Snowpack permittivity profile retrieval from tomographic SAR data

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Snowpack permittivity profile retrieval from

tomographic SAR data

Badreddine Rekioua, Matthieu Davy, Laurent Ferro-Famil, Stefano Tebaldini

To cite this version:

Badreddine Rekioua, Matthieu Davy, Laurent Ferro-Famil, Stefano Tebaldini. Snowpack permittivity

profile retrieval from tomographic SAR data. Comptes Rendus Physique, Centre Mersenne, 2017, 18

(1), pp.57-65. �10.1016/j.crhy.2015.12.016�. �hal-01277215�

Snowpack

permittivity

profile

retrieval

from

tomographic

SAR

data

Badreddine Rekioua

a

,

b

,

∗∗

,

1

,

Matthieu Davy

b

,

∗

,

Laurent Ferro-Famil

c

,

∗

,

Stefano Tebaldini

b

a_{IETRUMRCNRS6164,UniversitédeRennes-1,35000Rennes,France} b_{EcolemilitairepolytechniquedeBordjElBahri,16111,Alger,Algeria}

c_{DipartimentodiElectronica,InformazioneeBioingegneria,PolitecnicodiMilano,20133 Milano,Italy}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received24August2015 Accepted1December2015 Availableonline19February2016 PresentedbyM.DenisGratias

Keywords: Snowpack Snowpermittivity SAR SARtomography GB-SAR

Timedomainbackprojection

This work deals with 3D structure characterization and permittivity profile retrieval of snowpacks by tomographic SAR data processing. The acquisition system is a veryhigh resolution groundbased SAR system, developed and operated by the SAPHIR team, of IETR,UniversityofRennes-1(France).Itconsistsmainlyofavectornetworkanalyserand amulti-staticantennasystem,movingalongtwoorthogonaldirections,soastoobtaina two-dimensionalsyntheticarray.DatawereacquiredduringtheAlpSARcampaigncarried by the European Space Agency and led by ENVEO. Inthis study, tomographic imaging is performedusing Time DomainBack Projection and consists incoherently combining thedifferentrecordedbackscattercontributions.Theassumptionoffree-spacepropagation duringthefocusingprocessisdiscussedandillustratedbyfocusingexperimentaldata.An iterativemethodforestimatingtruerefractiveindicesofthesnowlayersispresented.The antenna patternis alsocompensatedfor. Theobtainedtomograms after refractiveindex correctionarecomparedtothestratigraphyoftheobservedsnowpack.

©2016Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

SAR imaging systems provide large-scale and high-resolution imaging capabilities, independently of day–night and weatherconditions.AconventionalSARprojectsa3D reflectivitymap ona2D plane.Thisprojection from3D to2D space

induces some effects likelayover andforeshortening [1].Forsome applications like urban area andnaturalenvironment monitoring,2D SARimagingcomesto beofa limitedinterestby omittingthedescriptionofthe verticalstructureofthe observed scene.This limitation is aserious shortcomingforthe specialcase ofsnow cover monitoringapplications, like hazardforecasting,waterresourcemanagement,andclimatechangemonitoring.Indeed,fortheseapplications,anaccurate descriptionoftheverticalstructureofthesnowcoverisofgreat interest[2].Manyworkshaveaddressedtheproblemof

*

Correspondingauthors.

**

Principalcorrespondingauthorat:IETRUMRCNRS6164,UniversitédeRennes-1,35000Rennes,France.

E-mailaddresses:badreddinerekioua@gmail.com(B. Rekioua),matthieu.davy@univ-rennes1.fr(M. Davy),Laurent.Ferro-Famil@univ-rennes1.fr

(L. Ferro-Famil),stefano.tebaldini@polimi.it(S. Tebaldini).

1 _{B.RekiouaisaPhdstudentoftheIETRofUniversityofRennes1(France)withascholarshipfromtheMilitaryPolytechnicalSchoolofBordjElBahri} (Algiers,Algeria).

http://dx.doi.org/10.1016/j.crhy.2015.12.016

1631-0705/©2016Académiedessciences.PublishedbyElsevierMassonSAS.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

58 B. Rekioua et al. / C. R. Physique 18 (2017) 57–65

Fig. 1. GB-SAR system of the SAPHIR team of the University of Rennes-1.

Table 1

MaincharacteristicoftheGB-SARsystem.

Freq. band X, Ku

Vertical resolution [cm] ∼4 (Ku band)

∼10 (X band) Range resolution [cm] ∼4 Azimuth resolution [cm] ∼2

Polarization VV, HV

assessingthephysicalpropertiesofsnowcoversusingSARdata.Mainly,SARdataareusedtoestimatesnowwetness[3–6], snow water equivalent [7–10], snow depth [11–13], snow density [14,15] or snow covered areas [16,17]. For the above cited works,one ormoreparametersofsnow coversareinvestigatedwithout addressingdirectlytheverticalstructure of the observed scenes whichmay relate into ambiguous interpretations or ill-conditionedestimation. In thiscontext, SAR tomography(TomSAR)representsapromisingtool formonitoringsnow andiceduetoits abilityofresolving 3D scatterer

contributionsandthereforeovercominglayoverandforeshorteningeffects[18,19].Inthiswork,thedescriptionofthe ver-ticalstructureofasnowpackusingTomSARdataisaddressedby retrievingtheverticalpermittivityprofilethatrepresents a key informationfor estimating thephysical parameters. The processed data were acquired using a Ground Based SAR (GB-SAR)systemdevelopedandoperatedbytheSAPHIRteam,oftheIETRoftheUniversityofRennes-1intheframework oftheEuropeanSpaceAgency(ESA)AlpSARcampaignledbyENVEOintheAustrianAlps.TheGB-SARsystemwasoperated at both Xand Kubands,allowing for resolutions within afew centimeters. The processing ofthe acquireddata isdone usingtheTimeDomainBackProjectionAlgorithm(TDBPA).Theresultingtomogramsshowan overestimationofsnowpack heightandgeometricaldistortions.Thesedistortionsarecausedbytheimplicitassumptionoffreespacepropagationduring thefocusingprocess.Inordertohaveacorrectinterpretationoftheobtainedtomograms,thefocusingalgorithmmusttake into account thecorrect permittivities,unknown apriori.An iterativeprocedure isproposed toestimate the correct per-mittivity foreach snowlayerandthecorrespondingthickness. Resultsarediscussedandcomparedtoin-situ stratigraphic

measurementsmadeavailablebyENVEO.

2. GB-SARsystemanddatavolumedescription

TheacquisitionsystemisaSteppedFrequencyContinuousWaves(SFCW)GB-SAR,developed,implemented,andoperated by theSAPHIRteamoftheUniversity ofRennes-1 (France).Itcomprisesfourrectangularhorn antennasfixedatdifferent elevationpositionsandconnectedtoaVectorNetworkAnalyser(VNA,seeFig. 1).ThefourantennasandtheVNAholdona metallicboxmountedona3 m-longrailequippedwithalinearsteppermotor.Eachantennacanoperateeitherasa trans-mitterorareceiverresultinginsixequivalentmonostaticantennapositionswithconstantspacingintheelevationdirection. Moreelevationpositionscanbeobtainedbychangingthepositionoftheantennaarrayonthemetallicboxorbychanging theelevationoftherail.ThegroundcoverageoftheGB-SARislimitedbytherailheightandtheantennas’radiationpattern. The systemanddatastoragearecontrolledbya laptopcomputer. Theparametersdescribingeachdataacquisitionarethe transmittedfrequency,thecombinationofthetransmittingandreceivingantennasaswellastheirazimuthposition. Acqui-sitions areperformedintheXandKufrequencybandswithabandwidth Bf

=

4 GHz.Thecorrespondingrangeresolution isgivenbyEq.(1):

δ

r

=

c

2 Bf

=

3

.

75

[

cm

]

(1)

The GB-SARis ahighresolution 3D imagingsystemwithafew centimeters resolutioninelevation, groundrangeand azimuthdirections.ItsmaincharacteristicsaresummarizedinTable 1.

The processeddatawereacquiredduring theESAAlpSARcampaignled byENVEOintheAustrian Alps.The snowpack measured attheRotmoos testsiteiscomposed ofastack ofhorizontalsnowslabs, witha totalheight ofabout137 cm. Its structure is depicted by the profile given in the panel on the right in Fig. 10, which provides the different physical parameters(seeTable 2)asafunctionofsnowdepth.

R Snow hardness description

H W [kg·m−2_{] Snow water equivalent}

ρ[kg·m−3_{] Snow density}

Fig. 2. TomSAR recording configuration.

3. Dataprocessing

3.1. 3Dfocusing

SARTomographyexploitsmultiplepassesoperatedovertheelevationdirectiontoachieveafocusingcapabilityalongthe elevationdirection[20](seeFig. 2).Theresponseofapointtargetatposition→−r seenbytheGB-SARisexpressedby:

s

(

→−r

−

→−p

,

k

)

=

exp

(

−

i k



− →_r

₋

→−_p

_)



→−r

−

→−p



σ

˜

(

− →_r

₋

→−_p

_,

_k

₎

_U

₍

→−_r

₋

→−_p

_,

_k

₎

₍₂₎

Herek

=

4_λπ isthe wavenumber,→−p isthetransmitting–receivingantennaspositionandU

(

→−r

−

→−p

)

isthe transmitting– receivingantennagain,

σ

˜

(

→−r

−

→−p

,

k

)

isthecomplexresponseofthetarget.

Assuming a constantmedium response over the frequencybandwidth andover theantennapositions, Eq. (2)can be writtenas: s

(

→−r

−

→−p

,

k

)

=

exp

(

−

i k



− →_r

₋

→−_p

_)



→−r

−

→−p



σ

˜

(

− →_r

₋

→−_p

₎

_U

₍

→−_r

₋

→−_p

₎

₍₃₎

Thetotalresponserecordedbyanantennacoupleatposition→−p isgivenbyEq.(4): S

(

→−p

,

k

)

=





s

(

→−r

−

→−p

,

k

)

d→−r (4)

GB-SARdata arefocused usingTimeDomain BackProjection(TDBP).The echoesrecordedby atransmitting–receiving antenna coupleare correlated withthe ideal response ofa target ata given position−→r0 to get the range-focused signal expressedbyEq.(5):

Sr

(

−→r0

,

→−p

)

=



Bk

S

(

→−p

,

→−k

)

exp

(

−

i k



−→r0

−

→−p

)

dk (5)

Here S

(

k

,

→−p

)

istheresponserecorded bya transmitting–receivingantennacoupleatposition→−p, k isthewavenumber, and



−→r0

−

→−p



isthegeometricdistancebetweentheinvestigatedpointandtheemitting–receivingantennacouple.

60 B. Rekioua et al. / C. R. Physique 18 (2017) 57–65

Fig. 3. Focused and multilooked tomogram. The panel on the left is the intensity image and the panel on the right is the coherency image.

Therequestedresolutioninthegroundrangeandelevationisthenachievedbycombiningcoherentlytherangefocused signalsatdifferentelevationpositionsanddifferentazimuthpositionsasfollows:

I

(

−→r0

)

=







Nel



nz=1 Naz



nx=1 Sr

(

−→r0

,

→−p

(

nz

,

nx

))







2 (6)

The coherencyoftheimage isdefinedastheratioofthecoherentsumofthe contributionsofthedifferentantennas

I

(

−→r0

)

tothenon-coherentsumoverelevationcontributions,whichisthesumoftheabsolutevaluesofthosecontributions:

C R

(

→−r0

)

=

√

I

(

→−r0

)



Nel nz=1







Naz nx=1Sr

(

− → r0

,

→−p

(

nz

,

nx

))





(7)

In the caseofa Stepped FrequencyContinuous Wave (SFCW)SAR, Sr

(

r→−0

,

→−p

)

is formulated asa discrete sumover the available wavenumberbins.ForNk equidistantwavenumberbinsrangingfromkmintokmax (withastepequalto



k),the range-focusedsignalcanbeexpressedas:

Sr

(

→r−0

,

→−p

)

=



Nk−1 nk=0 S

(

nk

,

− →_p

₎

_ei(kmin+nkk)−−→r0−→−p

=

ei kmin−−→r0−→−p



Nk−1 nk=0 S

(

nk

,

− →_p

₎

_ei nkkr−→−0−→−p ₍₈₎ Equation(8)maythenbeslightlymodifiedinordertoaccountforthefactthattheVNAdeliversbasebandsignals,i.e. signalswhosespectrumisdemodulatedaroundthenullfrequency:

Sr

(

→r−0

,

→−p

)

=

ei kc − − → r0−→−p Nk 2



nk=Nk2 −1 Smes

(

mk

,

→−p

)

ei mkk − − → r0−→−p ₍₉₎

where

{

Smes

(

mk

,

→−p

)

}

isthemeasuredresponsearoundthenullfrequencyandkc thecentralwavenumber.

The sumexpressed inEq.(9)is the IFFT of

{

Smes

(

mk

,

→−p

)

}

.This IFFT is computedovera regular distancegrid ranging

from0 to

(

Nk

−

1

)

× δ

r.Then,itisinterpolatedateachimagepositiondefinedbytherangedistance



r→−0

−

→−p



.

For 3D focusing,the imaginggrid isdefinedby a fixed azimuthslice in the elevation-groundrange plane.The set of usedantennasischosen insuch awaythat allinvestigatedpointsfallwithinthe

−

3 dBazimuthapertureoftheantenna pattern.

Fig. 3 showsa multilookedtomogram obtainedby applyingthe TDBP algorithmover 25 azimuthslices.The observed snowpack is characterizedby a multi-layered vertical structure.The strongest backscatteringcontributions are associated withinternalandbottomsnowlayerinterfaces,whereascontributionsfromsurfaceandnear-subsurfacearebarelyvisible on theintensityimage butwell highlightedonthecoherencyimage.Thesurface andthe near-subsurfacesnow interfaces appeartobehorizontal,whereasthedepthsassociatedwithinternalonesappeartodecreasewiththerange.Another inter-estingfeatureisthatthetotalsnowpackheightestimatedbythetomogramisabout160 cm,whilethein-situ measurements

give asnowheightof137 cm.Thisclearlybringsoutgeometricaldistortionsaffectingthetomogram.Thesedistortionsare causedbytheassumptionoffreespacepropagationusedduringthefocusingprocessandarediscussedinsection3.2.

3.2. Refractiveindexestimationfor3Dfocusing

WhenapplyingTDBPAunderafree-spacepropagationassumption,thedistance



r→−0

−

→−p



holdsforthegeometricdistance between the antennas coupleand the investigated point. Obviously,snow coveris a medium withrefractive indexn

(

z

)

higher than unity [21,22] andconsequently the electromagnetic distanced

(

→−r

,

→−p

)

is different fromthe geometric one. In orderto correctthefocusing process,thecorrectelectromagneticdistanced

(

r→−0

,

→−p

)

mustbeusedinstead ofthegeometric one. To quantifythe undergone distortions under a free spacepropagationassumption in a medium witha permittivity

Fig. 4. Illustrationoftheapparentpositionofatargetinamediumwithpermittivityhigherthanunityseenundertheassumptionoffree-spacepropagation. higher than unity, one mayconsider the configurationof Fig. 4. The medium is a stack of Nl homogeneous layers with permittivities

{

ni

}

.Theantennaatpositionr→−a

= (

xa

=

0

,

ya

=

0

,

za

=

H

)

T recordsechoesfromatargetatposition→−rt

= (

xt

=

0

,

yt

,

zt

)

T.Thetotalone-waytimedelayfromthetargettotheantennais:

τ

=

1 c0 − − →_r a



− →_r t n

(

z

)

dl (10)

InEq.(10),c0 isthefree-spacepropagationspeed,andn

(

z

)

istherefractiveindexofthelaterallylayeredmedium. ForthecaseofFig. 4,Eq.(10)canbewrittenas:

τ

=

1 c0

(

d0

+

Nl



l=1 nldl

)

(11)

whered0 isthedistancetraveledinthefreespace.

Focusing theSAR datausingthe geometric distanceto locate image points inthe rangedomain considers a constant velocityandaconstantincidenceanglewithinthemedium(seeFig. 4).Inotherwords,thesamepropagationtimedelayis expressedas:

τ

=

1 c0

(

d0

+

d1

)

(12)

Thus,thetargetisseenatanapparentpositionsatisfyingd₁

=



Nl

1 nldlandtheundergonedistortionscanbecomputed knowing the refraction angles

{θl}

and the permittivity profile

{

nl}. The higher the permittivities ofthe medium is,the higherthedistortionsare.Scattererslocatedatfarrangewillbeaffectedbystrongerdistortionsthannearertargets.

In the case of the AlpSAR campaign data, the permittivities of the observed snow covers have to be estimated. In this work, an iterativeprocedure forestimating refractive indicesof an horizontally layered medium isintroduced with applicationto snowpackpermittivity profile retrieval.Thisprocedure relieson thefact that the detected snowinterfaces appearhorizontalifthe correctrefractive indicesare usedinthe focusingalgorithm. Thisprocedure iteratively estimates indicesfromtoplayerstobottom onessince thetotalelectromagneticdistanceofatarget atelevationzt dependsonthe refractiveindexprofilen

(

z

)

,where z

≥

zt.Onemayconsideramedium withthreelayersandfourinterfaces,asdepicted inFig. 5.The firstinterface is theair–medium interface. Thesecond one separateslayer 1 from layer 2.Toestimate the refractiveindexoflayer 1,theGB-SARdataisfocusedfordifferentincreasingvaluesofrefractiveindexn1.Foreachrefractive indexvalue,thedistancetakingintoaccountn1 iscomputedandintroducedintheTDBPalgorithm.Theestimatedvalueof

n1ischoseninsuchawaythesecondinterfaceappearshorizontalinthefinaltomogram.Oncen1 isfixed,thevalueofn2 isestimatedbyfocusingGB-SARdatausingn1 asrefractiveindexforthefirstlayerandanincreasingvalueofn2 untilthe thirdlayerappearshorizontalonthefinaltomogram.Thisprocedureisrepeatedforeachlayeruntilallinterfacesaremade horizontal.

For each iteration of the correction procedure, the electromagneticdistance d

(

→−r

,

→−p

)

is computed. For this paper,the electromagneticdistancetakingintoaccountthedifferentrefractiveindexesiscomputedusingtheeikonalequation formu-lationwhichdescribesthewavefrontpropagationthrougha mediumdefinedbyan isotropicvaryingvelocitydistribution. AdetailedderivationoftheeikonalequationcanbefoundinRef.[23].Theeikonalequationis:



 ∇

d

(

→−r

,

r

a

)





2

=

n2

(

r

)

(13)

Equation(13)isnumericallysolvedusingthefast-marchingmethod[24].FortheAlpSARdata,thecomputationofthe correctelectromagnetic distanceon theimage grid is notperformed forall antennapositions. Since,the observed scene

62 B. Rekioua et al. / C. R. Physique 18 (2017) 57–65

Fig. 5. Illustration of the correction steps for a 3-layer medium.

is a laterally varying medium, the z-axis is a revolution symmetry axis. Hence, the distance from an antenna position

− →

ra

= (

xa

,

yz

,

za

)

T to agiventarget→−rt

= (

xt

,

yt

,

zt

)

T isnotchanged whenconsideringan antennaatposition

− →

r_a

= (

0

,

0

,

za

)

T andatargetatpositiondefinedbyEq.(14):

(

x_t

,

y_t

,

z_t

)

T

= (

0

,



(

xa

−

xt

)

2

+ (

ya

−

yt

)

2

,

zt

)

(14)

This remarkallowsustocompute theelectromagneticdistance fora tracknz onceovera finegrid,then thedistance

computation overthe Nx antennapositions belongingtothenz-thtrackisdone bylinearinterpolationovera grid

repre-sentingthefocusingpositions.

ThefinalresultafterrefractiveindexcorrectionisshowninFig. 10andcomparedwiththesnowstratigraphy.Adetailed discussionisgiveninsection4.

3.3. Antennapatterncompensationfor3Dfocusing

Acorrectinterpretationofthegeneratedtomogramsmusttakeintoaccounttheantennapatterneffectontherecorded backscatterechoes.Inourcase,theantennasarewidebandrectangularhornantennaswithaperturesofabout40°inboth elevationandazimuthdirections.Todescribethe antennapatterneffect,one mayconsiderthecomplexsignal amplitude foragivenimageposition→−r asfollows:

I

(

→−r

)

=



Kseiφs

σ

(

→

˜

−r

)

⊗

h

(

→−r

)

+



Kn



(

→−r

)

(15)

Here Ksand

φ

srepresentthegainandthephaseoftheSARsystem,

σ

˜

sisthecomplexbackscattercoefficientatposition

−

→_{r ,h istheimpulseresponseoftheSARsystemand}

_⊗

_{holdsfortheconvolutionoperation.Theterm}

√

_K

n



(

→−r

)

accountsfor

additivenoise.Tobeabletocompareintensitieswithinthesameimage,theeffectofthesystemgainmustbecompensated for.FortheAlpSARdata,

φ

sisconsideredasnullbecausethesystemisphasecorrectedbeforeacquisitionbycalibratingthe VNAandafteracquisitionbygeometricalcalibrationofantennaspositionsbasedontheideaof[25].Hence,thesystemgain isexpressedbyEq.(16):

Ks

=

Gtx

(θ

tx

, φ

tx

)

Grx

(θ

rx

, φ

rx

) λ

2

R4

α

s (16)

In Eq. (16), Gtx

(θ

tx

,

φ

tx

)

, Grx

(θ

rx

,

φ

rx

)

are the transmittingand receiving antenna gains, R is the target range and

α

s adjustsforallSARparameterscommonbetweenalltargets.Thefactor

α

s isunknownapriori forprocesseddata,hence,it doesnotaffectthecomparisonbetweentheintensitieswithinthesameimage.Itisworthnothingthatthecomputationof view angles foranytarget withrespectto antennadiffersin3D focusingfrom2D focusing.In 3Dfocusing, theelevation angles

(θ

tx

,

θ

rx

)

to be considered are not the geometricalones, butthose definedafterrefractive indexcorrection. If the

air–mediuminterfaceisdefinedatelevationz0,theangles

θ

tx,

θ

rxwillbetheoutanglesoftheelectromagneticraysonthe

surfacedefinedbyz0 (seeFig. 6).

Forthesyntheticaperture,an averageantennafootprintcompensationfactorovertheused antennapositions and fre-quencybinsiscomputed,resultinginFig. 7a.Using √1

Ks asthecompensationcoefficientgivesthetomogramofFig. 7b.The relevantremarkhereistheartefactoutatthelimitsofthe

−

30 dB apertureoftheantennawherethecompensationfactor istoohigh. Itisclearthattheintensityrecordedforimagepixelswheretheantennagain istoolowismorelikelytobe noiseandnotameaningfulintensityvalue.Amodified compensationfactortakingintoaccountthenoiselevelistherefore usedtoobtainFig. 8aand 8b:

β

=

√

Ks

Ks

+

σ

n2

Fig. 6. Definition of the angles accounted for to compute the antenna radiation pattern footprint on the focusing positions.

Fig. 7. Thepanelontheleftrepresentsthecoefficient√1

Ks usedtocompensatefortheantennapatternfootprintcomputedafterrefractiveindexcorrection.

Thepanelontherightisthetomogramobtainedafterantennafootprintcompensation.Thedisplayedtomogramislimitedoveradomainwhere√1

Ks<

30 dB toavoidthatthenoiselevelexceedthesignallevel.

Fig. 8. ThepanelontheleftisthemodifiedcompensationcoefficientofantennafootprintofEq.(17)andthepanelontherightistheresultofapplication ofthiscoefficient.Theusednoiselevelis−30 dB.

Here,

σ

2

n istheadjustingcoefficienttakingintoaccountthenoiselevel.IfKsisclosetoone,thecompensationfactorwill be

β

≈

_K1_s

≥

1.Inthiscase,agoodsignal-to-noise ratioisexpectedanditgivesahigherconfidencelevelto theintensity value.Conversely,ifKs istoosmallcomparedto

σ

n2,theintensityvalueismorelikelytobenoiseanditispenalized with afactor

β

≈

√ Ks

σ2

n

≤

1.Thevaluetakenfor

σ

2

n ischosenbasedontheanalysisoftherawdata,whereitisnoticedthatthe noiselevelisapproximatively

−

30 dB.

4. Interpretationoftheresults

4.1. Comparingtosnowstratigraphy

Tocomparetheobtainedtomogram withthe snowstratigraphy,the coherencyimage isexploitedsincethe coherency highlightsevencoherentbackscatteringwithlowintensity.Thetotalsnowheightobservedaftercorrectionislessthanthe totalsnowheightgivenbythesnowstratigraphy(seeFig. 10).Thelastinterface,aftercorrection,isplacedatheight15 cm andthusthegroundsurfaceisnotdetected.Thesnowstratigraphyshowsthatthefirstlayersarenew-fallendrysnowwith lowdensitiesandsmallroundedparticles.Forthiskindofsnow,therefractiveindexinthemicrowaveregiondependson thesnowdensityandisfrequencyandtemperatureindependent[21,22].Thisexplainsthelowrefractiveindicesfoundfor thefirstandthesecondlayers (1

.

1 and1

.

2 respectively) wheredensitiesareabout150and300 kg

·

m−3 forthefirstand thesecond layers,respectively.In[22],fordrysnow densitiesrangingfrom100to400 kg

·

m−3_{,refractive indicesranging} from 1

.

10 to 1

.

30 are found. For deeperlayers, the snow wetness, the densities andthe grain sizes increase. The snow

64 B. Rekioua et al. / C. R. Physique 18 (2017) 57–65

Fig. 9. Local incidence angle computed for the detected interface on the tomogram.

Fig. 10. Obtainedtomogramafterrefractiveindexcorrection.Thepanelontherightisthesnowstratigraphygivingdifferentphysicalparametersofthe snowpackasafunctionofdepth.Theestimatedrefractiveindicesaregivenforthedifferentlayers.Forthisfigure,theantennapatternisnotaccountedfor.

particlesarewell-bondedmeltedformswithpoly-clusters.Thesecharacteristicsexplain thehigherrefractiveindicesfound forthedeepestlayersnowslabs.Theestimatedvaluesofrefractiveindicesforthethirdandthefourthlayersare1

.

4 and 1

.

7 respectively.In[21,22],itisshownthatthesnowrefractiveindexincreaseswithsnowwetnessanddensityandranges from1

.

41 to1

.

61 at10 GHz.

Analyzingtheheightofthedetected interfacesandlayers,itcomesout thatthedetectedinterfacesandtheirpositions after correction correspondto changesin thephysicalparameters ofsome layers reportedon thesnow stratigraphy.The secondinterfaceisatransitionfromalayerwithalowdensityanddecomposingnew-fallensnowintoadenserlayerwith sphericalparticles.Thethirdinterface,withinthelimitsoftheresolutionofthesystem,canbeconsideredastheseparation between two layers with different particle shapes. The upper one contains mixedforms, while the lower one contains fairly wellbondedparticleswithsome melting.The fourthinterface,locatedat33 cmofsnow height,separatesanupper snow layer,withfairly bondedmeltingformsfromlower layerwithhigherwetness,larger grainsizes andmore particles aggregates.Thelastinterfacedefinesthelastlayerwithincreasingwetness,grainsizesupto4 mmanddepthhoar.

4.2. Backscatteredintensityanalysis

In Fig. 10,the effectofantennapatternbeforecompensation canbe clearly identifiedin groundrangesbetween1 m and 2 mcorresponding toa nullbetweenthe mainlobeandthe firstside lobeof theantennas. Aftercompensationfor the antennapattern(see Fig. 8b),low backscattered intensities areobserved on theshallowest snow interfaces.Thiscan be explained by the low changes in the refractive indexas well as the high local incidence angles on these interfaces (see Fig. 9).Indeed,themonostaticbackscatteringcoefficientfromarough surfacedependsontheincidenceangleswhere higherincidenceanglesgivelowerbackscatteredintensities[26,27].Fordeeperinterfaces,thehighbackscatteredintensities observed are morelikely duetothe low localincidenceangles ontheseinterfaces. Themaximum ofthelocal incidence angleisfoundtonotexceed42°forthelastinterface,whileitexceeds72°forthefirstone.

5. Conclusion

Inthiswork,experimentalresultsofTomSARappliedtosnow covercharacterizationhavebeenpresented.The GB-SAR systemwasoperatedtocharacterizetheverticalstructureofsnowpacks.Theinterfacesbetweendifferentsnowslabswere identifiedandtheinfluenceoffree-spacepropagationassumptionforsubsurfacefocusingwashighlighted.Aniterative pro-cedure was proposed forcorrecting thefocusing processby estimatinglayer permittivitiesandthicknessofthe observed snowpack.Forproperinterpretationofintensityinformation,anantennapatterncorrectionwasappliedtothetomograms.

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