Combination Of Geophysical And Geotechnical Data Using Belief Functions: Assessment With Numerical And Laboratory Data

2009·5.3 MB·

by  T. Dezert

JournalofAppliedGeophysics170(2019)103824
ContentslistsavailableatScienceDirect
Journal of Applied Geophysics
journal homepage: www.elsevier.com/locate/jappgeo
Combinationofgeophysicalandgeotechnicaldatausingbelieffunctions:
Assessment with numerical and laboratory data
T.Dezerta,b,⁎,S.PalmaLopesa,Y.Fargierb,c,P.Côtea
aIFSTTAR,GERS,GeoEND,F-44344Bouguenais,France
bCeremaDirectionterritorialeNormandie-Centre,F-41000Blois,France
cUnivLyon,IFSTTAR,GERS,RRO,F-69675Bron,France
a r t i c l e i n f o a b s t r a c t
Articlehistory: Theidentificationofthesubsoilconstitutivematerials,aswellasthedetectionofpossibleinterfacesandanom-
Received14December2018 alies,arecrucialformanysitecharacterizationapplications.Duringinvestigationcampaigns,complementary
Receivedinrevisedform13June2019 geophysicalandgeotechnicalmethodsareusuallyused.Thesetwosetsofmethodsyielddatawithverydifferent
Accepted16July2019
spatialscalesanddifferentlevelsofincompleteness,uncertaintyandinaccuracy.Inthiswork,amathematical
Availableonline22July2019
combinationofgeophysicalandgeotechnicalinformationisproposedinordertoproduceabettersubsoilchar-
acterization.Itisshownthatbelieffunctionscanbeusedforsuchafusionprocess.Aspecificmethodologyisde-
Keywords:
velopedinordertomanageconflictualinformationanddifferentlevelsofuncertaintiesandinaccuraciesfrom
Datafusion
Belieffunctions differentinvestigationmethods.Inordertotestandvalidatethismethodology,wefocusontheuseoftwose-
Geophysicaldata lectedmethods,ElectricalResistivityTomography(ERT)andConePenetrationTest.First,asyntheticmodel
Geotechnicaldata withartificialdataisconsidered,takingadvantageoftheresultsobtainedtoconductacomparativestudy(effect
Experimentaltestbench ofparametersandnoiselevel).Then,anexperimentaltestbenchisconsidered,inwhichatwo-layeredmodelis
Electricalresistivitytomography placed(plasterandsaturatedsands)andgeophysicalandgeotechnicaldataaregenerated,usingamini-ERTde-
viceandinsertiondepthvalues.Thisworkalsoaimsatprovidingabettergraphicalrepresentationofasubsoil
sectionwithassociateddegreesofbelief.Theresultshighlighttheabilityofthisfusionmethodologytocorrectly
characterizetheconsideredmaterialsaswellastospecifythepositionsoftheinterfaces(bothverticalandhor-
izontal)andtheassociatedlevelsofconfidence.
©2019ElsevierB.V.Allrightsreserved.
1.Introduction subsoilsistobeabletocombineacquiredgeophysicalandgeotechnical
data,whiletakingintoaccounttheirrespectiveuncertainties,inaccura-
Forsubsoilscharacterization,investigationcampaignsaresetup, ciesandspatialdistributions(Royetetal.,2013).Thecomplementarity
usually consisting of geophysical and geotechnical methods. These ofthesetwosetsofmethodsisoftenunderusedsincetheuncertainty
twofamiliesofmethodsarecomplementaryandareusedforvariousis- andinaccuracyassociatedwitheachmethodarerarelyconsidered.Fur-
suessuchasthecharacterizationofslopestability(CarisandVanAsch, thermore, the results are usually only graphically superimposed
1991;Merrittetal.,2014;Chambersetal.,2013;Abidinetal.,2012)the (Shaabanetal.,2013)insteadofbeingmathematicallymerged.
characterizationofpotentiallydangeroussites(Jamesetal.,2014),the Tocharacterizeasectionofsubsoilanditspotentiallyriskyareas,itis
characterizationofsitesatconstruction(Coker,2015)orthecharacter- essentialtodistinguishthedifferentmaterialsinplace.Thehorizontal
izationofriverembankments(Perrietal.,2014). andverticalinterfaces,aswellaspossibleanomalies,havetobelocated.
Ontheonehand,geophysicalmethodsarenon-intrusiveandpro- Forleveeembankment,asanexample,itisintheselocationsthatinter-
videphysicalinformationonlargevolumesofsoilsbutwithsignificant nalerosionislikelytodevelop,whichmayleadtothecompleterupture
potentialuncertainties.Theseuncertaintiesaredueinparticulartothe ofthelevee(Fosteretal.,2000).Suchasectioncharacterization,with
integrativeandindirectaspectsofthemethodsaswellastotheresolu- associated confidence indexes, could be included in failure hazard
tionoftheinverseproblems.Ontheotherhand,thegeotechnicalinves- models.
tigationmethodsareintrusiveandprovidemorepunctualinformation Theuseofbelieffunctions(Shafer,1976;Dempster,1967)anddif-
but also more accurate. An important issue for the assessment of ferentinformationcombinationrulestocombinegeotechnicalandgeo-
physicaldataisproposed.Thismakesitpossibletotakeintoaccountat
thesametimetheuncertainties,inaccuraciesandincompletenessof
⁎ Correspondingauthorat:IFSTTAR,GERS,GeoEND,F-44344Bouguenais,France.
datarelatedtoeachmethod.Inthefieldofgeosciences,belieffunctions
E-mailaddresses:[email protected](T.Dezert),[email protected]
(S.PalmaLopes),[email protected](Y.Fargier),[email protected](P.Côte). have already been used and provide interesting results for slope
https://doi.org/10.1016/j.jappgeo.2019.103824
0926-9851/©2019ElsevierB.V.Allrightsreserved.
2 T.Dezertetal./JournalofAppliedGeophysics170(2019)103824
instabilitymapping(Binaghietal.,1998;Althuwayneeetal.,2012),de- frompunctualgeotechnicalinvestigation).Uncertaintiescorrespondto
tectionofpreciousmetal(TangestaniandMoore,2002),groundwater degreesofconfidencethataregiventoavalue,whereasinaccuracies
(Mogaji et al., 2015) or flood susceptibility mapping (Tehrany and correspondtointervalsofvaluesthatcanbedirectlyassociatedwith
Kumar,2018).Toourbestknowledge,noworkhasbeenproposed,con- measurementerrorsrelatedtothemethod.Forexample,theuncer-
sideringthecombinationoftwosourcesofinformationwithdifferent taintyofmeasuringageotechnicalparameteridenticaltotheonemea-
spatialdistribution(spatializedandpunctual)andforaninvestigation sured in a borehole increases with the distance to that point. The
campaignintheverticalsection. inaccuracycanforitspart,beassociatedwiththeerrorbaroftheresult.
Here,aninnovativemethodofinformationfusiontocombineelec- Thebelieffunctionsallowtotakeintoaccounttheignoranceandincom-
tricalresistivitytomographyresultsandconepenetrometertestdata pletenessoftheinformation.Itisindeedpossibletograntcreditonall
is proposed.First, work on data obtained fromsynthetic models is thepossibleresultsinordertoquantifytheignorance.Forthereader
displayed.Theobtainedresultsallowtoconductacomparativestudy, eagertolearnmore,thetheoryisdetailedin(Martinetal.,2008).
evaluatingtheeffectofdifferentparameters(likethedatanoiselevel) ABayesianapproachaspartofasubjectiveprobabilityapproach
onthefusionresult.Thefusionmethodologyisthentestedfromdataac- (Cooke,1991)couldhavebeenconsideredforgeophysicalandgeotech-
quiredonatestbench.Inthiswork,thepotentialofsuchamethodology nicaldatacombination.However,themainlimitationofsuchanap-
isshownbyusinginsertiondepthdata,acquiredbyalaboratorypene- proachisthatprobabilitiesessentiallyrepresentuncertaintyandonly
trationcone,andelectricalresistivitydataacquiredbyaminiElectrical very poorly the level of inaccuracy. Moreover, in the probabilistic
ResistivityTomography(ERT)device.Thedepthofpenetrationdata modelingstage,thedifferentdecisions(events)areonlyrepresented
correspondstogeotechnicalinformationwhiletheelectricalresistivity onsingletons(i.e.singleevents)andarenecessarilyconsideredexhaus-
datacorrespondtogeophysicalinformation.Themainconcernisto tiveandexclusive.Theexclusivityisimpliedbytheassumptionofthe
highlighttheabilityofthisinformationfusionalgorithmtocharacterize additivityofprobabilities.However,thishypothesismaybetoostrong
theinterfacesbetweenmaterialsandtodiscriminatethreedifferent andlimittherepresentationoftheknowledge.Furthermore,witha
typesofmaterialswithvariationinthicknessofoneofthem,andto Bayesianapproach,itisdifficulttomodelthelackofknowledgeorthe
presentthevariationoftheresultsaccordingtothenumberandposi- knowledgethatisnotexpressedinprobabilitydistributions.
tionofthesimulatedboreholes. Inordertodefineandtousethebelieffunctions,itisnecessary(i)to
Themaincontributionsofthisworkareasfollows.First,thisnew setaframeofdiscernment,(ii)toassignbeliefmassvaluestotheevents
methodologymakesitpossibletotakeintoaccounttheuncertainties, ofthisframework(BasicBeliefAssignments-BBAs),(iii)tochooseafu-
inaccuraciesandincompletenessassociatedwiththedifferentmethods sionruleforcombininginformation;and(iv)torepresentthecombined
ofinvestigationused,proposingamodelingoftheBasicBeliefAssign- information.
ments(BBAs)specificallyadaptedtotheproblematic.Then,thepro- TheFrameofDiscernmentΘ(FoD)ismadeofallthepossibleevents
posed graphic representation is innovative since it allows both to abouttheproblemunderconcern,theelementsoftheFoDareexclusive
presentthedifferentgeologicalsetsthatwouldbepresentinthesubsoil andexhaustive,sothatfornevents:
andtheirlayout,whilepresentingtheconfidenceassociatedwiththese
results.Thismethodologyisparticularlysuitableforthecharacteriza- Θ¼fθ1;θ2;…;θng ð1Þ
tion of interfaces and anomalous zones, which may correspond to
areaswheretheriskofinstabilityispotentiallythegreatest.Thiswork Intheconsideredproblematic,thepossibleeventsoftheFoDcorre-
alsoallowstheimplementationofasmallphysicalmodeltovalidate spondtointervalsofvaluesofgeophysicalandgeotechnicalparameters
thefusionapproachwithrealdata. thatcanbeassociatedwithclassesofgeologicalmaterials(forexample,
Thisarticleisorganizedasfollows.InSection2apresentationofthe θ1=clays,θ2=sands…).Thespaceofbeliefmassfunctions,thesetof
approachoffusionusedinthemethodologyisgiven,whichintroduces allsubsetsofΘ,written2Θ,isfixedbyallthedisjunctionsandbythe
theuseoftheevidencetheoryandthecombinationmethodsusedhere. possibleconflictbetweenthesourcesofinformation(written∅)such
InSection3,asyntheticstudywillthenpresentthefusionapproach that:
fromartificialdata.Itwillalsopresentthecomparativeresultsassoci-
atedtotwoparametricstudies.Then,inSection4,apresentationof 2Θ¼f∅;fθ1g;fθ2g;fθ1∪θ2g;fθ3g;fθ1∪θ3g; ð2Þ
theinvestigationmethodsusedintheintroducedexperiment(labora- fθ2∪θ3g;fθ1∪θ2∪θ3g;…;fθ1∪θ2∪θ3∪…∪θngg
torypenetrationconeandminiERTdevice)isgiven.Finally,thetest
bench fusion results are presented in Section 5 and discussed in Asintheprobabilitytheory,thebeliefmassfunctionm isdefined,
j
Section6,inordertounderstandtheinterests,limitationsandperspec- forasourceofevidenceS (forj=1,2),attributedtoA(definedon
j
tivesofsuchamethodology. 2θ)in[0,1]suchthatthemorem(A)tendsto1andthemoretheconfi-
denceinAisimportant:
2.Fusionmethodology X
mðAÞ¼1 ð3Þ
j
2.1.Belieffunctionsandcombinationrules Aϵ2Θ
ThebelieffunctionshavebeenintroducedbyShafer(Shafer,1976) ThedifferencewiththeprobabilitytheoryliesinthefactthatAcan
in1976inthedevelopmentofhismathematicaltheoryofevidencein- representtheunionofseveralevents(forexample,eitherθ ORθ ).It
1 2
spiredbypreviousworksofDempster(Dempster,1967).Shafer'sthe- isthereforepossibletomodeluncertaintyandlackofknowledge.Forin-
oryisalsoreferredasDempster-Shafertheory(DST)intheliterature. stance,whennoinformationisavailableabouttheachievementofan
Thistheory(proposesamethodto)calculate(s)thebeliefandtheplau- eventmemberofΘ,onecansetm(Θ)=1,avoidingtheuniformdistri-
j
sibilityofanevent(hereasoilmaterialclass)fromdistinctsourceofev- butionthatwouldhavebeenconsideredinaprobabilisticscheme.Com-
idence(measureddata).Thepracticaladvantageofusingsuchatheory bination rules, as part of the belief functions theory, can thus take
liesinitsabilitytomanageinformationfromdifferentsources,associ- differentlevelsofuncertaintyandimprecisionsintoaccountaccording
atedwithvariableuncertaintiesandinaccuracies.Inthiswork,only tothesourceofinformation.Ifonlydefinedonsingletons,thebelief
twosourcesofinformationwillbeconsidered:geotechnicalandgeo- massfunctionissimilartoaprobabilitydistribution.
physical.Anotheradvantageofthistheoryisitsabilitytoassessthede- SmetsfusionapproachdevelopedinhisTransferableBeliefModel
gree of conflict between sources (ex: contradictory information (TBM)(Smets,1990)(i.e.conjunctivefusion)allowstheattributionof
between data obtained from large scale geophysical campaign and amassofbelieftotheconflict,outsidetheFoD,sothat(open-world
T.Dezertetal./JournalofAppliedGeophysics170(2019)103824 3
assumption): calculatedandiii)thetotalofpartialconflictingmasseshavetobe
redistributedproportionallyonnon-emptysets.So,formPCR6(Ø)=0
12
m12ð∅ÞN0 ð4Þ and∀A∈2Θ\{∅}:
" #
wherem12(·)denotesthecombinedBBAresultingfromthecombina- X m ðAÞ2m ðYÞ m ðAÞ2m ðYÞ
tionofinformationofsources1and2.Thebeliefmassresultingfrom mP1C2R6ðAÞ¼m12ðAÞþ m 1ðAÞþm2 ðYÞþm 2ðAÞþm1 ðYÞ ð9Þ
thefusionofinformationfromsource1and2iswritten: Y∈2Θ 1 2 2 1
X A∩Y¼∅
m12ðAÞ¼ m1ðXÞm2ðYÞ ð5Þ
X;Y⊆ΘjX∩Y¼A
2.2.ConstructionofBBAsfromgeophysicalandgeotechnicaldata
Andthelevelofconflictbetweenthetwoconsideredsourcesofin-
formationcanthereforebequantifiedby:
Beliefmasseshavetobeassignedtoeachconsideredeventofthe
X FoD,forbothsourcesofinformation.Thecombinationofthebelief
m12ð∅Þ¼ m1ðXÞm2ðYÞ ð6Þ massescanonlybeinitiatedafterthisstage.Inthefollowing,thegeo-
X;Y⊆ΘjX∩Y¼∅ physicalsourceofinformationwillbeidentifiedassource1andthegeo-
technicalsourceofinformationassource2.A2Dmodelassumptionwill
Withm (X)andm (Y)thebeliefmassesrespectivelyattributedto
1 2 bemade,correspondingtothexandzspatialaxes,sinceverticalsec-
eventsXandYbysources1and2.
tionsofsubsoilareconsidered.
AccordingtoShafer'sapproachandunlikeSmets'rule,Dempster-
Shafer'srule(DS)doesnotallowtheattributionofamassofbeliefto
2.2.1.Geophysicaldata
theconflict(closed-worldassumption):
Thediscretizationoftheconsideredsubsoilsection,aswellasthe
depthofinvestigationandtheresolution,dependontheacquisition
mDSðØÞ¼0 ð7Þ
12 methodused(Keareyetal.,2013).Itistheuserwhosets,usingthein-
versiontoolused,theshapeanddimensionsofthediscretizationgrid
Theconflictistherereallocatedthroughaclassicalnormalization
used.Itisaboutstartingfromthisdiscretizationandbeingabletoasso-
factor.ThemassofbeliefinA,mDS(A),resultingfromthefusionofinfor-
12 ciateforeachcell,massesofbeliefsforeacheventoftheFoD.
mationfromsources1and2iswritten: TheconstitutiveclassesoftheFoDarealsofixedattheendofthein-
X versionprocessbythegeophysicist,withthehelpofarepresentationof
1
mD12SðAÞ¼1−m12ð∅Þ X;Y⊆ΘjX∩Y¼Am1ðXÞm2ðYÞ ð8Þ tmhoeddailsctrliabsusetiso(nFiogf.t1hae).sTehteorfeinpvreesretendtagtieoonpihnytshicisalfovramluemsa,iknesthitepfoosrsmibolef
tohighlightthecenters,minimaandmaximaoftheeventsconsidered
Thedisadvantageofthismethodisthattheconflictbetweenthe inordertobeabletofixtheboundsoftheintervalsassociatedwith
sourcesisnolongerrepresentedanditispossibletoobtaincounterintu- theeventsoftheFoD.Thenumberofcellsofthesubsoilsectionarerep-
itiveresultsiftheconflictisimportantbecauseofthisnormalization. resentedaccordingtothegeophysicalparametervalues.Theinfimaand
Evenmoreproblematic,evenifthedistinctsourcesarebothinformative supremamustbefixedsothattheintervalsareofthesamewidthin
whateverthelevelofconflictis,Dempster-Shafer'sfusionprocesscan ordertoavoidtheappearanceofabiaswhencalculatingWasserstein
evennottakeintoaccountthesecondsourceofinformation(Dezert distances(detailedunder).ToassociatethebeliefmasseswiththeFoD
etal.,2012). events,theintervalsofinvertedvaluesofthephysicalparameter(in
On the other hand, the PCR6 (Proportional Conflict Redistrib- red,Fig.1b)areconsidered.Forsomegeophysicalmethods,theseinter-
ution No. 6) rule of combination (Smarandache and Dezert, 2009; valscancorrespondtothevalueobtainedattheendoftheinversion
SmarandacheandDezert,2005),consideringtwosourcesofinforma- withitsassociatedinaccuracy.
tion,onlytransferstheconflictingmasstotheeventsthatareactually Itisthennecessarytoassociatebeliefmassvaluesm (.)correspond-
1
impliedintheconflictandinproportionwiththeirindividualmasses ingtoeachelementof2Θ,foreachcelloftheinvertedsection.The
in order to preserve the specificity of the information. In order to masses are obtained from the calculation of Wasserstein distances
applythePCR6rule:i)thecombinationruledescribedinEq.(5)must (Tran and Duckstein, 2002), considering two geophysical intervals
be applied, ii) the total or partial conflicting masses have to be A=[a ,a ]andB=[b ,b ]withAandBbelongingtoR,Abeingthe
1 2 1 2
Fig.1.a)Modelclasses'distributionofthegeophysicalparametervaluesfromtheconsideredsubsoilsection,allowingtheselectionofthegeophysicalclassesinb).Theredinterval
correspondstoanintervalofinvertedvalues,fromonecellofa2Dsectionofsubsoil,usedforWassersteindistances'calculation.(Forinterpretationofthereferencestocolourinthis
figurelegend,thereaderisreferredtothewebversionofthisarticle.)
4 T.Dezertetal./JournalofAppliedGeophysics170(2019)103824
intervalcorrespondingtoaneventoftheFoDandBbeinganintervalof Kulhawy,1999):
invertedvalues(Fig.1b),Eq.(10):
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dWassðA;BÞ¼(cid:6)(cid:2)a1þ2a2(cid:3)−(cid:4)b1þ2b2(cid:5)(cid:7)2þ13"(cid:2)a2−2a1(cid:3)2þ(cid:4)b2−2b1(cid:5)2# Cv¼ nmes1h−1∑Qni¼m1eshðQ−QiÞ2 ð12Þ
ð10Þ
whereQisthegeotechnicalvalueofthereferencecellintheconsidered
Thiscalculationestimatesthedistancebetweentwointervalsac-
boreholeandQ thegeotechnicalvalueinthenearbyborehole.ForFig.
cordingtotheirsizeandthedistancebetweenthem.TheWasserstein i
2b,n =3hasbeenconsidered.Ifn =5or7,thecomputation
distancesarecalculated(usingalogarithmicscaleifthegeophysicalpa- mesh mesh
oftheC willtakeintoaccount5or7cellsinthenearbyborehole.In-
rameterrequiresit)betweentheinvertedvalueswithestimatedinac- v
deed,for two consecutive boreholes with similar values,at similar
curacies,andtheintervalsassociatedwitheachevent,chosenbythe depth,thedecayoftheconfidenceisslowerthanfortwoconsecutive
geophysicist.EachcellisfinallyassociatedwithastandardizedBBAre-
boreholespresentingradicallydifferentvalues.Thisdecayofbelief
spectingEq.(3).Thisway,themorethedistanceofageophysicalinter-
massiscarriedouttotheleftandtotheright,fromeachdrilling.
valresultingfromtheinversionis“close”tooneeventoftheFoD,the
If,betweentwoboreholes,themassofbeliefassociatedwithahy-
morethemassofbeliefassociatedisimportant,andreciprocally. pothesisAisb1(m (A)b1),thentheremainderofmasstobeallocated
2
tosatisfyEq.(3),isreportedontheproposition“anytypeofmaterial”
2.2.2.Geotechnicaldata
representedbytheunionofallevents,suchasEq.(13):
Forthegeotechnicalpart,theinformationproposedduringaninves-
tigationcampaignisspatiallypunctual(inthex-zplane)andoften
containedinverticalsoundingsmadefromthesurface.Itisaboutasso- m2ðθ1∪θ2∪θ3∪θ4Þ¼1−m2ðAÞ ð13Þ
ciatingmassesofbeliefwiththedifferenteventsoftheFoDforeachcell
oftheconsideredverticalsoundings.Forthis,thevaluesproposedat
eachdepthareconsideredwiththeassociatedinaccuracy,correspond- Consideringnboreholes,from1tonfromlefttoright:borehole1
ingtothemeasurementerrorthatcouldbeattributedtothemeasuring cannotbecomparedtoanyboreholetoitsleftneithercanboreholen
device(Fig.2a).Thus,asforthegeophysicalpart,intervalsofvaluesare becomparedtoanyboreholetoitsright.Indeed,foragivendepth,an
obtained. equalCvisconsideredforleftandrightdirections,forboreholeslocated
Thegeotechnicalmeshconsistingofasmanycellsindepthasthe atthebeginningandattheendofthesection.
numberofgeotechnicalvalues(Fig.2b)isgenerated.Amassofbelief
m (.)=1isassigned,inthedrillingpoints,totheeventscorresponding 2.3.Dimensioningofthemeshpriortothefusion
2
tothemeasuredgeotechnicalparameter.Avalueof1issetsinceweare
very confident in the information inside the boreholes unlike the Eachsourceofinformationimposesitsownmeshbutinorderto
spatializedgeophysicalinformation.Anewmeshisthenconstructed combinethebeliefmassesfromthegeophysicalinformationsource
(Fig.2c),accordingtothesizeanddepthoftheboreholes.Inorderto (source1)andthegeotechnicalsource(source2),itisnecessaryto
characterizetheentiresectionofthemodel,asdoesthegeophysical haveacommonmeshcontaining,foreachcell,thegeophysicalandgeo-
method,andtoassociatemassvaluestoeachnewcell(BBA),anexpo- technicalBBAs.Inordertonotalterthequalityoftheinformation,noin-
nentiallateraldecayofthebeliefmassisimposed,fromthedrilling terpolationiscarriedout.Itisdecidedtosuperimposethegeophysical
pointtothenearestboreholesothatthedecayrateisafunctionofthe discretizationgridresultingfromthe2Dinversiontothegeotechnical
valuesproposedbythenearbyborehole.Sothat,foraspecificdepth, division,dependingonthenumberandtheboreholepositions.Thus,
Eq.(11): anirregularmeshisobtainedbutwithoutanyapproximation(Fig.3).
3.Syntheticstudy
BBAðxÞ¼e−kCvxBBAð0Þ ð11Þ
Below,asyntheticstudybasedonartificialdataisproposedinorder
Withxbeingthedistancefromtheconsideredcelltothereference totestthisnewproposedmethodology.Itistheopportunitytoshow
borehole(x=0intheborehole),kadecayfactorfixedbytheusertoad- theimpactofdifferentlevelsofnoiseonthegeophysicalinformation
justthelateraldecayrate,BBA(x)thebeliefmassvaluesassignedtoeach aswellastheinfluenceofthelateraldecayfactork(Eq.(11))onthere-
eventoftheFoDforapositionx,withBBA(0)=1.C correspondstothe sultsofthefusioninordertobeabletochooseavaluefortheuseofsuch
v
coefficientofvariationexpressedinEq.(12),suchasusedin(Phoonand amethodologyfromrealdata.
Fig.2.Constructionofageotechnicaldiscretizationmeshfromtwoverticalboreholesacquisition(SD1andSD2).a)RepresentationofthegeotechnicalvaluesforSD1andSD2accordingto
thedepth.b)Theboreholesaredividedincellsassociatedwithbeliefmassequalto1fortheconsideredevent.c)Constructionofanewmeshaccordingtothesizeanddepthofthe
boreholes.
T.Dezertetal./JournalofAppliedGeophysics170(2019)103824 5
3.2.FoDandconsideredmodel
Forthissyntheticstudy,atwo-layermodelisconsidered,composed
ofmaterialsthatcanbelikenedtosiltsfortheupperlayerandclaysfor
theunderlyinglayer.TheFoDthereforecontainsthreematerialclasshy-
potheses,suchas:
Θ¼fθ1;θ2;θ3g ð14Þ
Withθ theeventcorrespondingtotheclayeymaterial,θ tothesilty
1 2
materialandθ tounknownmaterials.Thelatterisassociatedwith
3
theunionofthegeophysicalandgeotechnicalvaluerangesthatdo
notcorrespondtothoseassociatedwithθ andθ .Thiseventθ allows
1 2 3
usinacertainway,toquantifythelackofknowledgeoftheenviron-
mentsinceitdoesnotincludethetwofirstsets.Theconstructionof
theBBAsthenconsistsinassociatingthedataofthetwoconsidered
sourcestotheeventsoftheFoD.Fig.4showsthetwo-layermodel
basedoneventsfromtheFoD,usedforthissyntheticstudy.
Fig.3.Exampleofageophysicalmesh(inblack)andageotechnicalmesh(inred) 3.3.ConstructionofBBAsfromgeophysicalandgeotechnicaldata
superimposed to propose a new irregular mesh to carry out the combination
calculationsandpresentthefusionresults.(Forinterpretationofthereferencesto 3.3.1.Geophysicaldata
colourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)
TheelectricalacquisitionissimulatedwithaWenneracquisition
modeandwith96electrodesinterspacedfromonemeter.Anelectrical
3.1.Consideredmethods resistivityof100Ω.misconsideredfortheuppermaterialandaresistiv-
ityof30Ω.mfortheunderlyingone(Palacky,1987)(Fig.4).Electrical
Forthisstudy,theelectricalresistivitytomography(ERT)method acquisitionsaresimulatedwithdifferentnoiselevels(5,10and15%).
standsforthegeophysicalinformationsourceandtheConePenetrom- Theresultsofthisinversion(with10%noise,Fig.5a)allowtohighlight
eterTest(CPT)methodforthegeotechnicalinformationsource. thepresenceoftwolayersbuttheinterfacebetweenthesetwolayersis
ThebasicprincipleofDC-resistivitymethodsconsistininjectingan notperfectlyidentified.Avariationofthicknessinthecenterofthe
electriccurrentofknownintensity[A]bymeansoftwo“current”elec- modelisvisible.Theinterfaceisnotstraightforwardandanomalies
trodesandmeasuringavoltage[V]betweentwo“potential”electrodes. arepresentonthesurfaceeventhoughtheyarenotpartoftheinitial
Dependingontheelectrodelayout,thetopography,thepropertiesof model.
thematerialsandtheirdistribution,apparentresistivityvaluescanbe Fromtheseinversionresults,itispossibletodefinetherangesof
computed.Thedepthofinvestigationdependsonthespacingofthe electricalresistivitiesthatwillbeassociatedwiththedifferentevents
electrodes,theconfigurationoftheelectrodesandthenatureofthe consideredforthefusionprocess.Adistributioninmodalclassesis
soil(Loke,2011).Bygeneralizingthisprinciple,atwodimensional usedtovisualizethenumberofcells,inthediscretizedsectionofthe
(2D)ERTconsistsinaligningaseriesofelectrodesandacquiringa 2Dinversion,associatedwithspecificrangeofresistivities(Fig.5b).
largenumberofmeasurementsbasedonfourelectrodesconfiguration. Thisdistributionallowstohighlightthetwolargematerialclassesof
Theapparentresistivitydataacquiredaretheninvertedusinganinver- themodel.Thankstoit,theboundsoftheconsideredeventscanthus
sioncodeorsoftwaretoreconstructacomplete2D-sectionofelectrical bedefined(inΩ.m),sothattheintervalshavethesamelength(inlog-
resistivity[Ω.m].HeretheRes2Dinvsoftware(ver3.71.118)(Loke, arithmicscale):
2011)hasbeenused.
Inordertoobtainanartificialresistivitysectionofsubsoil,atwo θ1¼½25;45(cid:2)
stepsprocedureisfollowed.First,resistivitydataaresimulatedusing θ2¼½83;149:4(cid:2) ð15Þ
theRes2Dmodsoftware(Loke,2002),onthesectionthatwewantto θ3¼½13:89;25½∪(cid:2)45;83½∪(cid:2)149:4;268:92(cid:2)
consider. Second, apparent electrical resistivity values are inverted
withRes2Dinv,consideringaL1norm(Lokeetal.,2003)andanex- AsexplainedinII.2.,itispossibletoassociatebeliefmasseswitheach
tendedmodeldiscretization,toobtainthesyntheticinvertedsection cell of themesh thanks to thevalues resulting from the inversion.
ofelectricalresistivity. AspartoftheconstructionofgeophysicalBBAs,thevaluespresented
TheCPTmethodconsistsinpushingrodsintothesoil,atacon- Fig.6areobtained.Thisfigurehighlightstheassociationofthevalues
stantspeed,withaconicaltipattheend(ISO22476-1,2012,n.d.). ofFig.5awiththeeventsoftheFoD,Eq.(15).Thepresenceofatop
Thistestisoftenusedforthedeterminationofthesoilsmechanical layer(θ )andabaselayer(θ )canbedetected(Fig.6a).Itappears
2 1
resistanceproperties.Thetwomeasuredparametersarethetipre- thatthereisavariationinthethicknessofthelayersinthecenterof
sistanceq [MPa]andsleevefrictionf [MPa].Althoughthemethod themodel,buttheinterfaceisnotwellcharacterized.Moreover,thein-
c s
uses two parameters, only q will be considered as the study termediatevaluesofelectricalresistivityresultingfromtheinversion
c
parameter. (Fig.5a)betweenθ andθ layersinducetherepresentationofathird
2 1
Fig.4.RepresentationoftheeventsoftheFoDintheimposedmodelofsubsoilofthesyntheticstudy.
6 T.Dezertetal./JournalofAppliedGeophysics170(2019)103824
Fig.5.a)Subsoilsectiondisplayinginvertedelectricalresistivityvaluesfrom10%noisedataacquisitionandb)modelclasses'distributionofthecellspresentedina),accordingtothe
electricalresistivityvalues(Ω.m).
material(θ )whichhasnorealityinthemodelthathasbeenfixed.The followinganormaldistributiondefinedforeachevent.Meanq valuesof
3 c
beliefmassesaremaximumwhentheresistivityvaluescorrespondto 0.2and20MPaarerespectivelyusedtodefinethenormaldistributions
thecenteroftheresistivityclassessetforeachevent(Eq.(15)). ofthematerialclasses.Standarddeviationvaluesequalto10%ofthe
meanvaluesareassociated,echoingthe10%noiseusedforthegeophys-
icaldata.Keepingtheminimumandmaximumvalues,theserandom
3.3.2.Geotechnicaldata draws,makeitpossibletodefinethelimits,inMPa,oftheintervalsasso-
Concerningthesourceofgeotechnicalinformation,thesimulationof ciatedwiththeelements(i.e.materialclasses)oftheFoD:
fourverticalCPTsoundingsinterspacedfrom19misproposed(Fig.7).
20cmwideandupto15mdeepboreholesareconsidered,andavalue θ1¼½0:14;0:27(cid:2)
of q is recorded every 50 cm from the surface. An inaccuracy of
10−2cMPaonthemeasurementsisconsidered.Forafixednormalized θ2¼½13:5;23:5(cid:2) ð16Þ
frictionratioof3%,avalueofqcof20MPaisconsideredfortheupper θ3¼½0:1;0:14½∪(cid:2)0:27;13:5½∪(cid:2)23:5;100(cid:2)
siltymaterialandavalueof0.2MPafortheunderlyingclaymaterial,
asproposedintheRobertsondiagram(Robertsonetal.,1986). Theminimumandmaximumvaluesarefixedat0.1and100respec-
Inordernottohaveuniformvaluesofq forthematerialsandtotryto tivelybecausetheyaretheminimumandmaximumvaluesinRobe-
c
representthenoisyrealityofanacquisitioninthefield,valuesaredrawn rtson'sdiagram(Robertsonetal.,1986).
Fig.6.a)RepresentationoftheeventhavingthehighestbeliefmassaccordingtotheBBAconstructionfromgeophysicaldataandb)theassociatedbeliefmassvalues,consideringa10%
noise.Theblacklinerepresentthepositionoftheinterface.
T.Dezertetal./JournalofAppliedGeophysics170(2019)103824 7
Fig.7.2DsectionofsubsoildisplayingtrueERdistributionwithboreholespositionsinblackandassociatedtipresistanceverticalprofilesinwhite.
Therearetwotypesofsoundingresultsaccordingtotheirposition decayfactork(Eq.(11))influencingthelateraldecayrateofgeotechni-
(Fig.8).Oncethevaluesassociatedwiththemeshesofthesounding calinformation.
areobtained,itispossibletoassociatemassesofbelieftothewholesec- Fig.10showsthefusionresultswithdifferentvaluesofk(10−2,
tionbyextendingthegeotechnicalinformation,asexplainedinII.2.In 5.10−2,10−1,5.10−1and1)forasimulatednoiseof10%ontheacquired
theframeworkoftheconstructionofgeotechnicalBBAsandfork= geophysicalinformation.Noisewassetat10%sincetheelectricalresis-
0.1(Eq.(11)),theobtainedvaluesareproposedinFig.9.Thisfigure tivityclassesoftheFoDweredefinedfromthemodalclassesofthe
highlightsthefactthattheconfidenceismaximuminthesoundings. inverted10%noiseimage,Fig.5b.Foreachvalueofk,Fig.10aandbrep-
Thismethodallowsustocharacterizethematerialθ onthefirst5m resenttheresultsobtainedbySmetsfusionwhereasFig.10canddrep-
2
ofthemodelandthematerialθ from10to15mdeep. resenttheresultsobtainedbyPCR6fusion.WhileFig.10aandcshow
1
Thegreaterthicknessofthematerialθ inthecenterofthemodelis thematerialclasseshavingthegreatestmassofbeliefattheendof
2
alsowellcharacterized.Ontheotherhand,agreatdoubtappearsinyel- thefusionprocess,Fig.10banddcorrespondtothevaluesofthesere-
low(Fig.9a)incertainareas,notallowingthedeterminationofaspe- spective belief masses,between 0and1.Thesefigures,displaythe
cific material (θ ∪ θ ∪ θ ). For the model base area, this can be events(materials)potentiallypresentwithinthesection,aswellas
1 2 3
explainedbythefactthatthesoundingsstopat15mdepth.Regarding theirattachedlevelofconfidence.
theareasbetween5and10mtotherightandleftofthefirstandlast Thehigherthevalueofkis,thehighertherateofconfidenceingeo-
sounding,thisisrelatedtothefactthatforthesetwosoundings,the technicalinformationis.Thiscanbeseen,forexample,fromthelast
closestsoundingsproposedifferentvaluesatthesamedepths,thecon- boreholetotherightendofthesection(Fig.10b)orbetweenthe2nd
fidenceattributedtothepresenceofθ thereforedecreasesveryquickly and3rdboreholes(Fig.10d).Theincreaseofkimpliesthatfortwo
1
laterally.Thisdecayisalsohightowardstheedgesofthemodelbecause soundingsofferingsimilarvaluesatthesamedepth,theconfidenceas-
noothersoundingispresentattheendstoconstraintheinformation. sociatedwiththecorrespondingtypeofmaterialwilltendtodecrease.
Ontheotherhand,fortwosoundingsproposingdifferentvaluesat
thesamedepth,theincreaseofkwillhardlyhaveanyimpactonthebe-
3.4.Effectoflateraldecayfactorandnoiselevelonthefusionresults liefmassesassociatedwiththeselectedevents(e.g.between5and10m
ofdepthbetweentheboreholes1and2,Fig.10d).
Weexamineinthisparttheresultsofthefusionofbeliefmasses Withregardtothematerialclassesidentifiedafterthefusionpro-
established for the proposed synthetic model by varying the noise cess, the more k increases, the more the quantity of conflict de-
levelofthegeophysicalinformation,aswellasthevalueofthelateral creases(inred,Fig.10a).Thisisexplainedbythefactthatwhen
Fig.8.Examplesofthetwotypesofsimulatedsoundingswithtipresistancevaluesaccordingtotheinvestigationdepth.a)correspondstoboreholen°1and4onFig.7while
b)correspondstoboreholen°2and3onFig.7.
8 T.Dezertetal./JournalofAppliedGeophysics170(2019)103824
Fig.9.a)RepresentationoftheeventshavingthehighestbeliefmassaccordingtotheBBAconstructionfromgeotechnicaldataandb)theassociatedbeliefmassvalues.Theborehole
positionsareindashedlineswhiletheblacklinerepresentthepositionoftheinterface.
Fig.10.Representationoftheeventshavingthehighestbeliefmassina)andc)andtheirassociatedmassvaluesinb)andd),consideringa10%noise.Ini)k=0.01,ii)k=0.05,iii)k=0.1,
iv)k=0.5andv)k=1.FiguresontheleftsideareresultsforSmetsfusionwhilefiguresontherightsideareresultsforPCR6fusion.Thesoundingpositionsareindashedlineswhilethe
blacklinerepresentthepositionoftheinterface.
T.Dezertetal./JournalofAppliedGeophysics170(2019)103824 9
thereislittletrustinthegeotechnicaldata,thereislittleconflict SincetheclassesassociatedwithFoDelementswerefixedfromthe
withthegeophysicaldata.Inthemeantime,anincreaseinthepro- valueswith10%noise(Section3.3andFig.4b),itis“reasonable”to
portionofθ isobserved(Fig.10c)closetotheinterface.Thisobser- haveahigherconfidence(higherbeliefmasses)ontheseresultsthan
3
vationisexplainedbyalargermassattributedtotheunionofevents ontheresultswith5%and15%noise(Figs.11c,e,10iii.b,iii.d).Thefu-
andbygeophysicaldatawhichproposeintermediatevaluesatthe sionprocessallowstooverridethenoiseeffects,whetherthenoise
interfacelevel. levelis5or15%.ThiscanbeimputedtothecomputationofWasserstein
Inthefollowingofthiswork,anintermediatevalueofkwillbe distances,takingintoaccountthedatainaccuraciesandconsideringall
retained,equalto0.1.Withsuchparametervalue,agoodconfidence geophysicalclasses.
ininformationrepeatingbetweentwosuccessivesoundingsisob-
tained,butitalsoleavesroomfordoubtbyhavingenoughunknown 4. Setting upa test benchfor real geotechnical and geophysical
material(θ )attheinterfaces.Theobtainedfusionresultswithdif- acquisitions
3
ferentnoiselevelsaddedtothegeophysicalinformation(5,10and
15%)areshowninFigs.10iiiand11withk=0.1. 4.1.Materials
Thegreatertheamountofnoiseis,thelesscleartheinterfacespro-
posedbytheinversionare(Figs.5a,11.i.a,ii.a).Agreaternumberof Inordertobeabletoassessthevalidityofthedevelopedfusion
anomaliesarealsopresentwhenthenoiselevelincreases.Thenoise methodology,twomethodsofdataacquisitionwereretained:(i)a
levelfinallyimpactsthelevelofinaccuracyassociatedwiththegeo- mini-ERTdeviceactingasthegeophysicalsourceofinformationand
physicaldatausedinthefusionprocess.Largerdatainaccuraciesinduce (ii)alaboratorypenetrationconeactingasthegeotechnicalsourceof
widervaluerangesconsideredforcalculatingWassersteindistances, information.Beforesettingupthetestbench,itwasnecessarytoselect
whichinturncanbringtoconsiderbeliefmassesonmoreeventsof thematerialsthatcouldbeputinplaceinatankinordertocarryoutthe
theFoD. study. This selection implies that the materials used meet several
Fig.11.a)Subsoilsectiondisplayinginvertedelectricalresistivityvaluesfromi)5%noiseandii)15%noisedataacquisition.Representationoftheeventshavingthehighestbeliefmassin
b)andd)andtheirassociatedmassvaluesinc)ande).FiguresontheleftsideareresultsforSmetsfusionwhilefiguresontherightsideareresultsforPCR6fusion.Thesoundingpositions
areindashedlineswhiletheblacklinerepresentthepositionoftheinterface.
10 T.Dezertetal./JournalofAppliedGeophysics170(2019)103824
conditions in order to validate the methodology: they must have
(i)distinctelectricalresistivityranges,(ii)distinctpenetrationdepths
and(iii)acertainhomogeneityinthespacetolimituncontrolledanom-
alousvalues.
4.1.1.MiniERTdevice
Expresslyforthepurposesofthisstudy,aminiERTdevice(Fig.12)
hasbeensetup.Thisdeviceconsistsof48electrodesof6mmlength,po-
sitionedatregularintervalsofonecentimeter.Itcanbemovedalongthe
testbenchtomakemultipleacquisitionsandtocoveralongersection.
4.1.2.Laboratorypenetrationcone
ThelaboratorypenetrationconemethodisdescribedintheFrench
standard(NFP94–052-1,1995).Itconsistsinmeasuringapenetration
depthofacone,inmillimeters,subjectedtoitsownweight(Fig.13).
Thematerialsaretestedindividually,repeatedly,todetermineanaver-
agevalueandastandarddeviationofpenetrationdepthforeachmate-
rial.Thesevaluescanbeusedlaterinthestudytosimulatedifferent Fig.13.Laboratorypenetrationcone.
drillingpositionswithinthetestbench.Thismethodcanbelikenedto
theCPTmethodwhichisoneofthemostpopularinsitugeotechnical 4.2.FoDandBBAmodeling
tests.
4.2.1.FoDandtargetmodel
AFoDconsistingoffourelements(materialclasses)isconsideredso
4.1.3.Testbenchandusedmaterials that:
Forthevalidationofthemethodology,wewantedtobuildatest
benchthatcouldbeeasilysetupandcontrolled,withtwoorthree Θ¼fθ1;θ2;θ3;θ4g ð17Þ
layersandvariationoftheinterfacepositions.Fast-hardeningnatural
fine-grainedplasteraswellasHostunfinesand(Flavignyetal.,1990) Withθ theelementcorrespondingtotheplastermaterial;θ corre-
1 2
aretheretainedconstituents.Thesetwomaterialsmeetthethreecondi- spondingtosaturatedsand;θ correspondingtothehardandelectri-
3
tionslistedabove.TheywereplacedinatransparentPVCtankof100 cally insulating bottom of tank simulating a substrate and θ
4
×30×17cm3asshowninFig.14withanunderlyinglayerof5cmof correspondingtounknownmaterials,beingtheunionoftherangesof
plaster(settingtime=69h)overlaidbyalayerof2.5cmofwatersat- valuesnotcorrespondingtothoseassociatedwiththe3previouslyde-
uratedsand. scribedmaterials.Fig.15presentsthetargetmodelintheformofevents
Aformworkwasmadeduringtheplacementoftheplastersothata constitutingtheFoD,followingthedispositionofthematerialswithin
20cmlonganomalycouldbeinsertedin.Saturatedsandof7.5cmthick- thetestbench.Althoughthetankusedis1mlong,theERTacquisition
nessispresentinsteadofplaster.Thecontactbetweenthematerialsand onlycovereda83cmlongsection,onthecentrallineofthemodel,
thebottomofthetankisattheoriginofaninterfacethatwillbeinter- andallowedustoimageupto18cmofdepth.
estingtodetectwiththehelpofthemethodology.16kgofplasterwere
mixedwith8kgofwatertoobtainthematerialfinallyputinplace.The 4.2.2.ConstructionofBBAsfromgeophysicalandgeotechnicaldata
electricalresistivityoftheplasterwasmeasuredbeforeandafterthe Theelectricalacquisitionwascarriedouton83cmlong,onthecen-
placementofthesaturatedsandtoverifythatthepresenceofwater trallineofthemodel,withafirstacquisitionon47cm,andthreenext
hadanegligibleimpactontheelectricalpropertiesoftheplaster.
FortheHostunsand,15.82kgwerepluviatedin5.8kgofwater,
abovetheplastertoreachsaturation.Trialshadbeencarriedoutinad-
vance to determinethe proportions of water and sand required to
achievesuchastateaswellastovalidatetherepeatabilityofsuchinstal-
lationbypluviation.Thevaluesofelectricalresistivitiesandpenetration
depthsaredisplayedinTable1.
Fig.14.Transparencyviewofthetestbench.
Table1
Valuesofelectricalresistivityanddepthofpenetrationofthematerialssetupwithinthe
testbench.
Plasterbeforepluviation SaturatedHostunsands
Electricalresistivity(Ω.m)
Mean 31.28 78.15
Standarddeviation 3.23 11.18
Numberofmeasures 12 52
Penetrationdepth(mm)
Mean 0.11 17.31
Standarddeviation 0.04 1.61
Fig.12.MiniERTdevicewith48electrodesspaced1cm,adjustableheight,usedfor
Numberofmeasurements 8 10
electricalacquisitionsinthetestbench.