Estimating mineral abundances of clay and gypsum mixtures … · 2013-10-27 · Spectroscopy...

Icarus 277 (2016) 171–186

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Icarus

journal homepage: www.elsevier.com/locate/icarus

Estimating mineral abundances of clay and gypsum mixtures using

radiative transfer models applied to visible-near infrared

reflectance spectra

K.M. Robertson

∗, R.E. Milliken, S. Li

Dept. Earth, Environmental and Planetary Sciences, Brown University, Providence, RI 02912, United States

a r t i c l e i n f o

Article history:

Received 21 September 2015

Revised 20 April 2016

Accepted 25 April 2016

Available online 11 May 2016

Keywords:

Mineralogy

Spectroscopy

Radiative transfer

Mars, surface

a b s t r a c t

Quantitative mineral abundances of lab derived clay–gypsum mixtures were estimated using a revised

Hapke VIS-NIR and Shkuratov radiative transfer model. Montmorillonite–gypsum mixtures were used to

test the effectiveness of the model in distinguishing between subtle differences in minor absorption fea-

tures that are diagnostic of mineralogy in the presence of strong H 2 O absorptions that are not always

diagnostic of distinct phases or mineral abundance. The optical constants ( k -values) for both endmem-

bers were determined from bi-directional reflectance spectra measured in RELAB as well as on an ASD

FieldSpec3 in a controlled laboratory setting. Multiple size fractions were measured in order to derive a

single k -value from optimization of the optical path length in the radiative transfer models. It is shown

that with careful experimental conditions, optical constants can be accurately determined from powdered

samples using a field spectrometer, consistent with previous studies. Variability in the montmorillonite

hydration level increased the uncertainties in the derived k -values, but estimated modal abundances for

the mixtures were still within 5% of the measured values. Results suggest that the Hapke model works

well in distinguishing between hydrated phases that have overlapping H 2 O absorptions and it is able to

detect gypsum and montmorillonite in these simple mixtures where they are present at levels of ∼10%.

Care must be taken however to derive k -values from a sample with appropriate H 2 O content relative to

the modeled spectra. These initial results are promising for the potential quantitative analysis of orbital

remote sensing data of hydrated minerals, including more complex clay and sulfate assemblages such as

mudstones examined by the Curiosity rover in Gale crater.

© 2016 Elsevier Inc. All rights reserved.

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. Introduction

Numerous occurrences of a wide variety of hydrated minerals

ave been identified on the surface of Mars through a combina-

ion of orbiter, lander and rover observations. Hydrated sulfates

nd clay minerals are the two dominant classes of hydrated miner-

ls, both being more common in ancient Noachian and Hesperian

errains than in younger Amazonian terrains ( Poulet et al., 2005;

endrin et al., 2005; Bibring et al., 2006; Murchie et al., 2009;

hlmann et al., 2011; Carter et al., 2013 ). These minerals and their

ost rocks may record global climatic conditions and aqueous pro-

esses during the first billion years of Mars’ history ( Bibring et al.,

005; Chevrier et al., 2007; Murchie et al., 2009; Ehlmann et al,

011 ) as well as discrete, localized aqueous conditions during later

imes (e.g., Mangold et al., 2010 ). However, in order to relate min-

∗ Corresponding author. Tel.: 401-863-9663.

E-mail address: [email protected] (K.M. Robertson).

m

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ttp://dx.doi.org/10.1016/j.icarus.2016.04.034

019-1035/© 2016 Elsevier Inc. All rights reserved.

ralogical observations to potential environmental conditions it is

mportant to identify the specific mineral species as well as their

elative, and preferably absolute, abundances.

Initial orbital observations from NASA’s Mars Reconnaissance

rbiter CRISM and ESA’s Mars Express OMEGA visible-near infrared

VIS-NIR) imaging spectrometers suggested sulfates and clays were

argely segregated both temporally and spatially (e.g., Bibring et al.,

006, Poulet et al., 2005 ). However, subsequent global surveys and

etailed local studies suggest there are numerous locations where

hese minerals coexist (e.g., Wray et al., 2010; Milliken et al., 2010;

oach et al., 2010; Ehlmann et al., 2011; Noe Dobrea et al., 2012;

arter et al., 2013 ), either in discrete geologic units or possibly

ixed below the ∼18 m/pixel spatial scale of CRISM. Sulfate and

lay assemblages are commonplace in many terrestrial settings

e.g., Baldridge et al., 2009 ), and it is not unexpected that the same

ay be true of Mars, particularly for alluvial/fluvial and lacustrine

nvironments or rocks that have been subject to diagenesis.

Indeed, recent results from the Curiosity rover at Gale

rater provide excellent examples of strata that host both clay

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172 K.M. Robertson et al. / Icarus 277 (2016) 171–186

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minerals and sulfates, where both are believed to have formed

in situ ( Grotzinger et al., 2014; Vaniman et al., 2014 ). CheMin X-

ray diffraction measurements of rock powders drilled from the

Sheepbed mudstone indicate as much as ∼22% smectite and ∼3.6%

Ca-sulfate (bassanite and anhydrite) may be present ( Vaniman

et al., 2014 ). However, the mudstone is crosscut by numerous veins

and nodules filled with Ca-sulfate ( Grotzinger et al., 2014 ), thus

sulfates are likely a much more volumetrically important compo-

nent of the unit as a whole than indicated solely by the XRD re-

sults. Intriguingly, no hydrated minerals were observed in CRISM

orbital data for this region, likely due to the thin dust cover on

the outcrops, though spectral signatures of both clay and sulfate

minerals are present in other lower Mt. Sharp strata in Gale crater

( Milliken et al., 2010 ). In the Sheepbed mudstone, the clays have

been interpreted to be authigenic and the crosscutting, sulfate-

filled veins are indicative of a later diagenetic event. Therefore,

knowledge of the relative abundance of these phases can be used

to assess the relative contribution of these two distinct processes,

and similar concepts may apply to other clay and sulfate-bearing

assemblages on Mars.

Therefore, methods that allow for absolute or relative estimates

of clay and sulfate abundance from VIS-NIR reflectance spectra

can help to constrain the type and extent of processes that may

be responsible for the co-occurrence of these minerals. Assessing

how abundances of clay and sulfate vary within a stratigraphic se-

quence can also provide important insight into temporal changes

in local water chemistry, water–rock interaction, and paleoclimate.

However, uniquely identifying individual components in mixtures

of hydrated minerals using VIS-NIR reflectance spectroscopy can

be complicated by overlapping or non-unique absorption features

as well as mixing between strong and weakly absorbing (spectrally

‘neutral’) components. Qualitative methods that rely solely on the

positions and/or widths of individual absorptions may be prone to

missing the presence of volumetrically subordinate (yet geologi-

cally important) components, oversimplification of mineral identi-

fications, or misidentification of minerals.

Hydrated salts (e.g., hydrated sulfates and chlorides) are an ex-

cellent example of the difficulties associated with qualitative spec-

tral analyses, as VIS-NIR reflectance spectra for some of these

minerals lack unique diagnostic absorptions. Examples of ‘polyhy-

drated’ sulfates on Mars (e.g., Gendrin et al., 2005; Bibring et al.,

2006; Murchie et al., 2009; Carter et al., 2013 ) could represent

complex mixtures of various sulfates/hydrated phases (e.g., Cloutis

et al., 2006 ) or they could represent a single sulfate phase for

which weaker diagnostic absorptions are masked or muted.

Spectral ‘unmixing’ of VIS-NIR reflectance spectra using radia-

tive transfer models (RTMs) can help to constrain the mineralogy

of complex assemblages through the simultaneous fitting of re-

flectance values at many wavelengths. RTMs are also advantageous

in that they can provide estimates of the abundance and particle

size of each component in a mixture. However, as noted in pre-

vious studies (e.g., Sklute et al., 2015 ), widespread application of

these models to spectra of planetary surfaces has been somewhat

limited, primarily due to the lack of accurate optical constants

(real, n, and imaginary, k, components of the complex index of

refraction) for appropriate minerals, which are required inputs to

the models. This is particularly true for clay and sulfate minerals,

and though optical constants at VIS-NIR wavelengths have been re-

ported for montmorillonite ( Roush, 2005 ), gypsum ( Roush et al.,

2007 ), bloedite, epsomite, hexahydrite ( Dalton and Pitman, 2012 ),

and select Fe-sulfates ( Pitman et al., 2014; Sklute et al., 2015 ), they

are not readily available for many other hydrated minerals that are

of importance to Mars (e.g., nontronite, saponite, chlorite, bassan-

ite, kieserite, etc.) or for phases with variable H 2 O content. In addi-

tion, the application of these optical constants in radiative transfer

models for well-controlled mineral mixtures has been very limited,

articularly for minerals relevant to sedimentary environments and

ompositions (e.g., clays, sulfates, and other hydrated phases).

Two commonly used models in remote sensing applications for

lanetary bodies are those of Hapke (2005) and Shkuratov et al.

1999) , both of which are considered in this study. Previous lab-

ratory studies have tested the Hapke RTM for igneous materials

ith applications to the Moon and asteroids ( Mustard and Pieters,

987, 1989; Li and Milliken, 2015 ). Of relevance to Mars, laboratory

ixtures of nontronite and various igneous materials ( Ehlmann

t al., 2011 ) were recently analyzed using both Hapke (2005) and

hkuratov et al. (1999) models, and Stack and Milliken (2015) also

sed the Hapke model for a suite of clay and Mg-sulfate (epsomite)

ixtures. In addition, Poulet et al. (2009) applied the Shkura-

ov model directly to OMEGA spectra of clay-bearing regions on

ars. This study yielded promising results, but a detailed and well-

ontrolled laboratory study of how the Hapke and Shkuratov mod-

ls perform for clay and sulfate mixtures when starting with op-

ical constants is currently lacking. Although optical constants for

ontmorillonite and gypsum are available in the literature and

oth phases have been observed on Mars, to our knowledge these

ptical constants have not yet been tested in RTMs for laboratory

ixtures of clay and sulfate. This is of course a necessary first

tep in order to understand and quantify potential uncertainties

n modal mineralogy derived from such models when applied to

ctual reflectance spectra of Mars.

In this study we assess the efficacy of the Hapke model in esti-

ating modal mineralogy of montmorillonite–gypsum mixtures at

IS-NIR wavelengths. We examine the effectiveness of the model

n distinguishing subtle differences in moderate or weak absorp-

ion features that are diagnostic of mineralogy when they are

n the presence of stronger, non-unique H 2 O absorptions, as typ-

fy many CRISM spectra of Mars. In addition, we derive indepen-

ent estimates of the optical constants for montmorillonite and

ypsum using two different spectrometers and compare our re-

ults to previously published values. The effectiveness of the ASD

ieldspec3 spectroradiometer is evaluated as a tool for perform-

ng Hapke modeling and is directly compared to the birectional re-

ectance spectrometer. We present estimates of modal mineralogy

erived from the Hapke RTM for a suite of binary mixtures that en-

ompass a range in gypsum-to-montmorillonite ratios and particle

izes, and as a final step we compare these results to those derived

rom an implementation of the Shkuratov model. Though sedimen-

ary rocks on Mars are likely composed of more than two min-

ral phases it is important to begin validation of RTMs with sim-

le, well-controlled cases where variables can be minimized and/or

solated. The results presented here for binary sulfate–clay mix-

ures can provide a foundation for future application of Hapke and

hkuratov models to more complex lab mixtures, and ultimately to

eflectance spectra of sedimentary deposits on Mars.

. Methods

.1. Sample preparation

The gypsum used in this study was coarse and poorly sorted

ypsum sand collected near White Sands, NM, and the montmoril-

onite was sample SAz-2 obtained from the Clay Minerals Society

CMS). Though the CMS offers several different montmorillonite

amples, the SAz-2 variety was chosen for this work because the

ulk material arrived as large pieces, which allowed us to easily

rind, sieve, and control the particle size of this component.

The gypsum sand and montmorillonite chips were ground in an

gate mortar and pestle, wet sieved with H 2 O to remove clinging

nes, and dried in air to make two separate groups of particle size

ractions ( Table 1 ). Group 1 consisted of four particle size ranges

25–32 μm, 38–45 μm, 63–75 μm and 125–150 μm) and each

K.M. Robertson et al. / Icarus 277 (2016) 171–186 173

Table 1

Particle sizes used in Groups 1 and 2 along with their respective measurements and analyses. The bulk of the Hapke modeling

was performed on Group 1 whereas Group 2 was used to establish an empirical relationship between measured and modeled

‘effective’ particle size.

Table 2

Comparison of measured and modeled ‘effective’ particle sizes for the different size fractions of gypsum and montmorillonite for both RELAB

BDR and ASD spectra.

Gypsum

size fraction From microscope k optimization Modeled results using average k (BDR) Modeled results using average k (ASD)

125–150 μm 145 138 135 n/a

63–75 μm 85 69 69 75

38–45 μm 46 44 37 46

25–32 μm 39 39 39 40

Montmorillonite

125–150 μm 140 50 55 n/a

63–75 μm 62 42 40 40

38–45 μm 38 33 25 27

25–32 μm 31 25 25 24

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ample was analyzed with two different VIS-NIR spectrometers

see Section 2.2 ). Additional size fractions, Group 2 ( Table 1 ), were

enerated for both gypsum ( < 5 μm, 90–106 μm, 106–125 μm,

80–212 μm, 212–250 μm, 250–300 μm, 300–500 μm) and mont-

orillonite ( < 5 μm, 150–180 μm, 180–250 μm, 250–300 μm, chip)

nd were analyzed with the ASD spectrometer. This second suite

f particle sizes was created to further compare measured particle

izes with those estimated from the models for the pure mineral

ndmembers.

The gypsum and montmorillonite endmembers for each size

raction (Groups 1 and 2) were analyzed with an Olympus opti-

al microscope in order to assess the actual average particle size

or comparison with the assumed average particle size based on

he sieve size range ( Table 2 ). Analysis was performed at 100X

agnification with a pixel width of 0.15 μm allowing for the re-

iable identification of particles as small as 3 μm. Individual par-

icles were measured with image processing software where the

verage of the smallest and longest axes was taken as the average

article size. In all cases, a minimum of 500 particles were mea-

ured for the particle size determination.

Aliquots of the pure gypsum and montmorillonite were

eighed using a microbalance and mixed together by stirring and

haking to produce binary mixtures with 0, 10, 25, 50, 75, 90, and

00 wt. % gypsum for each size fraction in Group 1. These mix-

ures were kept under ambient conditions in the lab and measured

ith both spectrometers using the same procedures as for the pure

ineral endmembers.

.2. Spectral measurements

Reflectance spectra for all mixtures were measured in the NASA

eflectance Experiment LABoratory (RELAB) at Brown University

sing a custom built bi-directional reflectance (BDR) spectrometer

hat utilizes a monochromator, various light sources, and a combi-

ation of UV-VIS-NIR detectors. Details of this instrument and ba-

ic measurement procedures can be found in Pieters (1983) and

he RELAB user manual, available on the RELAB website (cur-

ently hosted at http://www.planetary.brown.edu/relab/ ). All sam-

les were measured using the standard RELAB on-axis viewing ge-

metry of incidence angle ( i ) = 30 °, emergence angle ( e ) = 0 °, and

hase angle ( g ) = 30 ° from 0.35 μm to 2.6 μm. Sample dishes were

pun during measurement to average out potential effects due to

article orientation, and pressed Halon was used as the reflectance

tandard.

Four reflectance spectra of each sample were also acquired us-

ng an ASD FieldSpec3 portable spectroradiometer. Unlike the BDR

easurements, samples were not spun during the ASD measure-

ents. Repeat measurements of each sample thus allowed us to

valuate potential uncertainties associated with sample hetero-

eneity, packing, particle orientation, and scattering effects. A fiber

http://www.planetary.brown.edu/relab/


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optic light source with a QTH bulb was used to illuminate the

sample and both the illuminating and receiving fiber optic bun-

dles have a field of view of ∼25 °, which is much larger than the

≤ 6 ° emergence angle for the BDR instrument. The fibers were po-

sitioned such that the central angle of each matched the viewing

geometry of the BDR measurements ( i = 30 °, e = 0 °, g = 30 °), over a

wavelength range of 0.35–2.5 μm allowing for a more direct com-

parison between the two types of spectral measurements. All sam-

ples were optically thick and the typical spot size was on the order

of 4–6 mm. A small Spectralon

® disk from Labsphere was used as

a reflectance standard.

2.3. Radiative transfer model

The implementation of the Hapke (2005) and Shkuratov et al.

(1999) radiative transfer models used in this study were devel-

oped by Li and Li (2011) and Li and Milliken (2015) which pro-

vide a more detailed description of the models, their parameteriza-

tion, and how they are solved using inverse methods. Briefly, RTMs

are typically used to describe how light intensity changes as it en-

ters, interacts with, and exits a specific medium. Hapke’s model

has been used in a number of previous studies for modeling spec-

tra of laboratory mixtures and planetary surfaces (e.g., Hapke, 1981,

2005; Li and Li, 2011; Lucey, 1998; Mustard, 1987; Poulet, 2004; Li

and Milliken, 2015 ). While Hapke’s model is designed for intimate

mixtures of particulates, the Shkuratov RTM treats particles as one-

dimensional layers. The Shkuratov model is computationally faster,

which is favorable for spectral un-mixing of large datasets or image

cubes acquired by instruments such as CRISM and OMEGA, but it

does not explicitly account for the viewing geometry dependence

of reflectance that is present in such image cubes ( Poulet et al.,

2002 ). In contrast, although the Hapke model may yield better es-

timates of modal mineralogy in some cases (e.g., Li and Milliken,

2015 ), it is at the expense of increased computation time.

In Hapke’s model, the reflectance can be described as a func-

tion of single scattering albedo ( ω( λ), where λ is wavelength),

solar incidence angle ( i ), emittance angle ( e ), back scattering

function ( B ( g , φ), where g is the phase angle and φ is the filling

factor), phase function ( P ( g )), and multiple scattering function ( H,

for both down-welling and up-welling radiance). Reflectance can

thus be described as;

R =

ω ave

4 π

μ0

μ0 + μ{ [ 1 + B (g) ] P (g) + H ( μo

◦ω ave ) H ( μ, ω ave } − 1

where,

ω ave =

∑

i

M i ω i

D i ρi

/ ∑

i

M i

D i ρi

in which ρ = solid density, M = bulk density, and D = particle

diameter. The particle diameter may be related to the optical path

length, < D > , as described below. The single scattering albedo of

an individual component, ω i , is a function of the optical constants

n and k through external and internal scattering terms ( S e and

S i , respectively) and the internal transmission factor, � ( Hapke,

1993 ). In its simplest form, � is approximated by Beer’s Law

( � = e −α·< D > ) and single scattering albedo is thus boiled down to

a transmission problem.

Therefore, each reflectance spectrum can be converted to a

single scattering albedo spectrum that is the weighted linear

combination of individual endmember single scattering albedo

spectra. The fundamental inputs to the model are typically the

optical constants, n ( λ) and k ( λ), and the solid density of each

endmember mineral. An estimate of porosity is also commonly

included. The final outputs of the model are an estimate of the

abundance and particle size for each endmember component. In

this study we assume B ( g ) = 0 and we adopt a constant value

or n for the BDR and ASD wavelength range. The real indices of

efraction can vary over the wavelength range (0.8 μm–2.5 μm)

sed in this study, but we adopt constant values of n = 1.51 for

ypsum and n = 1.52 for montmorillonite, which are averages

ased on estimates from Roush (2005 ) and Roush et al. (2007 ).

his is in contrast to the wavelength-dependent Kramers–Kronig

olutions for n used in other studies ( Roush, 2005; Roush et al.,

007; Sklute et al., 2015 ), which find that n may vary by up to

.03 (e.g., Roush et al., 2007 ). However, for this study we make

he assumptions that such differences are negligible and adopt

constant value over the wavelength range of interest. As will

e shown below, our use of a wavelength-independent n yields

esults extremely similar to those of Roush (2005) and Roush

t al. (2007) for these particular minerals, though we stress that

his simplification may not be appropriate for other minerals with

uch stronger variations in k (and thus n ).

We note that our implementation of the Hapke model is largely

imilar to that of Roush et al. (2007) and Sklute et al. (2015) ,

ith the exception that we estimate the phase function by us-

ng fixed values for the Legendre polynomial coefficients ( b = –0.4

nd c = 0.25, based on values for silicates from Mustard and Pieters

1989) and consistent with moderate forward scattering). As a test

f the sensitivity to these values, we varied the b and c parame-

ers by 0.3 and found that it changed the single scattering albedo

alues by only 1% (relative) and ultimately had a negligible effect

n the modeling results. Therefore, we conclude that an assump-

ion of moderate forward scattering is appropriate for the clay and

ypsum examined in this work.

In addition, we do not include the additional variable of the in-

ernal scattering coefficient, s , in the internal transmission factor

ecause appropriate values for this variable are poorly constrained

or different minerals and may be system-dependent (e.g., Roush

t al., 2007; Sklute et al., 2015 ). Omission of the internal scatter-

ng coefficient typically results in a lower single scattering albedo,

owever the particles in our study may be considered optically

hin (i.e. k � 1, D is ∼10 s microns), therefore the internal scatter-

ng effects are likely minimal ( Hapke, 1981 ) and we prefer to min-

mize the number of unknown parameters in our model. We note

hat the omission of the s parameter forces all observed spectral

ehavior to be accommodated in the imaginary part of the index

f refraction, k , when deriving the optical constants from the ra-

iative transfer model. All modeling was performed over a wave-

ength range of 0.8 –2.5 μm for both the ASD and BDR data. This

avelength range was selected because we were more interested

n testing the ability to model the subtle differences in the hydra-

ion bands than the slopes in the visible range, where the latter

ay be affected by the presence of Fe-bearing impurities (e.g., Fe-

xides in the clay samples).

.4. k-values for mineral endmembers

In this study we first use Hapke’s model to solve for the imag-

nary part of the index of refraction ( k ) by using the reflectance

pectra, a starting value for particle size, estimated porosity

1 – φ), and viewing geometry for the pure minerals as the inputs.

y measuring the reflectance spectra for a number of different par-

icle sizes of a given endmember (either gypsum or montmoril-

onite) it is possible to solve the equation using an inverse, least-

quares approach. A best fit is obtained when the differences be-

ween measured and modeled reflectance spectra are minimized

a tolerance of 10 −6 was used for each wavelength in this study

hen solving for k ), and the associated k values that satisfy this

or a single spectrum (or for spectra of all different particle sizes

imultaneously) are taken to be the best estimates for the imag-

nary part of the complex index of refraction. During the deriva-

ion of the k values, the particle size was allowed to vary within


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he sieved size range and smaller (given that smaller particles may

ave broken off of the clay aggregates during sample handling).

hen solving for a single k value for the different particle sizes

imultaneously, the code was run with a tolerance between mea-

ured and modeled reflectance values of 10 −6 and up to 10,0 0 0 it-

rations, whichever criterion was met first. The maximum number

f iterations was not reached during the inversion, indicating that

he tolerance between measured and modeled reflectance spectra

as reached. Because the k values derived from radiative transfer

odels of reflectance spectra represent some kind of average over

he different crystallographic axes and thus may differ from those

erived from transmission (absorbance) spectra, it is perhaps more

roper to think of the RTM-derived k values as ‘effective’ or ‘ap-

arent’ optical constants.

The effective optical constants for each endmember are then

sed as the input to the model, and the mineral abundance and

article size of each component in our gypsum-montmorillonite

ixtures can be solved using a similar inverse, least-squares ap-

roach. Again, a tolerance of 10 −6 was used (between measured

nd modeled reflectance spectra for each wavelength) and particle

ize was allowed to vary within or below the sieved size range

hen modeling each mixture spectrum. It is worth noting that

he particle diameter, D , may be related to the mean optical path

ength, < D > , by < D > = 0.9 D for spherical particles and < D > ≈ 0.2

for irregular particles ( Li and Li, 2011 ). These values are based

n lab results and Monte Carlo simulations taken from Shkuratov

nd Grynko (2005) . In their paper, Shkuratov and Grynko suggest

hat the particle diameter–path length relationship is dependent

n a small imaginary index of refraction. While they test differ-

nt values of τ (which inherently means different k values) they

o not give exact values for imaginary index of refraction. In our

ase, while there is a 2 order magnitude of difference in the imag-

nary index associated with the absorption features, the maximum

-value does not exceed 0.01 therefore, it can still be assumed

small’.

In addition to the irregular particle size assumption, the real

omponent of the optical constant ( n -value) is incorporated into

he path-length therefore, when using this model, any optical con-

tants derived using the spherical model should be scaled down

y (4.5/ n ) to account for these differences. Because modeled val-

es of k will be inherently linked to values of < D > (and by ex-

ension, D ), and n = 1.51 and 1.52 for gypsum and montmorillonite

espectively it can be seen that k values may differ by a factor of

depending on whether an assumption of irregular or spherical

articles is adopted. We estimate k -values under both assumptions

ut ultimately model the binary mixtures assuming that particles

re irregular, in accordance with tabular and platy crystals (appro-

riate for gypsum and montmorillonite) and with findings in pre-

ious RTM studies for other minerals ( Hiroi and Pieters, 1992; Li

nd Milliken, 2015 ). Therefore, it can be demonstrated that if the

olid density for each endmember is known then the unknown pa-

ameters are effectively the optical path length, < D i > , and bulk

ensity, M i , for each component, which can be solved for using a

east-squares approach and then easily converted to particle size

nd mass (or volume) fraction, respectively.

Imaginary refractive indices for the gypsum and montmoril-

onite were estimated independently using both the BDR and ASD

pectra. The k -values for Group 1 were determined by optimizing

he particle size input to the model in order to obtain a single,

onsistent k -value for all four particle sizes while simultaneously

olving for the effective particle size. A starting value for the parti-

le size was chosen for each size fraction (e.g., mean of sieve range

r mean determined from optical microscope measurements) and

hese values were simultaneously adjusted using a least-squares

pproach until a best-fit was achieved. The resultant output was

single k -value with a range of particle sizes for each particle size

ange. For gypsum, there was no range needed, and the min–max

article sizes to yield the same k -value were almost identical. The

lay needed a wider tolerance to accomplish this, giving a min–

ax particle size range. The values given ( Table 2 ) are the average

median) value for that particle range. Although such a solution

s non-unique, the resulting ‘effective’ particle sizes, which allow

he model to yield a consistent k value for a given mineral, can be

ompared to the sieve size range and the mean particle size deter-

ined from the optical microscope measurements.

As a separate experiment, the montmorillonite endmembers in

he 63–75 μm and 25–32 μm size fractions were systematically

ehydrated and measured with the ASD in order to observe the

ariation of k -values relative to the degree of hydration (amount

f interlayer H 2 O). Water content in hydrated minerals can have

large effect on estimates of optical constants, and for miner-

ls such as smectites the water content can be highly variable.

ndeed, the optical constants for montmorillonite determined by

oush (2005) were for samples measured in a purged environment

hat was then filled with nitrogen, thus the samples likely had

ecreased interlayer H 2 O. Aliquots of our montmorillonite sam-

le were heated in an oven from 25 °C to 350 °C and immedi-

tely transferred to the ASD Fieldspec3 for analysis. The dehy-

rated sample was then stored in a desiccator with drierite for 72 h

nd measured again to assess any potential changes in hydration

tate and to evaluate these effects on modeled k -values.

As a final step in this study we compared the results from the

apke RTM with those derived from the Shkuratov RTM. Our im-

lementation of the latter follows the procedures described by Li

nd Milliken (2015) , and we refer the reader to that publication for

more detailed description of the methodology. Importantly, when

stimating mineral abundances and particle sizes using the Shku-

atov model we rely on the effective optical constants that were

lso derived with that model; for consistency within the models

e do not use Hapke-derived optical constants in the Shkuratov

odel and vice versa ( Li and Milliken, 2015 ). Differences in the (ef-

ective) optical constants derived from the two RTMs are discussed

elow.

. Results

.1. Endmember particle sizes

The particle size distribution, mean particle size, and aspect ra-

io for both gypsum and montmorillonite, as determined from the

ptical microscope measurements, are shown in Fig. 1 for all par-

icle size fractions in Group 1. It is observed that the mean parti-

le sizes for gypsum are larger than the estimated values from the

ieve sizes, presumably due to the tabular nature of the gypsum

articles. The large aspect ratio can allow individual particles to

ass through the sieve lengthwise, skewing the particle size distri-

ution to larger values. In contrast, the mean particle sizes for the

lays are smaller than the sieve size range. Individual clay crys-

als are significantly smaller, and the particles we observe repre-

ent aggregates of many individual montmorillonite crystals. Even

hough the montmorillonite was wet sieved, rinsed and carefully

andled, there still remained some fine grains in the powder. The

ptical microscope used for the particle size distribution allowed

or particles as small as 3 μm to be detected and counted. There-

ore, the skew towards smaller values is possible due to the dis-

ggregation during and after sieving. This effect is not as appar-

nt for the 25–32 μm montmorillonite fraction, with the result be-

ng that the mean particle size for the 25–32 μm and 38–45 μm

ontmorillonite size fractions are very similar (31 and 38 μm,

espectively).

We note that the SAz2 montmorillonite is reported as having

levated iron content as described in the Clay Minerals Society.


Fig. 1. Particle size distributions for the different size fractions of (a) gypsum and (b) montmorillonite measured with an optical microscope. The mean particle sizes are

labeled.

Fig. 2. Reflectance spectra acquired with the RELAB BDR instrument for pure (a) gypsum and (b) montmorillonite for the four particle size fractions measured in Group 1.

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X-ray diffraction analysis of this sample revealed minor magnetite

present in the ‘as-shipped’ clays ( Chipera and Bish, 2001 ). Some

care was taken to handpick the iron oxide from the sample how-

ever some impurities still remain. We speculate that the presence

of these oxides may affect the overall albedo of the samples and

the spectral slope of the montmorillonite samples at wavelengths

< 1.3 μm (e.g., Fig. 2 ).

3.2. Bi-directional (BDR) reflectance spectra

Reflectance spectra of gypsum and montmorillonite endmem-

bers for the four size fractions in Group 1 are presented in Fig. 2 .

The absolute reflectance values for the gypsum endmembers are

consistent with trends typically associated with particle size vari-

ations, where reflectance values systematically increase with de-

creasing particle size. While the relationship between particle size

and overall reflectance is non-linear, spectra for the two small-

est size fractions (25–32 μm and 38–45 μm) exhibit similar re-

flectance values, in agreement with the measured particle sizes.

Spectra for the montmorillonite endmembers exhibit a similar se-

quence of increasing reflectance associated with decreasing parti-

le size ( Fig. 2 b). However, the absolute reflectance for the 38–

5 μm fraction is somewhat lower than might be expected con-

idering how similar the measured mean particle size is to that of

he 25–32 μm size fraction.

A change in the spectral slope at wavelengths < 1.3 μm is also

pparent between the different size fractions for the clay endmem-

ers ( Fig. 2 b). The slope appears to become shallower with increas-

ng particle size, which we speculate could be a result of lesser

pectral influence or decreased relative abundance of the darker

xide particles in the coarser samples. This minor variability in the

lope at short wavelengths has a surprisingly significant effect on

he modal abundance estimates for the coarser size fractions, a fac-

or that we discuss in more detail below.

The BDR reflectance spectra for the binary mixtures ( Fig. 3 )

how the expected behavior in the absorption band depth and po-

ition for all mixtures for each particle size (gypsum absorptions

ystematically weaken with increasing montmorillonite abundance

nd vice-versa). There is also a large drop in reflectance values for

he coarsest size fraction of gypsum ( Fig. 2 a), which results in a

ery small spread in the reflectance values for the 125–150 μm

lay–gypsum mixtures ( Fig. 3 d).


Fig. 3. Reflectance spectra of the binary mixtures of gypsum and montmorillonite for the (a) 25–32 μm, (b) 38–45 μm, (c) 63–75 μm, (d) 125–150 μm size fractions. The

weight fraction of gypsum is indicated in the legend as GXX. All spectra were measured with the bi-directional spectrometer at RELAB.

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.3. ASD reflectance spectra

The ASD reflectance spectra were scaled to the absolute re-

ectance of the BDR data at 1.09 μm in order to account for any

ifferences in absolute reflectance that may result from the ASD

aving a slightly different viewing geometry compared with the

ELAB BDR instrument (e.g., large cone angles). In addition, this

caling allows direct comparison of ASD unmixing results with BDR

esults using the same optical constants. A single representative

xample is shown in Fig. 4 for the 63–75 μm size fraction of the

0–50 gypsum–montmorillonite mixture. This plot shows the spec-

ral variability for the four repeat measurements of this mixture

ith the ASD compared with the BDR measurement of the mix-

ure. The spectra are very similar, with most discrepancies associ-

ted with subtle differences in the strength of H 2 O absorption fea-

ures near ∼1.4 and ∼1.9 μm. The observed differences between

ndividual ASD spectra and BDR spectra are likely due to variations

n the hydration level and/or orientation of particles at the time

f data collection. No attempt to control the ambient relative hu-

idity was done, therefore some variation in adsorbed water may

e expected between analyses. In addition, the BDR measurements

ere acquired while the sample dish was spinning, thus averag-

ng over particle orientations. In contrast, each ASD spectrum rep-

esents a static and different orientation of the sample dish, thus

ny slight particle orientation effects may affect the spectra.

Despite these different measurement techniques and instru-

ents, the differences between the ASD and BDR spectra were

inimal and did not have a significant effect on the un-mixing re-

ults. However, it is important to note that extreme care was taken

n the experimental setup of the ASD to attempt to reproduce the

llumination, geometry, and other measurement conditions of the

DR spectrometer. In this case, for relatively homogenous particu-

ate samples, the spectra are quite comparable, but given the large

OV of bare fiber of the ASD instrument this is unlikely to be true

f very heterogeneous or non-particulate samples.

.4. Derivation of optical constants

.4.1. Bi-directional spectra

Endmember spectra for the four size fractions in Group 1 were

sed to generate k -values for both phases. A least squares mini-

ization routine was independently applied to the Hapke model

sing the endmember BDR spectra and initial particle size esti-

ates for the four size fractions. The model was iterated and par-

icle sizes ( D ) were allowed to vary from 0 to the maximum mea-

ured value ( Fig. 1 ) as a means of adjusting the optical path length,

nd iteration proceeded until k values for each size fraction started

o converge. This optimization is necessary because, as demon-

trated above, the mean sieve size range was not always an ac-

urate predictor of the mean particle size or range. The end result

s a set of k ‘spectra’, one for each particle size, that are within

specified tolerance of each other for all measured particle sizes

Fig. 5 ).

As evident from Fig. 5 a, the k -values derived for gypsum over

he full wavelength range (0.8–2.5 μm) are extremely similar for

ll wavelengths for all particle sizes, indicating a nearly unique


Fig. 4. Example of the linearly scaled ASD data for the 63–75 μm size fraction of

the 50:50 gypsum: montmorillonite mixture. The largest variability is seen in the

1.4 μm and 1.9 μm H 2 O absorption bands (marked with arrows), possibly due to

small variations in surface-adsorbed H 2 O.

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solution within our tested parameter space. Convergence was

achieved where the min–max range for particle sizes was almost

zero. However, we found that with the montmorillonite spectra,

there was a lack of convergence and a wider particle range was

required to obtain a single k -value. Values of k ( λ) for the clay are

consistent at longer wavelengths ( > 1.3 μm), but a subtle difference

is observed below this value for the 25–32 μm and 125–150 μm

size fractions (see Fig. 5 b). This is also represented in the original

reflectance spectra ( Fig. 2 ) where there is an increase in the slope

at these wavelengths as the particle size decreases. The differences

in slope could be attributed to increasing spectral influence of the

iron oxide impurities or a scattering effect of unknown origin. This

discrepancy has a surprisingly large effect on the un-mixing of the

coarsest fraction, and is discussed below.

Fig. 5. Imaginary index of refraction values as determined from samples in Group 1 for (a

model. The particle sizes derived from the inverse model that was used to generate these

is observed at wavelengths < 1.3 μm for the montmorillonite, as discussed in the text. A

hydration bands.

Though the modeled k -values are technically non-unique solu-

ions (because they are dependent on < D > , which we are treating

s a variable), by constraining the particle size to be within the

nown or expected range it was found that the range in k -values

or a given wavelength was not large, particularly for the gypsum.

he particle sizes derived from the k -optimization are compared

ith the values determined from the optical microscope measure-

ents in Table 2 . The measured and optimized values are consis-

ent for the gypsum, whereas the particle sizes estimated for the

ontmorillonite are somewhat smaller than the measured values,

articularly for the two largest size fractions. Due to the nature of

he SAz-2 material, the particles contain significant porosity and

re actually aggregates of much smaller clay crystallites, thus they

re not solid particles. Therefore, the estimated particles sizes are

est thought of as ‘effective’ particle sizes that are representative

f average optical path length and not necessarily representative

f measured particle sizes or size distributions. In addition, as de-

cribed in Section 3.1 , disaggregation will result in a smaller aver-

ge particle size when compared to the sieved sizes. For this work,

uch ‘effective’ particle sizes are assumed related to measurable

article size in the same manner as for the gypsum endmember.

The average k -value solutions from the Group 1 particle size

uites were used in all of the un-mixing results. These single k -

alues were calculated using Hapke’s model assuming both spher-

cal and irregular particle shape and are compared to previously

ublished values in Fig. 6 . When assuming irregular particles,

he k -values for gypsum are higher than those of Roush et al.

2007) by a factor of ∼3 ( Fig. 6 a), which, as stated above, is simi-

ar to the expected difference between irregular and regular parti-

le shape models. Indeed, when assuming spherical particles our

esults are in excellent agreement, possibly because the Roush

t al. (2007) study followed the model of the Roush (2005) study

hat relied on equivalent spherical diameters for the particle size.

he agreement between our results and those of Roush et al.

2007) ( Fig. 6 ) also demonstrate that our use of a wavelength-

ndependent value for n is justified for gypsum.

By contrast, our estimated k -values for montmorillonite

how clear differences compared to those reported by Roush

2005) ( Fig. 6 b). Though our results under the assumption of

) gypsum and (b) montmorillonite assuming irregular particle shape for the Hapke

k values are labeled in parenthesis for each size range. Note that a large deviation

wavelength rage of 1–2.5 μm is used here to clearly observe the variation in the


Fig. 6. Estimates of optical constants ( k ) for (a) gypsum and (b)–(c) montmorillonite under different particle shape assumptions and water contents. The average k -values

are shown assuming both an irregular particle shape (black stippled line) and spherical particle shape (solid black line) and compared to the k -values measured by Roush

et al. (2007) for gypsum and Roush (2005) for montmorillonite. In (a) it is clear that the optical constants for the spherical gypsum case are almost identical to those of

Roush et al. (2007) , whereas in (b) our k -values are systematically higher in regions of strong absorption, including the 1.4 and 1.9 μm H 2 O absorption features. In (c) we

compare the k -values for a dehydrated SAz2 montmorillonite with those from Roush (2005) . Dehydration of the SAz2 montmorillonite improves the fit with the previously

published values, but differences still exist in the strength of H 2 O bands, the ∼2.2 μm Al–OH band, and the overall spectral slope. (For interpretation of the references to

color in this figure legend, the reader is referred to the web version of this article.).

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pherical particles are the same order of magnitude as the k -values

f Roush (2005) , there are notable differences in the overall slope

t longer wavelengths as well as in the depths of the H 2 O absorp-

ion features at 1.4 and 1.9 μm. This could be due to slight differ-

nces in clay chemistry between the samples used in these studies

SWy1 in Roush (2005) versus SAz-2 in this study), but it is more

ikely that the differences are due to sample texture and hydration

evel. Specifically, our SAz-2 samples consisted of single coherent

articles that were washed to remove clinging fines. In contrast,

he SWy-1 material is often supplied as a micromilled powder, and

ieving this material causes clumping that leads to ‘particles’ that

re actually aggregates of much smaller particles and crystallites

and the size of such ‘particles’ can be easily altered during sam-

le handling).

In addition, our samples were measured under ambient (mod-

rate to high humidity) conditions and those of Roush (2005) were

easured under purged nitrogen gas, thus our samples con-

ain significantly more interlayer H 2 O. This difference in H 2 O is

ikely a large factor in the observed differences in k -values at 1.4

nd 1.9 μm as well as the stronger spectral slope towards the

3 μm ‘hydration’ region. This is examined in Fig. 6 c, which com-

ares the k -values for our dehydrated montmorillonite with those

f Roush (2005) . Dehydrated montmorillonite spectra were ac-

uired via in-situ heating with the ASD instrument using the 25–

2 μm size fraction. This resulted in the expected decrease in the

2 O absorption band depths as well as a decrease in the slope

t short wavelengths for the k -values ( Fig. 6 c). The dehydrated

-values are in better agreement with those of Roush (2005) , but

here are still notable differences. Perhaps of most importance is

he difference in k -values near the diagnostic Al–OH absorption

t ∼2.2 μm, which will have important effects on modeled abun-

ances of montmorillonite in mineral mixtures. Nevertheless, the

act that our results are of the same order of magnitude as those

f Roush (2005) also provides confidence that our measurement

nd modeling approaches are yielding reasonable k -values.

While some differences may be expected due to different mea-

urement techniques, different instrumentation, and different mod-

ling approaches, these results highlight a fundamental difficulty of

etermining unique or standard optical constants for minerals like

lays and zeolites that can have highly variable amounts of H 2 O.

ny variability in the H 2 O content of sulfates, like gypsum, results

n a discrete phase transition with a new mineral structure con-

aining a fixed amount of H 2 O. Other hydrated phases, including

lays, zeolites, opal, and palogonite, can vary significantly in H 2 O

ontent yet maintain the same crystal structure, which means that

-values for the H 2 O absorptions are sample (and measurement)

pecific and cannot be uniquely tied to mineral abundance. Using

-values derived from a sample with an inappropriate H 2 O con-

ent (e.g., lab measurements where samples were too wet or too

ry compared to Mars-like conditions) can result in errors associ-

ted with modeled mineral abundances. While, not surprising, it is

undamentally important to ensure a proper representation of the

aterials measured in the orbital data. Future work in this area

hould look at the impact of using multiple k -values of variable

ydration levels for the same mineral phases.

.4.2. ASD spectra

A similar derivation of k -values was performed using the ASD

eflectance data for all particle sizes in Groups 1 and 2. Each sam-

le was measured multiple times with the ASD instrument to as-

ess repeatability and the extent to which factors such as particle

rientation may affect derived k -values. Fig. 7 shows the average

-values determined from the ASD spectra, as well as the standard

eviations based on the repeat measurements, for the two mineral

ndmembers. There is very little variation in the gypsum optical

onstants for all size fractions, and the average is effectively identi-

al to the BDR results, indicating the k -values for gypsum are very

obust. The ASD results for montmorillonite exhibited larger vari-

bility (up to ∼15%), much of which is associated with the H 2 O

bsorptions, but the average of the ASD values is still in extremely

ood agreement with the BDR-derived k -values. Despite the differ-

nt measurement conditions of the two instruments, this agree-

ent indicates that reliable k -values can be derived from ASD in-

truments if great care is taken to control illumination geometry

nd measurement conditions.

However, it is worth noting that the standard deviations of the

epeat ASD measurements (gray regions in Fig. 7 ) imply that a sin-

le ASD measurement for a given sample may not yield as accurate

f a result as the use of multiple measurements. Choosing one of

he extreme values did not improve the modeling results equally

t all wavelengths. Although an extreme may improve the spectral

t at some wavelengths, it is at the expense of other wavelengths

or certain samples. The average value, which we have focused on,

rovided the best overall fit when considering the full wavelength

ange.

.5. Estimates of particle size

The modeled ‘effective’ particle sizes for all size fractions in

roup 2, which consisted primarily of particles > 90 μm, are


Fig. 7. The average k -values (black lines) for (a) gypsum and (b) montmorillonite as measured using the ASD compared with the average k -values (red stippled lines) as

measured using the RELAB BDR spectrometer. The average k -values for the ASD spectra represent 5 repeat measurements for each of the 4 different size fractions, with the

standard deviation from the repeat measurements shown in grey. The average k -values between the two instruments for both phases are in good agreement, with some

minor variation observed in the H 2 O absorption bands for the montmorillonite. These estimates relied on the assumption of irregular particle shape. (For interpretation of

the references to color in this figure legend, the reader is referred to the web version of this article.).

Fig. 8. Comparison of measured particle sizes with modeled effective particle size, assuming irregular particle shape, as determined from particle size optimization for both

ASD data (black circles) and BDR data (red circles). (a) The gypsum values follow very closely to the 1:1 line (solid black line), whereas values for the (b) montmorillonite

show a distinct offset in the effective particle size. The latter may be due to the much smaller crystallites in the clay. (For interpretation of the references to color in this

figure legend, the reader is referred to the web version of this article.).

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compared with the measured particle sizes in Fig. 8 in order to es-

tablish an empirical relationship between actual particle size and

optical path length. The ‘effective’ particle sizes for gypsum are

similar to those determined from optical microscopy, even for large

particles ( Fig. 8 a). In contrast, and as expected, the modeled ef-

fective particle sizes for montmorillonite are significantly smaller

than measured values, particularly for particles > 50 μm. As noted

above, this is likely due the clay particles being aggregates of much

smaller crystallites with significant porosity, therefore the mean

optical path length ( < D > ) appears to decrease rapidly (and non-

linearly) as actual particle size increases. Additional study is war-

ranted to determine if this trend is true of clay minerals in general

r is specific to our particular montmorillonite sample, though we

uspect the former is likely correct.

.6. Radiative transfer modeling

The un-mixing results presented here are all based on the aver-

ge k -values for the gypsum and SAz-2 montmorillonite endmem-

ers derived from the BDR data (which are nearly identical to the

SD results, as shown in Fig. 7 ). The average k -values were then

sed for the mixture modeling in the ASD and BDR data set. For all

odels the particle sizes were allowed to vary along with modal

bundance (bulk density) and we used a least-squares approach


Fig. 9. Modeled and measured reflectance spectra and associated residuals for the 25–32 μm, 38–45 μm, 63–75 μm and 125–150 μm size fractions (rows a–d respectively)

with their respective χ2 values. The columns represent the G10, G50, and G90 modal abundances for the four different size fractions. Additional endmember slopes (negative

and positive) were used when modeling the 125–150 μm size fractions to account for the difference in slopes between the montmorillonite optical constants.

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o minimize the difference between measured and modeled re-

ectance spectra. Example spectral fits and associated residuals for

elect mixtures are presented in Fig. 9 with the average derived

article sizes shown in Table 2.

.6.1. Mineral abundances from BDR spectra

Estimated minerals abundances for the BDR data are shown in

ig. 10 for the four size fractions from Group 1. The model re-

ults for the 25–32 μm, 38–45 μm and 63–75 μm size fractions

how less than 5% deviation from the known values ( Fig. 10 a–c),

uggesting the model can adequately account for any subtle dif-

erences that may occur in the H 2 O absorption bands for the two

ineral endmembers. However, results for mixtures in the 125–

50 μm size fraction show large deviations in the clay-dominated

amples ( Fig. 10 d). As discussed above, the k -values for the 125–

50 μm montmorillonite samples were coincident with an effective

article size of 50 μm, which gave the closest match to the average

-values determined from the other three particle sizes ( Fig. 5 ).

A comparison between the k -values in Fig. 5 shows a sub-

le but distinct difference in the spectral slopes, with the

ost pronounced difference occurring at the shorter wavelengths

< 1.3 μm). This difference can either be attributed to differences in

he hydration level between the finest and coarsest grained frac-

ions, an uneven distribution of the iron oxide particles or due to

he incorrect grain size distribution (fine particles). Indeed, elec-

ronic absorptions due to iron are consistent with broad absorp-

ions at these wavelengths (e.g., Burns, 1993 ). Regardless of the

ause, it is apparent that these seemingly small differences in spec-

ral slope result in the model trying to increase the optical path

ength (and thus effective particle size) of the brighter mineral

gypsum) for the short wavelengths in the 123–150 μm size frac-

ions in order to match the observed higher reflectance values and

s such it results in an overestimation of the gypsum abundance In

ther words, it appears that the average k -values for the montmo-

illonite are too large (too absorbing) at the shorter wavelengths

ompared to the actual absorption properties of the 123–150 μm


Fig. 10. Measured versus modeled abundances for the four size fractions from Group 1 based on Hapke modeling of the RELAB BDR spectra. The 125–150 μm fraction

compares three different modeling experiments as described in the text. The red circles are for the uncorrected data, the grey circles present the modeled results using a

wavelength range that excludes shorter wavelengths, and the black circles present the modeled results for the slope-corrected data. (For interpretation of the references to

color in this figure legend, the reader is referred to the web version of this article.).

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montmorillonite sample and associated mixtures. To compensate

during the spectral fitting, the contribution from the more weakly

absorbing gypsum is over-modeled.

This disproportionate effect of the shorter wavelengths is con-

firmed by running the model without the < 1.3 μm wavelength

region. In this case (gray points in Fig. 10 d) the modeled min-

eral abundances are in better agreement with the measured values

and nearly all points are within 5% absolute. Removing the shorter

wavelength region may improve the unmixing results, and there

are no pertinent absorption features in this region for the binary

mixtures examined in this study, but it is still beneficial to account

for this problem in order to develop a more universal model that

can be applied to a wide range of mineral mixtures.

As noted above, it is possible that the variation in k -values for

the montmorillonite size fractions is due to the presence of impu-

rities in this mineral endmember (e.g., Fe-oxides). Additional end-

members were included in the model to account for any spectral

darkening and slight slopes in the coarser size fractions. Synthetic

optical constants were generated to simulate endmembers with

linear positive and negative sloped reflectance spectra, which can

help to account for variations in spectral slope not well-modeled

by the mineral optical constants. The mineral abundance results for

he models that used these endmembers were then normalized to

nclude only the gypsum and clay abundances. The results of these

slope-corrected’ values for the 125–150 μm size fractions are pre-

ented as red points in Fig. 10 d. Such linear slopes are not nec-

ssarily representative of the exact spectral properties of opaque

inerals such as magnetite and other oxides, which may have

hallow broad absorptions or slightly nonlinear spectral slopes in

his region (e.g., Morris et al., 1985 ), but it allows for a first-order

orrection of these effects. A more robust approach would be to

nclude optical constants of actual Fe-oxides, which will be exam-

ned in future work, but this simple correction produces results

imilar to those from the reduced wavelength range (gray points

n Fig. 10 d) on par with the three other size fractions.

.6.2. Mineral abundances from ASD data

The un-mixing results for the ASD mineral mixture spectra for

he four size fractions from Group 1 are presented in Fig. 11 . Abun-

ance estimates for the 63–75 μm and 38–45 μm size fractions are

early all within 5% of measured values, and though most mixtures

n the 25–32 μm size fraction were also within this range there is

general offset where most of the results underestimated the frac-

ion of gypsum fraction ( Fig. 11 a). The 125–150 μm fraction also


Fig. 11. Measured versus modeled abundances for the four size fractions from Group 1 based on Hapke modeling of the ASD spectra, including repeat measurements of each

sample. The 125–150 μm fraction presents the modeled results for the slope-corrected data.

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howed good results (using the slope-corrected model described

bove) except for the 25 wt% gypsum mixture, for which the gyp-

um was overestimated. Given the decent results for other mix-

ures in this size fraction we suspect this is a result of heterogene-

ty in the mixture or other uncertainties in the ‘known’ abundance

f the minerals within the field of view of the ASD.

Overall, the ASD results are very promising when compared to

he BDR data, even with the minor variability in the ASD mea-

urements associated with repeat measurements of the same sam-

les/mixtures. Though not shown here, the model fits to the ASD

pectra are good, similar to the BDR results ( Fig. 9 ), with the ma-

ority of the residuals associated with the H 2 O absorptions. The

article size estimates from these un-mixing models are compared

ith the values obtained from k -value derivation as well as the

easured values in Table 2.

.6.3. Comparison of Hapke with Shkuratov mineral abundances

The model results presented in Figs. 9–11 and discussed above

re all based on the Hapke model, and it is clear that this method

s able to correctly estimate mineral abundances within ∼5% of

easured values for nearly all mixtures and particle sizes. For

omparison, k -values for both the montmorillonite and gypsum

ndmembers were calculated using the Shkuratov model and were

ompared to the Hapke results, as well as those of Roush ( Fig. 12 ).

t is seen that the general spectral shape and properties of the

ptical constants are similar between the Hapke and Shkuratov

ethods, but the Shkuratov values are systematically lower, pos-

ibly related to the different im plementation of the phase function

etween the two RTMs. This trend is also shown in Roush et al.,

2007) , however it becomes reversed at longer wavelengths.

Mineral abundance estimates for both Hapke and Shkuratov

odels are presented in Fig. 13 . As with the Hapke-based results,

he Shkuratov estimates are largely within 5% absolute mineral

bundance, though there are larger deviations in the 50:50 mix-

ures for both the 25–32 μm and 63–75 μm size fractions. In gen-

ral, we observe that the Hapke model yields slightly better es-

imates of mineral abundance for the mixtures examined here.

dditional study is needed to determine if this is true for other

inerals and mineral mixtures, though recent results for model-

ng of eucrite and diogenite meteorites suggest the Hapke model

erforms better in silicate-dominated mixtures as well ( Li and

illiken, 2015 ). However, the Shkuratov model is computationally

ore efficient and the larger uncertainties may be acceptable in

ome applications where high accuracy is not necessary and pro-

essing speed is preferred.

.7. Irregular versus spherical particles

The model results described above were based on the assump-

ion of irregular particle shape such that < D > ≈ 0.2 D . This has the

ffect the k -values will be larger than when derived under the as-

umption of spherical particles ( Fig. 6 ). Though either assumption

ay be used, the choice may have an effect on the actual modeled

ineral abundances. This is because once the k -values are fixed


Fig. 12. Comparison of the optical constants for gypsum and montmorillonite calculated using the Hapke and Shkuratov models (assuming irregular particle shape). The

Shkuratov values are systematically higher than the Hapke results but are otherwise relatively similar in spectral shape and slope. A wavelength range of 1.25–2.5 μm is

highlighted here to allow discrimination of subtle changes in the hydration bands.

Fig. 13. Measured and estimated modal abundances of gypsum–montmorillonite mixtures measured on the BDR were modeled using the Hapke and Shkuratov models. The

Shkuratov model yields somewhat larger errors than the Hapke model, but many estimates are still within 5% of measured values. Average derived particle sizes for the

Shkuratov model are 63.9 μm, 33.8 μm, 33.3 μm for gypsum and 54.9 μm, 36.6 μm, and 31.3 μm for montmorillonite.

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the only way to adjust absorption strengths to match observed

values is to adjust < D > , which changes D and the modeled bulk

density, M , which ultimately affects the estimated mass or volume

fractions.

It is possible that in a complex mixture a change in < D > could

be balanced by a change in M to yield an accurate mass frac-

tion for a given component regardless of which particle shape

model is assumed, but this is not a given. Indeed, Li and Milliken

(2015) recently demonstrated that the irregular particle assump-

tion provided a better estimate between RTM-derived k -values and

transmission-derived values for a pyroxene. For the present study,

although the particle shapes are clearly non-spherical (particularly

for the tabular gypsum crystals), the k -values derived for gypsum

re more in line with previously published transmission measure-

ents (e.g., Roush et al., 2007 ) if it is assumed the particles are

pherical. Therefore, there are contradictory results as to whether

t is more or appropriate to assume spherical or irregular particles,

nd indeed the assumption may vary from case to case depend-

ng on the mineral particles in question and how they are oriented

ith respect to the incident photons.

Interestingly, for the materials examined in this study the dif-

erence between assuming spherical or irregular particle shape

eems to have little net effect on the cumulative results of the

odeled mineral abundances for the gypsum–montmorillonite

ixtures. This is demonstrated in Fig. 14 , which compares mod-

led mineral abundances for both assumptions (results for the


Fig. 14. Comparison of spherical and irregular Hapke un-mixing of the 38–45 um

size fraction. The spherical and irregular k -values were used for the respective

modeling. Un-mixing results yield similar abundance values, however, the spheri-

cal method yielded smaller particle sizes. Particle size results are shown where G irr

and G sph are the gypsum results for the irregular and spherical methods, while M irr

and M sph are the corresponding results for montmorillonite.

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8–45 μm size fraction are shown and are representative for the

ixtures suites as a whole). From this comparison it is clear that

odel results are often better under one assumption versus the

ther for a given mixture, but for the mixture suite as a whole

here is little difference between the two assumptions (that is, one

odel is not inherently better than the other for all or a majority

f the mixtures). This is because whether or not particles are as-

umed to be irregular or spherical has a relatively minor effect on

he corresponding modeled particle sizes ( D ), which in turn has lit-

le effect on the mineral abundances estimated for these mixtures.

erhaps this is not unexpected given other sources of uncertainty

n the model and the fact that the difference in k -values between

he two assumptions is less than an order of magnitude (only a

actor of ∼3).

However, the observation that the irregular particle assumption

ields more accurate results in some cases and not others (e.g., Li

nd Milliken, 2015 versus this work), and the (possibly fortuitous?)

esult that k -values for non-spherical gypsum particles are in bet-

er agreement with transmission derived values when they are

reated as spheres, deserves additional attention in future work.

pecifically, an assessment of whether or not a spherical or irregu-

ar particle shape assumption is appropriate or necessary in other

ases seems warranted, especially for other non-spherical parti-

les. In more complicated mixtures it is likely that some minerals

ay be better approximated by a spherical assumption (volcanic

r impact glass? olivine?). Therefore, it may be worth consider-

ng the use of mineral-specific relationships between D and < D >

o achieve the best possible estimates of mineral abundance in

ore complicated mixtures. Most importantly, when using Hapke

r Shkuratov RTMs it is critical to be self-consistent such that the

-values used as inputs to the model were derived using the same

odel and assumptions as what is being applied to the mixture

pectra. That is, Hapke-derived k -values are not appropriate for

hkuratov modeling (and vice versa) and RTM-derived k -values un-

er the assumption of irregular particles should not be used in a

odel that then assumes spherical particles (and vice versa).

. Conclusions

We have examined a suite of binary gypsum–montmorillonite

ixtures for a variety of particle sizes to understand the accuracy

f Hapke and Shkuratov radiative transfer models for estimating

ineral abundances when the imaginary indices of refraction are

ell-constrained. Key findings are as follows.

(1) Estimated values of k ( λ) derived from RTMs such as the

Hapke model are in good agreement when using RELAB BDR

spectral data and those acquired by an ASD FieldSpec in-

strument, provided care is taken when the latter is used for

this purpose. Specifically, illumination and viewing geometry

must be well-controlled (though need not be fixed at certain

values) and multiple measurements of the sample should be

acquired or samples should be spun during ASD measure-

ments to average out preferential orientation/scattering ef-

fects.

(2) Our estimates of k ( λ) for gypsum at NIR wavelengths are

in excellent agreement with results previously published by

Roush et al. (2007) under the assumption of spherical parti-

cle shape. This is despite the fact that instrumentation, sam-

ples, and modeling codes were different between these stud-

ies, demonstrating that these k ( λ) values for gypsum are ro-

bust at these wavelengths.

(3) Our estimates of k ( λ) for montmorillonite at NIR wave-

lengths are qualitatively similar to results previously

published by Roush (2005) under the assumption of spher-

ical particle shape. Some of the differences are further

reduced when our dehydrated montmorillonite is used to

derive k ( λ). Key sources of the remaining differences may

be the physical state of the particles, where our montmoril-

lonite samples consisted of discrete particles (composed of

smaller, cemented clay crystallites) that were washed to re-

move clinging fines, and remaining differences in hydration

state. Despite these differences our k ( λ) values are within

the same order of magnitude as those of Roush (2005) .

(4) Though previous studies have used Hapke and Shkuratov

RTMs to estimate values of k ( λ) for clays and sulfates, in

this study we have demonstrated that these k ( λ) values

are indeed appropriate for actual RTM modeling of simple

mixtures with components that are of relevance to Mars

and Earth. When using the RTM-derived optical constants to

model a wide variety of gypsum–montmorillonite mixtures

we find that estimated mineral abundances are typically

within 5% (absolute) of measured values for the samples ex-

amined here, although a slope correction was required for

the coarsest grain size due to issues with the montmoril-

lonite. Overall, the Hapke model provides slightly better ac-

curacy than the Shkuratov model for these mixtures. Future

work will focus on applying these optical constants to more

complex mineral mixtures as well as to gypsum and mont-

morillonite samples from different localities. The latter will

provide information as to how ‘universal’ these k ( λ) values

may be.

(5) Care must be used when relying on RTM-derived optical

constants for modeling of spacecraft and lab reflectance

spectra in order to understand which factors may be model-

specific. The optical constants used as inputs to an radiative

transfer model, for example, should be consistent with the

assumptions of that model, as factors such as particle shape

have the potential to introduce large uncertainties in min-

eral abundance if they are not consistent between the RTM

used to derive k ( λ) and the RTM used to model the mixture

spectra.

(6) When clay and gypsum were present in the mixtures at lev-

els as low as 10% they were still detected with the RTMs,

though the relative uncertainty in these modeled abun-

dances is accordingly higher. This suggests that CRISM and

OMEGA observations of simple clay and sulfate-dominated

rocks on Mars may have similar detection limits ( ∼10%),


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though this aspect certainly warrants additional research.

Future work will incorporate the optical constants reported

here with those of other clays and primary silicates (olivine,

pyroxene, feldspar) to model more complex mixtures that

are directly relevant to the mudstone compositions observed

in Gale crater by the Curiosity rover. Such higher fidelity

tests will ultimately provide a more detailed understanding

of the accuracies and limitations of deriving clay and sulfate

mineral abundances on Mars using RTMs at VIS-NIR wave-

lengths.

Acknowledgments

We would like to thank Dr. Takahiro Hiroi and the NASA RELAB

facility at Brown University for assistance with spectral measure-

ments. This work was funded in part by a grant to REM through

the NASA Mars Science Laboratory Participating Scientist Program.

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http://dx.doi.org/10.1126/science.1122659

http://dx.doi.org/10.1016/0016-7037(93)90182-V

http://dx.doi.org/10.1029/2012JE004145









http://dx.doi.org/10.1029/2011JE004036















http://dx.doi.org/10.1029/2011JE003837









http://dx.doi.org/10.1029/2009GL041870




http://dx.doi.org/10.1029/2009JE003342







http://dx.doi.org/10.1029/2012GL053180











http://dx.doi.org/10.1029/2003JE002179









http://dx.doi.org/10.1029/2007JE002920















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