 2 = SS:SS:LS, SOIL 3 = LS:SS:SS. Therefore the soils are a combination of 9 components (3 particle size separates for each mineral). To limit the number of parameters, we have chosen to model these soils by an intimate mixture of 6 components, using only two particle size separates for each mineral instead of three. This still results in optimizing a total number of 12 free parameters (2 particle sizes and 2 fractions for each mineral). Fits are performed assuming an intimate mixture. [18] Shown in Figure 8 are the best-fit reflectance spectra compared to the measured spectra of three soils. The actual particle sizes and fractional abundances of components in 5of12

6 the three soils are presented in Figure 9. The derived fractions agree with the actual values to within 10% for most components, and to within 15% in all cases, which is slightly better than the results obtained with the Hapke model [Mustard et al., 1993]. An important difference however is related to the role of grain size: the model used by Mustard et al. [1993] assumes correct grain sizes of the components, which are fixed parameters in this context. In our model, grain sizes are free parameters derived from a rough initial guess and therefore: (1) consistent fits do not strongly depend on assumptions about grain sizes, (2) the model returns a reasonably good estimate of grain size for each component. Overall, this is an important improvement over previous methods based on the Hapke model, which required information about end-members nature and particle size. [19] Our estimated fractions of the finest particles are however always in excess of the measurements. These discrepancies between actual and calculated fractions are similar to those derived using the Hapke model by Mustard et al. [1993], which attributed them partly to the complex settling of the samples preparations, and party to the behavior of fine particles. As suggested by Mustard et al. [1993], the finest particles (<25 mm) may coat the largest ones, so that their contribution is larger than expected. A solution to improve the modeling procedure would be to consider a particle size distribution instead of a single average size; however, this would require fitting a large number N of parameters, which would become rapidly unreasonable in terms of computing time. In fact, the addition of one end-member in an intimate mixture typically increases the computing time by a factor of 2. Moreover, to increase N leads to the problem of the optimization of the parameters and the unicity of the solution (enough independent spectral information must be present in the spectra so that the problem remains overdetermined). In the case of a large number of parameters, the downhill simplex optimization method can also become unstable; alternatives include genetic or Monte Carlo methods which are however pretty slow. For these reasons, it will be difficult to estimate more than 15 parameters from a single spectrum of a rocky sample or surface. We also notice a good stability of our model in this respect, since consistent results are obtained using only 2 grain sizes instead of the 3 actually present in the mixtures: particle sizes derived in this section always lie in the range of actual particle sizes, with little deviation from the average value. Besides, they are always much larger than the wavelengths used, therefore in agreement with the physical basis of Shkuratov s model which relies on geometric optics Application to Basalt Powders and Bulk Samples [20] The minerals most likely present at the Martian surface are related to basaltic/andesitic magmas and their alteration products, and include high- and low-calcium pyroxene, plagioclase feldspar, olivine, iron oxides, and sulfates [e.g., Singer et al., 1979; Mustard et al., 1997; Murchie et al., 2000; Hamilton et al., 2001; Minitti et al., 2002; Bandfield, 2002]. Although the UV-Vis-NIR range is particularly well suited to study the surface lithology from spectroscopic measurements, these different species have very different expressions in reflectance spectra, and may be Figure 7. Same as Figure 5 but for the large particle size sample. Calculations are shown with error bars and the actual mass fractions and average sizes are shown with diamonds. difficult to identify from a limited wavelength range. A specific problem is the detection and the quantitative estimation of components that are spectrally neutral in the UV/NIR. A well-known example of such a behavior is that of feldspar which is even difficult to detect in the NIR range, while it is expected to be the most abundant phase at least in pristine rock units. As mentioned in the introduction, detection of such units is possible, although in very limited areas only, from the coming NIR observations. For this reason, we model both powdered and bulk basaltic samples in this section. (In the case of Mars, the bright or dark surface dust is also expected to contribute to remotely sensed spectra, and must be taken into account. Particle size for dust and drift material on Mars range from 0.1 to 10 mm [e.g., Christensen and Moore, 1992]. The spectroscopic contribution of a mixture of submicron particles can be simulated by making another assumption about the representation of particles as mentioned in section 2.1. Such an approach is beyond the scope of this paper, but was used to model ISM spectra over dark regions of Mars [Poulet et al., 2003b].) [21] Three spectral types of material are responsible for the shape and intensity of the NIR spectra of basalts and their derived products: (1) A first type of materials exhibits marked electronic and/or vibrational absorption bands (pyroxene, olivine, hematite, phyllosilicates); (2) A second type 6of12

7 Table 1. Optimized Parameters for Enstatite (OPX) and Clinopyroxene (CPX) Laboratory Intimate Mixtures a Actual Fraction (OPX/CPX) Type of Mixture Computed Fraction, wt % (OPX/CPX) Computed Particle Size, mm (OPX/CPX) RMSD, % 0.75/0.25 Intimate / / /0.25 Areal / / /0.50 Intimate / / /0.50 Areal / / a The particle size range is mm. Intimate and areal mixtures are considered for the modeling. is transparent (plagioclase); (3) The third one is strongly absorbing (some iron oxides). Signatures of minerals of the first type will dominate the appearance of the spectra. However, the other two constituents influence the absorption bands [Crown and Pieters, 1987; Cloutis et al., 1990a, 1990b], the brightness [Crown and Pieters, 1987; Cloutis et al., 1990a, 1990b; Harloff and Arnold, 2001], the overall slope of the spectra toward the long wavelengths [Cloutis et al., 1990a, 1990b], as well as relative band ratios [Moroz and Arnold, 1999]. [22] Our tests are focused on a basaltic particulate sample measured by Harloff and Arnold [2001], which documents this sample. The sample is made of 5 major minerals listed in Table 2; the modal composition is recalled here in Table 3. We searched several databases (NASA RELAB, USGS [Clark et al., 1993] and JPL ASTER) for minerals which would match the modal compositions of the basalt, and also used extra minerals. The optical constants are derived, using the scheme in section 2.2, from the following data: plagioclase (andesine) is sample ID PF-CMP from RELAB (from C. Pieters); ilmenite is sample HS231 from USGS; clinopyroxene (diopside), orthopyroxene (enstatite) and olivine (forsterite) are RELAB samples from Pieters et al. [1993] used in section 3.2. [23] We first attempt to model the basalt spectrum with an intimate mixture of the 5 components present in the sample, then using different species of pyroxenes, feldspar and olivine listed in Table 2. The optimized (best-fit) fractions of different components are summarized in Table 3. They are clearly a very poor fit of the actual sample composition. In particular, the transparent plagioclase material is not identified in the fit, whereas a very large proportion of iron oxide is required to match the low reflectance and to reduce the pyroxene band depths. Three kinds of plagioclase with chemical composition close to the plagioclase present in the basalt gave identical results. Models incorporating pigeonite and/or augite instead of diopside completely failed to reproduce the spectrum (no convergence). [24] A general difficulty in this fit is related to the availability of the mineral spectra matching the exact composition of the sample under study. As described above, we need at least 3 different grain sizes to minimize the uncertainty on the derivation of the optical constants. The minerals also need to be as pure as possible (no contamination by oxides, clays, organics, or atmospheric absorptions), and properly documented. Currently, such data are sparse in the public NIR spectral libraries, and it proved impossible to match the exact chemical composition of the sample with appropriate end-members. However, we think that the mismatch obtained in our first tries is not related to the chemical composition of the end-members, but rather to the type of mixture involved, as further detailed below. [25] The most noticeable effect of increasing the plagioclase fraction is to increase the overall reflectance in the NIR. A good example is given by spectra of intimate mixtures of pure plagioclase and pyroxene acquired by Crown and Pieters [1987]: the two-component mixtures have a reflectance much larger than typical basaltic samples. Therefore a fitting procedure will tend to minimize the plagioclase fraction to match the low average spectrum level, since this has comparatively small effects on band depths. This problem is related to a bigger issue, namely: how do you get very dark rocks (basalts) by mixing mostly bright minerals? In principle, the answer is two-fold: (1) basalts also contain a significant contribution of opaque minerals, in the present case 5% ilmenite; (2) The grain size of basalt is quite fine, and the opaque components are even finer than the average grain size and dispersed throughout the rock (J. Mustard, personal communication, 2003). The smaller grains are known to have larger effects on the spectra. Gabbros derived from basalts by very slow cooling have much larger grain sizes and the opaque minerals are gathered into just a few grains. These rocks are much brighter, yet have the same composition. In this case, our first fits failed to reproduce the correct reflectance with a consistent modal composition. Any fitting procedure will in fact tend to correct the excess in brightness by decreasing the amount of plagioclase, increasing the fraction of iron oxides and/or using larger particle size [Crown and Pieters, 1987; Cloutis et al., 1990b]. Our interpretation is that direct applications of geometric optics models cannot handle particles as small as those involved in the opaque fractions of basalt. [26] According to conventional geometric optics, the samples of the finest particles (d l) should have the Figure 8. Reflectance spectra of soils (continuous line) compared to their best-fit spectra (dashed line). 7of12

8 Figure 9. Calculated proportions (top, solid lines) and particle sizes (bottom, thick lines) of each endmember for the three grabbroic soils when two particle sizes are assumed for each end-member. The actual average sizes and fractions are represented by diamonds. In bottom plots, the thin and dashed lines represent the actual size ranges of the two most abundant components. The thick lines corresponds to the two calculated particle sizes. Note that only one particle size was obtained for olivine in the case of Soil2. highest albedo, as mentioned above. However, the opposite is observed for very fine particles [e.g., Mustard and Hays, 1997]. Cooper and Mustard [1999] also observed a marked decrease in spectral contrast as particle size decreases below this range. Our modeling procedure is therefore not directly applicable for describing this regime, because the Shkuratov theory is based on geometric optics and assumes particles much larger than the wavelength. This limitation is common to other existing models as well. As particle size approaches the wavelength of light, the scattering behavior changes and no current photometric model can describe it properly. When very small particles are embedded in a matrix however, effective medium theories allow to work around this problem by computing the refractive index of the intramixture (matrix + embedded particles) and using the result as an end-member in further mixing with other mineral species [e.g., Bohren and Huffman, 1983; Wilson et al., 1994; Cuzzi and Estrada, 1998]. [27] In the present case, the matrix corresponds to plagioclase and the opaque component consists of ilmenite that most efficiently decreases albedo and masks the absorption features. As discussed below, a BSE image of the basalt 8of12

9 Table 2. Minerals Identified in the Basalt Sample and End-Members Used in the Modeling Mineral Group Basaltic Sample End-Members Feldpsar Labradorite, Andesine Andesine (2), Labradorite Pyroxene Augite, Pigeonite, Enstatite Diopside, Augite, Pigeonite, Enstatite Olivine Forsterite Fayalite, Forsterite Oxide Ilmenite Ilmenite bulk sample is provided by Harloff and Arnold [2001], where ilmenite grains appear mixed in the plagioclase matrix. Our modeling procedure was therefore run again using an intramixture of ilmenite and plagioclase, itself intimately mixed with pyroxenes and olivine. The results are shown in Figure 10 and Table 3. The derived abundances are greatly improved relative to the first try, with derived fractions within 5% of the actual ones for major components. The particle size of the plagioclase component is however difficult to adjust correctly because of its transparent nature. If labradorite is included in addition to andesine, the fraction of labradorite consistently tends toward zero, demonstrating the robustness of the model. This is to our knowledge the first successful attempt at modeling a basaltic powdered mixture from mineral end-members, matching the absolute reflectance level. In the present case, a few dark scatterers embedded in a bright matrix very efficiently reduce the resulting reflectance of the overall medium. [28] Only a few laboratory investigations of basaltic bulk samples (as opposed to powders) have been published [Yon and Pieters, 1988; Harloff and Arnold, 2001]. The paper by Harloff and Arnold presents an excellent comparison of the scattering properties of the bulk and powdered surfaces of basalt samples. Compared to powders, they found that the spectra of bulk surfaces are characterized by shallower bands and a usually bluer sloped continuum. As evidenced by images acquired in the backscattered electron mode of a scanning electron microscope, basaltic rocks are characterized by a matrix of randomly oriented plagioclase lathes in which other minerals are embedded. The scattering history of a photon in such a texture is similar in principle to its behavior in an intimate mixture. On the basis of this analogy, we suggest that the intimate/intra-mixture model used above can be applied to reproduce the spectrum of a bulk surface: again, we assume small oxide particles embedded in the plagioclase matrix. This assumption is tested by applying our method to the spectrum of the basaltic bulk sample from which the powder studied above has been extracted. The results are shown in Figure 10 and Table 4. As in the previous test, the estimates of the grain sizes are rough, but the modeling method actually allows us to derive accurate estimates of mineral fractions in a bulk sample. A comparison between Tables 3 and 4 shows that the main differences between powder and bulk moderating come from the derived grain size of pyroxene and the iron oxide content; in the case of the bulk, the smaller grain size reproduces the weaker absorption bands, while the larger iron oxide content diminishes the absorption bands. This last effect is qualitatively consistent with the interpretation of maturation effect in regoliths, where darkening is produced by very small inclusions of reduced iron [Pieters et al., 2000]. A consequence is that, in the frame of the present model, it is actually very difficult to distinguish between powder and bulk forms of basalt from spectral information only. In order to be able to discriminate the surface physical state (regolith versus bulk) from remote observations, geomorphological and thermophysical properties of the surface need be taken into account. In the case of Mars, such information is provided by high resolution cameras (MOC or HRSC) and thermal mappers (THEMIS) respectively. An example of combined analysis of NIR spectroscopy, thermal properties and geomorphological evidence has been performed on low-albedo regions observed by the ISM spectrometer on board Phobos-2 [Poulet et al., 2003b]. [29] A possible extension to this model would include surface roughness as a parameter. Harloff and Arnold [2001] identified its significant influence on the shape of the spectrum. Actually, variations in particle size for powder samples and variations in roughness for bulk samples produce the same spectral effects: the rougher the surface, the lower the reflectance; the NIR continuum slope also becomes bluer with increasing roughness. Harloff and Table 3. Known and Optimized Parameters for the Basaltic Powdered Sample a Mineral Name Modal Fraction, wt % Computed Fraction, wt% Computed Particle Size mm Intimate Mixture with Regular Plagioclase Plagioclase 65.7 ± Pyroxene 25.2 ± Olivine 3.8 ± Ilmenite 5.3 ± Intimate Mixture With Oxides Embedded in Plagioclase (Intramixture) Plagioclase 65.7 ± Pyroxene 25.2 ± Olivine 3.8 ± 0.5 <1 160 Ilmenite 5.3 ± <1 (fixed) a The grain size range is mm. The RMS residuals are 0.20% and 0.13%, respectively. 9of12

10 Figure 10. Upper frame: Comparison of the reflectance spectrum of a basaltic powder (thick line) with its model (thin line). The data/model ratio allows a rapid qualitative assessment of the accuracy of the fits. The 1.4, 1.9 and 2.3 mm water and hydroxyl bands due to the experimental conditions and the presence of a very small amount of phyllosilicates (<1%) are not modeled. Lower frame: Same as upper plot but in the case of a bulk surface. Arnold [2001] proposed an empirical model in which the mean ray path length (MRPL hereafter) is related to roughness: in a bulk sample, the more lateral passage events take place, the rougher the surface is, the larger the MRPL is (in other words, the ray is captured in the irregularities of the surface); in a particulate sample, the smaller the particle size is, the more multiple scattering events take place, the smaller the MRPL is. In any case, more theoretical and numerical examinations should help explaining the processes occurring at a solid surface Limitations of the Modeling Method [30] There are some limitations of our mixing model in the applications to the analysis of remote observations. [31] If inappropriate end-members are used to fit a mixture, larger fitting errors would occur in fractional abundance estimates. A first step in applying the unmixing model is therefore to detect the minerals possibly present in the mixture. To reduce the uncertainty related to endmember selection, and also to estimate the type of surface involved, the spectral mixture modeling is best used in conjunction with other analyses that provide information about thermophysical, morphological, or even magnetic properties of the surface. [32] Some parts of planetary surfaces are likely dominated by fine particles of size d l, which have weaker NIR absorptions and/or lower albedo [Mustard and Hays, 1997; Cooper and Mustard, 1999]. The Shkuratov theory is based on the geometric optics, where the particle size is larger than the wavelength. As particle size approaches the wavelength of light in the volume scattering region, the theory is thus no longer directly applicable to measurements, which implies we should use it with care. In particular, the fraction of mineral occurring with the finer particle size may be underestimated, because spectral reflectance decreases as particle size decreases. Mustard and Hays [1997] modeled the IR spectra of fine particle samples using a combination of Mie theory (to determine the single scattering albedo) and Hapke s model (to calculate the reflectance spectra). The results are qualitatively good but extracting reliable quantitative information on the physical and compositional properties in this optical regime remains an issue. A limitation of the Mie-Hapke model is that the Mie solution explicitly assumes particles separated by several particle radii, which is clearly not the case in soils. [33] Finally, we reiterate that the optical constants of common minerals are still rare in the Vis-NIR wavelength range, which could limit the field of application of our method. As explained in section 2.2, a way to avoid this problem is to derive optical constants from reflectance spectra of pure mineral samples. Because the uncertainty on this computation can be a source of large errors, the calculation of optical constants has to be made from PURE samples well characterized in terms of composition AND particle sizes. Consequently, these two conditions limit us in the choice of available end-members, and extensions of the Vis-NIR spectral libraries is required, even concerning usual minerals. 4. Conclusion and Future Applications [34] We have presented applications of our spectral nonlinear spectral mixing method, based on the Shkuratov model, to estimate complex indices of refraction in the NIR range and to invert spectra of controlled mineralogical samples. Four categories of samples have been used: simple Table 4. Known and Optimized Parameters for the Basaltic Bulk Sample a Mineral Name Modal Fraction, wt % Computed Fraction, wt% Known Grain Size, mm Computed Grain Size, mm Plagioclase 65.7 ± Pyroxene 25.2 ± Olivine 3.8 ± 0.5 < Ilmenite 5.3 ± <1 (fixed) a Intimate mixture with oxides embedded in plagioclase. The RMS deviation is 0.15%. 10 of 12

11 and complex laboratory mixtures of mafic minerals; powdered and bulk samples of natural basalts. These tests indicate that our spectral unmixing method can be used to determine the compositional and physical properties of a multi-component surface with similar confidence as previously existing methods: fractions are determined to better than 10% in most cases, similar to the MGM and Hapke model applied on the same samples. A noticeable advantage of the present method is to handle the components particle sizes as free parameters. Therefore the goodness of fit does not depend on a first guess of these parameters, in contrast to other methods. Besides, the estimates of particle size are in these tests accurate within a factor of 2, which is enough to discriminate between large categories of grains and therefore between formation processes. Finally, the type of mixture involved (areal versus intimate) can be tested. In this regard, what appears to be the first successful modeling of basalt from mineral end-members is derived using another type of mixture (intramixtures) to simulate small oxides particles embedded in the plagioclase matrix. A larger accuracy is expected when working with optical constants properly derived from laboratory measurements. Note that the spectroscopic measurements are still spare beyond 2.5 mm, so that the expansion of the library of optical constants (model-dependent or not) would assist with future modeling. [35] Bearing in mind the possible caveats, our deconvolution method can be a reliable tool to quantitatively estimate surface composition of planetary objects. It has been already used to investigate the surface composition of Martian low albedo regions as seen by ISM, which allowed to reject the presence of several minerals as well as to identify large quantities of iron oxides [Poulet et al., 2003a, 2003b]. 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