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Radiative Transfer Models for Reflectance Spectra 1

Icarus 137, 235–246 (1999)Article ID icar.1998.6035, available online at http://www.idealibrary.com on

A Model of Spectral Albedo of Particulate Surfaces: Implicationsfor Optical Properties of the Moon

Yurij Shkuratov and Larissa StarukhinaAstronomical Observatory, Kharkov State University, Sumskaya St., 35, Kharkov, 310022, Ukraine

E-mail: [email protected]

and

Harald Hoffmann and Gabriele ArnoldInstitute of Planetary Exploration, German Aerospace Center (DLR), Rudower Chaussee 5, D-12489 Berlin, Germany

Received September 9, 1997; revised June 29, 1998

A simple one-dimensional geometrical-optics model for spectralalbedo of powdered surfaces, in particular of lunar regolith, is pre-sented. As distinct from, e.g., the Kubelka–Munk formula, whichdeals with two effective parameters of a medium, the suggestedmodel uses spectra of optical constants of the medium materials.Besides, our model is invertible, i.e., allows estimations of spectralabsorption using albedo spectrum, if a priori data on the real partof refractive index and surface porosity are known. The model hasbeen applied to interpret optical properties of the Moon. In par-ticular, it has been shown that: (1) both color indices and depthof absorption bands for regolith-like surfaces depend on particlesize, which should be taken into account when correlations betweenthese optical characteristics and abundance of Fe andTi in the lunarregolith are studied; (2) fine-grained reduced iron occurring in re-golith particles affects bandminima positions in reflectance spectraof lunar pyroxenes and, consequently, affects the result of determi-nation of pyroxene types and Fe abundance by Adams’ method.c⃝ 1999 Academic Press

INTRODUCTION

To interpret optical data on atmosphereless celestial bodies,models of light scattering by regolithic media are needed. Re-goliths of lunar and planetary surfaces have a complicated par-ticulate structure including semitransparent and optically inho-mogeneous fragments and aggregates of rocks (minerals) andglasses of different sizes, from submicrometer to millimeterscales. At larger scales, the planetary surfaces have a stochas-tic relief. Mathematical description of such objects faces hugedifficulties and, therefore, a more or less strict theory of lightscattering for these objects has not been proposed yet. It may beexpected that this theory should be very cumbersome and shouldoperate with many parameters. Even in the geometrical-opticsapproximation, the problems remain very difficult. At present

there are two photometrical models close to each other that de-scribe observational data in first approximation (Hapke 1981,1986, Lumme and Bowell 1981a,b). The theories operate onlyfive parameters, but even this number is often too much, sincethey give ambiguous fittings to experimental data (Domingueand Hapke 1989). There is a “vicious circle”: the more parame-ters, the more exact the theory, but the more ambiguous the fitsto measured data. Therefore, heuristic models with small num-bers of parameters, but giving good fit to observations, seem tobe very attractive.The main purpose of this paper is to propose a simple model

of such a kind that describes spectral behavior of albedo ofregolith-like surfaces. Our model is based on geometrical-opticsapproximation. It has an important advantage—it is analyticallyinvertible, i.e., the wavelength behavior of average absorptioncoefficient of regolith material can be estimated from spectralalbedo data. Prototype elements of themodel can be found in theworks by Stokes (1904), Bodo (1951), Melamed (1963), Hapke(1981), Shkuratov (1982, 1987), Hiroi and Pieters (1992), andStarukhina and Shkuratov (1996).Spectroscopy of celestial bodies can be considered as a par-

ticular case of spectroscopy of a light-scattering object. Accord-ing to Rosenberg (1967), a general scheme of spectroanalyticexperiment includes the following stages: (1) measurement ofoptical characteristics (e.g., albedo) of the object as a functionof wavelength, (2) determination of optical parameters of itslight-scattering slab (e.g., volume coefficients of scattering andabsorption on the base of the radiation transfer theory), (3) de-termination of the optical characteristics of an “average” light-scattering element, (4) calculations of the optical constants ofthe surface material, (5) determination of physical and chemicalproperties.Solid surfaces of celestial bodies are objects with chaotic

structure. The surface particles display awide variety of physicalproperties. Therefore, the scheme cannot be entirely applied

235

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236 SHKURATOV ET AL.

to these surfaces. The usual approach is to find relationshipsbetween spectral and chemical/mineralogical parameters usinglaboratory measurements of analogs (in the case of the Moon,soil) of celestial bodies surface material. Thus, from the firststage of the spectroanalytic experiment a step is made directlyto the last one.This approach is rather fruitful. For example, the widely

known Adams (1974) pyroxene curve presents the relationshipbetween the type of pyroxene and position of the center of thed–d absorption bands formed by Fe2+ ions in the ranges near1 and 2 µm; the “titanium” curve enables one to estimate TiO2abundance in mature mare soils of the Moon using measure-ments of color index C(0.42/0.55 µm) (Charette et al. 1974).However, the empirical approach is useful only as a first step.The relationships between spectral and mineralogical parame-ters should be explained and the limitations of their applicationsshould be found. This requires a completion of all (or the mostof) stages in Rosenberg’s scheme, at least in a simplified form.This scheme is partially realized in the models mentioned

above (Hapke 1981, 1984, 1986, Lumme and Bowell, 1981a,b).They use the radiation transfer theory to calculate the contribu-tion of multiple scattering. However, as noted in Simons (1975),the theory is effective to estimate average optical characteristicsof the medium and not of the material constituting the parti-cles. Determination of the material parameters coming from theoptical characteristics of the scattering medium is not a trivialproblem. Moreover, the classical radiation transfer scheme can-not be consequently applied to regolithic surfaces. Indeed, thetransfer equation uses a small elementary volume of the scatter-ing medium. The volume should include a number of scatterersgreat enough, but this assumption is correct only for powderedsurfaces with rather high albedo.Often an “average” particle is considered (Hapke 1981) as

an elementary volume of particulate medium. The elementaryvolume is characterized by the albedo of single scattering. Toestimate this albedo in the geometrical-optics approximation,an approach based on a one-dimensional model of light scat-tering was used by some authors, e.g., Melamed (1963), Hapke(1981), and Shkuratov (1982, 1987). In this model multiple re-flections in a particle are considered as multiple scattering in aone-dimensional medium with some “effective” reflection co-efficients which are the result of averaging the usual Fresnelcoefficients over incidence angle.The main idea of our model is to apply the approach used for

single-particle-scattering albedo (Melamed 1963, Hapke 1981,Shkuratov 1982, 1987) to the albedo of a particulate surface(Bodo 1951, Shkuratov 1982, 1987, Starukhina and Shkuratov1996). We suggest changing scattering in a system of parti-cles (Fig. 1a) to scattering in an equivalent system of plates(Fig. 1b). Thus, we ignore all angle dependences of reflectance,assuming that the calculated one-dimensional reflectance can beidentified with the reflectance of three-dimensional medium ata small phase angle. Our laboratory spectral and photometricmeasurements of powdered color glasses (e.g., Shkuratov 1987)

FIG. 1. A scheme of light propagation: (a) through a particulate mediumand (b) through plates in a pile.

have shown that, in the limits of 10%, the measured albedoat the phase angle of about 50 can be identified with albedo(reflectance) A derived from the one-dimensional model. Thisalbedo can be used to estimate the brightness coefficient at ar-bitrary phase angles α: B= A · f (α), where f (α) is the phasefunction normalized at 50.

FORMULATION OF THE BASIC MODEL

Consider a powdered surface (a medium) composed of semi-transparent particles of arbitrary form and of sizes much greaterthan the wavelength λ. Let a parallel light beam be incident onthe boundary of the medium. Ray propagation in such mediumis a random branching process. The branching points are charac-terized by two values, namely, the reflection R(n∗, θ ) and trans-mission T (n∗, θ ) coefficients, where n∗ is the complex refractiveindex of the material (n∗ = n − iκ , where n and κ are the op-tical constants—the real and imaginary parts of the refractiveindex, respectively) and θ is the local angle of incidence on theparticle interface. The intervals between the branching pointsare characterized by optical density τ = 4πκS/λ, where S is theeffective (average) optical pathlength, κ and τ being equal tozero in the interstitials. In terms of the variables, the extinctionI/Io can be determined for any trajectory of a ray in the medium

SCATTERING THEORIES FOR SOLID PLANETARY SURFACES 315

brighter because more light escapes before being absorbed in-side the particle. s is usually assumed to be equal to zero (“clearparticles”) to avoid introducing an additional parameter. Thisimplies !H = exp(−8πk D/3λ). We point out that this entiredevelopment requires a particle radius larger than several timesλ—if not tens of times λ. It is this key assumption which is oftenviolated by the nominal model results.

2. To obtain the final expression for r of a homogeneousand semi-infinite monomaterial surface, the contribution of thefirst-order scattering is added analytically for arbitrary particleanisotropic scattering. An anisotropic phase function character-ized by one parameter (usually the asymmetry parameter) atleast is introduced at this step. For the multiple scattering term(with the single scattering term removed), an isotropic parti-cle phase function is assumed which reduces the calculationof the multiple scattering to the evaluation of the H-functionsof Chandrasekhar (1960). An approximation of these functionswas derived using the two-stream solution but shows large errorsfor bright material. In the Doute and Schmitt model (1998), theHapke single scattering albedo is used together with an improvedformulation of the Hapke bidirectional reflectance. In particu-lar, Doute and Schmitt numerically compute both the single anddouble scattering contribution with a phase function which canbe anisotropic. Cuzzi and Estrada (1998) combined the Hapkesingle scattering albedo with a Van de Hulst (1980) reflectanceapproach, which allows for anisotropic grain phase functionsat all orders of scattering. Other effects such as macroscopicsurface roughness (Hapke 1984), shadowing (Hapke 1986), andcoherent backscattering (Helfenstein et al. 1997) can be takeninto account in the calculation of r .

3. Two kinds of mixture can be considered: areal and homo-geneous. For areal mixtures, the reflectance of a surface com-posed entirely of some material component is first calculated;then these are linearly combined to represent the total reflectanceof a multicomponent “checkerboard” surface. For homogeneousmixtures, the averaging process is on the level of the individualparticle. In this case, two types of mixtures can be considered:(a) intimate or “salt-and-pepper” mixtures in which the singlescattering albedos are calculated for each component grain andthen averaged (Eq. (17) of H81) and (b) intramixtures, whichare simulated by calculating a weighted mean of the indices ofrefraction within each grain. Homogeneous mixtures are dis-cussed by Bohren and Huffman (1983), Wilson et al. (1994),and Cuzzi and Estrada (1998). Each grain, or regolith particle,has inclusions that are considerably smaller than both the grainand the wavelength.

2.3. Shkuratov Model

A geometrical optics model for albedo spectral dependenceof regolith-like surfaces was presented by S99. As was done forthe Hapke model, the formulation of the Shkuratov model canbe summarized by three steps:

1. The first step is to derive the albedo of a particle. Similarlyto the Hapke model, collimated light incident on a regolith grain

Te=1−ReRe

Ri

Ti=1−Ri

Ti=1−RiRi

Ri

Ri

S

−τW2 Te exp( ) Ti

W3 Te exp(−2τ) Ri Ti

Ti=1−Ri

Ti=1−Ri

W4 Te exp(−3τ) Ri RiTi

FIG. 1. Schematic diagram of the light propagation used in theShkuratov theory. Re is the average external reflectance coefficient, Ri is theaverage coefficient of reflection inside the particle of size S. The functions Wm

are the probabilities for the beam to emerge backward or forward at the mth scat-tering. A fraction Te = 1 − Re of the light enters the slab and is attenuated byabsorption. This effect is quantified by the internal-transmission factor exp(−τ ),where τ = 4πkS/λ. Following the first passage through the slab, a fraction Riis internally reflected and the remainder Ti = 1 − Ri is refracted through thesurface. The process continues as indicated by the arrows.

with randomly oriented facets is approximated by diffuse lightincident on one side of a planar slab with appropriate opticalproperties (Fig. 1). Multiple reflections in a particle are consid-ered as multiple scattering in this one-dimensional slab, withinternal reflection coefficients which are the result of averagingthe usual Fresnel coefficients over incidence angle. This methodprovides the fraction of light scattered by a particle into the back-ward hemisphere (rb) and into the forward hemisphere (rf) asgeometric series (Eqs. (9) of S99). These geometric series de-pend on the average external and internal reflection coefficientsRe and Ri and on the internal-transmission factor, which is givenby !S = exp(−τ ), where τ = 4πkS/λ. S is defined as the av-erage length of light propagation in a particle between internalreflections. For transparent particles, S should be equal approx-imatively to the average diameter of the particle. For nontrans-parent particles, it is hard to relate this parameter to measurablequantities. Because this part of the model of S99 is very similarin spirit to the comparable part of H81, S must then have thesame meaning as Hapke’s D. Again, as in H81, this step requiresS ≫ λ.

2. The second step consists of the derivation of the reflectanceof a particulate surface. The light propagation through the par-ticulate medium is simplified by the propagation through a half-infinite stack of layers. The summation of the multiple scatter-ing is done according to the principle of invariant imbedding.At this point, the model includes a dependence on porosity Pin the summation. The limitation of the model is that all angledependence of reflectance is ignored, so that the model is notappropriate for the analysis of “resolved” images of planetarysurfaces. Ultimately, the parameters of the model are the optical


Icarus 201 (2009) 69–83

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Quantitative compositional analysis of martian mafic regions using theMEx/OMEGA reflectance data1. Methodology, uncertainties and examples of applicationF. Poulet a,∗, J.-P. Bibring a, Y. Langevin a, J.F. Mustard b, N. Mangold c, M. Vincendon a, B. Gondet a, P. Pinet d,J.-M. Bardintzeff c,e, B. Platevoet c

a Institut d’Astrophysique Spatiale, Bâtiment 121, Université Paris-Sud, 91405 Orsay Cedex, Franceb Department of Geological Sciences, Brown University, Providence, RI 02912, USAc IDES, Bâtiment 509, Université Paris-Sud, 91405 Orsay Cedex, Franced Observatoire Midi-Pyrénées, 31400 Toulouse, Francee IUFM, Université de Cergy-Pontoise, RP 815, 78008 Versailles, France

a r t i c l e i n f o a b s t r a c t

Article history:Received 9 June 2008Revised 4 December 2008Accepted 11 December 2008Available online 31 December 2008

Keywords:Mars, surfaceMineralogyInfrared observationsSpectroscopy

The Mars Express Observatoire pour la Minéralogie, l’Eau, les Glaces et l’Activité (OMEGA) collected anunprecedented visible and near-infrared hyperspectral dataset covering the low albedo regions of Mars.We investigate the ability to infer modal abundance of surfaces of these regions from a radiative transfermodel developed by Shkuratov et al. [Shkuratov, Y., Starukhina, L., Hoffmann, H., Arnold, G., 1999. Icarus137, 235–246] and adapted to basaltic surfaces by Poulet and Erard [Poulet F., Erard, S., 2004. J. Geophys.Res. 109 (E2), doi:10.1029/2003JE002179]. From OMEGA measurements of mafic surfaces, we developseveral sensitivity tests to assess the extent to which the model can be applied to predict pyroxenecomposition (high-calcium phase and low-calcium phase), abundance of almost neutral components(plagioclase) in the near-infrared wavelength as well as grain sizes, by using a library of selected end-members. Results of the sensitivity tests indicate that the scattering model can estimate both abundancesand grain sizes of major basaltic materials of low albedo regions within uncertainties (±5 to 15 vol%).The model is then applied to data from dissected cratered terrains located in Terra Meridiani. The derivedgrain size including uncertainties is in the 50–500 µm range. This is consistent with the thermal inertiaand albedo of this region, which indicates a fine sand-sized surface with little dust. The abundances ofplagioclase (43–57%) and pyroxenes (35–45 ± 10%, including 11±5% of low-calcium phase) are in goodagreement with previous basalt-like compositions of low albedo regions from thermal infrared spectralmeasurements. The method presented in this paper will provide a valuable tool for evaluating the modalmineralogy of other mafic regions of Mars observed in the near-infrared wavelength range.

© 2008 Elsevier Inc. All rights reserved.

1. Introduction

The Observatoire pour la Minéralogie, l’Eau, les Glaces et l’Activ-ité (OMEGA) experiment is designed to study the composition andphysical properties of the martian surface thanks to Visible/Near-InfraRed (VNIR) reflectance measurements (Bibring et al., 2004).Using classical methods of spectral identification (spectral param-eter, Modified Gaussian Model, linear mixing), OMEGA has pro-vided consistent identification and spatial distribution of severalclasses of mafic minerals (Mustard et al., 2005; Bibring et al., 2005;Poulet et al., 2007). The Noachian crust is enriched in low-Ca py-roxene, with respect to more recent lavas flows in which high-Ca pyroxene dominates, whereas high relative concentration of

* Corresponding author.E-mail address: [email protected] (F. Poulet).

olivine is observed in dunes and eroded layers corresponding toancient lava flows or melt ejecta. However, only limited petrologyhas yet been derived and no quantitative information on miner-alogical and geochemical compositions was derived from OMEGAmeasurements. Previous remote sensing and in situ measurementsindicate that the mafic regions are dominantly basaltic, com-posed mostly of feldspar and pyroxene (Bandfield et al., 2000;Hamilton et al., 2001; McSween et al., 2004, 2006a, 2006b). Twomajor divisions in crustal composition were recognized on the ba-sis of their Thermal InfraRed (TIR) spectral signatures obtained bythe Mars Global Surveyor/Thermal Emission Spectrometer (TES) in-strument (Bandfield et al., 2000): (1) surface Type 1, which occursprimarily in the southern highlands, and (2) surface Type 2, whichis found primarily in the northern lowlands. Recently, a reassess-ment of the TES data by Rogers and Christensen (2007) indicatesthat mineralogical diversity of the low albedo regions occurs at

0019-1035/$ – see front matter © 2008 Elsevier Inc. All rights reserved.doi:10.1016/j.icarus.2008.12.025

[Icarus, 2009]Quantitative compositional analysis of martian mafic regions 71

Table 1Spectral library. Minerals are from different libraries (RELAB-Brown, SPECLIB-JPL,USGS) with the following exception of olivine fayalite and martian dust. Bold textdesignates end-members that comprise the minerals predominantly used in the fit-ting procedure. They define the so-called skeleton library. Analytical techniques areindicated when available.

Name group End-member Reference (library name, sample name,other information when available)a

Clinopyroxene Diopside (HCP) RELAB, PP-CMP-027High-calcium pyroxene from 1801volcanic bomb for “soil” mixtureexperimentWO46, ENS46, FS9 from microprobeanalyses

Augite USGS, NMNH120049WO34, ENS45, FS21 from XRD analyses

Pigeonite (LCP) RELAB, PP-EAC-042Low-calcium pyroxene frommicroprobes, wet chemistry, and XRF

Feldspar Labradorite SPECLIB, ts02bNa0.34, K0.01, Ca0.65, Si2.35, Al1.65, O8from XRD

Albite RELAB, PA-CMP-005Olivine Forsterite RELAB, PO-CMP-032

Fo90Fayalite Collected and analyzed by B. Platevoet

Fa99 from microprobe analysesOxides Magnetite USGS, HS78

Fe3O4 from XRDIlmenite USGS, HS231

FeTiO3 from XRDMartian dust Dust See text

a RELAB: Reflectance Experiment Laboratory; ASTER-SPECLIB: Advanced Space-borne Thermal Emission and Reflection Radiometer-Spectral library; USGS: US Geo-logical Survey.

and imaginary indexes can be calculated from laboratory mea-surements of their transmission spectra (Roush et al., 2007, andreferences herein). However, such experimental methods are hardto implement and cannot be applied for some end-members suchas the martian dust since it is a complex mixture of variousminerals of poorly known composition. Empirical methods existto numerically estimate the imaginary index of the constituentof a granular sample from its reflectance spectrum (Lucey, 1998;Shkuratov et al., 1999; Poulet and Erard, 2004; Roush et al., 2007).Using the invertible property of the Shkuratov model, the meanoptical properties of any material can be found if the a prioriknowledge of the real index (which may be considered constantin the 1–2.5 µm range) and the effective optical path length areknown (Shkuratov et al., 1999). This method, already used to de-rive optical constants of some mafic minerals in Poulet and Erard(2004), was successfully compared to other methods for materi-als with imaginary index values relatively low (Roush et al., 2007).Here, it is applied to laboratory reflectance spectra of mineral sam-ples well characterized in terms of composition and grain size(Table 1). The derived imaginary index values for the dark ox-ides (magnetite and ilmenite) are ranged between 0.001 and 0.01,which is much smaller than the measurements obtained from in-dependent measurements (Huffman, 1977). Using the Huffman’svalues would lower the abundances of the opaque oxide miner-als, which are expected to be of the order of a few percents atmost, but it should not affect the abundances of other materials.

The composition of the dust is not well known, so that we de-cided to include it in our modeling as an empirical end-member,whom the optical constants are calculated from OMEGA observa-tions. The reflectance spectrum we use for the martian dust is anaverage over several OMEGA pixels of several orbits that cover uni-form bright regions located in the northern hemisphere of Mars.The calculation of imaginary index for the dust component wasdone assuming an effective optical path of 5 µm. This value islarger than the upper limit of the grain size distribution of ae-

olian surface sediments on Mars, as inferred from Viking Landerdata and Pathfinder (Tomasko et al., 1999). However, considering alower value is excluded because it will break the principle of thegeometrical optics of the model.

A comparison of the spectral signatures of the end-membersis shown in Fig. 1A. The spectra are computational results. Thecalculation of the plagioclase spectrum incorporates inclusions ofmagnetite using effective medium theory (see Section 2.3). Thegrain size of a mineral modifies the absorption band depth (Fig. 1)and thus the detection limit and the relative abundance of the ma-terial in the mixture. The existence of an optimal particle size foreach absorption feature for a specific mineral of a given texturethat maximizes its band depth has been observed by many work-ers (e.g., Clark and Lucey, 1984; Hapke, 1993). For mafic material,the optimal particle size ranges between a few tens and a few hun-dreds of micrometers (Harloff and Arnold, 2001). The sensitivity tothe grain size will be estimated (see Section 3.2).

We note that our spectral library contains a rather small num-ber of end-members in comparison to other quantitative meth-ods, especially linear deconvolution (e.g., Hamilton et al., 2001;Combe et al., 2008). However, the major spectral types of mate-rial responsible for the shape and intensity of the NIR spectra ofbasalts and their derived products are represented:

(1) A first type of materials exhibiting marked absorption bands(pyroxenes, olivines). The 1- and 2-µm absorption bands ofthe pyroxenes result from crystal field transitions of iron inoctahedral coordination. The positions of these absorptionsvary systematically as a function of composition and crys-tal structure (Adams, 1974; Cloutis and Gaffey, 1991; Schadeet al., 2004). The presence of calcium, iron, and magnesiumin the crystallographic sites affects the locations and shapeof these distinct absorptions. The small number of pyroxeneend-members used in our study may be not representativeof the martian crust. In the presence of mixtures of pyrox-ene of intermediate compositions, it could be very difficult toreliably isolate the end-member components. However, anal-yses of OMEGA spectra using the Modified Gaussian Model(MGM) developed by Sunshine et al. (1990) shows that allaverage band positions and uncertainties for OMEGA spectraplotted within the range of the two major single phase pyrox-enes, namely Low-Calcium Pyroxene (LCP) and High-CalciumPyroxene (HCP) (Kanner et al., 2007). For this study, we arethus interested in calcium and iron substitutions, which areknown to have the most significant effects on the NIR bandposition by systematically shifting the 1- and 2-µm featuresto longer wavelengths with increasing abundance (Cloutis andGaffey, 1991).

(2) A second type corresponding to almost neutral (NIR spec-trally featureless) minerals, such as plagioclases; the pres-ence of such minerals plays a strong role by controlling theoverall albedo and by reducing the spectral contrasts due toabsorption bands of mafic minerals (see Section 3.1). How-ever, within this type, our analysis cannot discriminate be-tween plagioclase and other potential candidates such as somehigh-silica anhydrous phases. Some TES analyses revealed thathigh-silica phase(s) could be one of the significant mineralgroups for many low albedo regions. Primary high-silica vol-canic glasses are potentially indistinguishable from some sec-ondary amorphous or poorly crystalline high-silica materialsin the thermal infrared spectral region (Bandfield, 2002; Wyattand McSween, 2002; Kraft et al., 2003; Michalski et al., 2003;Morris et al., 2003; Ruff, 2004; Koeppen and Hamilton, 2005),which complicates the interpretation of high-silica phase(s)detected by the TES instrument. OMEGA detects hydrated min-erals including phyllosilicates in some specific geological units

Spectral Endmembers

Reflectance spectra are obtained from existing libraries and inverted to estimate values of k.

This requires knowledge of particle size!


72 F. Poulet et al. / Icarus 201 (2009) 69–83

Fig. 1. (A) Synthetic spectra of the major end-members used in the modeling. (B) Synthetic spectra of the LCP and HCP minerals for two different grain sizes. (C) Same as Bbut for forsterite and three grain sizes (10, 100 and 1000 µm). (D) Same as C but for fayalite.

that are disconnected from the low albedo mafic-rich regions(Poulet et al., 2005). Given the strong and unambiguous signa-tures of the phyllosilicates in the NIR, we can therefore excludethem from the spectral library with confidence. Actually, itis possible that the phyllosilicate abundances in the TES datacould be misrepresented if phyllosilicates on the martian sur-face have a spectral contrast much lower or higher than theaverage solid phyllosilicates in the Arizona State University li-brary (Rogers and Christensen, 2007). High spatial and spectralresolution reflectance data acquired by the Mars Reconnais-sance Orbiter Compact Reconnaissance Imaging Spectrometerfor Mars reveal the presence of H2O- and SiOH-bearing phaseson the martian surface, but they are found in very localized ar-eas, typically in finely-stratified deposits exposed on the floorof and on the plains surrounding the Valles Marineris canyonsystem (Milliken et al., 2008). The other high-silica phases arespectrally featureless in the NIR, so that they could be alsopresent and/or replace the role of the plagioclase component.However, we choose not to include high-silica end-members inour spectral library, because the nature and the abundance ofhigh-silica phases (from ∼10 to 30% depending on the anal-yses, e.g., McSween et al., 2003) are still controversial andbecause the significant abundance is mainly found in the TEStype II surface of the northern hemisphere that will not beconsidered in our further studies.

(3) A third one made of strongly absorbing minerals, such as ironoxides. In terrestrial basaltic materials, they are in general em-bedded in the plagioclase matrix, and their presence results ina decrease in both band depth and overall reflectance (Harloffand Arnold, 2001). These minerals (magnetite, ilmenite) areprimarily associated with basaltic soils and rocks on Mars (e.g.,Morris et al., 2006).

Further examples presented in the next sections will demon-strate that the end-members listed in Table 1 can satisfactorilyreproduce the spectra of the low albedo regions.

2.3. Choice of the end-type of surface

Assuming a homogeneous unconsolidated surface, many lowalbedo regions have a thermal inertia consistent with sand-sizedparticles (Palluconi and Kieffer, 1981; Edgett and Christensen,1991; Rogers and Christensen, 2003). One specific mixture canbe therefore considered: intimate or “salt-and-pepper” mixture inwhich the single scattering albedo in the scattering models is cal-culated for each component grain and then averaged (Poulet etal., 2002). However, numerical and laboratory experiments showthat an intimate mixture of the different pure components usuallypresent in basalt (plagioclase, pyroxenes, olivine, oxide) failed toreproduce the spectral characteristics of a basaltic sample (Pouletand Erard, 2004, and references therein). This problem is related tothe fact that the dominant minerals present in basalts are brightin the NIR. The color and darkness of basaltic rocks are stronglycontrolled by the proportion of iron oxides, especially magnetite(e.g., Faye and Miller, 1973). A successful modeling of basalt wasderived using another kind of mixture (intramixture) to consider:“dark basalt” where plagioclase grains are associated with inclu-sions of dark iron oxides (Poulet and Erard, 2004). When mixedwith the other major minerals present in basalt, these types ofplagioclase grains, whose scattering properties are calculated byeffective medium theories (Bohren and Huffman, 1983) can simu-late the reflectance spectra of basalts (Poulet and Erard, 2004).

3. Estimate of the abundance uncertainty

3.1. Sensitivity to the end-member

The formal error in mineral abundance derived using NIR spec-troscopic measurements under optimum laboratory conditions hasbeen estimated to be 5–10% (Poulet and Erard, 2004). This un-certainty is different for each mineral depending on the strength,width, and position of the spectral features, as well as the spec-tral properties of the other minerals present. Olivines and pyrox-enes have strong absorption bands in the wavelength range underconsideration and their abundances are well derived (<10%), espe-

The derived optical constants can then be used to forward-calculate what the reflectance spectra should look like for different grain sizes.

[Poulet et al., 2009]

Radiative Transfer Models for Reflectance Spectra 4Quantitative compositional analysis of martian mafic regions 73

Fig. 2. Effects of the admixtures. (A) Synthetic spectra of intimate mixtures of HCP mixed with varying concentrations of plagioclase (from 0 to 80% by step of 20%).A spectrum extracted from the mafic-rich DCT terrains is shown in red for comparison. (B) Same as (A) but with LCP instead of HCP. (C) Same as (A) but with olivine (Fo90)instead of HCP. (D) Synthetic spectra of three mixings of olivine (Fo90) with HCP in different proportions (0, 20, 40). The grain size is 50 µm for each mineral.

cially for grain size larger than several tens of µm. Plagioclase isspectrally flat in NIR, making it difficult to estimate relative abun-dance with an uncertainty better than 10%. These uncertaintieslikely increase on Mars owing to the lower signal to noise ratio ofOMEGA compared to laboratory instruments and the contributionof the atmospheric components. However, the presence of darkplagioclase plays a strong role by controlling the overall albedoand by reducing the spectral contrasts due to marked electronicabsorption bands of mafic minerals (Fig. 2). The shape and theband depths of the OMEGA spectra of mafic-rich terrains are notconsistent with those of spectra of pure mafic minerals (Fig. 2A),demonstrating that other nonmafic components such as plagioclaseare likely present. As mentioned previously, the TES spectra of thelow albedo terrains exhibit the signatures of plagioclase, which re-inforces our confidence to include this mineral as an end-membercapable to adjust the shallow mafic bands. Of special interest is theeffect of pyroxene when mixed with olivine (Fig. 2D). The presenceof pyroxene can be detected thanks to the 2 µm band, which is im-portant for petrological consideration.

Both HCP and LCP are identified in the surface composition ofthe low albedo terrains (Bibring et al., 2005). The presence of LCPwas not supported by the original TES analyses because this min-eral was modeled at best at the traditional TES detection limits(10–15%; Christensen et al., 2000). A test of the presence of thismineral and the inferred abundance is studied from one spectrumextracted from the DCT unit. The model fit is highly degraded(Fig. 3) when low-calcium pyroxene is removed as a component,demonstrating the high sensitivity of the model to this mineral.1

1 Small noticeable misfits between measured and best-fit spectra are observedat the shortest and longest wavelengths of the OMEGA data. Their causes are notwell understood but may result from an instrumental effect (small time-dependentdeviations from linearity that depends on the flux received by the instrument), at-mospheric effects, and/or precise end-member selection. But these discrepancies arestill inside the instrumental error bars of 2%.

One of the applications of this work is to determine the el-emental composition from modal mineralogy (companion paper).The mineral chemistry of the end-members is therefore essentialfor determining the petrogenesis of the pyroxene-bearing rocks.Moreover, most of SNC (shergottites, nakhlites, and chassignites)meteorites, which are assumed to have originated from Mars (e.g.,Treiman, 1995), exhibit diverse mineralogy including the two majorsingle phase pyroxenes HCP and LCP. The major LCP phase detectedin the SNCs is pigeonite (a part of the spectral library); their HCPphase is more often augite rather than diopside used in the previ-ous modeling of the DCT spectrum. Here we investigate the abilityto refine pyroxene composition by taking augite as the HCP end-member (Table 1). First-order examination of the spectral proper-ties of augite with the same particle size than diopside reveals twomain absorption bands similar to those of the diopside (Fig. 1).However, two small differences exist: (1) the band positions ofaugite are slightly shifted to shorter wavelengths by a few tensof nm, which is consistent with the lower Ca content in augite,(2) the band depths are more pronounced for augite (with a 2-µmband depth equal to 14% in comparison to 9% for diopside), whichtranslates to a larger absorption coefficient. When augite is usedas an HCP end-member, a smaller value of the HCP abundance isthen found in the modeling of the DCT terrain (Fig. 3, Table 2).However, we observe that the discrepancies are within the uncer-tainties and the ratio LCP/(HCP+HCP) is roughly the same. To bet-ter evaluate the numerical uncertainty associated with the choiceof the HCP end-member, the same sensitivity test is extended toseveral thousand spectra extracted from the DCT unit. Both HCPend-members provide acceptable solutions with similar low valuesof RMS. The effect of replacing diopside by augite on the modalmineralogy is shown in Fig. 4. The abundance of HCP decreasesby ∼2.5% when using augite, while the abundances of all otherend-members remain the same. The ratio LCP/(HCP + LCP) variesslightly: 0.29 ± 0.05 for augite in comparison to 0.25 ± 0.07 fordiopside. This difference is within the uncertainties of the model.

Quantitative compositional analysis of martian mafic regions 73

Fig. 2. Effects of the admixtures. (A) Synthetic spectra of intimate mixtures of HCP mixed with varying concentrations of plagioclase (from 0 to 80% by step of 20%).A spectrum extracted from the mafic-rich DCT terrains is shown in red for comparison. (B) Same as (A) but with LCP instead of HCP. (C) Same as (A) but with olivine (Fo90)instead of HCP. (D) Synthetic spectra of three mixings of olivine (Fo90) with HCP in different proportions (0, 20, 40). The grain size is 50 µm for each mineral.

cially for grain size larger than several tens of µm. Plagioclase isspectrally flat in NIR, making it difficult to estimate relative abun-dance with an uncertainty better than 10%. These uncertaintieslikely increase on Mars owing to the lower signal to noise ratio ofOMEGA compared to laboratory instruments and the contributionof the atmospheric components. However, the presence of darkplagioclase plays a strong role by controlling the overall albedoand by reducing the spectral contrasts due to marked electronicabsorption bands of mafic minerals (Fig. 2). The shape and theband depths of the OMEGA spectra of mafic-rich terrains are notconsistent with those of spectra of pure mafic minerals (Fig. 2A),demonstrating that other nonmafic components such as plagioclaseare likely present. As mentioned previously, the TES spectra of thelow albedo terrains exhibit the signatures of plagioclase, which re-inforces our confidence to include this mineral as an end-membercapable to adjust the shallow mafic bands. Of special interest is theeffect of pyroxene when mixed with olivine (Fig. 2D). The presenceof pyroxene can be detected thanks to the 2 µm band, which is im-portant for petrological consideration.

Both HCP and LCP are identified in the surface composition ofthe low albedo terrains (Bibring et al., 2005). The presence of LCPwas not supported by the original TES analyses because this min-eral was modeled at best at the traditional TES detection limits(10–15%; Christensen et al., 2000). A test of the presence of thismineral and the inferred abundance is studied from one spectrumextracted from the DCT unit. The model fit is highly degraded(Fig. 3) when low-calcium pyroxene is removed as a component,demonstrating the high sensitivity of the model to this mineral.1

1 Small noticeable misfits between measured and best-fit spectra are observedat the shortest and longest wavelengths of the OMEGA data. Their causes are notwell understood but may result from an instrumental effect (small time-dependentdeviations from linearity that depends on the flux received by the instrument), at-mospheric effects, and/or precise end-member selection. But these discrepancies arestill inside the instrumental error bars of 2%.

One of the applications of this work is to determine the el-emental composition from modal mineralogy (companion paper).The mineral chemistry of the end-members is therefore essentialfor determining the petrogenesis of the pyroxene-bearing rocks.Moreover, most of SNC (shergottites, nakhlites, and chassignites)meteorites, which are assumed to have originated from Mars (e.g.,Treiman, 1995), exhibit diverse mineralogy including the two majorsingle phase pyroxenes HCP and LCP. The major LCP phase detectedin the SNCs is pigeonite (a part of the spectral library); their HCPphase is more often augite rather than diopside used in the previ-ous modeling of the DCT spectrum. Here we investigate the abilityto refine pyroxene composition by taking augite as the HCP end-member (Table 1). First-order examination of the spectral proper-ties of augite with the same particle size than diopside reveals twomain absorption bands similar to those of the diopside (Fig. 1).However, two small differences exist: (1) the band positions ofaugite are slightly shifted to shorter wavelengths by a few tensof nm, which is consistent with the lower Ca content in augite,(2) the band depths are more pronounced for augite (with a 2-µmband depth equal to 14% in comparison to 9% for diopside), whichtranslates to a larger absorption coefficient. When augite is usedas an HCP end-member, a smaller value of the HCP abundance isthen found in the modeling of the DCT terrain (Fig. 3, Table 2).However, we observe that the discrepancies are within the uncer-tainties and the ratio LCP/(HCP+HCP) is roughly the same. To bet-ter evaluate the numerical uncertainty associated with the choiceof the HCP end-member, the same sensitivity test is extended toseveral thousand spectra extracted from the DCT unit. Both HCPend-members provide acceptable solutions with similar low valuesof RMS. The effect of replacing diopside by augite on the modalmineralogy is shown in Fig. 4. The abundance of HCP decreasesby ∼2.5% when using augite, while the abundances of all otherend-members remain the same. The ratio LCP/(HCP + LCP) variesslightly: 0.29 ± 0.05 for augite in comparison to 0.25 ± 0.07 fordiopside. This difference is within the uncertainties of the model.

Radiative Transfer Models for Reflectance Spectra 574 F. Poulet et al. / Icarus 201 (2009) 69–83

Fig. 3. Modeling of a spectrum extracted from the DCT unit of Terra Meridiani.(A) Spectra of the major end-members used in the fit procedures. (B) OMEGAspectrum (black line) compared to two best fit models (red and blue lines) anda model-derived spectrum (green line) for which the LCP component was excluded.The best fits in red and blue includes diopside and augite respectively as the HCPend-member. The values of the RMS are indicated.

Table 2Optimized parameters for the modeling of the DCT spectrum in using two kinds ofHCP mineral (see Fig. 3). The values of the ratio LCP/(LCP + HCP) are indicated.

Diopside Augite

Grain size(µm)

Abundance(%)

Grain size(µm)

Abundance(%)

Dust 5 0 5 0Olivine – 0 – 0HCP 150 33 100 27LCP 60 18 75 16PLG 150 49 70 53

RMS 0.21% 0.20%LCP/(LCP + HCP) 0.35 0.37

Consequently, although the HCP phase cannot be uniquely derivedfrom the remote measurements of OMEGA, the modal compositionusing major end-members of different phases (LCP, HCP, plagio-clase and dust) can be assessed.

3.2. Sensitivity to the grain size

In the tests done on laboratory basaltic samples, the inferredgrain sizes are accurate within a factor of 2, which was consideredto be enough to discriminate between large categories of grains(10s, 100s, 1000s µm) and therefore between formation processes

Fig. 4. Average modal abundances inferred from two simulations of 3000 spectraof DCT terrains. The two simulations differ in the HCP end-members (diamond:diopside, triangle: augite). The average RMS values are the same (0.27%).

(Poulet and Erard, 2004). However, grain size is an important pa-rameter that controls the shape and the depth of any absorption asmentioned previously and it therefore merits a special attention.The primary and spectrally neutral mineral plagioclase is actuallyexpected to be more sensitive to this parameter than any othermineral, because increasing the grain size (i.e. absorption) couldbe compensated by a decrease of the abundance of plagioclase,raising the question about the uniqueness of the final solution.The effect of grain size can be illustrated using models in whichall end-members have identical grain size, the optimization beingmade on this size with their relative abundance as a free param-eter (Fig. 5). First, it is important to note that the model fails toreproduce the pyroxene bands if all the grain sizes are forced tobe smaller than 50 µm or larger than 200 µm. This shows thatthe best fit of the grain size of the pyroxene phases is in the50–200 µm range. Second, variations of 5–10% are seen in the pla-gioclase abundance when the grain sizes vary from 100 to 200 µm.The abundances of the pyroxenes are also affected within 5% of thebest fit value, although the RMS values smaller than 0.30% are stillacceptable.

The purpose of the next set of numerical experiments is to fur-ther examine the effect of the grain size of plagioclase on the finalsolution and to determine the uncertainties on the inferred abun-dance of minerals. Fig. 6 shows the result of the fitting procedureof the DCT spectrum: the grain size of plagioclase is forced to be10, 100, 500 and 1000 µm while keeping free all the other pa-rameters. We remind that magnetite inclusions are embedded inthe plagioclase, but the content of this oxide mineral is the samefor each calculation. The model fails to reproduce the data spec-trum if the grain sizes are too large (500 and 1000 µm, cyan andgreen spectra respectively), because such large grain sizes reducetoo strongly the pyroxene bands. When the grain size is set atsmaller values than the best fit value of 150 µm (blue spectrum),the model-derived spectra (red and orange spectra) are satisfac-tory, but the RMS is significantly larger by about 30% than theRMS obtained for the best fit. Despite the NIR spectrally featurelesscharacter of plagioclase, these simulations show that its grain size,and thus its abundance, are constrained, with grain size within arange of a few tens to a few hundreds of micrometers and abun-dance within 40–60% range.

3.3. Sensitivity to the initial conditions

Due to the nonlinear formulation of the Shkuratov theory, theinversion problem has to be solved using an iterative approach.A downhill simplex technique is used here to find the RMS residual

Quantitative compositional analysis of martian mafic regions 75

Fig. 5. Sensitivity to the grain size. (A) The OMEGA spectrum (black line) is com-pared to three models. The best fit (red line) is obtained by assuming all theparameters free (see Table 2 for final results) and was already shown in Fig. 3. Ifthe grain size is forced to be 100 (blue spectrum) or 200 µm (green spectrum), thefits are slightly degraded. (B) The abundances of three major minerals correspondto the simulations shown in (A).

between measured spectrum and computed spectrum (Press et al.,2002). This optimization procedure depends on the initial condi-tions. We design another sensitivity test to discuss the effect of theinput parameters on the validity of the inferred modal mineralogy.We vary the starting grain size and abundance parameters for eachend-member and assess the consistency of resultant grain size andabundance solutions. The test is done on a large number of spectraextracted from the DCT unit of Terra Meridiani in order to betterevaluate the uncertainties. Comparison between the two simula-tions shows that the final average values of the abundances varyonly slightly with the values of the starting parameters (Fig. 7):28 ± 3% versus 28 ± 5% for HCP, 10 ± 4% versus 12 ± 3% for LCPand 52 ± 4% versus 47 ± 7% for plagioclase. Of special interest isthe increase of pyroxene abundances versus the pyroxene banddepth value. This expected compositional trend makes us confi-dent on the methodology and the values of the abundances. Theplagioclase abundance is less robust but the distributions and theaveraged values of abundances are still consistent within the un-certainties of the method (∼10%). By contrast, the initial conditionssignificantly affect the grain size parameter. As the input grain sizeparameter decreases or increases, the resultant solutions similarlyshifted to the same part of the parameter space. For instance, thefinal grain sizes of HCP are close to the initial value of 100 µm forsimulation 1, while the grain sizes of the simulation 2 that startedwith a 300 µm value, are ranged between 200 and 500 µm. Thisreveals that the starting parameters do have a significant effect on

Fig. 6. Effect of the grain size of plagioclase. (A) Data spectrum compared to fourmodel-derived spectra for which the size of plagioclase grains is fixed to 10 µm (redspectrum), 100 µm (orange), 500 (cyan), and 1000 µm (green). The blue spectrumindicates the best fit when all the parameters including the plagioclase grain sizeare free. In this case, the grain size of plagioclase is a free parameter. (B) Abundanceof the different minerals for the five fitting procedures shown in (A). RMS is con-sidered to be acceptable when it is smaller than 0.30%. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version ofthis article.)

the resultant solutions whatever the mineral. However, the rangesof the final acceptable solutions can be evaluated. Most of the finalvalues of the plagioclase grain size clearly plot in the 50–150 µmrange for the two simulations. The same trend is observed for theLCP component. The HCP grain size shows large standard deviation,but still in the range of a few 10s to a few 100s of micrometers asdiscussed previously.

The simplex algorithm calculates thousands of possible mix-tures before reaching a final solution. In order to illustrate therange of values considered in the fitting effort and provide anestimate of the uncertainty in the final results, we plot the RMSversus component abundances and diameters for several represen-tative pyroxene band depths (0.01, 0.02, 0.03, 0.04, 0.05 and 0.06)in Fig. 8. Apart from the grain size of the martian dust compo-nent that is initially fixed, the grain sizes are ranged in the 10sto a few 100s µm, which confirms the previous estimates. The pa-rameter space of the abundances is 40–55% for plagioclase, 25–35%for HCP, 10–15% for LCP, 5–10% for the dust, and less than 5–10%for olivine. These variations are also in good agreement with theuncertainties previously determined. For a given spectrum and forvalues of RMS lower than 0.35%, the excursions of the parameterare even smaller.

3.4. Sensitivity to the olivine mineral

Olivine has strong absorption bands in the wavelength rangeunder consideration, and its abundance should be well constrained.


Fig. 9. (A) Spectra of end-members used for the fits shown in (B). The olivine is forsterite with grain size forced to be 10 µm. (B) Spectrum and best fits without olivine (red),with an olivine abundance forced to be 10% (green) and 15% (blue). Grain size and abundance are indicated for each end-member. Values in parenthesis correspond to thebest fit (red spectrum) and are compared to the simulation with 10% of olivine. The arrow indicates the location where the spectrum is the most affected by the presence ofolivine. (C) Same as (A) but the grain size is here forced to be 100 µm. (D) Spectrum and best fits without olivine (red) and with olivine at 5% (blue).

Fig. 10. Two OMEGA observations of a dark terrain (354.7◦ E, 2.63◦ S) close tothe Opportunity landing site (solid lines) with similar photometric angles (nadirpointing, solar incidence angle of ∼50◦) but different amounts of dust in the at-mosphere (dustier conditions in black). Aerosol optical depths tau, interpolatedat 0.926 µm from Pancam measurements, are indicated (Lemmon et al., 2004;Lemmon, 2006). The surface spectra retrieved from these observations (dashedlines) are similar, which further validates our approach.

therefore applied to the OMEGA data cubes over Terra Meridianibecause of the proximity of the MER-B that measures the aerosoloptical depth thanks to Pancam observations. A test was done in

using two overlapping observations of Terra Meridiani obtainedwith different amounts of dust in the atmosphere (Fig. 10). Afterremoving the aerosol contribution, the derived surface spectra ofthe same region are similar, making us confident in the accuracyof our aerosol correction method. Aerosols are characterized by adecreasing continuum slope between 1 and 2.5 µm. The removalof their contribution increases the spectral signatures and modifiesthe general slope of the spectra (Fig. 11).

We now aim to estimate the uncertainties on the modal min-eralogy resulting from the aerosols contribution. For this investiga-tion, we model several thousands of aerosol-cleaned OMEGA spec-tra, whom the inferred mineralogy was previously derived fromuncorrected spectra (see simulation 1 of Fig. 7). Small differencesbetween the abundances and the grain size distributions exist, butthey are all within the derived uncertainty of the method (Fig. 12):29 ± 4% in comparison to 28 ± 3% for the HCP component, 11 ± 4%versus 10 ± 4% for LCP and 50 ± 6% versus 52 ± 4% for plagioclaseThis sensitivity test indicates that the application of our spectralmodel to OMEGA spectra uncorrected for the aerosol contributionproduces reliable values.

4. Application of the method to the DCT unit of Terra Meridiani

4.1. Derived mineralogy

We use the spectral model on OMEGA reflectance spectra ex-tracted from a part of the mafic-rich regions observed in TerraMeridiani, also referred as DCT unit (Arvidson et al., 2003). De-tailed mapping of the Terra Meridiani region shows that different


Fig. 6. Spectra extracted from different low albedo regions compared to their best fit model in red. Each spectrum corresponds to one OMEGA pixel. Left, from top to bottom:Syrtis Major Lavas (SML), Hesperia Planum (HP), Nili Fossae olivine-rich terrain (NFO), Echus Chasma (EC), Terra Tyrrhena (TT). Right, from top to bottom: Hesperia Planum(HP), LCP-rich outcrop (LCP1), olivine-rich dunes in Nili Patera (SMN2), Solis Planum (SL), Terra Tyrrhena (TT). For TT and HP, two spectra are shown to emphasize thespectral diversity of these regions. The RMS values are in the 0.11–0.28% range depending on the spectrum.

opal). Some of the TES analyses found that the total abundancesof these phases divide into 35% plagioclase and 15–20% high sil-ica phases. Given that the nature of the high-silica phases is still

controversial (see discussion in Poulet et al., 2009), we neverthe-less consider that our assumption to compare the summed TESabundances of spectrally NIR featureless plagioclase and high-silica

Fig. 6. Spectra extracted from different low albedo regions compared to their best fit model in red. Each spectrum corresponds to one OMEGA pixel. Left, from top to bottom: Syrtis Major Lavas (SML), Hesperia Planum (HP), Nili Fossae olivine-rich terrain (NFO), Echus Chasma (EC), Terra Tyrrhena (TT). Right, from top to bottom: Hesperia Planum (HP), LCP-rich outcrop (LCP1), olivine-rich dunes in Nili Patera (SMN2), Solis Planum (SL), Terra Tyrrhena (TT). For TT and HP, two spectra are shown to emphasize the spectral diversity of these regions. The RMS values are in the 0.11–0.28% range depending on the spectrum.

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