Deconvolution of mineral absorption bands an improved approach

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 95, NO. B5, PAGES 6955-6966,MAY 10, 1990
Deconvolutionof MineralAbsorptionBands'
An ImprovedApproach
JESSICAM. SUNSHINE,CARLEM. PIETERS,AND STEPHENF. PRATt
Departmentof GeologicalSciences,BrownUniversity,Providence,RhodeIsland
Although visible and near-infraredreflectancespectracontain absorptionbands that are
characteristicof the compositionand structureof the absorbingspecies,deconvolvinga complex
spectrumis nontrivial. An improvedapproachto spectraldeconvolutionis presentedhere that
accuratelyrepresentsabsorptionbandsasdiscretemathematicaldistributionsandresolvescomposite
absorptionfeaturesinto individualabsorptionsbands. The frequentlyusedGaussianmodelof
absorptionbandsisfirstevaluatedandshowntobeinappropriatefortheFe2+electronictransition
absorptionsin pyroxenespectra.Subsequently,a modifiedGaussianmodelis derivedusinga power
law relationshipof energyto averagebondlength. ThismodifiedGaussianmodelsuccessfullydepicts
the characteristic0.9-}xrnabsorptionfeaturein orthopyroxenespectrausinga singledistribution.The
modifiedGaussianmodelis alsoshownto providean objectiveand consistenttool for deconvolving
individualabsorptionbandsin the more complexorthopyroxene,clinopyroxene,pyroxenemixtures,
andolivinespectra.TheabilityofthisnewmodifiedGaussianmodeltodescribetheFe2+electronic
transitionabsorptionbandsin both pyroxeneand olivine spectrastronglysuggeststhat it be the
methodof choicefor analyzingall electronictransitionbands.
INTRODUC•ON
At visible and near-infrared wavelengths, reflectance
sP;ectraof mineralscontainabsorptionfeaturesthatare
characteristicof the compositionand crystalstructureof the
absorbing species [e.g.,Burns, 1970a; Marfunin, 1979;
Adams, 1974, 1975; Hunt and Salisbury,1970]. As such,it
is possibleto use remotely sensedreflectancespectra as a
basis for estimatingthe mineralogy of both terrestrial and
extra-terrestrialsurfaces[e.g., McCord et al., 1988]. Recent
advancesin technologyhave resultedin the developmentot
imaging spectrometerswhich allow the remotecollectionof
reflectancedata at high spectraland spatialresolutionover a
broad spectral range. Current and future exploration
programscall for an increaseduse of imagingspectrometers
from telescopic, airborne, and orbiting platforms to
determine the mineralogy of various terrestrial and extra-
terrestrialtargets.Thelargeinfluxof spectralreflectance
measurements from such missions requires a consistent
approachto the classificationof absorptionfeatures.
In order to characterize absorption features, it is
necessary to understand the physics that control the
absorption process. However, absorptionsin geologic
surfaces containing intimately mixed mineralogies are
complex and only partially understood. Electromagnetic
energy interacts with a particulate interface through a
combination of first surface (or specular) reflection
transmission,absorption,diffraction, and multiple scatteri•
from adjacent particles [e.g., Wendlandtand Hecht, 19(6
KOrtum, 1969]. Although reflectancespectraat visible ar•
near-infrared wavelengths contain diagnostic informatior.
deconvolving the composite spectral signals of multi-
component target surfacesto reveal individual components
and their modal mineralogyis nontrivial.
Copyright1990 by the AmericanGeophysicalUnion.
Papernumber89JB03621.
0148-0227/90/89JB-03621 $05.00
Avarietyof theoreticalandempirical'approacheshave
beendevelopedto addressthisproblem.Theuseof Hapke's
theoretical radiative transfer model to the extraction of modal
abundancefrom reflectancespectraof geologic materials
[Hapke, 1981; Clark and Roush,1984] hasbeentestedin the
laboratory [e.g., Johnsoneta!., 1983; Clark, 1987; Nelson
andClark,1988;Mustardand•Pieters,1989]andthefield
[e.g., Mustard and Pieters, 1987] andhasbeenshownto be
abletodeconvolvemixturespectratoWithinanaccuracyof
approximately5%. This methodrequiresdetailedknowledge
of the spectral reflectance properties of the end-member
componentsin mixturespectraandis thusmostsuccessfUl
for applicationsthat allowrepresentativesamplesto be
obtained.The practicalapplicabilityof the Hapkemodelto
regions where ground truth is tinavailable can be greatly
enhancedif it is utilized in conjunctionwith other methods
that can determineprobablemineralogicend-members.
Empirical approachesto the extractionof mineralogical
information from the absorption features in reflectance
spectrahave also been undertaken. A variety of successful
schemeshave been developedthat quantify observedtrends
in albedo and/or absorption characteristicsfor specific
minerals in carefully prepared laboratory mixtures [e.g.,
Clark, 1981;Cloutiset al., 1986;Gaffey, 1986]. While
these empirical studies are usefUl as diagnostic tools for
mixtures of specific minerals, they are limited to
applicationsthat only involve the mineralsstudied.
A more general method of spectral deconvolution is
presented here, which resolves spectra into individual
absorption bands by representing these absorption bands
with discrete mathematical distributions. Use of this
quantitative deconvolutionmethod for a spectrum of an
unknownmaterial is dependentonly the spectrumitself. It
thusprovidesan objectiveand consistenttool for examining
the individual absorptionbandsin spectra. Interpretationof
the resultant absorptionbands can then be made based on
independentempirical and/or theoreticalstudies.
Spectraof pure mineral separatesof olivine, pyroxene,
and mixtures of pyroxenes were obtained in order to tesl
6955

6956 SUNSHINEETAL.: DECONVOLUTIONOFMINERALABSORFrIONBANDS
these deconvolution methods. The Gaussian model of
absorptionbandsis discussedandevaluatedin detailin the
following sections. Although it has commonly been
invoked, the Gaussian model proves to be an inadequate
descriptionof the electronictransitionabsorptionsfoundin
pyroxenespectra.Subsequently,a modifiedGaussianmodel
is derived and shown to successfullymodel the electronic
transitionabsorptionsin pyroxene,pyroxenemixtures,and
olivine spectra. Preliminaryresultsfrom this study were
presentedby Sunshineet al. [1988, 1989].
GAUSSIANMODEL OFAB SORttriONBANDS
AND ITS ASSUMIrrIONS
The underlyingassumptionof the Gaussianmodel is that
absorption features observed in visible and near-infrared
spectra are composed of absorption bands which are
inherently Gaussian in shape. If this is true, spectral
featurescan be resolved into absorptionbandsby fitting
them with Gaussian distributions. Since Gaussian de-
convolution has produced results that often correlate with
specific real absorptionswithout requiring more complex
functions, this method has been used by several
investigatorsas a quantitativetool in spectralstudies[Burns,
1970b; Smith and Strens, 1976; Fart et al., 1980; McCord et
al., 1981; Clark, 1981; Singer, 1981; Clark and Roush,
1984; Huguenin and Jones, 1986; Gaffey, 1986; Roush and
Singer, 1986]. As pointedout by Roushand Singer [1986],
at the very least a Gaussiandeconvolutionshouldproducea
unique fit for a given mineral (under certain reasonable
restraints) and thus, even if not physically realistic, could
provide a useful aid in the analysis and classification of
absorptionfeatures.
Under the central limit theorem of statistics, a Gaussian,
or normal distribution, can be used to representall random
distributions given a large enough sample population. A
Gaussian distribution in a random variable x, g(x), is
expressedin terms of its center (mean) /A, width (standard
deviation) •, and strength(amplitude)s:
g(x)=s.exp{-(x-11)2}2CY2 (1)
In order to apply the central limit theoremto spectra,the
various electronic and vibrational processes that cause
absorption bands must each be randomly distributed in a
random variable x and result from a statisticallysignificant
numberof events. There are a sufficientnumberof photons
in incident electromagnetic radiation to satisfy the
assumption of a large sample population. Randomness,
however, is more difficult to justify but has been assumedtc
occur due to the effects of thermal vibrations and/or minox
variations in crystal structures. On the basis of the
examination of absorptionbands in transmissionspectra,it
has commonly been assumed that absorption bands are
distributedarounddiscreteenergies[e.g., Farr et al., 1980;
McCord et al., 1981]. If eachof theseassumptionshold, the
probability of an absorption can be considered to be
randomlydistributedin energy and an absorptionband can
be modeledwith a Gaussiandistribution(equation(1)), where
the randomvariablex is energy.
In transmissionspectra,absorptionbandsnot only occur
arounddiscreteenergiesbut alsoobeythe Beer-Lambertlaw:
I = I0 exp (-ad) (2)
where {z is the absorptioncoefficient and d is the optical
path length. At visible and near-infrared wavelengths,
absorption bands in reflectance spectra are expected to
behave similarly becausethe absorptionsprimarily occur as
radiation is transmittedthroughthe solid particles. In order
to representmathematicallythe reflectancespectraas linea•
combinationsof absorptionbands that occur arounddiscrete
energies,it is necessaryto carry out the Gaussianmodel of
absorptionbandsin naturallog reflectanceand energy.
In addition, observations of natural surfaces indicate that
absorption features in many reflectance spectra are
superimposed onto a base curve or "continuum" [e.g.,
McCord et al., 1981]. The physicalcausesof the continuum
are poorly understood. Often it is consideredto representa
combination of phenomena including first surface
reflectance,multiple scattering,wingsof strongultraviolet
absorptions,or absorptionsfrom materials with constant
spectral signatures [e.g., Burns, 1970a;Gaffey, 1976;
Pieters, 1978; Marfunin, 1979; Clark and Roush, 1984;
Huguenin and Jones, 1986; Yon and Pieters, 1988].
Althoughthe causesof continuaare difficult to determine,for
consistencythe continuausedin theseanalysesare linear
functionsof energy. In general,any reflectancespectrum
will contain several absorptionfeaturesand as such, the
Gaussianmodel of a spectrum consistsof multiple
distributions,superimposedonto a continuum.
EXPERIMENTAL APPROACH
Reflectancespectraof pyroxeneand olivine were usedto
assessthe validity of each deconvolution model. These
minerals were selected for analysis due to their relative
abundance among the rock-forming minerals of the solar
system and their well describedspectralcharacter[Burns,
1970a, b; Adams, 1974, 1975; Hazen et al., 1978; Rossman,
1980]. As seen in Figure 1, the spectrumof olivine is
dominatedby a complex absorptioncenterednear 1.0 gm,
while the pyroxene spectrum is dominated by its
characteristicabsorptionfeatures near 1.0 and 2.0 gm. In
both cases,these absorptionsare producedby spin-allowed
electronictransitionsof Fe2+in distortedoctahedral(M1and
M2) crystal field sites [Burns, 1970a]. Furthermore,the
centers of the characteristic pyroxene absorption bands
systematically vary with increasing iron and calcium
concentrationfrom 0.90 to 1.05 gm and 1.80 to 2.30 gm,
respectively [Adams, 1974; Hazen et al., 1978] and thuscan
be use to infer pyroxene composition.
The samplesused in this initial study are an enstatite
from Webster,North Carolina (Opx), a clinopyroxenefrom a
Hawaiian volcanic bomb (Cpx), and an olivine from
Hawaiian green beach sand (located near South Point). The
compositionsof theseminerals are given in Table 1. Hand-
picked mineral separatesof thesesampleswere crushedand
wet-sieved with ethanol into 45-75 }•m particle size
separates. In addition, seven mass fraction mixtures of the
two pyroxenes were created. Spectra of the olivine, the
pyroxenes, and the pyroxene mass fraction mixtures were
obtained from 0.325 to 2.600 }•m at 5 nm sampling
resolution using the RELAB bidirectional spectrometerand
are shown in Figures 1 and 2. The RELAB instrumentis
specifically designed to simulate, in the laboratory, the
viewing geometriesused in remote measurements[Pieters,

SUNSHINEETAL.:DECONVOLUTIONOFMINERALABSORPTIONBANDS 6957
0.80
0.60
0.40
0.20
0.00
45- 75 micronparticles
, , • I , , , I , , , I , , , I , , , I , ,
0.30 0.70 1.10 1.50 1.90 2.30
Wavelength In Microns
0.80
OrthopyroxeneSpectrum
0.60
o 0.40
0.20
45-75 micronparticles
0.00 ' ' ' • ' ' ' • ' ' ' • ' ' ' • ' ' ' • ' '
0.30 0.70 1.10 1.50 1.90 2.30
b WavelengthInMicrons
Fig. 1. (a) The reflectancespectrum(45-75 Ixmparticles)of the
olivineusedin this study. The olivinespectrumis dominatedby a
multiple absorptionfeature centerednear 1.0 Ixm. (b) The
reflectancespectrum(45-75 Ixmparticles)of the orthopyroxeneused
in this study. The pyroxene spectrumis dominatedby the
characteristicabsorptionfeaturescenterednear1.0 and2.0 !xrn.
1983]. A 30ø incident angle and a 0ø emission angle
(measured from the vertical) were chosen for these
measurements.
Analysis of these spectra was made possible by the
TABLE1. ChemicalAnal•rsisofMineralsUsedin ThisStudy
Clinopyroxene Orthopyroxene Olivinea
PP-CMP-21 PE-CMP-30 PO-CMP-25
SiO2 49.69 56.23 38.90
TiO2 0.79 0.02 0.03
AI203 6.40 1.02 0.46
F%O3 (asFeO) (asFeO) (asFeO)
FeO 6.29 8.53 10.7
MgO 15.14 32.81 47.20
CaO 20.92 0.41 0.41
Na20 0.40 0.01 0.00
K20 0.01 0.00 0.02
MnO 0.12 0.19 0.22
Total 99.76 99.22 98.71
Wo45.59 Wo0.78
En45.89 En87.03
Fs8.52 Fs12.19
Fo89
FromSinger[1981].
developmentof an enhancedGaussianfitting package. This
relatively fast routine, based on the nonlinear least squares
algorithm developed by Kaper et al. [1966] permits the
simultaneousanalysis of an entire spectrum. From initial
estimatesprovided by the investigator,the fitting procedure
iteratively adjusts the model parameters(Gaussian widths,
strengths, and centers, as well as continuum slope and
intercept)until a negligibleimprovementin the overall fit is
obtained. Additional Gaussian distributions can be added as
deemed necessarybased on an examination of the residual
error between the measuredand the modeledspectra. The
mathematicsof this procedure,as it specifically appliesto
reflectancespectra,are presentedin detail in the appendix.
A fit is consideredcomplete when the residual error,
definedas the root meansquare(RMS) error of the difference
between the model and the actual data (see the appendix,
equation(A14)), is of the same order of magnitudeas the
observational uncertainties. The observational errors in the
laboratory spectraused here are approximately0.25%. As
such,the fitting routineis terminatedwhen the RMS error of
1.20
1.00
o 0.80
0
o 0.60
r'r' 0.40
0.20
PyroxeneMixture Spectra
45-75 micronparticles
0.00
0.30 0.70 1.10 1.50 1.90 2.30
WavelengthIn Microns
Orthopyroxene
85/1 50px/Cpx
75/25 Opx/Cpx
60/40 Opx/Cpx
50/50 Opx/Cpx
40/60 Opx/Cpx
25/75 Opx/Cpx
15/85 Opx/Cpx
Clinopyroxene
Fig.2. Thereflectancespectra(45-75pmparticles)ofthepyroxenemassfractionmixtures.Theyaredisplayedfrom
bottomtotopwithincreasingorthopyroxenefractionanda 5%offsetin reflectancebetweeneachsuccessivespectrum.

6958 SUNSHINE ET AL.: DECONVOLUTION OFMINERAL ABSORPTIONBANDS
the residualis of order of magnitudeof tenthsof a percent,
or more likely, when the improvementof the RMS residual
error in several successive iterations is negligible (of
magnitude10'5orless).
INADEQUACIESOFTHEGAUSSIANMODELFOREI.FCWRONIC
TRANSmON ABSORPTIONBANDS
One goal of the investigation reported here is to
determineif Gaussiandeconvolutionsaccuratelyrepresent
physical phenomenon and to examine whether Gaussian.
analysis provides a consistent and unique method for the
extraction of compositional information from reflectance
spectra.
Thecomplexityof theFe2+ absorptionfeaturesfoundin
olivine spectra make them natural candidatesfor Gaussian
analysis. An olivine spectrum,the continuum,the Gaussian
distributionsthat comprise its fit, and the residual error of
that fit are shownin Figure 3a. Although the analysis is
0.00
-0.80
-1.60
-2.40
I FIMSerror=4.312•
-3.20
0.30 0.70 1.10 1,50 1.90 •.30
Fig. 4. The Gaussianmodel of the orthopyroxenereflectance
spectrum(45-75 }amparticles)usingonly singledistributionsfor the
absorptionnear 1.0 and2.0 !xm. This modelproducesa poorfit and
an unacceptablyhigh residualerror.
0.00
-0.80
-1.60
-2.40
' ' ' I ' ' ' I ' ' ' I ' ' ' IGaussian Model of Oilvine
_
RMS error = 0.645X
-3.20 , , • I • , , I • , , I , , , I
30.00 24.00 1 8.00 1 2.00 6.00
a Wavenumber[cm-1/1000]
carried out in natural log reflectance and energy, because
reflectance spectra are often displayed as a function of
wavelength,all subsequentdata will be shownin naturallog
reflectanceand wavelength. This causesthe continuumthat
is a linear functionof energyto appearcurved(Figure3b).
Gaussiananalysisof an olivine reflectancespectrumreveals
that the 1.0-I.tm absorptionfeature can be representedby
three Gaussian distributions [Singer, 1981]. This result is
expectedfrom transmissionstudiesthat indicatethatthe 1.0-
I.tm olivine featurearisesdue to threeindependentabsorption
bands [Burns, 1970b]. Unfortunately, olivine does not
provide a goodtest of the model becauseit requiresthree
distributions, and thus nine free parameters,all within the-
1.0-lxm region. Given this much flexibility, almost any
functioncouldprovidea reasonablefit regardlessof whether
or not it physically representsthe three absorptionbands
that comprisethe absorptionfeature.
0.00 - -
-0.80
• 0.00
-.6o
-o.8o
-2.40 or"
• -1.60
-3.20 , , , I , , , i , , . i , . , I , , I , , -•
A better test would be to model absorptionfeaturesthat
b
Fig. 3. (a) The Gaussianmodelof theolivinereflectancespectrum
(45-75 }xmparticles). In thisandall subsequentfiguresthemodel
componentsare (from bottom to top): the modeled spectrum
superimposedon to the reflectancespectrum,the continuum(dashed
line) that is a linear function of energy (wave number), the
individualGaussiandistributionsthat comprisethe model and the
residualerrorbetweenthe modelandthe actualspectra(offset 10%
for clarity). (b) The Gaussian model of the olivine reflectance
spectrum(45-75 !xmparticles). As in Fig. 3a butdisplayedas a
functionof wavelength.This causesthe continuumthat was a linear
functionof energyto appearcurved.
ssian Model of Orthopyroxene
. _
'0.30 0.70 1.10 1.50 1.90 2.30
WavelengthInMicrons
Z
RMS error = 0.611
-3.20 ' ' • ' ' ' • ' ' ' • ' ' ' • '
0.30 0.70 1.10 1.50 1.90 2.30
Fig. 5. The Gaussianmodelof the orthopyroxenereflectance
spectrum(45-75 !xmparticles)usingtwo Gaussiandistributionsfor
bothof the absorptionfeaturesnear 1.0 and2.0 !xm. Whilethis
producesan acceptablefit to the features,it is contraryto
expectationsthattheseabsorptionsareeachcausedby primarilyone
electronictransitionabsorptionband.

SUNSHINEETAL.:DECONVOLUTIONOFMINERALABSORFFIONBANDS 6959
are believed to result from single absorptionbands, such as
the absorption features near 1.0 and 2.0 •m in the
orthopyroxenespectrum. However, as shownin Figure4, a
Gaussiananalysisof the orthopyroxenereflectancespectrum,
using single distributions for each of the dominant
absorptionsproduces an obviously poor fit with an RMS
residual error of 4.32%. In order to reduce the residual error
to the order of magnitudeof the observationalerrors, it is
necessaryto use two Gaussiandistributionsfor both of the
absorptionfeaturesnear 1.0 and 2.0 •m (Figure 5). While
this producesan acceptablefit to the features,it is contrary
to our expectation that these absorptionsare each caused
primarily by one electronictransitionabsorptionband.
These apparently spuriousGaussians,suggestthat the
Gaussianmodel may not reflect the actualphysicalprocesses
of absorption. As suggestedby Singer [1981], this may be
a productof attemptingto model an inherentlyasymmetric
feature with symmetricGaussiandistributions. The Gaussian
model may nonetheless provide a consistent tool for
compositionalanalysisof spectra. In order to examine this
potential, a Gaussiananalysisof the spectraorthopyroxene,
clinopyroxeneandpyroxenemixtureswas performed.
The Gaussian deconvolution of the orthopyroxene
spectrum (Figure 5) consists of nine discrete Gaussian
distributions, including two Gaussians in each of the
dominant absorption features, superimposed onto a
continuum. Similarly, the Gaussian deconvolution of the
clinopyroxenespectrum,shown in Figure 6, also consistsof
a continuumandnine Gaussiandistributions. Here again,the
1.0-and 2.0-I.tm absorption features required multiple
Gaussians.The model fits to both of the pyroxenesproduce
RMS residual errors on the order of 0.5%.
On the basis of the Gaussian deconvolutions of these two
pyroxene end-member spectra, each of the seven mass
fractionpyroxenemixturespectraaremodeledusingGaussian
distributions with centers fixed at the wavelengths
determinedfor the pure orthopyroxeneand clinopyroxene
end-members. Although eachend-memberspectrumconsists
of nine Gaussians,the first four distributionsoccur at very
similar wavelengthsfor both end-members. These centers
are too close to distinguish in the mixture spectra and as
0.00
-0.80
-1.60
-2.40
-3.20
' ' ' I ' ' ' I ' ' ' I ' '
ConstrainedGaussianModelof'75'•• '
RMSerror=0.632•. . . I . . . I . . • I • • , I • . Ii
0.30 0.70 1.10 1.50 1.90 2.30
'''i'''I'''I'ConstrainedGaussianModelof'25'Y.' ' ''Cpx
• 0.00
• -0.80
o-1,6o///
• -240
Z
-3.20
0.30 0.70 1.10 1.50 1.90 2,30
b WavelengthInMicrons
Fig. 7. (a) The Gaussianmodelof the 75% orthopyroxene25%
clinopyroxenemixture reflectancespectrum(45-75 !am particles).
(b) The Gaussian model of the 25% orthopyroxene 75%
clinopyroxenemixture reflectancespectrum(45-75 Ixm particles).
These models are constrained to have the Gaussian distributions with
centersfixed at the wavelengthsdeterminedfor the pyroxeneend-
memberspectra. Under suchconstraints,thesemodelsof pyroxene
mixture spectra reflect the proportion of clinopyroxene in each
sample.
aussianModel of linopyroxene
o.oofg .
.• -1.60
• -2,40•// -
•-- I/ RMSerror=0.444•320
0.30 0.70 1.10 1.50 1.90 2.30
Fig. 6. The Gaussianmodel to the clinopyroxenereflectance
spectrum(45-75 •tm particles). Note that multiple Gaussian
distributionsarerequiredfor bothof theabsorptionfeaturesnear1.0
and2.0 pro.
such,eachmixture spectrumis modeledwith 14 (rather than
18) Gaussianssuperimposedonto a continuum. A Gaussian
analysisof the mixturespectraundertheseconstraintsyields
RMS residual errors on the order of 0.5%. The constrained
model fits to the 75/25 Opx/Cpx and 25/75 Opx/Cpx
mixturespectraareshownin Figure7, asexamples.
While this constrainedGaussian analysis produces
acceptablefits, to truly be useful,Gaussiananalysismustbe
able to extractaccurateresultsfrom the spectraof pyroxene
mixtureswithoutpredeterminedknowledgeof their spectral
components. To assessthe feasibility of this, the mixture
spectra were modeled with the constraints on the Gaussian
centers removed. For example, with 14 unconstrained
Gaussiansthe resultingfit to the 75/25 Opx/Cpxspectrum,
shown in Figure 8, is still excellent. However, the Gaussian
parmeters of this unconstrainedmodel bear no resemblance
to thoseof the known pyroxeneend-membercomponents.
As such,the Gaussiananalysisof mixture spectradoesnot
reveal any useful compositionalinformation.
The inability of the Gaussian model to accurately

6960 SU'NS• ET AL.: DECONVOLUTION OFMn',,?ERALABSORPTIONBANDS
f'''I'''I'''I'''5I'''I''UnconstrainedGaussianModelof 7 • Opx 25• Cpx
• -0.80
n"
• -1.60
-2,40
FIMS error = 0.403X
-3.20 , , , , , , , • , , , I , , , • , , , • ,
0.30 0.70 1.10 1.50 1.90 2.30
Fig. 8. The Gaussianmodel of the 75% orthopyroxene25% clino-
pyroxenemixturereflectancespectrum(45-75 }xmparticles)with the
constraintson the band centersremoved. While this producesan
acceptablefit, this Gaussiandeconvolufionbearsno resemblanceto
the deconvolufionsof the known pyroxeneend-members.
characterizepyroxenemixture spectrais not surprising,since
it requires multiple Gaussian distributions in order to
accurately describe single electronic transition absorption
bands. Using spuriousGaussiandistributionsto model the
pyroxene end-memberspectraresultsin 44 free parameters
(14 Gaussian distributionseach with three free parameters
and a continuum with two) for each pyroxene mixture
spectrum.Thisover-abundanceof parametersnotonlyslows
theprocessingtimeconsiderably,but increasingthenumber
of freeparametersalsoincreasesthelikelihoodof obtaining
a nonuniquesolution.
Theseproblemswere foreseenby Kaper et al. [1966].
Kaper et al. pointedout that mathematically,a discrete
spectrumcontainingobservationalerrorshas,in general,no
uniqueGaussiande.convolution.Thenonlinearleastsquares
minimumizationprocedureused for Gaussiandeconvolution
(see the appendix)merely requiresthat a relative, not an
absolute,minimumbe obtained.It is quitepossibleto start
from different initial estimates and arrive at different relative
minima. In addition,while minimizingthe RMS residual
errorprovidesan objectivemeasurementfor determiningthe
"best"fit, it is in generalalwayspossibleto reducetheRMS
residualerrorby includingadditionalGaussians.Kaperet al.
do however,suggestthatif thesignalto noiseratiois high
andthecomponentsarewell separated,theremayin factbe a
uniquesolutionandrequiringthatthe RMS residualerrorbe
of the order of the observational errors is sufficient.
It thereforeappearsthatin its presentform, theGaussian
modeldoesnot correspondwith actualphysicalabsorptions
nor providea feasibleapproachto extractingcompositional
informationfromunknownsamples.
MODIFIED GAUSSIANMODEL
If a mathematicalmodelcan be developedwhich can
describea singleabsorptionbandwith onlyonedistribution,
many of the problems of the Gaussian model could be
overcome. Such a model would decrease the number of
distributionsnecessaryto describea spectrum,make the
spectraldeconvolutionmorestable,andbe morephysically
realistic. Given the limitations of the Gaussian model for
pyroxene electronic transition absorptions,it is appropriate
to reexamineits numerousassumptionsto determinewhether
such a model can be derived. One fundamentalassumption
of the Gaussianmodel is that energy is the random variable
for all absorptions (charge transfer, electronic transition,
molecular bending, stretching, and vibrational phenomena).
However, for electronic transition absorptions,the energy of
absorption is a function of the distortion and the average
ligand-ion bond length of the crystal field site [Burns,
1970a]. In a crystal field site, the averagebondlength will
vary due to random thermal vibrationsand variationsin the
crystal site. Thus, the random variable for electronic
transition absorptionbands is not the energy of absorption
but the averagebond length.
With such an understanding,one can invoke the central
limit theorem of statistics and consider the average bond
lengthsto be Gaussiandistributed. Since the averagebond
length determines the energy of absorption, a Gaussian
distribution of average bond lengths will map into a
distributionof absorptionenergies.The crystal field theory
descriptionof electronic transition absorption[e.g., Burns,
1970a; Marfunin, 1979], suggeststhat the absorptionenergy
(e) and the averagebond length (r) are related by a power
law:
eo• r n (3)
Utilizing this power law relationship (equation (3)), a
Gaussiandistributionof averagebondlengthscanbe mapped
into a "modified" Gaussian distribution of absorption
energiesm(x):
{-(xn-lln)2}m(x)=s. exp (4)
Changingthe exponentof (xn-#n) correspondsto alteringthe
symmetryof the distribution, i.e. the relative slopeof the
right and left wings.
The bestvalue of the exponentn in equation(3) can be
derived empirically. The 0.9-gm orthopyroxeneabsorption
featureis chosenfor thisderivationsinceit is thoughtto be
dominated by a single electronic transition band.
Furthermore, in order to remove the complications of
multiple scattering,a single-crystaltransmissionspectrumis
usedto derivethe appropriatevalueof n. The RMS residual
error of a single modified Gaussiandistribution fit to the
0.9-gm absorptionfeatureof the [i-transmissionspectrumof
the Bamble enstatite,using several different values of n, is
shown in Figure 9a. The transmissionabsorptionband
appearsto be best fit using a modified Gaussiandistribution
with n -- -1.0.
A similar test for the 0.9-}.tm absorptionfeature in
reflectancespectrumof the Websterite enstatiteis shown in
Figure 9b. As with the transmissionabsorptionfeature,
n---1.0 results in the smallest RMS residual error. A
comparisonof the fits of various distribution models to the
0.9-}.tmreflectanceabsorptionbandis shownin Figure 9c.
The Gaussianmodel (n = 1.0) producesan unacceptablefit,
but a progressiveimprovementoccurs(e.g., n =-0.2) until
the best fit is achieved(n = -1.0).
Using singlemodifiedGaussiandistributionsof the form
{-(x-l-it-I)2}m(x)=s. exp (5)

SUNSHINEET AL.: DECONVOLUTION OFMINERAL ABSORFHON BANDS 6961
0.08
0.07
O.06
0.05
o.o•
0.03
0.02
O.Ol
Residual Error of Models:
TransmissionSpectrum
n= 1.0
Gaussian Model
d n=-0.2ß
ß
n=- 1.0
Modified Gaussian Model
ß I ' I ' I ' I '
0 1 2 3 4
0.08
0.07
0.06
0.05
o.o•
0.03
0.02
0.01
b
Residual Error of Models:
ReflectanceSpectrum
n= 1.0
Gaussian Model
•= ßtn n=-0.2
n=- 1.0
Modified Gaussian Model
ß i ß i ß i ß i ß i
-1 0 1 2 3 4
1.00
0.50 --
--
--
_
-1.oo -
--
-1.50 '
0.60
0.00
o
'• -o.5o
z
I I I I I '
0.9 micron Orthopyroxene
Absorption Band
I I I I I
0.90
Residual Errors
n= 1.0
(GaussianModel)
n = -0.2
n = -1.0
(Modified GaussianModel)
'+' ReflectanceSpectrum
(data points)
1.20
Fig.9. Theresidualerrorbetweenthe0.9-ginorthopyroxeneabsorptionfeaturemodelsusingdifferentvaluesof n. (a)
The [;-transmissionspectrum,(b) the 45-75 !xmparticlesizereflectancespectrum,and(c) the <45 gm particlesize
reflectancespectrum.In all easesn•. -1.0producestheoptimumfit. Whilen = -1.0 doesnotproducethelowestRMS
residualerror in all cases,the differencebetweenthe RMS residualerrorof the modelwith n = -1.0 andthe lowestRMS
residualerror is negligible.
producesextremelyaccuratefits to the 0.9-gm pyroxene 1/r dependenceas the Coulombicpotentialenergyof the
absorption band in both transmission and reflectance crystalfield site.
spectra. Modified Gaussiandistributionsthusappearto be a
more physically realistic model of electronic transition
absorptionbands.
On the basis of these results, it appears that the
absorption energy (e) and the averagebond length (r) are
inverselyrelated and that equation(3) shouldread
eo,r-1 (6)
It is interestingto note that this relationshiphas the same
RESULTSOFTHEMODIFIEDGAUSSIANMODEL
To test this new model further the multiple featuresin
extendedpyroxenetransmissionandreflectancespectrawere
fit usinga versionof thefittingroutinewhichdeconvolves
eachspectruminto modifiedGaussiandistributions(seethe
appendix).Thefit of thecomplete[•-transmissionspectrum
of the Bamble enstatite,obtainedwith the modified Gaussian

6962 SUNSHINEET AL.: DECONVOLUTION OFMINERAL ABSORFFIONBANDS
t ModifiedGaussianModelofOrthopyroxeneModifiedGaussianModelofClinopyroxene
.o0.00 •0,oo'"'•• •....-0.80 • -o.8o .•• • _. ....
• -1.6o-1.60
• -z4o-2.40
-3'20'' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '• -3'2•.3•' 0.7• 1.10 1.50 1.90 2.300.30 0.7 0 1.10 1.50 1.90 2.30
Fig. 10. The modifiedGaussianmodelof the orthopyroxene
transmissionspectrum.Only single distributionsare required for
both of the absorptionfeaturesnear 1.0 and 2.0 Ixm.
model is shownin Figure 10. This deconvolutionrequires
only one modifiedGaussiandistributionfor each of the
absorptionfeaturesnear 1.0 and 2.0 I,tm. Nonetheless,the
fit is quiteremarkable.The modifiedGaussianmodelof the
orthopyroxene(Websterire)reflectancespectrmnalsorequires
only one distribution for each of the characteristic
absorptionfeaturesand also producesan excellentfit as
shownin Figure11. Similarly,themodifiedGaussianmodel
of theclinopyroxenereflectancespectrmnis shownin Figure
12. Here again, the absorptionfeature near 2.0 Ixm is
describedby a singlemodifiedGaussiandistribution.
An essential test of the modified Gaussian model is
whether it can correctly identify and characterize
superimposedabsorptionsin mixturespectra.Unconstrained
modifiedGaussiananalysesof the 75/25 and25/75 Opx/Cpx
mixture spectraare shownin Figure 13. Note that the
distributionsresultingfrom this unconstrainedfit correspond
directlyto thosefoundin the spectraof the orthopyroxene
and clinopyroxene used to create these mixtures. As
expectedfrom the proportionof orthopyroxenein the
mixture, the orthopyroxene features are significantly
' ' ' I ' ' ' I ' ' ' I ' ' ' I ' ' ' I '
fModifiedGaussianModelofOrthopyroxeneo 0.00
• -0.80
• -1.60
.
•-2.40 1
-3.20
0.30 0.70 1.10 1.50 1.90 2.30
Fig. 11. The modified Gaussianmodel of the orthopyroxene
reflectancespectrum(45-75• particles).Onlysingledistributions
arerequiredfor bothof theabsorptionfeaturesnear1.0and2.0 •trn.
Fig. 12. ThemodifiedGaussianmodelof the clinopyroxene
reflectancespectrum(45-75Ixmparticles).
stronger than those of the clinopyroxene in the 75/25
Opx/Cpx mixture, while the clinopyroxenefeaturesdominate
the 25/75 Opx/Cpx mixture.
The entiresuiteof 45-75 I•m particle size orthopyroxene,
clinopyroxene,and mass fraction mixtures is also fit using
•.'''I'''I'''I'''I'''I'aussianModel of 75% Opx 25% Cpx
• 0.00 '
• -0.80
o -1.60
"• -2.40
0.30 0.70 1.10 1.50 1.90 2.30
a WavelengthInMicrons
dified GaussianModel of 25 Opx Cpx
• 0.00
• -o.8o
ø•1-1.6o
'• -2.401-
• [] UNCONSTRAINED
•/ , , , RM•error=0.•39X-320
ß 0.30 0.70 1.10 1.50 1.90 2.30
d WavelenqthInMicrons
Fig.13. (a) The Gaussiandistributionmodelof the 75%
orthopyroxene25% clinopyroxenemixturereflectancespectrum(45-
75 •m particles).(b) The modifiedGaussianmodelof the 25%
orthopyroxene75% clinopyroxenemixturereflectancespectrum(45-
75 I•m particles). Unlike the Gaussianmodel, thesefits are
unconstrained,yet still reflect the proportionof clinopyroxenein
each sample.

SUNSHINEETAL.: DECONVOLUTIONOFMINERALABSORFFIONBANDS 6963
0.00 --
-
-
-
-0.40 -
.
-0.80 -
-
.
-
-1.20 -
_
-
-
-1.60 --
-
-
-
-2.00
0.80
x
x
PrimaryAbsorptionBandsin PyroxeneMixtureSpectra:
Unconstrained Modified Gaussian Model
+
+
x A
Clinopyroxene
15/85 Opx/Cpx
25/75 Opx/Cpx
40/60 Opx/Cpx
50/50 Opx/Cpx
60/40 Opx/Cpx
75/25 Opx/Cpx
85/15 Opx/Cpx
Orthopyroxene
, , , I • , • I • , • ,
1.20 1.60 2.00 2.40
Fig.14. Therelativeintensitiesof themodifiedGaussiandistributionsthatdescribetheprimaryabsorptionbandsin
thepyroxeneandpyroxenemixturespectra(45-75!amparticles).Notethemonotonicvariationinbandstrengthwith
proportionof clinopyroxenein the sample.UnliketheGaussianmodel,thecenterof thedistributionsarenot
constrainedto occurat thewavelengthsdeterminedfor theorthopyroxeneandclinopyroxeneend-members.
unconstrained modified Gaussian distributions to determine if
the new model reveals systematic and useful trends that
correspondto the relative proportionof orthopyroxeneand
clinopyroxenefound in each sample. Shown in Figure 14
are the strengthsand band centersof the primary absorption
bands derived for the entire suite of pyroxene mixture
spectra. The band centers for each sample spectrum are
determined independently yet occur at almost identical
wavelengthsfor all mixture spectra. In addition,there is a
monotonic variation in band strength which correspondsto
the modal mineralogy of the sample. It should be noted
that, unlike the Gaussiananalysis,the modified Gaussianfits
to the pyroxenemixture spectraare not constrainedto have
the bands centers occur at the same wavelengths as those
found for the spectraof the pyroxeneend-members.The fact
that the bands centersof the mixture spectra are consistent
with the bands centers of the two pyroxene end-member
spectralendsfurthercredenceto the model.
On the basis of the success of the modified Gaussian
model of electronic transition absorptionbands in pyroxene
spectra,a similar approachwas appliedto the more complex
electronic transitionsin olivine spectra. Given the overlap
of absorptionsin the olivine spectrum, it is not surprising
that the olivine spectrum is also easily fit with modified
Gaussiandistributions(Figure 15). Yet, as can be seenby
comparingthe GaussianandmodifiedGaussianmodelsof the
olivine reflectance spectrum (Figures 3 and 15), the two
methodsprovide different assignmentsof absorptionbands
to the complex 1.0-lxm feature. However, since the modified
Gaussian model is shown to be more appropriate for
pyroxeneelectronictransitionabsorptionbands,theresults
from the modified Gaussianmodel are expectedto provide a
0.00
-0.80
-1.60
' ' ' I ' ' ' I ' ' ' I ' ' '
Modified Gaussian Model of Olivine
I ! ! ! i ! ,
-2.40
-3.20 ' ' '
0.30
I
0.70
RMS error = 0,655%
, , I , , , I , , , I , , , I ,
1.10 1.50 1.90 2.30
Fig. 15. The modified Gaussianmodel of the olivine reflectance
spectrum(45-75 Ixmparticles).

6964 SUNSttINE ET AL.: DECONVOLUTION OF MINERAL ABSORPTIONBANDS
more appropriatecharacterizationof the absorptionbandsin
olivine spectra.
CONCLUSIONS
Deconvolving spectra into modified Gaussian
distributions (equation (5)) is a physically realistic and
practicalmodel for characterizingthe absorptionfeaturesof
both transmissionand reflectancespectra. The ability of the
modified Gaussianmodel to fit single electronic transition
absorptionbands with only one distribution representsa
significantimprovementover the Gaussianmodel. Modeling
single absorptionbandswith single distributionsis not only
more physically realistic, but requiring fewer distributions
decreasesboth the likelihood of nonuniquesolutionsand the
processing time.
Since absorptionband assignmentsprovide information
about the nature of absorption sites, it is of the utmost
importancethat the correctdeconvolutiontechniquebe used
to describe absorptionsin crystals. The ability of the
modified Gaussianmodel to describethe absorptionbandsin
both pyroxeneand olivine spectrastronglysuggeststhat it
be the method of choice for analyzing electronic transition
bands.
The success of the modified Gaussian model for electronic
transitionabsorptionbandsis quite encouraging. It provides
an objective and consistent method for deconvolving
individual absorptionbandsfrom the spectraof pure samples
and mixtures;as such,the analyticalcapacityof the modified
Gaussian model has great potential. It can provide an
invaluable too1 for quantifying compositional trends from
laboratory data and in identifying mineral constituentsin
remotelyacquiredspectra.
Although this study has shown that the Gaussianmodel
is an inadequate description of electronic transition
absorption bands, models for other absorptionphenomena
must be critically evaluated and, if necessary, alternate
models must be derived. As is the case for the modified
Gaussian model of electronic transition absorptionbands,
formulation of new models will undoubtedlyfocus on the
determinationof the randomvariable of absorptionand how
it relatesto energy.
APPENDIX: MATHEMATICS OFTHE GAUSSIAN MODEL
Mathematically, the Gaussian model of spectral
convolution can be describedby a nonlinear least squares
analysis. The detailsof thisprocedurehavebeenadaptedfor
reflectance spectra based on the mathematics outlined by
Kaper et al. [1966].
A spectrum,the percentreflectancemeasuredat rn discrete
wavelengths,is first definedas a discretefunction,
y(x/c), k=l to rn (A1)
The spectrumis then approximatedas a superpositionof
GaussiandistributionswitheachindividualGaussian,g(xk)i .
expressedin termsof its center(mean)/ti , width(standard
deviation)•i, andstrength(amplitude)si:
g(xk)i=si'exp{-(xk-Ili)2}2ai2 (A2)or
n
Y(Xk)• G(xk)= • g(xk)i
i= 1
(a3)
Sinceeachg(xk)i hasthreeparameters,!•i , •i, si , andthere
are a totalof n Gaussians,the spectrumis fully describedby
3n parameters,or in vector form as a single 3n vector of
parameters,
P -- (Pl ,P2 ..... P3n) (A4)
The accuracyof the approximationis determinedby the
residualdifferencebetweenthe actualspectrumand its model
andis measuredbasedon a sumof squaresof the residuals,S,
where
m
s-= [y(xD- g(xD]2
k=l
(AS)
To producea bestfit to the spectrum,thesumof squaresof
theresiduals(A5) mustbe of thesameorderof magnitudeas
the observationalerrors. This is determinedby a least
squaresminimization of S (A5) with respectto the model
parametersp (A4). This is accomplishedby solvingthe
normal equations
V S = 0 (AOa)
where
V • , , ooo
•SP1 •SP2 •SP3n
and
a g(xk)i (Xk-/ti)
a g(xk)i (Xk_/li)2
a g(xk)i
•Ssi si
or, from (A5),
m
[y(xD-G(x,)]VG(xk)=0
k=l
(A6b'
Thesenormalequations(A6),forma setof 3n equationswith
3n unknowns,p.
However, since G = G(x•:)is itself nonlinearin p, the
normal equations (A6) are also nonlinear and therefore
cannot be solved directly. Instead, an iterative processis
usedto approximatep. Startingfromaninitialestimate,Po,
a sequenceof vectors{pj}(j denotingthejth iteration)are
calculatedsuchthat they convergeto a vectorp thatsatisfies
(A6). Formally,G = G(xk) maybe expandedintoa Taylor
seriesinApj,where
Apj=pj+1-Pj
and
= + +... (^8)
Here,thehigherordertermsaredisregardedaspj approaches
p. Under this expansion,the normal equations(A6b) can
be rewritten as

SUNS• ET AL.: DECONVOLU'IIONOF• ABSORFHONBANDS 6965
or
m
Z [Y(Xk)-GJ(xk)-VGfixk)ApJ]VGJ(xk)=0
k=l
m
k=l
(A9a)
where C = ax + b, a is the slope, and b is the intercept of
the continuum. The only effect this has on the mathematics
outlined above is to add two additional model parameters,
continuum slope and intercept to the problem. Doing so
requiresthat
P -- (Pl ' P2..... P3n+2)
and
m
k=l
(A9b)
Invokingmatrix notation,the left-handsideof (A9b) canbe
written as
where
DjApj (A10)
O nm
6j(xD ß
The right-handsideof (A9b) canbe expressedin termsof the
sumof squaresof theresiduals(A5) as
where
-1/2 V Sj
m
VSj=-2Z [Y(Xk)-Cj(Xk)jlVCj(Xk)k= 1
where
and
8
8
•P3n+2
8
--= 1. (A17)
8b
Thus, applyingthis leastsquaresprocedure((A1)-(A14))
using equations (A16) and (A17), a spectrum can be
deconvolvedinto multiple Gaussiandistributionswhich are
superimposedonto a continuum.
The nonlinear least squaresanalysis for deconvolvin&
spectrawith modified Gaussiandistributions(see modified
gaussianmodel section) remains essentially as indicated
above. However, the Gaussian distribution (A2) must be
(A11)replacedwithamodifiedGaussiandistributionoftheform
_(xk-l_lti-1)2}m(xk)i=sißexp 2•yi2
Combiningequations(A10) and (All), equation(A9b)
becomes
(A18)
DjApj=- I/2V Sj
or
-1
Apj=Dj (- 1/2VSj) (A12b)
Equation(A12b)isthenusedforthenextiterationofp with
Pj+I = Pj +Apj (A13)
andthe entireprocessis repeatedfor j =j+l. Terminationof
the procedure takes place when there is a negligible
improvementin the root mean square(RMS) error of the fit,
where
RMS--dS/m (A14)
However,thisprocedurehasnot yet accountedfor the fact
that reflectance spectra, as indicated earlier, should be
modeledin natural log reflectanceand energy. In order to
accomplishthis we must transform equation (A3) into a
discrete function in energy (x = energy) and log spaceso
that
tny(xp a(x) = g(xDi (A5)
i= 1
One final adjustmentis necessaryto accountfor the fact
the Oaussians are superimposedonto a continuum. To
include the effects of a continuum, the discrete energy
spectrumshouldbe modeledas
InY(Xk)• G(xk)+C = •g(xk)i +C (A16)
i=1
(A12a) In addition,thepartialderivatives(A6a) mustberedefmedas
8 m(xk)i(xk-l_#i-1)#f2
8 m(xk)i (xk-llti-1)2
8 m(xk)i
= (A19)
8si si
Acknowledgments. We wish to thank G. Rossman for
generouslyprovidingpyroxenetransmissionspectra Detailed and
thoughtful reviews by P. Helfenstein and J. Mustard led to
significant improvementsin this manuscript. The microprobe
facility at Brown University is supportedby funds from the Keck
Foundationand is operatedby J. Devine. All reflectancespectra
wereobtainedusingRELAB, a multiuserfacility supportedby NASA
grant NAGW-748. The financial supportfor this research,NASA
grantNAGW-28, is greatlyappreciated.
REFERENCES
Adams,J. B., Visible and near-infrareddiffusereflectancespectraof
pyroxenesas applied to remote sensingof solid objectsin the
solarsystem,J. Geophys.Res.,79, 4829-4836, 1974.
Adams, J. B., Interpretation of visible and near-infrared diffuse
reflectancespectraof pyroxenesand otherrock formingminerals,
in Infrared and Raman Spectroscopyof Lunar and Terrestrial
Materials, editedby C. Karr, pp. 91-116, Academic,San Diego,
Calif., 1975.

6966 SUNSHINE ET AL.: DECONVOLUTION OF MI•N'ERAL ABSORFEION BANDS
Bums, R. G., Mineralogical Applications to Crystal Field Theory,
224 pp., CambridgeUniversityPress,New York, 1970a.
Bums,R. G., Crystalfield spectraandevidenceof cationorderingin
olivine minerals, Am. Mineral., 55, 1608-1632, 1970b.
Clark, R. N., Water frost and ice: The near-infrared spectral
reflectance 0.65-2.5 !•m, J. Geophys.Res.,86, 3087-3096,
1981.
Clark, R. N., Deconvolution of reflectance spectra of mineral
mixtures into component abundanceand grain size (abstract),
EOS Trans. AGU, 68, 464, 1987.
Clark, R. N., and T. L. Roush, Reflectance spectroscopy:
Quantitativeanalysistechniquesfor remotesensingapplications,
J. Geophys.Res.,89, 6329-6240, 1984.
Cloutis, E. A., M. J. Galley, T. L. Jackowski, and R. L. Reed,
Calibrations of phase abundance,composition,and particle size
distributionsfor olivine-orthopyroxenemixturesfrom reflectance
spectra, J. Geophys.Res.,91, pp. 11,641-11,653, 1986.
Fart, T. G., B. A. Bates, R. L. Ralph, and J. B. Adams, Effects of
overlappingoptical absorptionbandsof pyroxene and glass on
the reflectancespectraof lunar soils, Proc. Lunar Planet. Sci.
Conf., 11th, 719-729, 1980.
Gaffey, M. J., Spectral reflectancecharacteristicsof the meteorite
classes,J. Geophys.Res.,81, 905-920, 1976.
Gaffey, S. J., Spectral reflectance of carbonateminerals in the
visible andnear-infrared(0.35-2.55 microns): Calcite, aragonite,
and dolomite, Am. Mineral., 71, 151-162, 1986.
Hapke, B., Bidirectional reflectance spectroscopy,1., Theory, J.
Geophys.Res.,86, 3039-3054, 1981.
Hazen, R. M., P.M. Bell, and H. K. Mao, Effects of compositional
variation on absorptionspectraof lunar pyroxenes,Proc. Lunar
Planet. Sci. Con., 9th, 2919-2934, 1978.
Huguenin,R. L., and J. L. Jones,Intelligent informationextraction
from reflectance spectra: Absorption band positions, J.
Geophys.Res.,91, 9585-9598, 1986.
Hunt, G. R., andJ. W. Salisbury,Visible andnear-infraredspectraof
minerals and rocks, I, Mod. Geol., 1, 283-300, 1970.
Johnson,P.E., M. O. Smith, S. Taylor-George,and J. B. Adams,A
semi-empiricalmethod for analysisof the reflectanceof binary
mineral mixtures,J. Geophys.Res.,88, 3557-3561, 1983.
Kaper, H. G., D. W. Sinits, U. Schwarz, K. Takakubo,and H. Van
Woerden, Computer analysis of observed distributions into
gaussian components, Bull. Astron. Inst. Neth., 18, 465-487,
1966.
K6rtum, G., Reflectance Spectroscopy,336 pp., Springer-Verlag,
New York, 1969.
Marfunin, A. S., Physics of Minerals and Inorganic Materials, 340
pp., Springer-Verlag,New York, 1979.
McCord, T. B., R. N. Clark, B. R. Hawke, L. A. McFadden, P. D.
Owensby,C. M. Pieters,and J. B. Adams,Moon: Near-infrared
spectralreflectance: A first good look, J. Geophys.Res.,86,
10,833-10,892, 1981.
McCord, T. B., et al. (Ed.), Reflectancespectroscopyin planetary
science:Reviewsand strategyfor the future,NASA Spec.Publ.,
SP-493, 40 pp., 1988.
Mustard, J. F., and C. M. Pieters, Abundance and distribution of
ultramafic microbreccia in Moses Rock Dike: Quantitative
application of mapping spectroscopy,J. Geophys.Res.,92,
10,376-10,390, 1987.
Mustard, J. F., and C. M. Pieters, Photometricphasefunctionsof
common geologic minerals and applications to quantitative
analysisof mineralmixturereflectancespectra,J. Geophys.Res.,
94, 13,619-13,634, 1989.
Nelson, M. L., and R. N. Clark, Applicationof radiative transfer
theoryto the spectraof mineral mixtures(abstract),Lunar Planet.
Sci. Con., 19th , 846-847, 1988.
Pieters,C. M., Mare basalttypeson the front sideof the Moon: A
summaryof spectral reflectancedata, Proc. Lunar Planet. Sci.
Conf., 9th., 2825-2849, 1978.
Pieters, C. M., Strength of mineral absorptionfeatures in the
transmittedcomponentof near-infraredreflected light: First
resultsfrom RELAB, J. Geophys..Res.,88, 9534-9544, 1983.
Rossman,G. R., Pyroxenespectroscopy,in vol. 7, Rev. Mineral.,
Pyroxenes, edited by C. Prewitt, pp. 93-115, Mineralogical
Societyof America, Washington,D.C., 1980.
Roush,T. L., and R. B. Singer,Gaussiananalysisof temperature
effectson the reflectancespectraof maficmineralsin the 1-•tm
region, J. Geophys.Res.,91, 10,301-10,308, 1986.
Singer,R. B., Near-infraredspectralreflectanceof mineralmixtures:
Systematiccombinationsof pyroxenes,olivine, and iron oxides,
J. Geophys.Res.,86, 7967-7982, 1981.
Smith,G., andR. G. J. Strens,Intervalence-transferabsorptionin
somesilicate,oxideandphosphateminerals,in ThePhysicsand
Chemistryof Mineral and Rocks,editedby R. G. J. Strens,583-
612, JohnWiley, New York, 1976.
Sunshine,J. M., C. M. Pieters,andS. F. Pratt,Gaussiananalysisof
pyroxenereflectancespectra(abstract),LunarPlanet. Sci. Conf.,
19th, 1151-1152, 1988.
Sunshine, J. M., C. M. Pieters, and S. F. Pratt, Mathematical
deconvolutionof mineral absorptionbands (abstract),Lunar
Planet. Sci. Conf., 20th , 1087-1088, 1989.
Wendlandt,W. W., andH. G. Hecht,ReflectanceSpectroscopy,298
pp., Wiley-Interscience,New York, 1966.
Yon, S. A., and C. M. Pieters,Interactionsof light with rough
dielectric surfaces: Spectral reflectance and polarimetric
properties,Proc. Lunar Planet. Sci. Conf.,18th., 581-592, 1988.
C. M. Pieters,S. F. Pratt, and J. M. Sunshine,Departmentof
GeologicalSciences,Brown University,Providence,RI 02912.
(Received April 20,1989;
revised October 10, 1989;
acceptedDecember5, 1989.)

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