Machine learning for automatic identiﬁcation of new minor species Journal of Quantitative Spectroscopy & Radiative Transfer

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Journal of Quantitative Spectroscopy & Radiative Transfer

journalhomepage:www.elsevier.com/locate/jqsrt

Machine learning for automatic identiﬁcation of new minor species

Frédéric Schmidt

^a^,^∗

, Guillaume Cruz Mermy

^a

, Justin Erwin

^b

, Séverine Robert

^b

, Lori Neary

^b

, Ian R. Thomas

^b

, Frank Daerden

^b

, Bojan Ristic

^b

, Manish R. Patel

^c

, Giancarlo Bellucci

^d

, Jose-Juan Lopez-Moreno

^e

, Ann-Carine Vandaele

^b

aUniversité Paris-Saclay, CNRS, GEOPS, Orsay, 91405, France

bBelgian Institute for Space Aeronomy (BIRA-IASB), Avenue Circulaire, Brussels 3 B-1180 Belgium

cSchool of Physical Sciences, The Open University, Milton Keynes, MK7 6AA,U.K.

dINAF-Istituto di Astrofsica e Planetologia Spaziali, Rome, ITALY

eInstituto de Astrofísica de Andalucía CSIC, Spain

a rt i c l e i nf o

Article history:

Received 4 May 2020 Revised 14 August 2020 Accepted 28 September 2020 Available online 29 September 2020 Keywords:

Spectroscopy Atmosphere Data mining Machine learning Unsupervised Source separation

Non-negative matrix factorization

a b s t r a c t

One ofthe main diﬃculties toanalyze modernspectroscopic datasets isdue tothe large amount of data.Forexample,inatmospherictransmittancespectroscopy,thesolaroccultationchannel(SO)ofthe NOMADinstrumentonboardtheESAExoMars2016satellitecalledTraceGasOrbiter(TGO)hadproduced

~ 10millionsofspectrain ~ 20000acquisitionsequencessincethebeginningofthemissioninApril 2018until15January2020.Otherdatasetsareevenlargerwith ~ billionsofspectraforOMEGAonboard MarsExpressorCRISMonboardMarsReconnaissanceOrbiter. Usually,newlinesarediscoveredaftera longiterative processofmodel ﬁtting and manual residualanalysis.Herewe proposeanew method basedonunsupervisedmachinelearning, toautomaticallydetectnewminor species.Althoughprecise quantiﬁcationisoutofscope,thistoolcanalsobeusedtoquicklysummarizethedataset,bygivingfew endmembers(”source”)andtheirabundances.

The methodologyisthe following:weproposed away to approximatethedataset non-linearityby a linearmixtureofabundanceandsourcespectra(endmembers).Weusedunsupervisedsourceseparation informofnon-negativematrixfactorizationtoestimatethosequantities.Severalmethodsaretestedon syntheticand simulation data.Ourapproach isdedicatedtodetect minorspeciesspectra rather than preciselyquantifyingthem.Onsyntheticexample,thisapproachisabletodetectchemicalcompounds present inform of100 hidden spectraout of10⁴,at1.5 timesthe noise level. Results onsimulated spectraof NOMAD-SO targetingCH₄ show that detectionlimits goes inthe rangeof100–500ppt in favorableconditions.ResultsonrealmartiandatafromNOMAD-SOshowthatCO2 andH2Oarepresent, asexpected,butCH₄isabsent.Nevertheless,weconﬁrmasetofnewunexpectedlinesinthedatabase, attributedbyACSinstrumentTeamtotheCO2magneticdipole.

1. Introduction

In modern exploration science, one has to face a major chal- lenge:howtolearnsomethingnewfromanalyzingalargedataset collectionwhiletakingintoaccountwhatwealreadyknow.Ifthe currentknowledgeoverridestheanalysis,thediscoveryofnewel- ementsmaybe diﬃcult.Usually,intheﬁeld ofspectroscopy,one can compare laboratory spectra, model and observation spectra.

Goingbackandforthleadstodiscoveryofnewlinesbyidentifying unexpectedresidualsintheobservationdata(notexpectedbythe model).Sometimes,initialidentiﬁcationoflinescanbewrong.As

∗Corresponding author.

E-mail address: [email protected] (F. Schmidt).

an example,spectroscopicevidenceofatmospheric CO₂ icecloud wasreportedafterthediscovery ofan emission spikeata wavelength of4.3mm fromMariner 6and7infrared probings ofthe bright martian limb [16], but this spectral feature wasmistaken foraresonantscatteringbandofCO₂ ﬂuorescence[22].

Foronesinglespectrum,onecanusesimulationalgorithm(see for instance [9]). For large datasets, simplest ideas would be to scrutinize average spectra, or potential band depth distribution.

Unfortunately,inthecaseoflowsignal-to-noiseratio(SNR,deﬁned assignal /standarddeviationofnoise), suchmethodsfail(aswill beillustrated inthetoy example).Analyzingresiduals aftermod- elingisagoodmethodbutitrequiresalotofwork.

Severalstatisticaltoolswithvariousapproacheshavebeenpro- posed,such asthePrincipalComponentAnalysis(PCA)[12,28],or

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F. Schmidt, G.C. Mermy, J. Erwin et al. Journal of Quantitative Spectroscopy & Radiative Transfer 259 (2021) 107361

Independent Component Analysis (ICA) [8,30], but mostof them require a human operator to pick endmembersand trends since those methodsarenothingmorethanachangeofrepresentation.

Furthermore, noneof these methods guarantees positivity ofthe component (which are sometimes also called source), which can be problematicduringthe interpretation.Recently, advanced ma- chinelearningmethodsbasedonnon-negativematrixfactorization havebeenproposed[6,13,17,20,25,29].Thisapproachiscompletely different fromPCA/ICA: eachsource ispositive andrepresentsan endmember/atrend.Asourceisnotonespectrumextractedfrom thedatasetbutastatisticalreconstruction.Byusingthisapproach, the human operator doesn’t have to identifyendmembers/trends anymore, since theyare automaticallypicked by thealgorithm in form of source.Furthermore when there are statistical / spectral correlationsbetweensourcesPCA/ICAfailsbecauseitassumesor- thogonality/independence,whichisnotthecasefornon-negative matrixfactorization.

Basedonthisnewapproach,weproposeatool:

• togiveanoverviewandquicklysummarizealargeandcomplex spectroscopicdatasetwithsimplevariables

• todetectpotentialnewspectroscopicfeatures(unexpectedmi- norspecies,newabsorptionlines,...)

• tobeperformedinafullyblindway(withoutpriorinformation onneitherthespectra,northeabundances).

The target observation type of this study is solar occultation. This measurement principle has been proposed asearly as 1900, an interesting review was publishedby Smith andHunten [31]. Several recent instruments used this technique to investi- gatethecompositionoftheEarth’s(SCIAMACHY/ENVISATBovens- mann et al.[4]), Mars’ (SPICAM Bertauxet al. [2]) orVenus’ atmospheres (SPICAV Bertauxetal.[3]). Herewe will focusonthe recentNOMADinstrument[33],andespeciallytheSOchannel,de- signed tostudytheMartianatmosphereandits tracegases, such asmethane.IndeedthepresenceofCH₄onMarsisaveryhottopic fortheplanetarysciencecommunity[14,19,23].Inthepresentarti- cle,weproposetoapplythetoolforpotentialCH₄detection.Nev- ertheless,theapproachcanbeextendedtoothertypesofspectro- scopicmeasurements.

2. Dataset

We propose here to focus on the Nadir and Occultation for MArsDiscovery(NOMAD)instrumentonboardESA’sExoMarsTrace GasOrbiterandespeciallytheSolarOccultation(SO)channel[33]. NOMAD isa compact,high-resolution, dualchannel IR spectrometer (SOandLNO)coupled withahighlyminiaturized UV-visible spectrometer (UVIS),capableofoperating indifferentobservation modes:solaroccultation,nadirandlimb.

The SO channel operatesatwavenumbers from2320cm⁻¹ to 4550 cm⁻¹ (wavelength 2.2 to 4.3 μm), using an echelle grating withagroove densityof4lines/mmina Littrowconﬁgurationin combinationwithanAcousto-OpticTunableFilter(AOTF)forspec- tral order selection. The width of the selected spectral ranges is recordedby320spectels(spectralelement)andvariesfrom20to 35cm⁻¹dependingontheselecteddiffractionorder.Thedetector isanactivelycooled HgCdTeFocalPlaneArray.SOachievesan instrument lineproﬁle resolutionof 0.15cm⁻¹, corresponding toa resolvingpowerl/Dlofapproximately25000.Alldetails ofthein- strumentareavailablein[27]and[34].Theorderswiththemaxi- mumsensitivitytoCH₄are:119,134and136.Wewillusethedata fromthe beginning ofthemission inApril2018 until 15January 2020, incalibrationversion1p0a.Duetotemperaturechange,the spectral registrationvaries,producinga shiftup to ~ 10 spectels.

We correcteditbyaligning thefulldatasetto areferencespectra (arbitrarily choosen withthe maximum band depth ofwater) by

cross-correlation. No sub-spectel resamplinghas been performed but a simple shift. When the calibration will be improved, this step willmost probablybe replaced by a routine correction. The dataare available on the ESA/Planetary ScienceArchive aftera 6 monthsembargoperiod.

3. Method

Inthissection,weﬁrstdescribethedatapretreatmentrequired fornon-negativematrixfactorizationpurposefollowedbythedata miningmethod.

3.1. Datapretreatment

After calibration, the NOMAD SOspectra are in transmittance T=I/I₀, depending onwavenumber

ν

^, ^with^I ^the ^observed ^light

intensitytroughtheatmosphereandI₀thesolarspectrameasured outsidetheatmosphere.

Assumingthattheatmosphereishomogeneous,andthatmulti- plescatteringandrefractionarenegligible[4,31],theopticaldepth

τ

^is ^a ^linear combinationof E(

ν

⁾ ^the ^total extinction,and

^the

slantcolumndensity,foreachchemicalspeciesi:

τ ( ν )

=−logT

( ν )

≈

NS

i=1

Ei

( ν )

.

ⁱ⁺^MC

( ν )

⁽¹⁾

withN_S,thetotalnumberofspeciesandMC(

ν

⁾^a^modeled^contin-

uumdescribedbelow.

Theslantcolumndensity

^is^directly^related^to^the^total^num-

berofparticlesN(s)alongthelineofsights:

=

N

(

^s

)

^ds ⁽²⁾

Whiletheextinctionbygasisusuallyhighlystructured,absorp- tionbyparticles,scatteringbymoleculesandparticles,andalsore- ﬂectionatthesurfacearebroadbandfeatures. Suchlargefeatures aremodeled bya continuum MC(

ν

^), ôften^takenâsâ polynomial, thatisfilteredout.

The problem with this continuum removal rationale is that whentheopticaldepthislarge,theSNRisdecreasedandthenoise effectoncontinuumremovalampliﬁed(seeSup.Mat.).

Instead ofusing thisrationale, we propose to ﬁrst correctfor thecontinuumC(

ν

⁾ⁱⁿ^thetransmittancespace:

T^∗

( ν )

=T

( ν )

−C

( ν )

⁽³⁾

Thenconvertthespectraintoabsorbance:

X

( ν )

=1−T^∗

( ν )

⁽⁴⁾

Theﬁnalstepisthelinearmixture:

X

( ν )

≈

NS

i=1

Si

( ν )

.Ai (5)

with S_i(

ν

⁾ ^the ^source ^spectra ^and ^Ai the spectral abundance. In thisdescription, the physical meaning of S_i(

ν

⁾ ând Âi islost but the apparent SNR isdramatically increased, which ismuch more importantforour analysis.Nevertheless,assumptions required in Eq.(1)areusuallynot relevant.Radiativetransfermodelusedfor precisequantificationishighlynon-linear.

One has to consider that this unsupervised linear unmixing problemisalreadyverydiﬃcultformachinelearning.Solvingnon- linear model in a unsupervised way is a research area that is clearlynotsolvedyet.Inaddition,wewouldliketofocusonspec- tral detection, rather than quantiﬁcation. Thus, we will focus on S(

ν

⁾^much^more^than^A^.^We^will^show^that^for^linear,^but^also^non-

linearsimulationandrealdata,meaningfulS(

ν

⁾ ^can ^be^retrieved.

Duetonon-linearity,Amaydiffersigniﬁcantlyfromtruth,butthe

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big tendencies should berespected. After the quick-lookanalysis, estimatingS_i andA,onemustgo backtothe realdata.Themost trivialstrategy istopickthe spectraXout ofthecollection,with thehighestabundanceofaselectedsourceS_i.

Inthefollowing,wewillusethecontinuumestimationC(

ν

⁾^us-

ing asymmetric least square [7], with parameters :

ν

smooth=10³ andp=1−10⁻²,10numberofiterations.

3.2. Nonnegativematrixfactorization

Foracollectionofspectra,Eq.(5)canbewritteninmatrixform X_kj≈S_ki.A_ij,withithesourceindex(from1toN_S),jtheobserva- tionindex(from1toN_O)andkthewavenumberindex(from1to N_ν).Thus,onehavetoestimateSandA,byminimizingtheobjec- tivefunction:

F =

^X⁻^S^.^A

² ⁽⁶⁾

with

^.

^,^the^Frobenius^norm^(usual^L2norm).

Several algorithms have beenproposed to solve thisproblem, subjecttopositivity(bothSandAarenon-negative).Suchproblem is calledNonnegativeMatrixFactorization (NMF).Thisconstraint is important to keep the physical meaning, but also to promote sparsityofS(asignalissparsewhenmostofthevaluesareclose tozeroexceptseveralnon-zerovalues).LetS˙ andA˙ betheestima- tionofthosequantities.

MUWe proposetousetheMultiplicativeUpdates(MU)ofLee andSeung[20]acceleratedbyGillisandGlineur[13].Weusedthe convergence parameter

α

MU=1. Other alternative algorithms are possiblebutgiveequivalentresultssincetheyminimizethesame cost function.Thisalgorithm hastheadvantageofvery fastcom- putationtimebuttheresultmaydependoninitialization.

BPSS2 We propose to test another kind of algorithm: the Bayesian Prior Source Separation [6,24], that has beenoptimized [29], hereafter calledBPSS2. This algorithm has the main advan- tage to account for extra constraint : the sum-to-one or sum- lower-than-oneontheabundances (

iA_{i j}=1) thatalsopromotes sparsity of S. This algorithm, based on Monte Carlo approach is much moretimeconsuming. Oneapproachtoreduce thecompu- tation time is to select only relevant spectra out of the dataset [25], but then the statistics may be biased [29]. Thanks to the advances of computer capabilities, we propose to treat the full dataset. Thiskindofalgorithm isveryslow butsincethe formu- lationisBayesian,itconvergetowardanuniquesolution.

psNMF In orderto regularize the problemof Eq.(6), one can add anextrapenalizationtermtoenforcesparsityonA(onlyfew nonzeroselementsinA)[18]:

F =

^X⁻^S^.^A

²⁺

λ

^A

1 (7)

With

^.

1,theL₁norm.Theﬁrsttermiscalleddataattachment term(the usual squareddifference).The second iscalledregular- izationterm.Theproblemwiththisapproach,isthathyperparam- eter

λ

îs^not ^known ând^has^to ^be^tuned ^manually. Â^recentâp-

proach has beenproposed to solve thisproblem inthe Bayesian framework [17].The main idea isto encompass all variables and hyperparametersinauniqueproblemthatisestimatedwithvaria- tionalupdateprinciple.Wewillreferthisalgorithmtoprobability sparse NMF(psNMF). Thisalgorithmhastheadvantage tohavea reduced computationtimeand nohyperparametertuning. Italso has a regularization termto avoid strongdependence ofthe ini- tializationontheﬁnalsolution.

In order to estimate the precision of the reconstruction, we usedtheRootMeanSquareDifferenceRMSD:

RMSD=

X−S˙.A˙

2

^X

⁽⁸⁾

With

^.

^,^the^mean.

Oncethesourcesareestimated,wequantifytheirrelevancefor the global dataset. From the total reconstruction X˙_{k j}=S˙_ki.A˙_{i j}, for alli,wecan estimatethecontributionofsourcei,that istosay:

X˙ⁱ_{k j}=S˙_ki.A˙_i_j.Thus,therelevanceofsourceiisdeﬁnedas:

Rⁱ=

|

^X^˙ⁱ−X˙

|

^X^˙

⁽⁹⁾

ThisdeﬁnitionisconvenientsincethesumofallRⁱisone(this property isonly presentwhen sources andabundances are positive)andwecaneasilyestimatethe%contributionofeachsource inthe ﬁnal reconstruction. One hasto note that relevance is not ameasureofpresenceornotofaminorspecie(forinstanceCH₄) but a measure of how important is the source over the dataset.

Major species,should always havea larger relevance than minor species.Inthefollowing,weplotallsourcesresultsbydecreasing orderofrelevance.

3.3. Banddepth(BD)

Weusedthefollowingbanddepthdeﬁnition,differenceofthe geometricmeanoftworeferencewavenumbersinthecontinuum, comparedtotheband:

BD=X

( ν

l

)

^ν^ν^c^r⁻⁻^ν^ν^l^l.X

( ν

r

)

^ν^ν^r^r⁻⁻^ν^ν^c^l −X

( ν

c

)

⁽¹⁰⁾

withXtheobservedspectraintransmittance,

ν

^c ^the^wavenumber

of the centerof band,

ν

l the wavenumber of thereference level ontheleft (smallerwavenumber),

ν

r thewavenumberoftheref- erencelevelontheright(largerwavenumber).

4. Synthetictests

Wesimulatedseveralsyntheticobservationsindifferentcondi- tions,tomimicthecaseofNOMAD-SO.Theﬁrstsectiondescribes a simpletoy modelexample andthesecond one presentsexten- sive testsof thistoy modelwithvarious cases.Byhiddenspectra, hidden compounds and hidden CH₄, we always refer to a spectral datasetwithadominantmajorcomponent(herewater)andami- norspecie(hereCH₄).Thegoaloftheproposedapproachistopick upasource,containingCH₄only.

4.1. Toyexample

4.1.1. Syntheticdataset

Inordertodemonstratetheusefulnessofourmethod,wepro- poseherea toyexampleinavery diﬃcultcase.Wewill seethat usualmethodfailsdetectionbutourmethodisabletodetectthe hiddencompounds.

Forthistoyexample,wesimulatealinearmixtureofN_O=10⁴ observations spanned over N_ν=320 spectels (see Fig. 1) simi- larto order 136 ofNOMAD-SO.Each spectrum is amixture of a spectraofwatervapor S_H₂_O (coming fromoneactual sourceesti- matedfromrealdatausingpsNMF)andtheoreticalmethaneS_CH₄ fromVillanuevaet al.[35],with corresponding abundancesA_H

2O, A_CH

4:

X=SH2O.AH2O+SCH4.ACH4+n (11) Thenoise nisassumedto be aGaussian process witha standard deviation of

σ

=0.001 and no bias: n=G(⁰,

σ

)^. ^All ^spectra contain pure water vapor with a coeﬃcient following A_H₂_O= 5/6.

β

(¹,10)+1/6.U(⁰,1), a mixture of beta (

β

⁾ distribution for 5/6 ofthesample andan uniform(U) distribution for1/6 ofthe sample. This process mimics well the water vapor band depth

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Fig. 1. Synthetic dataset containing 10 ⁴spectra with various abundances of H 2O and 100 containing CH 4at 3- σlevel of the noise. In blue the reference spectra S H2Oof H 2O (coming from actual data analysis). In red the reference spectra S CH4of CH 4(from theoretical data).

Fig. 2. Water vapor Band Depth distribution (left) in the real observation (right) modeled by the toy example.

distribution (BD, seedeﬁnitionin Section 3.3) of thereal dataset (see Fig. 2). As the baseline of S_H

2O is not zero, we also mimic baseline correction errors. In addition 100 spectra out of 10,000 contain methanewithACH₄=1,suchthat thebanddepth ofSCH₄

is at 3-

σ

^level.^Please ^note ^that ^the ^model ^to ^generate ^the ^data

is not fulfillingthe sum-to-one constraint, but fully fulfillingthe positivityconstraint. Giventhedefinednoiseandsignallevel,the RMSDexpectedforaperfectreconstructionofthesignal(andnot thenoise)is0.16.

TheﬁnalsyntheticdatasetisrepresentedinFig.1.

Inordertocheckthequalityoftheestimation,wesimplycom- putethecorrelationcoeﬃcientbetweenS_CH

4 andtheestimatedN_S sourcesS˙,using:

Q=corr

SCH4,S˙:i

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The ith source with the maximumcorrelation is identiﬁed to CH₄ contribution. The value to the maximum correlation is used asmetrictoassessthequalityoftheretrieval.

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Fig. 3. (left) Histogram of Band Depth at 3067.2 cm ⁻¹from the dataset containing 100 CH 4at 3- σ level out of 10 ⁴spectra. (right) 100 spectra with the maximum Band Depth at 3067.2 cm ⁻¹speciﬁc of CH 4. Signal is dominated by water and by noise. No speciﬁc signature of CH 4is visible.

4.1.2. Results

By plotting the 10,000 samples of the dataset, one is able to identifyeasilytheH₂Obands.Nevertheless,wecannotobservethe target CH₄ in theaveragespectrum,even at3-

σ

^level,^because^it

islostinthebaselinechanges.

The second simpletool fordetectionwouldbe the analysisof thebanddepth.Fig.3(left)showsthehistogramofthemainCH₄ band that exhibits no sign of the presence ofCH₄ (no asymme- try inthe positive part).Fig. 3(right) representsthe 100 spectra with themaximum CH₄ BDat 3067.2 cm⁻¹. Again, no particular elementscanbeusedtoarguefordetection.

Fig.4representstheresultsfromthenon-negativematrixfactorization using psNMF algorithm. One can clearly identify both H₂OandCH₄ sources.Since those2chemicalcompounds arenot correlatedinabundance,(A_H

2OandA_CH

4 areindependent),twodif- ferentsourcespectraareidentiﬁed.Please notethattherelevance ofsource4isverylow(0.4%),meaningthatonly0.4%ofthevari- abilityinthedatasetisduetoCH₄,averylowvalue,asexpected forminorspecies.

Inthiscase,thecorrelationcoeﬃcientbetweenestimatedabun- dances A˙₄_: andtrueonesA_CH

4 is0.73. Sincethequantiﬁcation of abundance isa morediﬃcult problem,we will notpayexcessive attentiononthisparameter.

4.1.3. Convergenceandcomputationtime

We settheMUalgorithmconvergencetorelativedifference of the cost function <10⁻⁸ anda maximum runningtime of 1000 seconds.ForpsNMF,wesettherelativedifferenceofthecostfunc- tion to <10⁻⁷ anda maximum iterationto 2000.ForBPSS2, we compute a minimum burn in of 1000 iterations and after that whenthelong termstatistics(1000lastiterations)oftheMarkov Chainisclosetotheshorttermstatistics(100lastiterations),con- vergenceisconsideredtobereached.Thenanother1000iterations arecomputedtoestimatetheﬁnalsolutionstatistics.

Werunthe3identiﬁedtools10timesonthesamedatasetwith differentnoise realization,andcompute meanandstandarddevi- ationfromthese10experiments.ResultsarepresentedinTable1. Onecanclearlyseethattheeveniftheconvergenceisset,thereis a highvariabilityinMUresults,duetothelackofregularization.

Onthisparticularexample,thebestisclearlypsNMFalgorithm.

Table 1

Results (mean and standard deviation) from 10 realizations of a toy synthetic example with N S= 5 (in agreement with next section on synthetic tests), N O= 10 0 0 0 , N ν= 320 and 300 CH 4spectra hidden at a level of 1 std of the noise. Quality is computed as a correlation coeﬃcient (see Eq. (12) ). RMSD is computed from Eq. (8) . Computation time is expressed in second.

MU psNMF BPSS2

Quality Q 0.35 ±0.12 0.822 ±0.005 0.41 ±0.06 RMSD relative error 0.1455 ±2 . 10 ⁻⁶ 0.1461 ±5 . 10 ⁻⁶ 0.1468 ±3 . 10 ⁻⁴ Computation time (s) 13 ±8 46 ±9 413 ±21

TheRMSDiscomputedforallcasesandshowninTable1.We canobservethat thevalue isalmost equivalent,around 0.146,for allmethodbutMUisslightlybetter,duetothefactthat thecost function hasnoother term. MUalgorithm isjustminimizing the reconstruction.As acomparison,theRMSDexpectedforaperfect reconstructionofthe signal(andnot thenoise)ofthistoy exam- pleis0.16.With5 sources(signiﬁcantlymore thanthe3 sources deﬁnedinthistoyexample),noiseisalsoencompassedwithinthe approximatedlinearmodel,asexpected.

The quality Q is the only parameter to assess the quality of the algorithm todetect minor specie (hereCH₄). Inthis particu- lartoyexample,psNMFseemstobethebestalgorithm,providing asourcecorrelatedwithgroundtruthCH₄ withacorrelationcoef- ﬁcientupto0.8. Wewillextensivelytest thisperformance inthe nextsection.

Wealsoestimatethecomputationtimeona2.9GHzIntelCore i7with16GoDDR3RAMasanexample.Allalgorithmsareimple- mentedin©Matlabusingparallelizedmatrixcomputation.Results, presentedin Table 1,demonstrate that MUis fasterthan psNMF but both are clearlyless resources consuming than BPSS2. From thecomputationtimeandeﬃciency,weexcludedBPSS2fromthe nexttests.

4.2. Extendedsynthetictests

Fortheﬁrstsetoftests,weusedthesametoymodeldescribed inSection4.1,exceptwith100CH₄ spectrahiddenatalevelof2 and3standarddeviationofthenoise(thisnumberiscalled“factor above noise level”). In order to haverobust results,we made 10 realizationsandaveragedtheresults.

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Fig. 4. Results of the psNMF algorithm for N S = 4 . Sources 1 and 3 are identified to the level with significant noise contribution, source 2 is identified to H 2O (correlation coef. with groundtruth 0.99), and source 4 is CH 4(correlation coef. with groundtruth 0.98). Relevance is computed from Eq. (9) .

Fig. 5 represents the results as a function of the number of sources NS. It presents two quality indicators of the results: the average correlation coeﬃcient Q (see Eq. (12)) and the fraction of realizationwithacceptableresults (withQ >0.5). Wecan ob- serve that the psNMF is always better than MU on average at cost of an higher variability (higher standard deviation). Adding sources seems to always increase the detection until reaching a plateau around N_S=5. Adding more sources will not drastically increase/decrease the source estimation. Nevertheless, it requires more computationtime for a larger number of source (approxi- matelyx2between3and9sourcesbutthecomputationtime al- waysstaysbelow200seconds).

For thesecond set oftests, we usedthe sametoy model,ex- cept with50and100 CH₄ spectrahiddenatalevel of0.7,1,1.2, 1.5, 2.0,2.5and3standarddeviationofthenoise(thisnumberis called “factorabove noiselevel”). In orderto haverobust results, we made10 realizationsandaveraged theresults.Results are always withRMSD< 0.18with an average ~ 0.16.RMSDfromthe noiselevelis0.16whatevertheexperiment(theCH₄islowenough so thatit’scontributionto RMSDisnegligible), sothereconstruc- tionisinaverageasexpected.

Fig.6presentstwoqualityindicatorsoftheresults:theaverage correlation coefficient Q(seeEq.(12)) andthefractionofrealiza- tion withacceptable results (withQ > 0.5).Both indicatorsindi- catethatthemethodpsNMFclearlyoutperformsMUathighfactor abovenoiselevel.Fromourvisualinspectionoftheresults,wede- fine thedetectionlimitwhen atleast50%oftheresultsare with Q > 0.5(correlationcoefficient > 0.5).This definitionisdebat- ablebutthereisnoabsolutewayofdefiningit.Fig.6showsthat thedetectionlimitisat1.5factorabovenoiselevelfor100hidden spectracase,around2for50hiddenspectra.Belowthislimit,none

ofthemethodisabletodetecttheCH₄spectrafromthenoise.For 20hiddenspectra,evenatafactorabovenoiselevelof3,noneof themethods isabletodetect theCH₄ spectra.Onecanalso note thatthepsNMFislessstablesincethestandarddeviationismuch larger.

5. SimulationofNOMAD-SO 5.1. Simulationdataset

This second dataset has been generated with the most pre- cisedirectmodel,takingintoaccount thefullnon-linearradiative transfer and instrumental effects to produce synthetic transmittance,highlycomparablewithactualobservations.Synthetictrans- mittances were made forreal NOMAD-SOobservation ﬁlesusing therelevantgeometryandinstrumentparameterstoattempttoin- cludethevariabilityinherentinthetruemeasurements.

Modelatmospheres foreach occultation were developedfrom theGEM-Marsgeneralcirculationmodel[5,26].Theoutputofthe modelwere providedfor1 Martiandayevery 10solarlongitude, and48timestepsper Martianday.Atmosphericproﬁles were de- velopedforeachoccultationby interpolatingthemodeltempera- tureandpressuretothesolarlongitude,localsolartime,latitude, longitude,andtangentaltituderelativetotheareoid.

Toconstructthesimulatedtransmittancespectra,thehighres- olutionirradianceswerecomputedforeachoccultationassuminga sphericallysymmetryandthetangentatmospheredevelopedfrom GEM-Mars for several different abundance of methane and water,which were simulatedasconstant volume mixingratios. The spectroscopic data for methane and water were taken from HI- TRAN 2016 using CO₂ broadening [10,11,15]. The instrument for-

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Fig. 5. Results of the MU and psNMF algorithm for N S= 3 to 9, N O= 10 0 0 0 , N ν= 320 , as a function of the number of source. The average Q of 10 realizations of the best estimated source (thick lines and standard deviation in thin lines) and the fraction of acceptable results (with Q > 0.5). (left) with a factor above noise level of 2 (right) with factor above noise level of 3.

Fig. 6. Results of the MU and psNMF algorithm for N S= 5 , N O= 10 0 0 0 , N ν= 320 , as a function of the factor above noise level. The average Q of 10 realizations of the best estimated source (thick lines and standard deviation in thin lines) and the fraction of acceptable results (with Q > 0.5). (left) with 100 hidden CH 4spectra (right) with 50 hidden CH 4spectra.

ward model wasthen applied to each simulation by considering theAOTFbandpass,instrumentInstrumentLineShape (ILS),blaze function, specteltowavenumbercalibration,andthecontribution oflightcomingfromthemainorderandnearbyorders[27,33,34]. The ﬁnal synthetic transmittancespectrais theratio ofthislow- resolutionirradiancetothetop-of-atmospherelowresolutionirra- diance.

The AOTF/echelle instrument was modeled using the latest available calibration[1,21],consideringorderadditionfrom+/−2 nearby orders (5 total). The spectral calibration of NOMAD-SO varies because it is affected by the instrumenttemperature, and is provided foreach individual NOMADspectra.The 320spectels

covertherange3056.1 cm⁻¹ to3080.4cm⁻¹ witha wavenumber stepof0.0763cm⁻¹.

Nosimulation ofdusthasbeenperformed.Due tothelimited spectralrangeonasingleorder,about25cm⁻¹,themajoreffectof dustandotheraerosolsisrelativelyﬂatbaseline,whichweremove atthepre-treatment ofthe spectra.When dustis opticallythick, then non-linearity may appear that are out of the scope of this simulation.

Thesimulationdatasetconsistof12,486spectra,simulatingob- servationsoforder136inthesameconﬁgurationasthe106solar occultationsactuallyobservedfromMaytoDecember2018.

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Fig. 7. Results of the psNMF algorithm for N S= 5 on simulation dataset, averaged over 10 noise realizations, for different noise levels (0.001 and 0.0001) and different fractions of hidden CH 4(1%, 5%, 10%, 100%). Hidden CH 4are taken within the same orbital sequences. The left panels represent results for 10 ppm of water vapor and the right ones for 100 ppm of H 2O. From top to bottom, we show: a) Fraction of the 4 main CH 4peaks detected in the best source ; b) Mean distance to the expected center in spectel and c) Abundance of CH 4in the source αCH4. Please note that the absence of plotted data means that no source was successfully detected.

Table 2

Simulation parameters. Fraction of CH 4is fraction of spectra containing methane hidden in the simulation dataset.

CH 4[ppt] H 2O [ppm] fraction of CH 4[%] noise level Value 0; 100; 500; 1000 0; 10; 100 1; 5; 10; 50; 100 0.001; 0.0001

Weaddtothedatasetarandomnoisewithstandarddeviation of 0.001 and0.0001 in orderto simulate the instrumental noise (correspondingtoSNRof100and1000approximately).

WehidespectracontainingCH₄ inafractionofthetotalnum- berofspectrafrom1%to100%inarandommanner.Inrealobser- vation,CH₄maybespatially/temporallycoherentbutthenumber ofscenariosisinﬁnite.Wefeelthattherandomcaseisinteresting enoughtobetested.Onehastonotethatcontrarilytotheprevious toymodelofSection4,hereabundancearequantitativeabundance intheatmosphere.

ThesimulationparametersaresummedupinTable2. 5.2. Detectionlimits

WeappliedthepsNMFmethodwithN_S=5,whichisthemost promisingonefromtheprevious analysis.Wecomputetheanaly-

sis10timesfor10differentrandomnoiserealizationsandaverage theresultsinordertopresentrobustconclusion.Weselectapure CH₄ andapureH₂Ospectra(notedP_CH

4 andP_H

2O)fromthesimu- lationasreferencespectra.

5.2.1. Methodstoanalyzetheresults

Themaindifferencewiththetoymodelsectionin4isthatH₂O andCH₄ maybe highly mixedinthe sources. Simple correlation coeﬃcienttopickthebestsourceisthusnoteﬃcientenough.We proposehereanotherapproachtoestimatethebestsource.

ForeachestimatedsourceS˙_:_i,weanalyzeitasalinearmixture ofP_H

2OandP_CH

4:

S˙_:_i=PH2O.

α

H2O,i+PCH4.

α

CH4,i (13) ThisproblemiscalledsuperviseddetectionalgorithmsinceP_H

2O

and P_CH₄ are known, contrary to the general one, presented in Eq.(5),wheresourcespectraarenotknown.Thesourcei^∗withthe maximum

α

CH₄,i∗ isselectedasthebesttarget CH₄ source,called bestsourcehereafter.

Wethenproposetousethreeindicatorsofgooddetection:

• Fraction of the 4 main CH4 peaks detected (at 3057.7, 3063.4, 3067.2and3076.6cm⁻¹).Thisiscomputedusingthepeakde- tection algorithm from ©Matlab on both simulation and best

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Fig. 8. Results of the psNMF algorithm for the diffraction order 119 for N S = 5 . The sources 1, 3 and 5 are identified to CO 2(shift of 0.01 for clarity). The source 2 is identified to the background level (continuum misestimation). The source 4 is identified to H 2O. No source seems to be related to CH 4.

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Fig. 9. Results of the psNMF algorithm for the diffraction order 134 for N S= 5 . The source 1 is identified to the background level (continuum misestimation), the sources 3, 4 and 5 are identified to H 2O. The sources 2 present unmodeled lines that are not present in the spectroscopic database. These lines has been first detected in the ACS instrument data and attributed to CO 2magnetic dipole transition [32] . No source seems to be related to CH 4.

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Fig. 10. Results of the psNMF algorithm for the order 136 for N S= 5 . The source 1 is identiﬁed to the level background (continuum misestimation), the sources 2, 3, 4 and 5 are identiﬁed to H 2O, either directly either from the adjacent orders. No source seems to be related to CH 4.

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sourcewithatoleranceof2spectels,i.e.detectedpeakscanbe 2spectelsoff theexpectedone.Thepeakmustbewithamax- imumamplitudelargerthan1/1000themaximumofS˙_:_i∗tobe considered signiﬁcant. Please note that even there are only 5 possiblefraction(0,0.25,0.5,0.75and1),sinceweaverageon 10realizations,anynumbercanappear.

• Meandistanceto the expectedcenter.Mean distancein spectel betweentheCH₄peaksdetectedinthebestsourceandtheref- erenceone.

• AbundanceofCH₄ inthesource.

α

CH₄ (fromEq.(13)),whichde- scribestheamplitudeoftheCH₄ peaksinthebestsource.

5.2.2. Analysisoftheresults

Fig. 7 summarizes all the results.Fraction of the4 main CH₄ peaksdetected inthemostrelevantsourcehasalwaysastandard deviation < 0.43 and a mean value of 0.06 over the 10 realizations. The Mean distance tothe expected centerhas always a standard deviation < 0.40 anda mean value of 0.07 over the 10 realizations.TheabundanceofCH₄ inthesource hasalwaysa standarddeviation < 0.05andameanvalueof0.005overthe10 realizations.

This ﬁgure shows that the detection limits clearlydepend on CH₄density,butalsoonthefractionofhiddenCH₄andnoiselevel, as expected. Abundance of CH4 in the source

α

CH₄ maximum is 25%,meaningthatinanycasesH₂Oisdominatingthebestsource andsobothCH₄ andH₂Oarepresentineachbestsource.Thisis becauseCH₄isaminorspecie(asexpectedfromtheconditionsof oursimulation),itsabsorptionbandgenerallyfollowstheair-mass, asH₂Odoes.SothereisnoparticularsourceforCH₄only.

When morethan twolinesaredetected,we canconsideritas a detection.This limit isreached forCH₄ ≥ 500 pptfor10 and 100 ppm of H₂O. Nevertheless,the detection limitslies between 100and500pptinthecaseof10ppmofH₂Ovaporsincethede- tection isperfect(100%ofthe4mainCH₄peaksdetected)occurs forafractionofCH₄5to50%.Interestingly,theoptimumdetection isnot when100%ofthespectracontainsCH₄,butmorebetween 5–50%. This behavior is due to the statistics that is richerwhen alsoCH₄ islackingincertain spectra.When100%ofspectracon- tain CH₄,thestatisticalvariabilityofthedatasetismainly dueto airmass (atmosphereis assumedto be well mixed). Soboth CH₄ and H₂Oare varying together andthere is less statistics to base thedetectionon.

Noiseleveldoesnot affectﬁrstthefractionofthe4mainCH₄ peaksbutincreasesthespectral shiftofthebandcenter.Inaddi- tion,itclearlyaffectstheabundanceandthusthebanddepth.

In conclusion, fromthis simulationanalysis, one could expect detectionlimitsofCH₄ intherange100–500pptwhen operating infavorableconditions.

6. Realdataanalysis

Inthissection,wereporttheresultsofactualNOMADdata,fo- cusingondiffractionorderswithpotentialCH₄ lines:119,134and 136,areshownrespectivelyonFigs.8,9and10.Weusedthe821 ingress andegress transitorbitsfororder119, 2358orbitsforor- der134and703fororder136.WeﬁlterspectrawithSNR > 100.

ResultsarecomparedwithNOMADsimulations[35]usingthecal- ibrationpipeline. Thisprocess addsghost lines fromadjacentor- ders,asinrealdata.Table3summarizestherelativeerrorandthe number ofspectra.Theapproach hereistocompute the analysis withpsNMFusingN_S=5inagreementwiththeprevioussection.

Please remindthat ourapproachis fullyblind:nospectral information hasbeen included inthe analysis(nothing aboutH₂O, CO2orCH4).

Forall orders,sources ofH₂Oare estimated,asexpected. Also asourcepresentingaresidualofthecontinuum isalwayspresent.

Table 3

Number of spectra N Oand RMSD relative errors for 4 to 10 no. of sources N Sre- sulting from the analysis of all observations of NOMAD data up to 15 January 2020, using the psNMF algorithm. RMSD is computed from Eq. (8) .

119 134 136

N O 134,045 365,985 140,064

N S= 4 0.476 0.575 0.634

N S= 5 0.456 0.553 0.609

N S= 6 0.442 0.553 0.585

N S= 10 0.410 0.484 0.544

Duetonon-linearitiesoftheradiativetransfer,theacquisitionpro- cess (temperature dependence) and the wavenumber shift, the molecularspeciesappearssometimesindifferentsources.

Order136givesthe 1sourcerelatedto thebackgroundand4 sourcesrelatedtoH₂O.All4sources ofwaterhavethepeaksbut withdifferentrelativeintensitiesandwavenumbershift.

Fororder119,bothCO₂andH₂Olinesareidentiﬁed(seeFig.8).

Since those two components are uncorrelated, separatedsources arefoundbythealgorithm.

Interestingly, order 134 presents a source with unexpected lines. The main linesare atpositions : 3016.70, 3017.07, 3018.12, 3019.54, 3020.90, 3022.25, 3023.60, 3024.96, and 3027.29 cm⁻¹. TheselineshasbeenalsodetectedintheACSinstrumentdataand attributedtoCO₂ magneticdipoletransition[32].Furtheranalysis shallbedonetocomparebothNOMADANDACSdata.

Solar lines are never appearing in the sources. Theyare self- corrected by the calibration since we don’t use a referencesolar spectrabutthesolarobservationduringthetransitwhenthetan- gentaltitudeissohighthatthereisnomartianatmosphere(typi- cally > 200km).

NoneoftheanalyzedorderspresentssourcesrelatedtoCH₄.

7. Discussionsandconclusion

We implemented a new strategy to analyze spectroscopic datasets. This strategy is fully unsupervised, so that anykind of absorptionbandscanbediscovered.Theamountofpriorinforma- tionrequiredisthusverylow.Thecomputationcanbedoneona regular hardwareforthemostcommondatabaseandwithin rea- sonableamountoftime(~ 100,000spectra).

Weillustratetheapproachfortypicalatmosphericspectroscopy.

Weﬁrstputforwardasynthetictest,basedonsimplelinearmix- ingtogiveatoyexampleandtoidentifythebestpromisingalgo- rithm.ThepsNMFclearlyoutperformedMUandBPSS2.

Then we proposed a simulation, based on realistic radiative transfer andinstrumental effects, applied on NOMAD-SOspectra.

The detection limitsgoes below500 ppt in favorableconditions, withreducedH₂Oandlownoise level.The samerangeofdetec- tionlimitsisreachwithusualapproachofmodelﬁttingatamuch highercomputationcostandanalysiseffort.Giventhesimplicityof use,thistoolmayberelevanttohandlelargeandcomplexdatasets atﬁrstglance.Asaperspective,analysisofresidualsafterthenon- linearretrievalofthedatamaylowerthedetectionlimits.Onecan thentestiftheresidualsaresimplyGaussiannoise,oriftheymay containinterestingfeatures.

Interestingly,a molecular specie not well mixed in theatmo- spherecanbemosteasilydetectedwithourapproach.

Thelastsectionpresentedtheresultsoftheapplicationonreal NOMAD-SOdata,using orders119, 134 and136,selectedasthey arerepresentativeofthebaselinestrategyofmeasurementsinNO- MAD, allowing characterizationofH₂Oandpotential detectionof CH4.TheoutcomeisthatnoCH4hasbeenidentiﬁed,butH2Oand CO₂ aredetected.Interestinglyanewsetofspectrallineshasbeen discoveredintheNOMADdata.Theselineshasbeenﬁrstdetected

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intheACSinstrumentdataandattributedtoCO₂magneticdipole transition [32]. We thus conﬁrm their presence withour current analysis.

One way togo back to the data isto pick the realdata with the highestsource contribution A˙.Our quicklook analysisis thus only a starting point of a more complete scientiﬁc analysis. This second step will requiremuch moreprior information (chemical compounds, fundamentalspectroscopic constants, radiativetrans- fermodel,...).

Future work should apply the proposed approach to other datasets, such asother NOMAD-SOorders,or other spectroscopic datasets (including hyperspectral images) from laboratory mea- surements, groundbasedtelescopes orspace-born spectrometers.

The approach is generic enough to treat datasets that can be at ﬁrstorderapproximatedtoalinearmixture.

DeclarationofCompetingInterest

Theauthorsdeclarethattheyhavenoknowncompetingﬁnan- cialinterestsorpersonalrelationshipsthatcouldhaveappearedto inﬂuencetheworkreportedinthispaper.

CRediTauthorshipcontributionstatement

Frédéric Schmidt: Conceptualization, Software, Methodology, Writing -original draft. GuillaumeCruz Mermy: Software,Writ- ing - review &editing. JustinErwin: Writing -review & editing.

Séverine Robert: Writing - review & editing. Lori Neary: Writ- ing-review&editing.Ian R.Thomas:Writing-review&editing.

Frank Daerden:Writing -review&editing.BojanRistic:Writing - review & editing.ManishR.Patel: Writing - review & editing.

Giancarlo Bellucci: Writing - review & editing.Jose-Juan Lopez- Moreno:Writing -review &editing.Ann-CarineVandaele:Writ- ing-review&editing.

Acknowledgments

We acknowledge support from the “Institut National des Sci- ences de l’Univers” (INSU), the "Centre Nationalde la Recherche Scientiﬁque" (CNRS) and "Centre National d’Etudes Spatiales"

(CNES) through the "Programme National de Planétologie" and the ExoMars TGO programs. The NOMAD experiment is led by the Royal Belgian Institute for Space Aeronomy (BIRA-IASB), as- sistedbyCo-PIteamsfromSpain(IAA-CSIC),Italy(INAF-IAPS),and the UnitedKingdom(OpenUniversity).Thisprojectacknowledges fundingbytheBelgianSciencePolicyOffice(BELSPO),withthefi- nancialandcontractualcoordinationbytheESAProdexOffice(PEA 4000103401,4000121493), bySpanishMinistryofScienceandIn- novation (MCIU) and by European funds under grants PGC2018- 101836-B-I00 and ESP2017-87143-R (MINECO/FEDER), as well as by UK Space Agencythrough grants ST/R005761/1, ST/P001262/1, ST/R001405/1andST/R001405/1andItalian Space Agencythrough grant2018-2-HH.0.ThisworkwassupportedbytheBelgianFonds delaRechercheScientifique-FNRSundergrantnumber30442502 (ET-HOME). The IAA/CSIC team acknowledges financial support fromthe StateAgency forResearchofthe SpanishMCIUthrough the Center ofExcellenceSeveroOchoa award fortheInstituto de Astrofísica de Andalucía (SEV-2017-0709). US investigators were supported by the NationalAeronautics andSpaceAdministration.

Canadian investigators were supported by the Canadian Space Agency.

Supplementarymaterial

Supplementary material associated with this article can be found,intheonlineversion,atdoi:10.1016/j.jqsrt.2020.107361.

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