Sensor Array Signal Processing

SENSORARRAYSIGNALPROCESSINGSENSORARRAYSIGNALPROCESSINGPrabhakarS.NaiduCRCPressBocaRatonLondonNewYorkWashington,D.C.1195/DisclaimerPage1Monday,June5,20003:20PMLibraryofCongressCataloging-in-PublicationDataNaidu,PrabhakarS.Sensorarraysignalprocessing/PrabhakarS.Naidu.p.cm.Includesbibliographicalreferencesandindex.ISBN0-8493-1195-0(alk.paper)1.Singalprocessing–Digitaltechniques.2.Multisensordatafusion.I.Title.TK5102.9.N352000621.382'2—dc2100-030409CIPThisbookcontainsinformationobtainedfromauthenticandhighlyregardedsources.Reprintedmaterialisquotedwithpermission,andsourcesareindicated.Awidevarietyofreferencesarelisted.Reasonableeffortshavebeenmadetopublishreliabledataandinformation,buttheauthorandthepublishercannotassumeresponsibilityforthevalidityofallmaterialsorfortheconsequencesoftheiruse.Neitherthisbooknoranypartmaybereproducedortransmittedinanyformorbyanymeans,electronicormechanical,includingphotocopying,microfilming,andrecording,orbyanyinformationstorageorretrievalsystem,withoutpriorpermissioninwritingfromthepublisher.TheconsentofCRCPressLLCdoesnotextendtocopyingforgeneraldistribution,forpromotion,forcreatingnewworks,orforresale.SpecificpermissionmustbeobtainedinwritingfromCRCPressLLCforsuchcopying.DirectallinquiriestoCRCPressLLC,2000N.W.CorporateBlvd.,BocaRaton,Florida33431.TrademarkNotice:Productorcorporatenamesmaybetrademarksorregisteredtrademarks,andareusedonlyforidentificationandexplanation,withoutintenttoinfringe.©2001byCRCPressLLCNoclaimtooriginalU.S.GovernmentworksInternationalStandardBookNumber0-8493-1195-0LibraryofCongressCardNumber00-030409PrintedintheUnitedStatesofAmerica1234567890Printedonacid-freepaperPrologueAnarrayofsensorsisoftenusedinmanydiversefieldsofscienceandengineering,particularlywherethegoalistostudypropagatingwavefields.Someexamplesareastronomy(radioastronomy),medicaldiagnosis,radar,communication,sonar,nonrestrictivetesting,seismology,andseismicexploration(see[1]fordifferentapplicationsofthearraysignalprocessing).Themaingoalofarraysignalprocessingistodeducethefollowinginformationthroughananalysisofwavefields:•(a)Sourcelocalizationasinradar,sonar,astronomy,andseismology,etc.•(b)Sourcewaveformestimationasincommunication,etc.•(c)Sourcecharacterizationasinseismology•(d)Imagingofthescatteringmediumasinmedicaldiagnosis,seismicexploration,etc.Thetoolsofarraysignalprocessingremainthesame,cuttingacrosstheboundariesofdifferentdisciplines.Forexample,thebasictoolofbeamformationisusedinmanyareasmentionedabove.Thepresentbookaimsatunravelingtheunderlyingbasicprinciplesofarraysignalprocessingwithoutareferencetoanyparticularapplication.However,anattemptismadetoincludeasmanytoolsaspossiblefromdifferentdisciplinesinanorderwhichreflectstheunderlyingprinciple.Intherealworld,differenttypesofwavefieldsareusedindifferentapplications,forexample,acousticwavesinsonar,mechanicalwavesinseismicexploration,electromagneticwavesinradarandradioastronomy.Fortunately,allwavefieldscanbecharacterizedunderidenticalmathematicalframework.Thiscommonmathematicalframeworkisbrieflysummarizedinchapter1.Herewehavedescribedthebasicequationsunderlyingdifferentwavefieldsandthestructureofarraysignalsandthebackgroundnoisewhenthenoisesourcesfollowsomesimplegeometricaldistribution.Thetopicscoveredarewavefieldinopenspace,boundedspaceincludingmultipathpropagationandlayeredmedium.Alsocoveredistheweakscatteringphenomenonwhichisthebasisfortomographicimaging.Inchapter2westudydifferenttypesofsensorconfigurations.Theemphasisishoweveroncommonlyuseduniformlineararray(ULA),uniformcirculararray(UCA).ManypracticalsensorarraysystemscanbestudiedintermsofthebasicULAandUCAsystems(cylindricalarrayinradarandsonar,crossarrayinastronomyandseismology).Likesensors,thesourcescanalsobeconfiguredintheformofanarray.Thesourcearrayisusefulinsynthesizingadesiredwavefrontand/orwaveform.Inchapter3weexaminetheissuesconnectedwiththedesignof2Ddigitalfiltersforwavefieldanalysis.Sincethepropagatingwavefieldspossesssomeinterestingspectralcharacteristicsinfrequencywavenumberdomain,forexample,thespectrumofapropagatingwavefrontisalwaysonaradialline,itisnaturaltotakeintoaccountthesefeaturesinthedesignofdigitalfiltersforseparationofinterferingwavefields.Specifically,wecoverindetailthedesignofafanfilterandquadrantfilter.Also,theclassicalWienerfilterasanoptimumleastsquaresfilteriscoveredinthischapter.Thethemeinchapters4and5islocalizationofasource.Inchapter4wedescribetheclassicalmethodsbasedonthefrequencywavenumberspectrumoftheobservedarrayoutput.WestartwiththeBlackmanTukeytypefrequencywavenumberspectrumandthengoontomodernnonlinearhighresolutionspectrumanalysismethodssuchasCapon’smaximumlikelihoodspectrumwhichisalsoknownasminimumvariancedistortionlessresponse(MVDR)beamformerandmaximumentropyspectrum.Localizationessentiallyinvolvesestimationofparameterspertainingtothesourceposition,forexample,azimuthandelevationangles,range,speedifthesourceismoving,etc.Inthelasttwodecadesahostofnewmethodsofsourcelocalizationhavebeeninvented.Weelaboratethesenewapproachesinchapter5.Theseincludesubspacebasedmethods,useofman-madesignalssuchasincommunicationandfinallymultipathenvironment.Quiteoftenlocalizationmustbedoneintherealtimeanditmaybenecessarytotrackamovingsource.Adaptivetechniquesarebestsuitedforsuchtasks.Abriefdiscussiononadaptiveapproachisincluded.Inchapter6welookintomethodsforsourcewaveformseparationandestimation.Thedirectionofarrival(DOA)isassumedtobeknownorhasbeenestimated.WeshalldescribeaWienerfilterwhichminimizesthemeansquareerrorintheestimationofthedesiredsignalcomingfromaknowndirectionandaCaponfilterwhich,whileminimizingthepower,ensuresthatthedesiredsignalisnotdistorted.Wealsotalkabouttheestimationofdirectionofarrivalinamultipathenvironmentencounteredinwirelesscommunication.Thenexttwochaptersaredevotedtoarrayprocessingforimagingpurposes.Firstly,inchapter7welookatdifferenttypesoftomographicimagingsystems:nondiffracting,diffractingandreflectiontomography.Thereceivedwavefieldisinvertedundertheassumptionofweakscatteringtomapanyoneormorephysicalpropertiesofthemedium,forexample,soundspeedvariationsinamedium.Forobjectsofregularshape,scatteringpointsplayanimportantroleingeometricaldiffractiontheory.Estimationofthesescatteringpointsforthedeterminationofshapeisalsodiscussed.Inchapter8westudythemethodofwavefieldextrapolationforimaging,extensivelyusedinseismicexploration.Therawseismictracesarestackedinordertoproduceanoutputtracefromahypotheticalsensorkeptclosetothesource(withzero-offset).Asuiteofsuchstackedtracesmaybemodeledasawavefieldrecordedinanimaginaryexperimentwhereinsmallchargesareplacedonthereflectorandexplodedatthesametime.Thezero-offsetwavefieldisusedforimagingofreflectors.Theimagingprocessmaybelookeduponasadownwardcontinuationofthewavefieldorinversesourceproblemorpropagationbackwardintime,i.e.,depropagationtothereflector.Allthreeviewpointsareverybrieflydescribed.Thebookisbasedonacourseentitled“DigitalArrayProcessing”offeredtothegraduatestudentswhohadalreadytakenacourseondigitalsignalprocessing(DSP)andacourseonmodernspectrumanalysis(MSA).Ithasbeenmyconvictionthatastudentshouldbeexposedtoallbasicconceptscuttingacrossthedifferentdisciplineswithoutbeingburdenedwiththequestionsofpracticalapplicationswhichareusuallydealtwithinspecialtycourses.Themostsatisfyingexperienceisthatthereisacommonthreadthatconnectsseeminglydifferenttoolsusedindifferentdisciplines.Anexampleisbeamformation,acommonlyusedtoolinradar/sonar,whichhasaclosesimilaritywithstackingusedinseismicexploration.Ihavetriedtobringoutinthisexpositionthecommonthreadthatexistsintheanalysisofwavefieldsusedinawidevarietyofapplicationareas.Theproposedbookhasasignificantlydifferentflavor,bothincoverageanddepthincomparisonwiththeonesonthemarket[1-5].Thefirstbook,editedbyHaykin,isacollectionofchapters,eachdevotedtoanapplication.Itrapidlysurveysthestateofartinrespectiveapplicationareasbutdoesnotgodeepenoughanddescribethebasicmathematicaltheoryrequiredfortheunderstandingofarrayprocessing.ThesecondbookbyZiomekisentirelydevotedtoarraysignalprocessinginunderwateracoustics.Itcoversingreatdepththetopicofbeamformationbylinearandplanararraysbutconfinestolinearmethods.Modernarrayprocessingtoolsdonotfindaplaceinthisbook.ThethirdbookbyPillai[3]hasaverynarrowscopeasitdealswithingreatdetailonlythesubspacebasedmethods.ThefourthbookbyBouvetandBienvenu(Eds)isagainacollectionofpaperslargelydevotedtomodernsubspacetechniques.Itisnotsuitableasatext.Finally,thepresentbookhassomesimilaritieswithabookbyJohnsonandDudgeon[3]butdiffersinoneimportantrespect,namely,itdoesnotcovertheapplicationofarraystoimagingthoughabriefmentionoftomographyismade.Also,thepresentbookcoversnewermaterialwhichwasnotavailableatthetimeofthepublicationofthebookbyJohnsonandDudgeon.Duringthelasttwodecadestherehasbeenintenseresearchactivityintheareaofarraysignalprocessing.Therehavebeenatleasttworeviewpaperssummarizingthenewresultsobtainedduringthisperiod.Thepresentbookisnotaresearchmonographbutitisanadvancedleveltextwhichfocusesontheimportantdevelopmentswhich,theauthorbelieves,shouldbetaughttogiveabroad“picture”ofarraysignalprocessing.Ihaveadoptedthefollowingplanofteaching.Astheentirebookcannotbecoveredinonesemester(about35hours)Ipreferredtocoveritintwopartsinalternatesemesters.Inthefirstpart,Icoveredchapter1(exclude§1.6),chapter2,chapters4,5and6.Inthesecondpart,Icoveredchapter1,chapter2(exclude§2.3),chapter3(exclude§3.5),chapters7and8.Exercisesaregivenattheendofeachchapter.(Thesolutionguidemaybeobtainedfromthepublisher).1.S.Haykin(Ed),ArraySignalProcessing,PrenticeHall,EnglewoodCliffs,NJ,1985.2.L.J.Ziomek,UnderwaterAcoustics,ALinearSystemsTheory,AcademicPress,Orlando,1985.3.S.U.Pillai,ArraySignalProcessing,Springer-Verlag,NewYork,1989.4.M.BouvetandG.Bienvenu,HighResolutionMethodsinUnderwaterAcoustics,Springer-Verlag,Berlin,1991.5.D.H.JohnsonandD.E.Dudgeon,ArraySignalProcessing,PrenticeHall,EnglewoodCliffs,NJ,1993.6.H.KrimandM.Viberg,Twodecadesofarraysignlprocessing,IEEESignalProc.Mag.,July1996,pp.67-94.7.T.Chen(Ed)Highlightsofstatisticalsignalandarrayprocessing,IEEESignalProc.Mag.,pp.21-64,Sept.1998.PrabhakarS.NaiduFebruary,2000Prof,DeptofECE,IndianInstituteofScience,Bangalore560012,India.SensorArraySignalProcessingContentsChapterOneAnOverviewofWavefields1.1Typesofwavefieldsandthegoverningequations1.2Wavefieldinopenspace1.3Wavefieldinboundedspace1.4Stochasticwavefield1.5Multipathpropagation1.6Propagationthroughrandommedium1.7ExercisesChapterTwoSensorArraySystems2.1Uniformlineararray(ULA)2.2Planararray2.3Broadbandsensorarray2.4Sourceandsensorarrays2.5ExercisesChapterThreeFrequencyWavenumberProcessing3.1Digitalfiltersintheω-kdomain3.2Mappingof1Dinto2Dfilters3.3MultichannelWienerfilters3.4WienerfiltersforULAandUCA3.5Predictivenoisecancellation3.6ExercisesChapterFourSourceLocalization:FrequencyWavenumberSpectrum4.1Frequencywavenumberspectrum4.2Beamformation4.3Capon'sω-kspectrum4.4Maximumentropyω-kspectrum4.5ExercisesChapterFiveSourceLocalization:SubspaceMethods5.1Subspacemethods(Narrowband)5.2Subspacemethods(Broadband)5.3Codedsignals5.4Arraycalibration5.5Sourceinboundedspace5.6ExercisesChapterSixSourceEstimation6.1Wienerfilters6.2Minimumvariance(Caponmethod)6.3Adaptivebeamformation6.4Beamformationwithcodedsignals6.5Multipathchannel6.6ExercisesChapterSevenTomographicImaging7.1Nondiffractingradiation7.2Diffractingradiation7.3Broadbandillumination7.4Reflectiontomography7.5Objectshapeestimation7.6ExercisesChapterEightImagingbyWavefieldExtrapolation8.1Migration8.2Explodingreflectormodel8.3Extrapolationinω-kplane8.4Focusedbeam8.5Estimationofwavespeed8.6ExercisesAcknowledgmentThethoughtofformalizingthelecturenotesintoatextoccurredtomewhenIwasvisitingtheRurhUniversitaet,Bochum,Germanyin1996asaHumboldtFellow.Muchofthegroundworkwasdoneduringthisperiod.IamgratefultoAvHFoundationwhosupportedmystay.ProfDr.J.FBoehmewasmyhost.Iamgratefultohimforthehospitalityextendedtome.Manyofmystudents,whocreditedthecourseonArraySignalProcessinghavecontributedbywayofworkingouttheexercisescitedinthetext.Iamparticularlygratefultothefollowing:S.Jena,S.S.Arun,P.Sexena,P.D.Pradeep,G.Viswanath,K.GaneshKumar,JobyJoseph,V.Krishnagiri,N.B.Barkar.Mygraduatestudents,Ms.A.VasukiandMs.ABuvaneswari,havesignificantlycontributedtochapter7.Dr.K.V.S.Harireadthemanuscriptatanearlystageandmademanyconstructivesuggestions.IwishtothanktheCRCPressInc.,inparticular,Ms.NoraKonopkaandMsMaggieMogckfortheirpromptnessandpatience.FinallyIoweadeepgratitudetomyfamily;mywife,MadhumatiandsonsSrikanth,SridharandSrinathfortheirforbearance.Imustspeciallythankmyson,Srinathwhocarefullyscrutinizedpartsofthemanuscript.DedicationThisworkisdedicatedtothememoryofthegreatvisionary,J.R.DTatawhoshapedtheIndianInstituteofScienceformanydecades.ChapterOneAnOverviewofWavefieldsAsensorarrayisusedtomeasurewavefieldsandextractinformationaboutthesourcesandthemediumthroughwhichthewavefieldpropagates.ItisthereforeimperativethatsomebackgroundindifferenttypesofwavefieldsandthebasicequationsgoverningthewavefieldmustbeacquiredforcompleteunderstandingoftheprinciplesofArraySignalProcessing(ASP).Inanidealisticenvironmentofopenspace,homogeneousmediumandhighfrequency(whererayapproachisvalid),athoroughunderstandingofthewavephenomenonmaynotbenecessary(thosewhoareluckyenoughtoworkinsuchanidealisticenvironmentmayskipthischapter).Butinaboundedinhomogeneousmediumandatlowfrequencieswherediffractionphenomenonisdominating,thephysicsofthewavesplaysasignificantroleinASPalgorithms.InthischapterouraimisessentiallytoprovidethebasicsofthephysicsofthewaveswhichwillenableustounderstandthecomplexitiesoftheASPproblemsinamorerealisticsituation.Thesubjectofwavephysicsisvastandnaturallynoattemptismadetocoverallitscomplexities.§1.1TypesofWavefieldsandGoverningEquations:Themostcommonlyencounteredwavefieldsare:(i)Acousticwavesincludingsoundwaves,(ii)Mechanicalwavesinsolidsincludingvibrationsand(iii)Electromagneticwavesincludinglight.Thewavefieldsmaybeclassifiedintotwotypes,namely,scalarandvectorwaves.Inthescalarwavefieldwehaveascalarphysicalquantitythatpropagatesthroughthespace,forexample,hydrostaticpressureisthephysicalquantityinacousticscalarwavefields.Inavectorwavefield,thephysicalquantityinvolvedisavector,forexample,thedisplacementvectorinmechanicalwaves,electricandmagneticvectorsinelectromagneticwaves.Avectorhasthreecomponentsallofwhichtravelindependentlyinahomogeneousmediumwithoutanyexchangeofenergy.Butataninterfaceseparatingtwodifferentmediathecomponentsdointeract.Forexample,ataninterfaceseparatingtwosolidsapressurewavewillproduceashearwaveandviceversa.Inahomogeneousmediumwithoutanyreflectingboundariesthereisnoenergytransferamongcomponents.Eachcomponentofavectorfieldthenbehavesasifitisascalarfield,likeanacousticpressurefield.1.1.1AcousticField:Acousticfieldisapressure(hydrostatic)field.Theenergyistransmittedbymeansofpropagationofcompressionandrarefactionwaves.Thegoverningequationinahomogeneousmediumisgivenbyρd2φ∇2φ=(1.1a)γφ20dtφγwhere0isambientpressure,isratioofspecificheatsatconstantpressureandvolumeandρisdensity.Thewavepropagationspeedisgivenbyγφκc=0=(1.1b)ρρwhereκiscompressibilitymodulusandthewavepropagatesradiallyawayfromthesource.Inaninhomogeneousmediumthewaveequationisgivenby1d2φ1=ρ(r)∇⋅(∇φ)c2(r)dt2ρ(r)(1.2a)1=ρ(r)∇()⋅∇φ+∇2φρ(r)Afterrearrangingthetermsin(1.2a)weobtain1d2φ∇ρ(r)∇2φ−=⋅∇φ(1.2b)c2(r)dt2ρ(r)whererstandsforpositionvector.Theacousticimpedanceisequaltotheproductofdensityandpropagationspeedρcandtheadmittanceisgivenbytheinverseoftheimpedanceoritisalsodefinedintermsofthefluidspeedandthepressure,fluidspeed∇φAcousticadmittance==pressurejωφNotethattheacousticimpedanceinairis42butinwateritis1.53x105.1.1.2MechanicalWavesinSolids:Thephysicalquantitywhichpropagatesisthedisplacementvector,thatis,particledisplacementwithrespecttoitsstationaryposition.Letdstandforthedisplacementvector.Thewaveequationinahomogeneousmediumisgivenby[1,p142]∂2dρ=(2µ+λ)graddivd−µcurlcurld∂t2whereµisshearconstantandλisYoung’smodulus.Intermsofthesetwobasiclameconstantswedefineothermorefamiliarparameters:µ+λα=(2)Pressurewavespeed:ρµβ=Shearwavespeed:ρλPoissonratio:σ=2(µ+λ)2Bulkmodulesκ=(µ+λ)3Theaboveparametersareobservablefromexperimentaldata.Adisplacementvectorcanbeexpressedasasumofgradientofascalarfunctionφandcurlofavectorfunctionψ(Helmholtztheorem)d=∇φ+∇×ψ(1.3a)φandψsatisfytwodifferentwaveequations:∂2φ∇2φ=1α2∂t2(1.3b)1∂2ψ∇×∇×ψ=−∇.ψ=0β2∂t2where∇×isacurloperatoronavector.Theoperatorisdefinedasfollows:eeexyz∂∂∂∇×F=det∂x∂y∂zfxfyfz∂f∂f∂f∂f∂f∂f=(z−y)e+(x−z)e+(y−x)e∂y∂zx∂z∂xy∂x∂yzwhereex,eyandezareunitvectorsinthedirectionofx,y,andz,respectivelyandfx,fyandfzcomponentsofvectorF.Thescalarpotentialgivesrisetolongitudinalwavesorpressurewaves(p-waves)andthevectorpotentialgivesrisetotransversewavesorshearwaves(s-waves).Thep-wavestravelwithspeedαands-wavestravelwithspeedβ.Thecomponentsofdisplacementvectorcanbeexpressedintermsofφandψ.From(1.3a)weobtain=d(dx,dy,dz)∂φ∂ψ∂ψd=+z−yx∂x∂y∂z∂φ∂ψ∂ψd=+x−z(1.4)y∂y∂z∂x∂φ∂ψ∂ψd=+y−xz∂z∂x∂yψ=(ψψψwherex,y,z).Insolidswemustspeakofstressandstraintensors.Anelementofsolidisnotonlycompressedbutalsotwistedwhileanelementoffluidisonlycapableofbeingcompressedbutnottwisted.Wehavetousetensorsforcharacterizingthephenomenonoftwisting.WeshalldefinethestressandstraintensorsandrelatethemthroughHooke’slaw.Astresstensorisamatrixofninecomponentssssxxyxzx=ssxysyyszy(1.5a)sxzsyzszzThecomponentsofthestresstensorrepresentstressesondifferentfacesofacube(seefig.1.1).Astraintensorisgivenbyeeexxyxzxε=exyeyyezy(1.5b)exzeyzezzThefirstsubscriptreferstotheplaneperpendiculartotheaxisdenotedbythesubscriptandthesecondsubscriptdenotesthedirectioninwhichthevectorispointing.Forexample,sxxisastressinaplaneperpendiculartothex-axis(i.e.,y-zplane)andpointingalongthex-axis.zysxzsxysxxxFigure1.1:Anelementofvolume(cuboid)andstressesareshownonafaceperpendiculartothex-axis.Thestresscomponentsondifferentfacesofacuboidareshowninfig.1.1.Thetorqueonthecuboidshouldnotcauseanyrotation.Forthis,wemusthave=sxysyxandsimilarlyallothernondiagonalelementsinthestressmatrix.Thus,smustbeasymmetricmatrix.Thecomponentsofastrainmatrixarerelatedtothedisplacementvector∂d∂d∂dε=x,ε=y,ε=zxx∂xyy∂yzz∂z∂d∂dε=ε=x+y;xyyx∂y∂x∂d∂dε=ε=y+z;(1.6)yzzy∂z∂y∂d∂dε=ε=z+x.zxxz∂x∂zFinally,thestressandstraincomponentsarerelatedthroughHooke’sLaw:=ρα2ε+ρα2−β2ε+εsxxxx(2)(yyzz)=ρα2ε+ρα2−β2ε+εsyyyy(2)(xxzz)=ρα2ε+ρα2−β2ε+ε(1.7)szzzz(2)(xxyy)==ρβ2εsxysyxxy==ρβ2εsyzszyyz==ρβ2εszxsxzzxUsingequations(1.4),(1.6)and(1.7)wecanexpressallninestresscomponentsintermsofthescalarandvectorpotentialfunctions,φandψ.Forexample,itispossibletoshowthatsxxisgivenby∂2φ∂2φ∂2φ∂2ψ∂2ψs=ρα2+ρ(α2−2β2)(+)+2ρβ2(z−y)xx∂x2∂y2∂z2∂x∂y∂x∂zAgeneralsolutionof(1.3b)maybegivenby∞1−2+2−2+φ(x,y,z,ω)=Φ(u,v,ω)euvkαzej(uxvy)dudv(1.8a)π2∫∫4−∞∞−2+2−2ψω=1Ψωuvkβzj(ux+vy)(x,y,z,)2∫∫(u,v,)eedudv4π−∞(1.8b)ωωwherek=,k=.Φ(u,v,ω)andΨ(u,v,ω)arerespectivelytheααββFouriertransformsofthedisplacementpotentialsφandψevaluatedonthesurfacez=0.Furthermore,ψmustsatisfyzerodivergencecondition(1.3b).ThiswillplaceadditionalconstraintsonΨ(u,v,ω),namely,Ψω+Ψω−2+2−2Ψω=jux(u,v,)jvy(u,v,)uvkβz(u,v,)0(1.8c)Recallthatthepressurewaves(p-waves)travelatspeedαandtheshearwavestravelatspeedβ,whereαisgenerallygreaterthanβ.Thedisplacementvectorisinthedirectionofthegradientofthescalarpotentialbutitisinthedirectionofcurlofthevectorpotential(1.4),thatis,normaltothevectorpotential.Thusthereisafundamentaldifferenceinthenatureofpropagationoftheshearandthepressurewaves.Theshearwavesarepolarized;thedisplacementvectorisalwaysperpendiculartothedirectionofwavepropagation.Thedisplacementvectorexecutesamotiondependinguponthephasedifferencebetweenthecomponentsofthedisplacementvector;alinewhenthephasedifferenceiszero,acircularpathwhenthephasedifferenceis900,orarandompathwhenthephasedifferenceisrandomlyvarying.Thesefactorsplayanimportantroleinthedesignofsensorarraysystemsandprocessingofvectorpotentialsignals.1.1.3ElectromagneticFields:Inelectromagneticfieldstherearetwovectors,namely,electricvectorEandmagneticvectorH,eachwiththreecomponents,thusasixcomponentvectorfield.Thebasicequationsgoverningtheelectromagnetic(EM)fieldsaretheMaxwell’sequations(inmksunits),∂B∇×E=−∂tFaraday’slaw