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TheresearchdescribedinthisreportwassponsoredbytheUnitedStatesAirForceunderContractF49620-91-C-0003.urtherinformationmaybeobtainedfromtheStrategicPlanningDivision,DirectorateofPlans,HqUSAF .

ISBN:-8330-2376-4

Copyright1996R A N DA llrightsreserved.opartofthisbookmaybereproducedinanyformynylectronicrechanicaleansincludingphotocopying,recording,orinformationstorageandretrieval)withoutpermissioninwritingfromR A N D .

R A N Disanonprofitinstitutionthathelpsimprovepublicpolicythroughresearchandnalysis.AND'spublicationsdonotnecessarilyreflectthe opinions orpoliciesof itsresearchsponsors.

Published1 9 9 6 by R A N D1700Main Street,P.O.B ox 2138,SantaMonica,CA 90407-2138R A N DU R L :ttp://www.rand.org/

T oorderR A N Ddocumentsortoobtain additionalinformation,contactDistribution Services:Telephone:310)451-7002;

Fax:310)451-6915;Internet:[email protected]

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T H I SO C U M E N TSE S T Q U A L I T Y A V A I L A B L E .H EC O P Y F U R N I S H E DT O DTTCCONTAINED AI G N I F I C A N TU M B E RF P A G E SH I C HOO T R E P R O D U C E LE G I B LY.

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ISBN:0833023764Technicalrp t#:RAND/MR-737-AF OrderNo./Price:$15.00Catalogingsource:CStmoR CStmoR Geographieareacode:n-us L C call# :UG743. 1 8 31996Personalname:Isaacson,JeffreyA .Title:Estimationandpredictionof ballisticmissiletrajectories/JeffreyA .Isaacson,DavidR .Vaughan.Publicationinfo:SantaMonica,C A R A N D ,1996.Physicaldescription:xxvii,70 p.:ill.,maps 23cm .Note:"ProjectA irForce."Note:Includesbibliographical references(p .69-70).Securitycontrols:U NC L A S S I F I E DAbstract:Toexaminethecapabilitiessatellitescanbringtobearinatheatermissiledefense(TMD)environment,th eauthorsdescribeamethodology,basedonKaimanfiltering,fo rth eestimationandpredictionof ballisticmissiletrajectoriesandthenapplyth emethodologytoanotionaltheaterballisticmissile.

O neusefulapplicationisinestimatingth euncertaintyassociatedwithth elocationof amissilelaunch.eterminingmissilelocationuncertaintyatanypointalongth etrajectoryisanotherapplication.Filtersoptimizedfor randomerrorsaloneaswellasrandom plusbiaserrorsareoutlined.Harnessedinatheaterof operations,thetypeof informationdescribedinthisreportcanbeusedtoenhanceth ecapabilityof activeandpassivedefensesandattack operations.Ctrct/Grnt/Proj/Task:A ir Force;F49620-91-C-0003;R C N147J;R C N 3550Relatedpublications:SupersedesRAND/DRR-526-AF . Subject:BallisticmissiledefensesUnitedStatesPlanning. DTICdescriptor:Guidedmissiles.Personalname:Vaughan,David.Corporatename:ProjectA ir Force(U.S.).ForceModernization andEmploymentProgram.Corporatename:R A N DCorporation.Corporatename:UnitedStates.A irForce.Relatedentry:RAN D/DRR-526-AFAuthordepartment:DefenseandTechnology PlanningResearchunit:ProjectA irForceDistributioncode:3 Subjectbibliography:MI L I TA R YS TR A TE GY A N DTA C TI C SDistribution date:19960426;19960426Librarydeposit date:9960426

Project:C4I SpaceProgram:ForceModernizationand Employment MISC1:9604

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^_ProjectA IR F O R C E 19 46-1996

E S T I M A T I O N A N D P R E D I C T I O N O F

B A L L I S T I C M I S S I L E T R A J E C T O R I E S JeffreyA.IsaacsonDavidR.Vaughan

PreparedfortheUnitedStates Air ForceApprovedforpublicrelease;distributionunlimited

RAND

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P R E F A C E

Th isreportdocumentsanalysisoriginating frommorecomprehen-siveR A N Dresearch to establishaninvestmentstrategy fo rU.S.spacesystemsandconceptsofoperationfo rcounteringcriticalmobiletargets.heworkwasonductedwithintheProjectA IR F O R C E Force ModernizationandEmploymentprogram,undertheauspicesoftheC4I/Spaceproject,fo rthe A irCombatCommand.The studydescribes an analyticaltoolusefulin establishingfiguresofmeritfo rsatellitesin anotionaloperational settingin which ballisticmissiledefensesareemployed. frameworkfamiliartosystem de-signersisdescribedpedagogically,anditsutilityinderivingopera-tionalimplicationsisdemonstratedfo roneinterestingcase.hereportshouldbeusefultodecisionmakersandanalystswithintheU.S .A irForcendheDepartmentofDefense,swellsothersgenerallyconcernedwiththeatermissiledefensearchitecturesandoperationaleffectivenessanalysis.P R O J E C T A IR F O R C E ProjectA IR F O R C E , divisionofR A N D ,istheA irForcefederallyfundedresearchanddevelopmentcenter( F F R DC )fo rstudiesandanalyses.tprovidestheA irForcewithindependentanalysesofpolicy alternativesaffectingthedevelopment,employment,combatreadiness,andsupportofcurrentandfutureaerospaceforces.e-

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iv EstimationandPrediction ofBallisticMissileTrajectories

searchisperformedin threeprograms:trategy,Doctrine,and ForceStructure;ForceModernizationndEmployment;ndResourceManagementandSystem A cquisition.ProjectA IR F O R C Eiscelebrating50yearsofservicetotheUnitedStates A irForcein1996 .Project A IR F O R C E beganin March1946as ProjectR A N DatDouglasA ircraftCompany,undercontracttotheA r m y A irForces.w oyearslater,theproject'scontractand person-nelwereseparatedfromDouglastoformanew,privatenonprofitinstitutiontoimprovepublicpolicythroughresearchandanalysisfo rthe publicwelfareandsecurityofthe UnitedStatesthe founda-tionofwhatis knowntoday as R A N D .

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C O N T E N T S

PrefaceFigures Tables xSummary iAcknowledgments xvA cronyms xviiChapterO ne I N T R O D U C T I O NT MDDevelopmentIsUnder W ay

SatelliteSensorsSupport T MDBattleManagementOrganizationoftheReportChapterT w oTHE O R E TI C A L U N D E R P I N N I N G S 1 LinearEstimationandPrediction1 LinearApproximationto NonlinearSystems4ChapterThreeESTIMATION A N DPREDICT ION O F BALLISTIC MISSILE T RAJECT ORI ES 9 GeometryofMissile-SensorEngagement9 FilterMethodology 5NotionalResults 2

Launch PointUncertainty (LPU)2MissileLocation Uncertainty(MLU)7Revisit T i m eSensitivities 8

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vi Estimation andPrediction of BallisticMissile Trajectories

ChapterFourT HE E F F E C T O F BIASE R R O R S 1 SuboptimalTreatmentofBias 2

NotionalResults 5OptimalTreatmentofBias 0

NotionalResults 3ChapterFiveC O N C L U D I N G R E M A R KS 9 A ppend ix :MISSILE T R A J E C T O R I E SO N T HE E A R TH' SS U R F A C E 5Bibliography 9

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F I G U R E S

S. 1 . SatelliteSensorsSupportBoth Forward andTargetArea Defenses ii5.2.eometryofT B MTrajectoryand Sensors iv 5.3.PU s(= 2)fo r T w oSensors withRandomErrors..vi5.4.ensitivity ofL PU (= 2)to RandomErrorvi5.5.ensitivity ofML U (= 2)to RandomErrorvii5.6.ensitivity ofL PU (= 2)to Revisit T i m e(TwoSensors) viii5.7.ensitivity of ML U = 2)to Revisit T i m e(TwoSensors) ix 5.8.PU s(= 2)fo r T w oSensors withRandom and BiasErrors(O p t ima l Filter) x5.9 .ensitivity of L PU (= 2)to BiasErrorTw oSensors;OptimalFilter) xi S.10. Sensitivity ofML U = 2)to BiasError(TwoSensors;OptimalFilter) xiS.U. Sensitivity ofL PU (= 2)to Revisit T i m eTw oSensors;O p tim a l Filter) xi iS.12. Sensitivity ofML U (= 2)to Revisit T i m e(TwoSensors;O p tim a lFilter) xiii1 . 1 .hirty-ThreeNationsPossessT BMs1.2.oreSystemsEmphasizeTargetA rea1.3.atelliteSensorsSupportBoth ForwardandTargetAreaDefenses3.1 .otionalSensorin GeosynchronousO rbit03.2.arth-CenteredCoordinates03.3.otatedCoordinates 1 3.4.iltedCoordinates 2

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viii Estimation andPredictionof BallisticMissile Trajectories

3.5.pinningCoordinates 23 3.6.oost-Phase ofNotional Missile 27 3.7.otional100-sec-Burn MissileTrajectory 28 3.8.eometryofTB M TrajectoryandSensors 29 3.9 .oost-Phase MeasurementSequence 30 3.10.ensitivityofL PU = 2)to RandomError(OneSensor) 34 3.11 .PU sI= 2)fo r T w oSensors withRandomErrors.. 35 3.12.ensitivityof L PU (= 2)to RandomError(TwoSensors) 36 3.13.PU s{ =2)fo raSymmetricExample36 3.14.ensitivityofML U (= 2)to RandomError(OneSensor) 38 3.15.ensitivityofML U (= 2)to RandomError(TwoSensors) 39 3.16.ensitivityof L PU (= 2)to R evisit T i m eTw oSensors) 40 3.17.ensitivityofML U (= 2)to RevisitT i m e(TwoSensors) 40 4.1.PU s(= 2)fo r T w oSensors withRandom andBiasErrors(SuboptimalFilter) 46 4.2.ensitivityofL PU (= 2)to BiasError(TwoSensors;SuboptimalFilter) 47 4.3.ensitivityofML U (= 2)to BiasError(TwoSensors;SuboptimalFilter) 47 4.4.ensitivityof L PU (= 2)to RevisitT i m eTw oSensors;Suboptimal Filter) 48 4.5.ensitivityofML U (= 2)to RevisitT i m e(TwoSensors;SuboptimalFilter) 48 4.6.PU s(= 2)fo rT w oSensors withRandom and BiasErrors(Op t imalFilter) 53 4.7.ensitivity ofL PU ( = 2)to BiasError(TwoSensors;Optimal Filter) 54 4.8.ensitivityof ML U (= 2)to BiasError(TwoSensors;O p t i m a lFilter) 54 4.9.ensitivity ofL PU (= 2)to Revisit T i m eTw oSensors;OptimalFilter) 55

4.10. Sensitivityof ML U (= 2)to RevisitT i m e(TwoSensors;OptimalFilter) 56 A.l. Great CirclesontheEarth'sSurface66

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T A B L E S

1 .1 DevelopmentProgramsComplicateEffortsto CurtailT B MProliferation 52.1 EstimationSequence 154.1PU s{1=2)fo rT w oSensorsProcessedin Stereo(SuboptimalFilter,inkm 2) 49 4. 2L U s(I= 2)atApogeeforT w oSensorsProcessedinStereo(SuboptimalFilter;EquivalentSphericalRadi i inkm ) 504. 3PU si= 2)fo r T w oSensorsProcessedin Stereo(Op t ima lFilter,in k m 2) 56 4.4L U s(I= 2)atApogeeforT w oSensors Processedin Stereo(Op t ima lFilter;EquivalentSpherical Radi iin km ) 57 5. 1PU s(= 2)fo r T w oSensorsProcessedin Stereo(in k m 2) 60 5. 2L U st= 2)atApogee for T w oSensorsProcessedin Stereo(Equivalent SphericalRadiiin km )61

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S U M M A R Y

Thirty-threeations, numberofwhichactivelypursuepoliciescontrary toU.S.interests,possessTBMs.oreover,theexportablesupply of T B M scontinuesto grow through worldwidedevelopmentefforts,andmissilesofincreasedrangeand payloadcouldfind theirw ay intotheweaponsinventoriesofmanynationsduringthenextdecade.oupledwithaconcomitantspreadof weaponsofmassdestruction(WMD),suchT BMscouldpermitastrikecapabilitythatcouldthreatenregionalbalances, .S .llies,reven .S .orcesdeployedoverseas.hus,althoughtherearediplomaticeffortstocurtailmissileproliferation,1heUnitedStateshasundertakenanambitiousesearchndevelopmentffortnheatermissiledefense(TMD).Activedefenses,passivedefenses,attackoperations,andcommand,control,communications,andintelligence(C3I)form thefour"pil-lars"of theU.S.theaterdefenseprogram.2stheatermissilede -fensesarefieldedatthedecade's end,satellitesensorswill likely playanimportantsupportingrole.ow mightthesesensorscontributeto C3Iin the T MD environment?xTheMissile TechnologyControlRegime(MTCR)isonesucheffort. reatedin1987,theMTCR controlsthetransferoftechnologies that couldaidtheunmanneddeliveryof a500-kilogram payloadover a300-kilometer distance.or abriefdescriptionof theM T C R ,seeBallisticMissileDefenseO rganization,BallisticMissile Proliferation:An Emerging Threat,A rlington,Virginia:ystem Planning Corporation, 1992,p p .64-65.2C 3Iis ina sense thefoundationsupportingthese pillars,ratherthanapillar itself.

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xii EstimationandPredictionof Ballistic MissileTrajectories

Consider th enotionalmissile launch depictedinFigureS.l. satel-litesensor in positiontoview a boosting TBM3caninprinciple pro-videusefulinformationtoavariety of theaterdefenseplatforms.ygatheringinformationonth eTBMtrajectory,forexample,a"for-wardtrack"of th emissilecanbe derived,enabling th etimean dlo- cationof missileimpactto be estimated.f relayedtoth etargetareainatimely manner,appropriate passivedefensivemeasuresm ay be employed.naddition,th eforwardtrackca nincludeestimatesof th emissilepositionasafunctionof timealong thetrajectory.uch RANDMfl737-S.J

w ForwardtrackSatell itesensor ^ \

Detection,backtrack and forwardtrack

Fighterorattackaircraft

^>Sis>2

' ATACMS

ATACMS=Army TacticalMissileSystemP G W =Precision-guidedweapon

TE L=Transporter-erectorlauncher^acktrack Fighter orattack aircraft

3ABGWor-. >\. submunit ions - Shelter orotherhideplaceiport^ Launc

Figure S.lSatellite SensorsSupport Bot h Forward an d Target AreaDefenses

3Toimplifyuriscussion, eseheermsatelliteensor"oepresentspaceborneplatform capableofdetectingmissilesduringth eboost-phaseonly.SensorscapableofdetectingTBMsafterboosterburnout(e.g.,BrilliantEyes-typesystems)are no tconsidered here.

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xiv Estimation andPrediction of Ballistic Missile Trajectories

tude,34.77 longitude.herelevantgeometry isillustratedinFigureS.2,whereth esatellitesarepositionedin geosynchronousorbit at 0latitude, an d1 5 an d 75 eastlongitude,respectively.10T hetw osatellitessample th e trajectory independently,each measur-ingtw oangles(whicharesubjecttorandom errorsandbias)at 20-secintervals(the assumed revisittime1 1).obegin filtering,w emustspecifyth enitialovariancefthetatestimaterrorbeforemeasurement. 12Weassumean initial1 uncertainty inlaunch lati-tudean d longitude,20 uncertaintyin launch heading,20-secuncer-taintyinlaunch time,1-km uncertaintyinlaunchaltitude,an da1

RANDAOT737-S.2Missilelaunch:Iranto Israel

Figure S.2GeometryofT B M TrajectoryandSensors

10Al though w eillustratethemethodologyfor anotionalmissilelaunchandsatelliteconfiguration,theformulasandequationsderivedaregenerallyapplicabletoawiderangeof threatscenariosandsensorconstructs.11In asensitivityexcursion,w elaterexaminetheeffectsonthetrajectoryanalysisofvarying therevisittime.12Se e Chapter T w o for adetaileddescriptionof theestimationsequence.

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Summary xv

uncertaintyinloftangle.orsimplicity,w eassumecloudsdonotpresenta viewingproblem,andignoreeffectsof early boosterenginecutoff.13N O T I O N A L R E S U L T S :R A N D O M E R R O R S O N L Y O neusefulapplicationofthefilteringmethodologyisinestimatingtheuncertaintyassociatedwiththelocationofaTB Mlaunch.sdepictedinFigureS.3,launchpointuncertainty(LPU)maybere -ducedignificantlybyprocessinghemeasurementsrombothsatellitessequentially(i.e.,stereoprocessing).Here, 2orre-spondson7percentonfidenceevel.14)hePUerivedmonoscopicallyfromeachseparatesensorisalsoshown,indicatinghowadifferent viewinggeometrymayleadto differentresults.Intheabsenceofmeasurementerrors,six measuredangleswoulduniquely determine thesix-dimensionalstatevectorw eareestimat-in g(assuming our templateisexact).inceeachmeasurementpro-videstw oangles,only threemeasurementswouldberequiredtode -terminethe state. FigureS.4illustratesthe = 2launch pointuncertaintyasafunctionoftimeforvariousrandomerrorsinmeasurementangle100,30 ,and10microradians)andstereoprocessing. sisclearin al lcases,aprioriuncertaintiesreeducedmostapidlyyheirstew measurements,ndatalowerpacehereafter.Measurementsoccurttimesndicatedbydotsnheigure.)sxpectedonintuitiverounds,oreover,hePUerivedfterheinalmeasurementhasbeenmadescalesroughlyasthesquareoftherandom error.Determiningtheuncertainty associatedwithmissilelocationatanypointalongitstrajectoryisanotherusefulapplicationof thetech-nique.issilelocationuncertainties(MLUs)15fo rtw osensorspro-13Thatis,w eassumefull-burntrajectoriesthroughout.ote,however,thatthetimeat burnoutis stilluncertain,owing to missile launchtimeuncertainty.14

A discussionof probabilitiesanduncertaintyellipsesis found in Chapter Three.15ML U isthevolumeof anellipsoidthatsurroundstheestimatedtargetpositionandcontainstheactualtargetwithsomespecifiedprobability.

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xvi Estimationan dPrediction of Ballistic Missile Trajectories

4 .8 sq km (stereo)

5km HANDAOT737-S.3

18.6sq km (15-E )

5km

14.3sq km (75-E )

30-microradianrandom error

FigureS . 3 L P Us(1=2)fo r T w oSensors with Random ErrorsRANDMB737-S.4

Zerobiaserror

4 00Time(sec)

80

Random error(m icroradians)100 30 10

100

Figure S.4Sensitivity of L P U (I=2 )to R a n do mError(Two Sensors)

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Summary xv

cessedstereoscopicallyareshowninFigureS. 5fo rvariousrandomerrors.Here,in threedimensions= 2correspondsto a74percentconfidencelevel.)sillustrated,theuncertaintyvolumeincreasesmonotonicallyuntilthelatterpartofthetrajectory,whentheML U turnsover.16A sapointofreference,asphereof62-km radiusen-closes avolumeofroughly106k m 3.) In general,decreasing therevisittimeallowsmore measurementsto bemadeand,consequently,providesmoreinformationaboutthemissiletrajectory.igureS.6illustratestheL PU fo rvariousrevisittimes,spanningtherangeof 2.5-40sec. tlatetimes,notethattheL PU scalesroughlylinearlywith the numberofmeasurements.

RANDMR737-S.S

E o

10 0 2 00 30 000 Time(sec) 50 0 Random error(m icroradians)60 0Figure S.5Sensitivity ofM L U (= 2 )to Random Error( T w o Sensors)16B yexaminingthetrajectorieswithperturbedlauncht imes,altitude,andloft,onefindsfo r theexampleathandthatthedeviationfrom thenominalbaselinetrajectorybeginsto decreaseatanaltitudeof approximately30,000ft.hiseffect,manifestedin thedecreasinguncertainty58 0secintotheflight,isrelatedtoboththeatmosphericdegradationof the missilevelocity uponreentryandourchoice of a minimumenergytrajectory to perturbabout.

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xviii Estimation andPrediction of BallisticMissile Trajectories

RANDMH737-S.61 00

2 0 400Time(sec) 10 0FigureS.6Sensitivity of LPU (I= 2)to Revisit Time(TwoSensors)FigureS.7showstheeffectofrevisittimeonmissilelocationuncer-tainty. sisevidentfrom thisplot,anorder-of-magnitude reductionin revisittimegenerates more thanan order-of-magnitudereductionin uncertaintyvolume. N O T I O N A L R E S U L T S :R A N D O M A N D B I A S E R R O R S A tthispoint,w e have considered afilteroptimizedfo rrandomerrorsalone.nmanyituations,owever,iasrrorsominatehemeasurementuncertainty,ndmustthereforebeccountedfor.Therearetw opossibilities:1)examinetheeffectofbiaserrorsontheexistingfilter optimizedfo rrandomerrors,and(2)designafiltertoccountorheiasrrorsxplicitly.eeferoheseformulationsassuboptimal andoptimal,respectively.17I7Itisaturaltosk w hyonewouldbotherusingauboptimalormulation.fredesigninganexisting filteroptimizedforrandomerrorsaloneisnotdesirable,thesuboptimalapproachallowsth eeffect of biasonthat filterto beexamined,albeit asanafterthought.ee ChapterF o urfo radetaileddescriptionof bothapproaches.

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Summa r y xix

RANDMB737-S.7

1 00

Seconds

2 00 30 000 Time(sec) 500 60 0Figure S.7Sensitivity of ML U i= 2) to RevisitTime(TwoSensors)

Whentreatingbiassuboptimally,hefilterappliesgainsindeed,sometimeslargegains tothesystembyconsidering randomerrorsalone. saresult, whenthe effectsofbiasareexamined,theymaybelargebecausetheyareamplifiedby largegains.n anoptimalformu-lation,onthe otherhand,thefilter knows biaserrorsar epresentandcanadjustthesegainsaccordingly.Nonetheless,whenthebiasisnotdominant(i.e.,biaserrorislessthanorcomparabletotherandomerror),onewouldexpectboth approaches to yieldsimilarresults.18F orthenotionalTB Mlaunchdescribedabove,FigureS.8illustratesthelaunchpointuncertaintyinthepresenceofbiastreatedopti-mally). elative to FigureS.3,largerlaunchpointuncertaintyellipses areobtainedwithbothsourcesof errorpresent.18Keep in mind,however,that thesuboptimaltreatmentof biascangoawry evenin caseswheretherandomandbiaserrorsar ecomparable.fthefilterapplieslargegains duringtheestimationsequence,resultsobtainedtreating biassuboptimally m ay differmarkedly from thoseobtainedwith anoptimalformulation.n all cases,though,theformerapproachwouldoverestimatetheerror.

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xx Estimation an dPrediction of Ballistic Missile Trajectories

RANDMA737-S.8-r10km

16.8sq km (stereo)

47.5sq km (15-E )

39.6sq km (75-E )

J--1030-microradianrandom an dbiaserrors

10km

Figure S . 8L P Us(i= 2)fo r T w o S ensorswith Random an d BiasErrors(OptimalFilter) FigureS.9illustratesthetimeevolutionof thelaunchpointuncer-taintyfo rthecaseofa30-microradianrandomerrorand100-,30-,and10-microradianbiaserrors,respectively.nlikethecasewithrandomerrorsloneFigure.4),hePUderivedfterthefinalmeasurementdoesnotscaleasthesquareoftheerror.oreover,theimportanceofbiasisapparentinthelargedifferencebetweenthe30-and100-microradiancases. Themissileocationuncertainty,llustratednFigure.10,lsoshowsqualitativedifferencesfromestimatesconstructedintheab -senceofbias(FigureS.5).nparticular,curvescharacterizedbydif-ferentbiasesappearto coalescelatein thetrajectory.Finally,assensorrevisittimesarevaried,theL PU derivedafterthefinalmeasurementisrelatively insensitivetorevisittimeinthecaseof30-microradianrandomandbiaserrors(FigureS.ll).neffect,a

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Summary x x i

1E+06 1E+05-1E+04 -

=3DL

RANDWB737-S.9

2 0 4 00Time(sec) Biaserror(microradians)100Figure S.9 Sensitivity ofL PU (I=2)toBias Error(TwoSensors;OptimalFilter)RANDMR737-S.J0

30 000 Time(sec)

50 0

Biaserror(microradians)1003010

60 0

FigureS.10 Sensitivity ofML U (I= 2)to Bias Error(TwoSensors; OptimalFilter)

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xxii Estimationan dPrediction of Ballistic Missile Trajectories

a.

RANOMB737-S.frIU U 30-microradianrandom an dbiaserrors

vV"~" * r"* * " *4 10

1 I -L

2010,5,2.5

2 0 4 00Time(sec) 80 100 Figure S . 11Sensit ivityofL P U (I= 2 )to Revisit Tim e (TwoSensors;OptimalFilter) pointiseachedinthefilteringsequencewheredditionalmea-surementscontainingunknownbiaserrorsprovideinformationoflimitedutility.h isunderscoresthepointthatincreasingthedatacollectionratewillnotreducethelaunchpointuncertaintysignifi-cantlyunlessrandomerrors dominatethe measurementprocess. h isis becausestatisticsalonedo not "beat down"theeffectsofbias. l-thoughtheinsensitivityto revisittimeisnotapparentinthecaseofmissilelocationuncertainty(FigureS.12),thespreadinML U valuesasrevisittimesvariedseducedelativetoheunbiasedase(FigureS.7).O ntheotherhand,thisisnottosuggestthatrevisittimeiswhollyunimportantinthepresenceof biaserrors.ntheeventofearlyboosterenginecutoff,fo rexample,sizableuncertaintiesin burnoutvelocity coulddominatetheerroranalysis w ithorwithoutbiasef- fects.yusingthegeneralmethod describedhereinwhichcanac-

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Summary xxiii

RANDMH737-S.72

o 1E+04

Seconds

30 000 Time(sec)Figure S . 12Sensitivity of ML U (= 2)to Revisit Time(TwoSensors;Opt imal Filter)

commodate,ndthusestimate,theeffectsof earlyboosterenginecutoffashortrevisittimeouldimproveourknowledgeofthemissile burntime,and,consequently,ourstatevectorestimate.C O N C L U D I N G R E M A R K S A stheatermissiledefensesarefieldedatthedecade'send,satellitesensorswilllikelyrealizenimportantT MDbattlemanagementfunction.Waging"informationwarfare" will requireincreasingly so-phisticatedC3Ietworksthatcanpiecetogetherthemultifariouspacketsof informationrequiredto effectbattlespacedominance.nthisregard,imelytransmissionthroughoutthetheateriscentral.B utsuccessfulbattlemanagementrequiresmorethanconnectivityalone:hequalityofthenformationbeingtransmittedspara-mount. hus ,ourprimaryfocusin thisstudyhasbeen ondescribingaechniquewhosepplicationannrinciplerovideuchinformation in theT MDoperational environment.

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xxiv Estimation andPredictionof Ballistic MissileTrajectories

Harnessedinatheaterofoperations,thetypeofinformationde -scribedhere canbeusedto enhance the capabilityofactivedefenses,passivedefenses,andattack operations.tisthusimportantfor theA irForce to modelandunderstandsuch enhancement in operationalterms,so thatpersonnelcanunderstandthetrade-offsavailablebe -tweenrevisittime andhighaccuracy.ndeed,the useofmodelsthatcancapturetheoperationaleffectsofthesetechnicaldetailsseemsimportantfo ranydecisionsinvolving theacquisitionofspace-basedsensorsystems. sthedatapresentedheredemonstrate,thisises-pecially trueofsensorswithshortrevisittimesandsmallmeasure-menterrors,atleastinsofaras ournotionaltrajectoryanalysisiscon-cerned.tisimportanttoremember,however,thatbiaserrorscanbesignificant,andperhapsevendominant.ncludingtheireffects(optimallyincertaincircumstances)isthereforecentraltothesuc-cessofanymethodologyseekingtoestimateandpredictballisticmissiletrajectories.oreover,echniquestoreduceoreliminatesucherrors,whereapplicable,shouldbegivendueconsideration.

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A C K N O W L E D G M E N T S

T heauthorswishtothankMichaelJacobsandHowardHoltzoftheAerospaceCorporationfo rsharingtheirinsightsontrajectoryesti-mationndrediction;A N DolleaguesamesonomondMichaelD.Millerforthoughtfulreviews;HerbertHoover,Mario Juncosa,ndMoiraRegelsonorelpfuluggestions;ichardBuennekefo rgraphicalassistance;JuneKobashigawafo rmanuscriptpreparation;ndinally,StephenGuariniorinvaluableFortranprogrammingupporttheproject'soutset.eedlessoay ,responsibility fo ranyerrorsoromissionsis our own.

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A C R O N Y M S

A T A C M S A r m y Tactical MissileSystem B MD Ballistic missiledefenseC3I Command,control,communications,andintelligenceC O N O P S Concepts ofoperationDoD Department ofDefenseE R I NT Extended-rangeinterceptorJSTARS JointSurveillanceTargetAttack RadarSystem km KilometerL PU Launchpointuncertaintyurad MicroradianMEADS MediumExtended A irDefenseSystem ML U MissilelocationuncertainlyMTCR Missile Technology ControlRegimeO DS OperationDesertStorm PA C PatriotAdvancedCapability PG W Precision-guidedweaponR V Reentryvehiclerpm RotationsperminuteSA M Surface-to-airmissile SDI StrategicDefenseInitiative TB M Theaterballisticmissile TB MD TheaterballisticmissiledefenseTEL Transporter-erector-launcher T H A A D TheaterHigh-A ltitude AreaDefenseT MD TheatermissiledefenseTMD-GBR Theatermissiledefense ground-basedradarW MD Weaponsofmass destruction

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ChapterO neINTRODUCTION

A ttheoutsetofOperationDesertStormODS),activelydefendingagainst ballisticmissileattack w asnotanew idea.ndeed,theU.S.A irForcehadbegunexaminingthetechnicalfeasibilityofballisticmissiledefenseBMD)searlyas9 46withprojectsWizardandThumper,beforemanyrelevanttechnologieswerematureenoughto offermuchhopefo rsuccess. ecognizingthesimilaritybetweenai rdefenseandmissiledefense,theU.S .A r m yenteredtheB MDarenain 1955, whenitbegan developingNike-Zeus,anuclear-tipped inter-ceptorbasedontheNike-Herculesanti-aircraftsystem.y1958,aninterservice competition fo rthe B MD missionw as wellunder way.1 A tthesametime,newtechnicalissuesarosethatcalledtheB MDmissionintoquestion:ouldradarsdiscriminatebetweenreentryvehicles(RVs)anddecoysabovetheatmosphere?Wouldthesystem become saturatedif R Vsarrived atcloseintervals?W as guidancead -equatetobringtheinterceptortowithinthekillradius?ouldthesystemfunctionproperlyinanuclearenvironment?2 Largelybe- 1V.N .chwartz , as tandPresent:he HistoricalLegacy,nA.CarterndD.N .Schwartz(eds.),BallisticMissileDefense,Washington,D.C.: heBrookingsInstitution,1984,p p .331-332.2W enotethatthecontextualsettingofearly B MD work w as muchdifferentthanthatof today,andthusresearcheffortsfaceddifferent problems.orexample,thenuclearthreatmandatedlowleakagelevelsandrequiredsystems tobefunctionalinanuclearenvironment.oreover,trategiccompetitionwiththeSovietUnionoftenmadetechnicalissues(e.g.,decoy discrimination)hard tosettle.Whilem anyof theseissuespersist,thepresentcontextrenderstheirresolutionlesscrucial.

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Estimation andPrediction of BallisticMissileTrajectories

causeoftheseconcerns,Nike-Zeusproductionstagnatedthroughoutthe Eisenhower years.3By1963,technologicaladvancesintheareasof computing,radar,and propulsionestablishedthe feasibility of anendoatmosphericin -terceptor,Nike-X,whichcouldinprinciplediscriminatebetweenR Vsanddecoysbydiscerningdifferencesintheirinteractionswiththeatmosphere.ithphased-arrayradars,moreover,thesystemwouldbelessvulnerabletosaturation.espitetheseadvantages,Nike-XlatercalledSentinel)becamevulnerableto newsetofstrategic considerationsfirstraisedbytheMcNamara Pentagon:heprospectthatmissiledefensescouldstimulateadestabilizingarmsracewith theSoviet Union . hus,Sentinelw as suspendedin1969 by theNixonadministration,ndalthoughit srevisedB MDprogram(Safeguard)w asinitiallyfunded,by May1972theUnitedStatesandtheovietUnionhadstablished reatyaimedtimitinghedevelopmentofballisticmissiledefensesoverylowlevels.ycongressionaldirective,Safeguardw as terminatedin fiscal1976.4O nMarch23,983,aspeechbyPresidentRonaldReaganbroughtB MDto thefore ofpublicconsciousnessandsetin motionan exten-sive research and developmenteffortknownasthe StrategicDefenseInitiative(SDI).arnessingnewtechnologicalachievements,SD IsoughttoprovideadefensiveumbrellashieldingtheUnitedStatesfromtrategicttack.nhensuingyears,heonceptualndtechnical feasibilityof B MD w as revisitedinanewcontext,althoughmanyssuesemainedunresolved.5utecausefost,hewarmingofuperpowerelationsnheate980s,ndong-standingconcernshatB MDouldundermine relativelystablestrategicbalance,the initialfervorassociatedwithSD I waned by thedecade'send.Nevertheless,thinkingabout missile defensew as alivein1990,albeitfocusedprimarilyonprotectingthe U.S .homeland.63 Ibid.,pp .332-333.4Ibid.,p p .334-344.5A broadcollection of essaysonthissubjectisfoundin A .CarterandD.N .Schwartz,1984.6DefendingagainstconventionallyarmedSovietmissilesinEuropew as oneexcep-tion.ndeed,hadthisnotbeenan issue inthe mid-1980s,thePatriot missiledeployed inO DSm ay not have hadan ycapability to engageTBMs.

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Introduction

R efocusingthisthinkingon protectingU.S.forcesandalliesin oper-ationaltheatersarguablybeganontheseconddayofODS,whenmodifiedScudmissileslandedonT elAviv.lthoughfew peoplewereinjuredintheinitialattacks,thespectreofchemicalweaponsthreatenedtodrawIsraelintotheGul fconflict,potentiallyunder-miningasomewhatfragilecoalitionofArabstatesalliedwiththeUnitedStatesagainstIraq.tbecameapparent,consequently,thattheaterballisticmissile(TBM)usecouldexactaheavy tollin thepo-liticalarena,if notthe operational one.A sthewar continuedon,however,the tollof TB Mstrikesontacticaloperationslsobecamepparent.Scud-hunting"with-15Es,F-16s,A-10s ,A-6Es,B-52s,andJSTARSaircraftdivertedthousandsofairortieswayromothermissions.econnaissanceircraft(U-2/TR-lsandRF-4Cs)werealsoshifted.7AlthoughthedefensiveperformanceofPatriotmissilesprovidedapositivepsychologicalfactor,tbecamecloudedincontroversy8andontributedtohesubstantialpropertydamagenflictedbythe8modifiedcudslaunchedduringthewar.9inally,28U.S.oldierswerekilledin Dhahran,Saudi A rabia,whenasingleTB Mstrucktheir barracks.In large measure,theO DSexperiencegalvanizedU.S .interestin the-atermissiledefenseTMD),inpartbecauseoftheworld'ssizableinventoryofballisticmissiles.hirty-threenations, numberofwhichctivelypursuepoliciescontraryto .S .nterests,possessTBMs.SeeFigure1.1. )10SecretaryofDefense,Conductofth ePersianGulfWar:inalReporttoCongress,Washington,D.C.: .S .GovernmentPrinting O ffice,April1992,pp .224-226.8Forexample,ee ,T.A .ostol,LessonsoftheGul fW arExperiencewithPatriot," InternationalSecurity,Vol.16,N o. 3,Winter1991 /92,p p .119 -171 ;R .M.Stein,"Patriot A T B MExperienceinheGul fWar,"InternationalSecurity,Vol.6, o.,Winter1991 / 92 ,ddendum; .M.SteinandT.A .ostol,Correspondence:atriotExpe-rienceintheGulf War,"InternationalSecurity,Vol.17,N o.1 ,Summer1992,p p .1 9 9 -240.9SeeSecretaryofDefense,1992,p p .226-227;S.Fetter,G. N.Lewis,andL .Gronlund,"Why WereScudCasualties SoLow?"Nature,28 January1993,p p . 293-296. 10Thesemissilesar einserviceandhavem ax i m u m rangesof 200kilometersor greater. Theformer SSR"nigure.1ncludesnlyAzerbaijan,elarus,eorgia,Kazakhstan,Russia ,andUkraine.eeD.Lennox,"BallisticMissilesHi t N ew Heights," Jane's DefenceWeekly,30Apri l1994,pp .24-28.orabroaderdiscussionof ballistic missileproliferation,seeJanneE.Nolan,rappingsof Power: allisticMissiles inth eThirdWorld,Washington,D.C.: h eBrookingsInstitution,1 9 9 1 .

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4 EstimationandPrediction of BallisticMissileTrajectories

FranceU nitedKingdom

PolandCzechRepublic

SlovakiaBulgarialgeriaomerHungarygyp tSSRRomania L ibya

U nitedStates

AfghanistanIranIraq IsraelSaudiArabia Syria UnitedArabEmirates Y e m e n

ArgentinaChina ^NorthKorea

India SouthKorea Pakistan Vietnam

Figure1.1Thirty-Three NationsPossess TBMsPerhapsmoreimportant,worldwidedevelopment effortscontributetotheexportablesupplyof TBMs,manyofwhichmayrealizemaxi-mumrangesin excessofIraq's650-km 1 1A lHussein(seeTable1.1) .Coupledwitha concomitantspreadofweaponsofmassdestruction(WMD),uchB Msouldnable trikeapabilityhatmightthreatenregionalbalances,U.S.allies,orevenU.S.forcesdeployedoverseas.Theevolving securityenvironmentcontainselementsthatarepotentiallyworrisometbest;tworst,heyredownrightthreatening.nD.Lennox,1994 .

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Introduction

Table1 .1 DevelopmentPrograms Complicate EffortstoCurtailT B M Proliferation

Country Missile Range(km) Payload(kg)Iran Mushak200 20 0 500SouthKorea NHK/A (HyonMu ) 300 300Pakistan Hatf2 300 500Iran CSS-7/M-11 variant 300 500India Prithvi SS-350 350 500Pakistan Hatf3 600 500Iran Iran700(ScudC) 700 500Libya,Iran A lFatah 9 50 500Taiwan Tien Ma (SkyHorse) 9 50 500NorthKorea,Iran Labour-1 (Nodong1 ) 1000 1000China,Iran M1 8(Tondar-68) 1000 400Spain Capricornio 1300 500North Korea,Iran Labour-2(Nodong 2) 1500 1000China,Iran DF-25 1700 2000North Korea Taepo-Dong1 2000 1000India Agni 2500 1000North Korea Taepo-Dong 2 3500 1000S O U R C E :.Lennox,"BallisticMissilesHitN ew Heights,"Jane's DefenceWeekly,30 April1994,p p .24-28.T M D D E V E L O P M E N T IS U N D E R W A Y Notwithstandingdiplomaticeffortstocurtailmissileproliferation,

12itsourprisehathenitedtatesasndertakennambitiousesearchndevelopmentffortnheatermissiledefense.obetterunderstandtheT MDmission,anotional"cradletorave"B MdeploymentsequencesllustratednFigure.2 ,alongwithheCoreSystems"lannedyheDepartmentfDefense(DoD)(discussedbelow). 12T heMissileTechnologyControlRegime(MTCR)isonesucheffort.Createdin1987,theMTCR controlsthetransferoftechnologies that couldaidth eunmanneddeliveryofa 500-kilogram payloadovera300-kilometer distance.or abrief descriptionof theM T C R,se eBallisticMissileDefenseO rganization,Ballistic MissileProliferation:AnEmerging Threat,A rlington, Virginia:ystem Planning Corporation, 1992,p p .64-65.

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6 EstimationandPredictionofBallistic MissileTrajectories

R A N D J W ? 7 3 7-r . 2

Figure1.2CoreSystemsEmphasize Target AreaFol lowingits manufacture/assemblyina production facility,theno-tionalmissileistransportedtoaprelaunchsite,whichmaybediffi-culttolocateanddestroy.henadeploymentorderisgiven,theTB Mmovesonatransporter-erector-launcher(TEL)tothelaunchsite,wherethemissileiserectedandfired.ollowingaperiodofpoweredflightin whichrocketfuelburnswithabright signature,themissile proceedsonaballistictrajectory,definedin large measure by its velocity and position atburnout. tthispoint,the missileisonitsw ay toimpact,andthemobileT ELmaybefleeingtoapostlaunch"hidesite"orresupplydepot.ItisconvenienttodifferentiatebetweenopportunitiestocounterT BMsinthetargetareaandtheforwardarea(Figure1.2).argetarea defenseseitherattemptto interceptthe incomingmissile1 3 near13A sthesituationdictates,reentry vehicles m ay betheappropriatetargets,ratherthanthemissilesthemselves.

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Introduction

thepointofimpact(thePatriotmissileintheGulf W arisafamiliarexampleofsuchasystem)orrelyonpassivemeasuresin thetargetzone(seekingshelter,donningprotectiveclothing,etc.).h eT MD CoreSystemscurrentlyplannedbyDoD emphasizeinterceptionfo rtargetareadefense,withhreeeparateinitiatives:A C - 3with extended-rangenterceptor,RIN T ) ,14NavyA reaT BMD,15ndTheaterHigh-AltitudeAreaDefenseTHAAD)withT MD-G BR) .16Thesesystemsarescheduledfo rinitialdeploymentin1998 ,9 9 9 ,and2001,respectively,atatotal costof about$2 5billion.17Forwardareadefenses,ontheother hand,wouldtargetthemissilewhileitis boostingorascending,therebyprovidingcapabilityagainstT B M swithfractionatingpayloads.18Productionfacilities,prelaunchsites,esupplydepots,ndheTEL itselfcouldalsobeargeted.Attack operationsof thissortand,indeed,forwardareadefensesin generalwouldlikelyemployaircraft,owingtotheneedtoreachintoheorwardrea.19 Althoughorwardreaevelopment14PatriotAdvancedCapabiIity-3withxtended-rangenterceptor)s,oughlyspeaking,anewandimprovedPatriotmissile.xistingPatriotlaunchersandradarswillbe modified. 15Navy Area TBMD(formerlyknow nasNavyLower-Tier)willu se StandardBlock IV A missilesdeployed onroughly 50 AEGIScruisersand destroyers.hip-based radars willbemodified to accommodatetheT MD mission.16T H A A DwithT MDground-basedradar)saground-based,upper-tierdefensesystem requiring new missilesandnewradarsfor targetacquisitionandfirecontrol. 17Se eCongressionalBudget O ffice,The Futureof TheaterMissile Defense,Washington,D.C.:.S .GovernmentPrintingO ffice,June994,p. xv .heabovecostincludesestimatesoffundsappropriatedbefore1995.18Boost-phasenterceptorsrenefthreeAdvanced-CapabilityT MDystemscurrentlybeingexaminedby DoDNavyTheater-WideTBMDformerlyknow nas NavyUpper-Tier)andtheMedium Extended A irDefenseSystem[MEADS]formerly knownasCorpsSAM)aretheothers.ecauseof budgetaryconstraints,itisexpected thatonlyoneof thesewilleventuallyproceedtodevelopment.eeCongressionalBudgetO ffice,1994,p.xiv.19SpecialOperationsForces(SOF)deployed in the forward areacouldsupport attackoperationsbyrelayinginformationaboutTB Mlaunchestostrikeaircraft.uring ODS,SO F groupsinfactprovided vital information about Iraqi missiles.eeSecretaryof Defense,992,.226;andD. .Waller,he Commandos ,N ew York:imon& Schuster,1994,pp .335-351.

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8 Estimation andPredictionof BallisticMissile Trajectories

programsaveoteceivedheighestrioritywithinoD ,promisingconcepts ofoperation( C O NO P S )have beenidentified.20

S A T E L L I T E S E N S O R S S U P P O R T T M D B A T T L E M A N A G E M E N T Activedefenses,passivedefenses,and attackoperationsas describedaboveform three ofthefour"pillars" oftheU.S .theaterdefensepro-gram.hefourthcommand,control,communications,andintelli-genceC3I)isinasensethefoundationsupportingthesepillars,rather thanapillaritself.ow mightsatellitesensorscontributeto C3I in theT MDenvironment?Considerthenotionalmissile launchdepictedin Figure1.3. satel-litesensorinpositiontoview aboostingT B M21aninprincipleprovideusefulinformationtoavarietyof theater defenseplatforms.Bygatheringinformation onthe TB Mtrajectory,fo rexample,a"for-wardtrack"ofthemissilecanbederived,fromwhichthetimeandlocationofmissileimpactcanbeestimated.f relayedtothetargetareainatimelymanner,ppropriatepassivedefensivemeasuresmaybeemployed.naddition,theforwardtrackcanincludeesti-matesofthemissilepositionasafunctionoftimealongthetrajec-tory.u ch estimatescouldbeusedtocuesearchradarsof activedefensesystems,andperhapsprovidefire-controlquality"launchbaskets" fo rTB Minterceptors.22Similarly,aTB M"backtrack"tothelaunchpointprovidedbysatel-litesensorscouldsupportattackoperations withaircraft or ground-launchedmunitions.During theGulf War,Scudlauncherscouldbemovedwithinminutesofmissilefiring,andafter15minutes,could20Se eD.Vaughanetal.,Evaluationof OperationalConcepts forCounteringTheaterBallistic Missiles,SantaMonica,Calif.:AND ,WP-108,1994. 21T oimplifyuriscussion,w euseheermsatelliteensor"oepresentspacebornelatformapablefdetectingmissilesuringheoost-phasenly. SensorscapableofdetectingTBMsafterboosterburnoute.g., rilliantEyes-type systems)are notconsideredhere.22Inthecaseof boost-phase/ascent-phaseintercept,timeconstraintsm ay limittheutilityof satellite-based information.

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Introduction

RANDMH737-7.3

Forwardtrack ATACMS=Army TacticalMissileSystem P G W =Precision-guided weapon

TE L =Transporter-erectorlauncherFighteror

attackaircraft

Figure1.3Satellite Sensors Support Both Forwardan dTargetArea Defensesbe anywherewithinninemilesofthelaunchpoint,underscoringtheimportanceoftimelyresponse.23ydetectingandrackingtheTB Mduringboost-phase,24however,thespacebornesystemscon-sideredherehavethepotentialtosupplyinformationfo rsuchare-sponse,ndtoosonearlygloballyonanessentiallycontinuouscoveragebasis.O R G A N I Z A T I O N O F THEREPORTThisreportdescribestheoperationalimplicationsofanestablishedanalyticalprocedurewhich,appliedtonotionalsatellitemeasure-ments,suppliesinformationto abattle managementfunctioncentral23Secretaryof Defense,1992,p. 224.24Dependingonthetypeof missile,boost-phasestypicallylastbetween30and1 20sec.ee CongressionalBudget O ffice,1994,p .5 .

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10 EstimationandPredictionofBallisticMissile Trajectories

toth etheatermissiledefensemission.hapterT w odescribesth etheoretical underpinnings of th eapproach,knownaslinearfiltering.Theequations of aKaiman filter optimizedfor random measurementerrorsarederivedfo rboth linearsystemsandnonlinearsystemsinth elinearapproximation.helatterareappliedtoanotionalTBMlaunch against IsraelinChapter Three,with an emphasison analyz-ingaunchointuncertaintyndmissileocationncertainty.ChapterFourdiscussesth eeffect of measurementbiason thisfilter,onafilteroptimizedforbothrandom andbiaserrors,an donth etrajectoryanalysisinbothcases.inally,ChapterFiveofferssomeconcludingremarks.

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Chapter T w oTHEORETICALUNDERPINNINGS

Thischapterbrieflydescribestheformalismof aKaimanfilteropti-mizedforrandomerrors.quationsarederivedfo rboth linearsys-temsand nonlinearsystemsinthelinearapproximation.L I N E A R E S T I M A T I O N A N D P R E D I C T I O N Consideraphysicalsystemwhosecharacteristicsmaybefullyde -terminedatanytimebythestateofthesystem,x. 1oradynamicalsystem,sucha vectormightcontaintheposition,orientation,time,velocity,acceleration,and/oranyotherparametersrelevanttode -scribingits state.f measurementsonsuch asystem (in theabsenceof errors)yieldobservationsthatare proportional to thestatevector(in thematrixsense),thesystem willobeythelinearrelation

z=Hx + v,2.1)where

z-dimensionalmeasurement vector,x-dimensional statevectorofsystem,Hknown(p x n)-dimensionalmatrix,veasurementerrorsin z(p-dimensional).2.2)

xOurnotationissimilartothat of A .E.BrysonandY.-C.Ho,AppliedOptimalControl,N ew York:HemispherePublishing Corporation,1975.

1 1

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W eassumethemeasurementerrorsarerandom,with vanishingex -pected value:

E(v)= 0. (2.3)Denotetheestimateofthestatebeforemeasurementbyx,andde -finethe errorcovariance ofthemeasurementanderrorcovarianceofthestatebeforemeasurementby

R=E|wT

andM= E(x-xHx-x)1

(2.4)

(2.5)respectively.ssumingxandvtobeindependentvectorsobeyinggaussianstatistics,theprobabilitydensityp(x,v)isproportionaltoexp(-J),where J is thequadratic form

< = r x-x)TMT 1 (x-x)+ (z-Hx )TR "1 (z -Hx )O necanshowthat Jis minimized by thevector

x=x+PH TR_1 (z-Hx),withPthe errorcovariance ofthe stateaftermeasurement:

P = E[(X-X)(X-X)T].

(2.6)

(2.7)

(2.8)A s esult,= xnd=v= z -Hieepresenthemaximumlikelihoodestimate"ofthestatevector,in that theymaximize(x ,v)giventhemeasurementz.2nother words, isthemost likely statevectorresultingin themeasurement z,given thestatisticalpropertiesofxandv.32 Ibid.,p .357.3Se eA.Gelb(ed.),AppliedOptimalEstimation,Reading,Massachusetts:h eAnalyticSciences Corporation,1974,p. 103.

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Theoretical Underpinnings 13

I tisastraightforwardexerciseto verify that satisfiesP=|M-1+ HTR-1H

=M-MHT (HMHT+R)"1HM .2.9)Ifthestatevectorisofgreaterdimensionthanthemeasurementvector(i.e.,if n>p ),Pismoreeasily obtainedfrom thelatteroftheaboveequations.otethatthisequationalsopredictsthatmea-surementsdecreasetheuncertaintyinourknowledgeof thestate(i.e.,TPx

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14 EstimationandPrediction ofBallistic Missile Trajectories

wheresequentialestimates arelinkedviax i+1 X ; Xjisgiven)2.14)

P i=(Mi-1+ HiTRfHi)=Mi-iHiTfHiMiHiT+ R il HjM; 2.15)

andMi+1 Pi,M jisgiven).2.16)T h eabovesetofequationsconstitutesaparticularformofaKaimanfilter.5quations(2.13)-(2.16)may beusedtorefineaninitialesti-mateofthestate(Xj)andit scorrespondingerrorcovariance(MJthroughtheuseofinformationobtainedthroughthemeasurementprocess.heestimationsequenceisrepresentedin Table2.1 .NotethatthematricesH;and ;mustbespecifiedto run the filter.A sformulated,thefilterisoptimizedforrandomerrors,whichareuncorrelatedfrom measurementtomeasurement.nChapterFour ,w ewillinvestigatetheeffectof"bias"errors,whicharecorrelated.L I N E A R A P P R O X I M A T I O N T O N O N L I N E A R S Y S T E M S

F ew physicalsystemsarelinearinthesenseofEq.2.1);mostaredescribedby thenonlinearequation

z= h(x)+ v,2.17)5R .E.Kaiman,"A N ew Approach to Linear Filteringand Prediction," Trans.ASME,Vol.82D,1960,p.35.More generally,thestatevectorhasa know n transitionmatrix( < I > ) ,aknow nprocessnoisedistributionmatrix(T),ndisaffectedbyarandomprocessnoise vector(w):

Xi +l=*ixi+riwi-Sincetherear enodisturbancesto thestatein ourformulation(i.e.,noprocess noise),w em ay chooseourstatevectortoomprisenitialvalueata,nwhichas ebecomestheidentitymatrix.ith thischoice,thedynamicsofthephysicalsystem ar emanifestednhemeasurementprocessandapturedmathematicallyinthedefinitionof theH-matrix.ee A .E. Bryson and Y.-C.Ho ,1975,p p .359-361.

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Theoretical Underpinnings 1 5

Table 2.1 Estimation Sequence

B e f o r e Measurementf t e r MeasurementxlfMxJ.PJx2=x1,M2 P 12,P x3 x2,M3 P23,P

whereh isadifferentiablefunctionofx.n the eventthatsufficientaprioriknowledgeofthestatevectorisobtainable,Eq.(2.17)canbeexpandedinaTaylorseriesaboutaninitialestimateofthestate.Denotethisestimateby,andthemeasurementtowhichitcorre-spondsbyz.Expandingaboutthisestimateto linearorder,oneob-tains

-- 3z z-z= 9 x (x-x)+ v=H(x-x)+ v.2.18)A saresult,the developments oftheprecedingsectionca nbeappliedifw esimplyshiftthestateandmeasurementvectorsby appropriateconstantvectors.learly,thematricesM,P,ndRreunaffectedby thisredefinition[seeEqs.(2.4),(2.5),and(2.8)].h eestimateofthestate,ontheotherhand, willbegivenby

x=x + PH TR-1(z-z-H(x-x)),2.19)whichreducesto x=x + PH TR"1(z-z)2.20)if w eidentifyx=x.Inthelinearapproximation,then,itiseasytoverifythatthetimeevolutionofthefilter is governedby

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16 Estimation andPredictionofBallisticMissileTrajectories

XjX;+PjH^Ri^CZi-z,-H J C X J-x)),2.21)where H, i. ,2.22)1 3x x=xandsequential estimatesarelinked via

x j+1 X j,x x xisgiven)2.23)\-iP + HiVH^

=M,-MjHjfHjMiHi1"+R j) H;Mj(2.24)

andMi+i=Pj,M iisgiven).62.25)Finally,rewritingEq.(2.18)inthe sequentialform Zj-Z;=Hj(x-x)+vi(2.26)whereZ ;is the measurementvectorat time indexi correspondingto

theexpansionvectorx,anddefiningej X j-x 2.27)

Eq.(2.21)canbe rewrittenas e, (I-KiHi)ei_1+KjVj ,2.28)wheretheKaimangain matrix isdefinedas K, P j H^R i-1.2.29)

6Al thoughnotimplementedinthepresentwork ,itisalsopossibletou sethestatevectorestimateaftermeasurement(x )toupdatetheexpansionpoint(x),iteratinguntilconvergenceisachieved.

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TheoreticalU nderpinnings 1 7

A saresult,sincethecovarianceattheithstageis given byT " PjSE e (2.30)

onecanshowthat theerror in thestatevector(e;)andtheestimateX ;ar euncorrelated.n ChapterThree,w ewillapply this formulationto theproblemofestimatingandpredicting theaterballisticmissile trajectories.

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Chapter ThreeE ST I MA T I O N A N DP R E D I C T I O N O F BALLISTIC MISSILE T R A J E C T O R I E S

In thischapter,w eapply theforegoingdiscussionto thecaseofbal-listicmissiletrajectories.W ebeginwitha descriptionof themissile-sensorengagement.G E O M E T R Y OFMISSILE-SENSOR E N G A G E M E N T A sdepicted in Figure3.1,ournotional sensor spinsclockwise(i.e.,in theright-handed sense)aboutanaxisoriginatingatthecenter of theearthandextendingoutwardthroughtheequator.uch ageometrymaybeusedtodescribesatelliteviewingfromgeosynchronousor -bits.T omodelthemeasurementprocess,it isusefultoerecta coordinatesystemmovingwiththenotionalsensor.onsiderfirstasphericalcoordinatesystemcenteredontheearth,asillustratedinFigure3.2.In thesecoordinates,thesensorlocationisdescribedby apositionvectorwithcomponents(r,0,< & ) . susual,thesphericalsystemisrelatedto ordinaryCartesian coordinatesthroughthetransformation

x= rsin 0co s 4 > y= rsin0sin

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20 Estimation an dPredictionof Ballistic Missile Trajectories

RANDMfl737-3.7

Figure 3.1Notional Sensor in Geosynchronous OrbitRANDMR737-3.2

Figure3.2 Earth-Centered Coordinates

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Estimationand Prediction of Ballistic MissileTrajectories 2 1

T otransform thissystem intoonerotatingwiththesensor,w em a k easequenceofcoordinate transformations.irst,translate theorigin of theCartesiansystem avectoramountr.ext,rotatethe( x ,y,z)systemanamountO aboutthez-axis(seeFigure3.3)using thema-trix relation ^

\ cos< & 001 V (3.2)Noworienthe'-axiswithheadialdirectionyotatingnamount0-nil aboutthey' -axis:

'x'A sin 0 0 co s00 10- co s 0 0 sin 0 y " Az'j (3.3)RANDMH737-3.3z=z

Figure 3.3Rotated C oordinates

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22 Estimation andPredictionofBallisticMissile Trajectories

Next,tiltdownward anamountr,againaboutthey' -ory" -axes:(3.4)cosTJ^z"^

T h eresultis depicted in Figure3.4./_v RANDMR737-3.4

rx'") fcosT 0 -sinrYx""jy"z'" 0 1 0sin T 0 co s r

Figure 3.4Tilted Coordinates A sthesensorspins,avectorin a coordinatesystemwhoseoriginliesattheensorpositionotatesounterclockwisei.e.,nheeft-handedsense)withrespectto the spinaxis(seeFigure3.5).ntermsofthecoordinates above,therotatingvectorp obeys1

p=r"' co sc o t +hfii F'')(l-co sc o t ) + (r"'xn)sin c o t (3.5)wherendefinestheaxisof(counterclockwise) rotation.1H.Goldstein,Classical Mechanics,R eading,Massachusetts: ddison-Wesley,1980,p. 164.ememberthataclockwiseotationfthecoordinatesystemppearsasacounterclockwise rotationof the vector.

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EstimationandPrediction of BallisticMissile Trajectories 23

A saresult,applyingEqs.(3.1)-(3.5), w eobtainth efollowing relationsforth epositionof an object viewedinacoordinatesystem thatro-tatesas in Figure3.5:X "'cosrot+ (x"'cosr+ z"'sinr)cosr(l-coscot)-y'"sin T sin c o tY "'coscot+ (x'"sinr-z"'cosr)sincotZ "'coscot + (x"'cosr+z"'sinrjsinr(l-coscot

+y'''cos rsinc o t ,(3.6)

where

an d

x"' = x" cosT -z" sinT y 'zlit II' Sx" sin r+z"cosrx"=xsin0cos + ysin0sin +zcos0y" =-x sin < & + ycos< E > z" =-xcos0 cos< & -ycos sin 0+ z sin 0

(3.7)

(3.8)RANDMR737-3.S

r'" = (x'",y'",z'")

Figure3.5SpinningCoordinates

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24 Estimation andPrediction of BallisticMissile Trajectories

Thus ,in addition to the positionsofthesensor andmissile,therota-tionrateandtiltofthesensorneed to bespecified toadequatelyde -scribethe engagementin thisformalism.With Equation(3.6)athand,definethe angles

ZX (3.9)

and

o c h tan-i X (3.10)representing "vertical"and"horizontal"anglesin thesensor'sframe,respectively.T heboresightofthesensorpointstowardtheearthalongaaythroughheoriginoftheY -Zcoordinateplanei.e.,througho c v=ah=0)].Whenanobject passesthroughthesensor'sfieldofview,h 0.W e maythereforeuse therotationphaseangleQ . = c o tto defineanotherangle representingthe rotationalpositionofthesensorwhen the objectpassesby .2SettingEq.(3.10)to zero,andusingEq.(3.6),w e obtain

Q .= tan-lz"'co sT-x"'sin T

(3.11)(Loosely speaking,Qepresentstheangular positionofahandonaclock,wheretheface representsthe disk ofthe earthas seenfrom thepositionof thesensor.)ntermsof thefilteranalysis,thetw osetsofangles( o cv,ahndav,C l)areequivalent.3nlessotherwisestated,w ewillassume2B ydefiningameasurementtooccurwhenj,= 0,heensorismoreproperlydescribedasaverticalslitwithnohorizontalextent.ot ethattheslitisalignedtoward thenorthw h en=0. 3Thatis,measurement vectorstaken as

SM:; -iyieldthesameresults.ote,however,thatbymathematicalconventionthetan'functionassumes values between-90and+9 0degrees,andsocannotrepresent angles

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EstimationandPrediction of BallisticMissileTrajectories 25

(V-2J (3.12)

ItisconvenienttoparameterizetheerrorinQ .ntermsoftheerrorinc h.ifferentiatingq.3.10)holdinghemissilepositionconstant),settingah 0,and definingthequantityD os 2 + cosr+ -sinTx co sr(l-co s Q (3.13)V -^7^sinrsin2

w efind5ah D - sin Q. sin r co srx cos 2

With alittlealgebra,w emayrewritethe aboveas 8ah -8Q(sin r+tanavco s r)

(3.14)

(3.15)

F I L T E R M E T H O D O L O G Y T ocalculatetheH-matrix,w edefineatemplatefo ragivenmissilefromitsrange-altitudedata,whichareobtainedbymodelingthemissile'sflightintheatmosphereofaspherical,nonrotatingearth.4Thistrajectoryisusedasabaselinefrom whichperturbationsandultimately,theH-matrixelementsaregenerated.nthefield,sen-sor measurements would beobtainedfromtheactualmissileunderobservation;here,oimplyestimatetheerrorsonemightexpectusingthefilter technique(as opposed toestimatingthe statevector),inthesecondandthirdquadrants.napplyingEq .3.10),thisisnotaproblemformostpracticalgeometriesbecauseX isusually negative.u ch isnotthecasefor Eq. (3.11),so that specialcaremustbetakeninapplyingthisequation. ne solutionisto u se Eq.(3.10)todeterminew h ena measurementoccurs(i.e.,w h e n%=0) ,andthensubstitutetherelevantcoordinates[usingEq.(3.6)]intoEq .(3.11)tofindQ ..4Forintermediate-andshorter-rangemissiles,neglectingrotationaleffectsisusually justifiable.

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w esimulatethisprocessbytakingthemeasurementsonthetem-platetrajectory.T he(constant)statevectorisdefinedby

(x=

VxeJ(3.16)

where

*3X4x5

=aunchlatitude,=aunchlongitude,=aunch heading,=aunchtime,=aunchaltitude,

x g = loft.anglecharacterizing missilepitch-over.5FromEq.(2.18),w ecanobtaintheH-matrixby perturbingthestatevectorelementsandexaminingthechangesonthemeasurementvectorz.nthis manner,fo rthecaseofatwo-dimensionalmeasure-mentvector[i.e.,withcomponents(zx,z2)] ,heelementsofHaregiven by

H= 9zj 3zj dzl 3zj 3zj 3zj9xj dx 2 3x, 3x 4 3x 5 3xfi3z2 3z 2 3z 2 3z2 3z 2 3z23x, 3x? 3x, dxd 3xc 3x ,6 7

(3.17)

5thisparameterizesafamily oftemplatesbasedonearlypitch-overfollowedby zeroangleofattackfor theremainderof theflighttrajectory.or egenerally,w ecouldhaveassumedan yfamilyof flighttrajectorytemplatesforwhich variationoftheloftangleisdeterminedby aone-parameterfamily ofsteeringfunctionsappliedearlyin theboost-phase.

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EstimationandPrediction of Ballistic Missile Trajectories 27

where partialderivativeswith respectto one statevectorelementarecalculated holdingother elementsconstantandevaluatedattheini-tialguess = x .6Variations in launch position(xx,x2)and heading(x3)arestraightforward,simplychangingthegeometricrelationshipbetweenmissileandsensor.Variationsin launchtime(x4),ontheotherhand,movethepositionofthemissileforwardorbackwardin its timehistory.f onewereto imagineasequence ofbeads ona wirerepresentingpointsalongthetrajectory(seeFigure3.6),such varia-tionscouldbedescribedbyslidingthebeadsbackwardorforwardalongthewire.T he"beads"shownin Figure3. 6representtrajectorypointsplottedveryfiveeconds.)aryinglaunchltitudex5)causesmorethanaverticaltranslationofthetrajectory,sincedragdependsontheatmosphericdensity,anapproximatelyexponentialfunctionofaltitude.inally,toallow fo rplanarvariationsin the tra-jectorytemplate,w evarytheloftanglex6)duringthepitch-over

RANDMB737-3.6

0> a3

2 50Range(km) 75 Figure3.6Boost-PhaseofNotional M issile6T heomplexityofthisproblememandsthattheseerivativesbealculatednumerically.

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28 EstimationandPredictionof BallisticMissileTrajectories

phaseof flight.oftangleisidenticalto theangleofattack,definedwithrespecttotheinstantaneousvelocityvectorofthemissile.tlaunchandpriortopitch-over,themissilevelocityisintheverticaldirectiondefinedlocally).s6increases,roughlyspeaking,theverticalspeedofthemissileisconvertedto horizontalspeed,so thatsmallloftanglesesultinloftedtrajectoriesndlargeoftanglescausetrajectoriestodepress.In what follows,w econsidertheestimation/predictionproblemfo rthe caseofanotional TB M whosetrajectoryis depictedin Figure3.7.A s the figureillustrates,thismissilehasaboost-phaseof100-secdu-rationandatotalrangeof1200km .oran initialguess,w e assume alaunchnranat34.01atitude,7.40ongitude)with 63 heading,7impactingT el Avivat32.05 latitude,34.77longitude. herelevantgeometryisillustratedinFigure3.8,wheretheatellites

C D"O3

RMW3MR737-3.7 JUU

20 0

100 -'Burnout(100sec)

0 i 1 1 1 1 1 _13 0 0 600

Range (km)900 1200

Figure3. 7 Notional100-sec-Burn M issileTrajectory

7n 0 represents du enorth and90,du eeast.

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EstimationandPrediction of BallisticMissileTrajectories 29

RKR0MR737-3.8Missilelaunch:Iranto Israel

Figure 3.8Geometry of TB M Trajectoryan dSensorsarepositionedingeosynchronousorbitat0latitude,and1 5and75astongitude,espectively.8T hequationsescribingtrajectoriesonaspherical earth maybefoundin the Appendix .)A sdepictedin Figure3.9,the tw osatellites independently samplethemissileboost-phase,achmeasuringtw oanglesz,,z2)t20-secintervals(theassumedrevisittime9 ).incethenotionalTB Mtakes42se ctoreachanaltitudeof10km ,a sensorunabletose e throughacloudlayeratthisaltitudewould ,roughlyspeaking,bedeniedtw o8Geosynchronousorbitaboutasphericalearthoccursatan altitudeof roughly 35,800km asmeasuredfrom theequatorialurfaceequivalentto radiusvectorabout42,200km inextentasmeasuredfrom thecenterof theearth).orsatellitesat0 latitude,thediskof theearthsubtendsahalf-angleof roughly8.8,sothata4.4tiltangle witha4.4fieldof view coversthedisk completelyasth esensorrevolvesaboutits spinaxis.9Inasensitivity excursion,w elaterexaminetheeffectson thetrajectoryanalysisof varying the revisitt ime.

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30 EstimationandPredictionof Ballistic Missile Trajectories

RANDMB737-3.9

Boosterengine /?burnout

frMeasurementsby sensorat15 'E frMeasurementsbysensorat 75* E

Cloudpw"v deck-~->N J -~-S J r-

Figure 3.9Boost-Phase M easurem entS equencemeasurements.orsimplicity,w eassumecloudsdonotpresentsuchaproblem,andignoreeffectsofearly boosterengine cutoff.10First,considerthe satellitepositioned at75 longitude.TheH-matrixcorrespondingto eachmeasurementmaybecalculatednumerically,utilizingEqs.3.1)-(3.10)nd3.12).ortheexampleathand,w eapproximatederivativesbydifferencesusingastep

5x='o.oi^ 0.01 0.01 1 0.01 v0.01

(3.18)

10Thatis ,w eassumefull-bum trajectories throughout. ote,however,thatth etimeat burnout isstilluncertain,owing toanuncertainty in th e missile launch time.

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Estimationan dPredictionof Ballistic Missile Trajectories 3 1

withanglesmeasuredndegrees,imeinseconds,ndaltitudenkilometers.yvaryingthestatevectorelementsindependentlybytheamountsillustratedabove,thechangeinthemeasuredanglesmaybecalculatedandderivativesdetermined.ivemeasurementsoccurduring boost-phaseinthisexample,atroughly1 8,38 ,58,78 ,and98secafterlaunch.The20-secperiodicityreflectstherevisittimeofthesensor.hemotionofthemissileisnegligibleherebe-causeitisviewedfromgeosynchronousaltitude.)otethatat8sec,theH -matrixreadsH ^.OxlO-2 -6.2X10" 2 5.4xl0~6 -2 .7XKT4 1 .2xl(T3 2.3xl0"5

1 .0. 9X10-1 5.2X10-5 7.0X10-4 3.4xl0~5 -8.0x10"^'(3 .19)

whereasjustpriortoburnout(9 8sec),itisgivenby8.0xl0~2 -6.3xl0"2 9.0xl0-4 -3.4xl0~3 4.3xl0-3 -4.1xl0-3

1 .0.7X10-1 8.9xl0-3 2.4X10-2 8.4xl0"3 -9.7xl0"2(3.20)

ThisillustratesthattheH -matrixis time-dependent.Onewouldnotexpectmatrixelementscorresponding tochangesinlaunchposition(i.e.,thefirsttwocolumnsofH)tovaryappreciablyduringthemissileflight,ince,neffect,hesechangesamounttosliding the"wire"trajectory asawholeover theearth'ssurface.emayestimate theseusingorder-of-magnitudeapproximationstothemissile-sensorengagement:

RG E OAav RE A R T H All= >f L; L_ RE A R T H _Q-1 (32lR G E OAccv E A R T H A L j 3xj 3x 2 R GEO

andiRE A R T H A~E A R T H' I=>i^2_(^2__j 322)1RE A R T H A~E A R T H A LJxx3x 2

with heatitudeXj), heongitudex2),E A R T Hmearth'sradius,andGE0thegeosynchronousaltitude.Recal lthat(z,z2)=

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(av,Q).]imilarscalingargumentsmaybeconstructedforothermatrixelements, althoughtheseexhibitamore complicated geomet-ricandtemporaldependence.1 1 OnceHiscalculated,thecovariancematrixofrandomerrorsRisconstructed.ecausew echoosetodefineQintermsof5ahseeEq.(3.15)],R willexhibitaslighttime-dependence,asthefollowingillustrates:or a30-microradian random error,the R-matrixreads

Rfoxer 0 ) at1 8sec, whereas (3.23)0 2.3X10-4jR (3. 0x1 (T 60.2 x1 (T 4 at9 8sec.3.24)Byspecifying aninitialguessforP[i.e.,MjseeEq.(2.25)],w emayrunthe filteralgorithm.W enextdescribesomenotionalresults.

N O T I O N A L R E S U L T S Launch PointUncertainty( L P U )Determiningtheuncertaintyassociatedwithmissilelaunch locationisausefulexampleoftheKaimanfiltertechnique'sutility.escribethe launchpositionwiththevector

w= Wjvw2 y (3.25)wherew ^ W j )reaunchatitudendongitude,espectively.Writing the aboveas

w =F x ,3.26)uInparticular,becausetheprevalenteffectsofloftanglevariationsar emanifestedlaterinthetrajectory,thematrixelementscorrespondingtothesevariationsvary bymorehanw ordersfmagnitudeve rheourseftheboost-phase.hus,measurementsoccurringearlyintheboost-phasear emuchlesssensitivetothesetypesof variationsthanmeasurementsoccurring nearmissileburnout.

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Estimation andPredictionof BallisticMissileTrajectories 33

where

itiseasy to se ethatF= 10 0 0 0 00 1 0 0 0 0)'

W= E trlT(W -w)(w-w) F PF T Pll Pl2 V*2i(3.27)

(3.28)r22jLaunch pointuncertaintyisthusdeterminedfrom a2x2submatrixcomposingP. T h eprobabilitythatw lieswithintheellipse

(w-w)TW_1(w-w)= 2isgivenby1 2

2 Jexp f o\r rdr=1 -ex p

(3.29)

(3.30)or0.393,0.865,and0.989 fo r^=1,2,and3,respectively. Considerthe notionaltrajectorydiscussed previously(Figure 3.7).A saninitial estimate of thecovariance,w eu se

Mfl00000040 0000000000001 (3.31)

corresponding(attheone-sigma level)to a1uncertaintyin launchlatitudeand longitude,20 uncertainty in launchheading,20-secun-certainty in launchtime,1-km uncertaintyin launchaltitude,anda1uncertaintyin loftangle.or asinglesatellitepositionedat0 lati-tude,75longitude,Figure3.10illustrates thet= 2launchpointun-12A .E.Brysonand Y.-C.Ho,1975,pp .310-311.

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34 Estimation andPrediction ofBallisticMissile Trajectories

RANDMB737-3.H)

4 00Time(sec) 80 Random error(m icroradians)10 010 0Figure3.10SensitivityofL P U (i= 2 )to RandomError(OneSensor)certainty as afunctionoftimefo r variousrandomerrors(100,30,and10microradians,respectively).13In the absenceofmeasurementerrors,six angleswoulduniquelyde -termineheix-dimensionaltateectorerestimating(assumingourtemplateisexact).inceeachmeasurementprovidestw oangles,onlythreemeasurementswouldberequiredtospecifythestate.Consequently,as isclearin allcasesabove,aprioriuncer-taintiesar ereducedmostrapidlyby thefirstfew measurementandataslowerpacethereafter.Nextconsiderasecondsatellitepositionedat0latitude,15longi-tude.rocessingitsmeasurementssequentiallywiththoseofthefirstsatellitesi.e.,stereoprocessing)maysignificantlyreducethelaunchpointuncertainty.igure3.11depictstheL PU fo rthiscase,derivedafterthelast measurement hasbeenmade. heL PU calcu-latedmonoscopically fromeachseparatesensorisalsoshown,indi-13W eassume< X h= 5avfor simplicity[seeEqs.(3.9)-(3.15)].

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RANDMfi737-3.)J5km 18.6sq km (15-E )

5km

14.3sqkm (75-E )4 .8sq km (stereo)


Figure 3.11LPUs(t= 2)for Tw oSensors with Random Errorseatinghowadifferentviewinggeometrymayleadtodifferentre-sults.14nall cases,a30-microradianrandomerrorisassumed.Figure3.12illustratesthe =2 launchpointuncertaintyasafunc-tionoftime forvarious randomerrors(100,30,and10microradians)inthecaseofstereoprocessing.T hesecondsensor providesmea-surementsatroughly 2,22,42,62,and82sec.) sexpectedonintu-itivegrounds,theL PU derivedafterthefinalmeasurementhasbeenmade scales roughly as the squareofthe random error.Finally,onsiderasymmetricexample,wherethesamemissileislaunchedfrom 35latitude,45longitudeheadingduenorth.fthesensorsrotateintheoppositedirectionrelativetoeachother,w ewouldexpectthesymmetryoftheproblemtomanifestitselfintheresults.A sFigure3.13illustrates,thisis indeedthecase. 14A shownater, perfectlysymmetricxampleesultsndenticalPUvaluescalculatedfrom each sensor.

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36 Estimation an dPredictionof Ballistic Missile Trajectories

RMDMR737-3.12

20 400Time(sec) Random error(m icroradians)Figure3.12Sensi t ivi tyofL P U (I=2 )to R a n do m Error(Two Sensors) RANDAOT737-3.13

T km

4 .6 sq km (stereo)

14.0sq km (15-E )

5km

14.0sq km (75-E )


Figure 3 . 1 3 L P U s(t=2)fo r a Symmetric Example

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Estimation andPrediction of BallisticMissile Trajectories 37

MissileLocationUncertainty( M L U )Determining theuncertaintyassociatedwithmissilelocationatanypointalongitstrajectoryisanotherusefulapplicationof thetech-nique.15Describetheinstantaneousmissile positionattimetbythevector

y(t) y2M y3(t)j(3.32)

wheretheelements(y1(y2,y3)arereferencedtoaCartesiancoordi-natesystemcentered atthecenteroftheearth.Wemaychoose they3-directionto intersectthepoles,andtheyi-y3plane to intersectGreenwich . )Althoughy(t)isanonlinearfunction ofthestatevector,w emay expandto linearorderaboutaninitialestimateof thestate[see Chapter T w o ,Eqs.(2.17)-(2.22)].n similar fashion,w efind- 3y (x-x)sG(x-x).

T hecovarianceofy(t)willthereforebegiven by0= E[(y-y)(y-y)T ]= G P G T,

(3.33)

(3.34)oncetheG-matrixisdetermined. (Thismaybeaccomplishednu-merically,usingaprocedure similarto thatusedin determiningH. )The probabilitythat y lies withintheellipsoid

(y-y)T0-1(y-y)= 2isgivenby1 6 Jexp r2dr=erf fl -L\ ex p

(3.35)

(3.36)or0.199,0.739,and0.971fo r1 = 1 ,2 ,and3,respectively.15

Themathematicalframework developedfor analyzingML U couldbeappliedmoregenerallytootheruncertaintiesforexample,missilevelocity(inthreedimensions),impactpoint(in tw odimensions). 16A .E. Bryson and Y.-C.Ho,1975.

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38 EstimationandPrediction ofBallistic MissileTrajectories

Figure3.14depictsth emissilelocationuncertainty asafunctionof timealongth etrajectoryin th ecaseof 100-,30-,an d10-microradianrandom errors,respectively. sillustrated,th euncertainty volumeincreasesm onotonically untilth elatterpartof th etrajectory,whenth eML U turnsover.17A sapointofreference,asphereof 62-km radiusencloses a volumeof roughly1 0 6km 3.) esultsfortw osensorsprocessed stereoscopicallyareshownin Figure3.15.Revisi tT i m e SensitivitiesIn general,decreasingth erevisit t imeallowsmoremeasurementstobe madeand,consequently,moreinformationtobe obtainedabout

RANDMW37-3.M

10 0

Random error(m icroradians)

2 00 30 000 Time(sec) 50 0 600 Figure3.14SensitivityofM L U (= 2 )to Random E r r o r(OneSensor)

17B yexaminingthetrajectorieswithperturbedlaunchtimes,altitude,andloft,on efinds fo rtheexampleathand thatthedeviationfrom thenominalbaselinetrajectorybeginsoecreasetareentryltitudefapproximately0,000t.h isffect,manifestedin thedecreasinguncertainty580se cintotheflight,isrelatedtobothth eatmosphericdegradationofthemissilevelocityuponreentryandourchoosingaminimum-energy trajectory toperturbabout.

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RANDMfl737-3.r5

2

30 000 Time(sec) Random error(microradians)Figure 3.15Sensitivity of ML U I= 2)to Random Error(TwoSensors) themissiletrajectory.igure3.16illustratestheL PU fo rvariousre-visittimes,spanningtherangeof2.5-40sec. tlatetimes,notethatthe L PU scalesroughly linearly withthenumberofmeasurements.Finally,Figure 3.17showsthe effectof revisittimeonmissile locationuncertainty. sisevident fromthisplot,anorder-of-magnitudere-ductioninrevisittimegeneratesmorethananorder-of-magnitudereductionin uncertaintyvolume.

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RAHDMR737-3.16100

E cr a.

0.1


2 0 J _ 400 Time(sec) 80 10 0Figure3.16Sensitivityof L P U {= 2 )toRevisitTime( T w o Sensors)RANDMfl737-3.T7

1E+04

100 2 00 30 000 Time(sec)

50 0

Seconds

60 0

Figure3.17Sensitivity ofM L U (t= 2 )to RevisitTime(TwoSensors)

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ChapterFourTHEEFFECTOFB I A S ERRORS

Whencorrelationtimes1areshorter thanthetimescalesof themea-surement sequence, w emayapproximate theerrorsas uncorrelated,whichw ehavedonethusfa rinourtreatmentofaKaimanfilterop-timizedfo rrandomerrors.nmanysituations,however,biaser-rorswhicharecorrelatedfrommeasurementtomeasurementdominatetheuncertainty.nsuchcases,onemustexplicitly accountfo rtheireffectsontheuncertaintyanalysis. Considerthelinearsystem

Zj-Z;=Hi(x-x)+Vi+bi,4.1)wherex ,;,H ;,andsareas before(see Chapter Two) ,andbj= bias error in measurementofZ ;(p-dimensional). (4.2)

W eassumethatexpectedvaluesofboth therandomandbiaserrorsvanishi.e.,(v,)=E(b ;)=0),buthatbiasermsxhibitorre-lationsfrom measurement to measurement:E(bibjT)*0.4.3)

Althoughsensorbiasesmaybeeliminated(tothe extentpossible)byrepeatedcalibration,someslowly varying sensor errorsmayremain.lrrhecorrelationtime,z,measuresroughlythemeantimebetweentw osuccessivemaxima(orminima)ofsomefluctuatingfunction,f(t).ssuch,xcharacterizestherateatwhich f(t)varies.eeF .Reif,Fundamentalsof StatisticalandThermalPhysics,N ew York:McGraw-Hill,1965,p. 561.

41

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42 Estimation andPredictionofBallistic MissileTrajectories

If thesearesmallenough,their effectsmaybeinsignificantor toler-able,althoughignoringtheminthefilterdesignanderroranalysisprocedurerunstheriskofproducingmisleadinguncertaintyesti-mates.ere,tw oviewpointsarepossible:1)theestimateof themeasurementerrorvarianceiscorrect,butitisnotrecognizedthatanelement ofitresultsfrombias,or(2)thevarianceestimateiscor-rect fo rthe uncorrelated measurementerrors,butan additionalbiaserrorispresent.nanycase,theeffectofbiasonafilteroptimizedforrandomerrorsalonecanbeexamined.However,ifthebiaserrorsarelargeenough,furthercalibrationorredesignof thefilter w hereinbiaseffectsaremodeledexplicitly maybedesirable.nwhatfollows,w eexaminethefollowingtw oquestions:Whateffectwilltheadditionalbiasterm(in Eq.4.1)haveontheexistingfilteroptimizedfo rrandomerrorsalone?owdoesthiscomparewithresultsobtainedusingafilter designedto accountoptimally fo r both random andbiaserrors? SUBOPTIMALT R E A T M E N T O F B I A S2T oaddressthefirstquestion,itisusefulto adoptthenomenclatureof linearsystemsanalysisasdevelopedattheendofChapterTwo .3A ttheithstageofthefilteringprocess,definethedifference betweenthestateofthesystemanditsbestestimate(aftertheithmeasure-ment)as[(Eq.(2.27)]

e;sxj-x. 4.4)T helinearsystem evolvesaccording to

ei+i A i+1ej+ Ki+ivi+i+K i+1bi+1,4.5)2Herew eexaminetheeffectof biaserrorsonanexistingfilteroptimizedfo rrandomerrors.hereis,consequently,no reasontoexpectou rresultsto beoptimal.or thisreason,aswellas fo rbrevity,in whatfollows w e referto thistreatmentas suboptimal.It is natural toask w h yonewouldbotherusing asuboptimalformulation.Theanswerishatfredesigninganxistingfilteroptimizedfo rrandomrrorslonesotdesirable,hesuboptimalapproachl lowsheffectsfbiasonhatfiltertoeexamined,albeit as anafterthought.3For imilarutmoreeneralreatmentfhisroblem,ee .riedland,"Treatmentof BiasinRecursiveFiltering,"IEEETransactionsonAutomaticControl,Vol.AC-14,N o. 4, A ug us t1969 ,p p .359-367.

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T he EffectofBias Errors 43

whereA ; =I-KjH;Kj =P ; Hj R j (4.6)

and;rstheerror covariancepredicted by the filterin theabsenceofbias[seeEq.(2.28)].xplicitly writing therandomandbiascontri-butionsto theerroras

w efindei+lei+l +ei+l

ei+ir-Aj+1e;K i+1vi+1ei+i -Ai+1ej+Ki+ibi+1

(4.7)

(4.8)T orecoverthenotionofestimatesbeforeandaftermeasurement(seeTable2.1),w especifyan initialguesse0rwhose square expectedvalueis the covarianceof the statebeforethe firstmeasurement;thatis,

MsE ' reo eo (4.9)Inthisway,recursive applicationof Eq.(4.8)yields

eirAieor+K1VIe3r=A3e2r+K3V3 (4.10)enr Anen-ir+K nvn

Proceedingsimilarly fo rthebiascontribution,onefinds44 Choosinggb=0isequivalenttorequiringthecovarianceof thestatevectorbeforethefirstmeasurement(i.e.,Mj)toresultentirely from randomerrors.incethesedrive thefilter,thischoiceis appropriate.

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44 Estimationan dPredictionof Ballistic Missile Trajectories

e Kxb ,e2b A 1 5+K2b2e3b= A 3e2b+K3b34.ll)en =A nen_ ;L + Knbn

Collecting terms fo rb ; b(a constant),onemaywriteenb= [ K n+ A nKn_!+ A nA n_iK n_ 2+ -+A nA n_iA n_2 A^Jb

(4.12)sothattheeffectofbiasontheerrorcovarianceofthestatewillbegivenby

P nb-Eele = nE[bbT] F nT (4.13)wherethetotalcovariance ofthestateafterthe nthmeasurementis

Pn Pnr+Pn\4.14)Thus ,specifying

B ; E(bibiT)= E(bbT)4.15)enablesonetoaccountfo r(constant)biaseffects.inally,notethat%alsotimeevolves linearly,as onemightexpect: * P i+ 1-A i+1* F i+ Kj+i (4.16)Inmanyapplications,thebiaswillnotbeconstant.ndeed,inourownformulationutilizingthemeasurementvector(z1(z2)=(av,Q),constantbiasrrorsassociatedwithav,a)ranslateintoime-dependenterrorsassociated withQ..nthiscase,however,w emaymodifytheformalismin astraightforwardway.et ei+ i _A i+1ei K i+1oi+1bi+1 (4.17)

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T heEffectof Bias Errors 45

where

ibi= , A,ih V (u \ (4.18)andg;isatime-dependentfactor.ThenEq.(4.16)issimply modifiedas

i=A+1^i+Ki+1a+1.4.19)Inthecaseofmultiplesensorsprocessedsequentially,w emayfur-thermodify theformalismtoallow fo rmultiplebiases.n thecaseoftw osensors,write

ei+ib=A i+1eib+K i+1o - i+1bi+1+Ki+iC i+iC i+1,4.20)whereqsthebiasoftheadditionalsensor.roceedingasabove,w edefineej bi+XjC;.4.21)

A s s u m ethemeasurementsareprocessedin analternating manner,withonesensormeasuringatoddvaluesofiandtheotheratevenvalues.or odd i ,the bias contributionswillobeywhereasfo reveni,

Xj AJXJ.JXj AjXj.x+ KiOj

(4 .22)

(4.23)

NotionalResultsF or the example discussedin ChapterThree,theeffectof biasontheexistingfilterisillustratedinFigure4.1,whererandomandbiaser -rorsarebothassumedtobe0microradians.helaunchpointuncertainlyellipses obtained ar elargerthanin Figure3.11 .

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46 EstimationandPrediction of Ballistic Missile Trajectories

RAND/WR737-*1 10km

47.5sq km (15-E)

-1 039.6sq km (75-E )17.2 sq km (stereo)

J--1030-microradianrandom an dbiaserrors

10km

Figure 4.1LPUs(t= 2)for Tw o Sensors with Random an d BiasErrors(Suboptimal Filter)Figure4.2illustratesthe timeevolutionof thelaunchpointestimatefo rthecaseof a 30-microradian randomerror and100-,30-,and10-microradianbiaserrors,respectively.nlikethecasewithrandomerrorsalone(Figure3.12),theL PU derivedafterthefinalmeasure-mentdoesnotscaleasthe squareoftheerror.Moreover,the impor-tanceofbiasisapparent inthelargedifference betweenthe30 -and100-microradiancases. hemissilelocation uncertaintyshowssimi-larqualitativebehavior(seeFigure4.3).Theeffectofvaryingrevisittimein thepresenceofrandomandbiaserrorsisillustratedin Figures4.4and 4.5,respectively.ncontrastto theunbiasedcase(Figure3.16),Figure4. 4indicatesthattheL PU de -rivedafterthefinalmeasurementisrelativelyinsensitivetorevisittimewhenboth30-microradian randomand biaserrorsarepresent.In effect,apoint isreachedin thefiltering sequencewhere additionalmeasurementscontainingunknown biaserrors provideinformationof limitedutility.h isunderscoresthepointthatahighrateof datacollectionwillnotreducethelaunchpointuncertainty significantlyunlessrandomerrorsdominatethemeasurementprocess.ndeed,

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TheEffect of Bias Errors 47

RANDMW37-4.2

E

DL

2 0 400Time(sec) 80 Biaserror(microradians) 100Figure 4.2Sensitivity ofL P U (= 2 )to BiasError(Two Sensors; Subopt imalFilter)RANDMH737~f.3

100 2 00 30 000 Time(sec)

500

Biaserror(microradians)10 0

600

Figure4.3Sensitivity ofM L U (=2)to Bias Error( T w oSensors;Subopt imalFilter)

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48 Estimation an dPrediction of Ballistic Missile Trajectories

RANDMR737-4.4 10 0

4 00Time(sec)Figure 4.4SensitivityofL P U ((= 2 )to Revisit Tim e(Two Sensors;Subopt imal Filter)RANDMH737-4.5

10 0 2 00 30 000 Time(sec)

50 0

Seconds

600

Figure 4.5SensitivityofM L U (I= 2)to R evisit T i m e(TwoSensors;Subopt imalFilter)

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The Effect of Bias Errors 49

statisticsalonedonot"beatdown"theeffectsof bias. lthough theinsensitivitytoevisittimeisnotapparentintheasefmissilelocationuncertaintyFigure4.5),hespreadinML U valueswithvaryingrevisittimeisreducedbyroughlyanorderofmagnituderelativeto theunbiased case. Inummary,w ehowlaunchpointndmissileocationuncer-taintiesatapogee(327se cintothetrajectory)in Tables4.1and4.2,respectively,orvaryingrevisittimesi.e.,umbersofmeasure-ments)andbiaserrors(treatedsuboptimally).otethatwhenthebiaserror islarge,increasing thenumberofmeasurementsca nleadto largeruncertainties.Althoughcounterintuitive,suchbehaviormightbeexpectedfromasuboptimal formulation,especiallywhenthebiaserrorsar elarge.nsuchatreatment,thefilterappliesgainsindeed,sometimeslargegains tothesystemby consideringrandomerrorsaloneseeEq. (2.29)]. saresult,whentheeffectsofbiasareexamined,theymaybelargebecausetheyareamplifiedbylargegains.fthefilterw as awarethatbiaserrorswerepresent,itcouldadjustthesegainsac -cordingly.Theseresultsunderscoretheimportanceofproperly accountingfo rbiasundersuchircumstances.ex tw eexaminenlternativetreatmentthatremediesinconsistenciesbyincorporatingthebiasterms intothefilterdirectly.

Table4. 1 L P U s(= 2 )fo r T w o SensorsProcessed in Stereo (Suboptimal Filter3,in km 2)

NumberofBiasError(nrad)

RevisitRate(sec) Measurements 0 1 0 30 1 00

40 5 10 .9 12 .4 23.7 150.720 1 0 4.8 6. 2 17 .2 142.71 0 20 2. 4 3. 8 15.1 143.45 40 1 .2 2. 7 14 .2 145.72.5 80 0. 6 2. 1 13 .9 147.9

a30-microradian random error.

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5 0 Estimation and Prediction of B a l l is t i c M i s s i le Trajectories

Table 4.2MLUs(I= 2)at Apogee for T w o SensorsProcessed in Stereo(SuboptimalFilter3; Equivalent Spherical Radii in km )

N u m b e rof BiasError(urad)

RevisitRate(sec) Measurements 0 10 30 10 040 5 30.5 31.0 33.8 45.720 10 22.4 23.4 28.5 49.710 20 15.8 17.1 22.2 38.05 40 11.2 13.0 17.6 29.52.5 80 8. 0 10.1 14.1 23.5a30-microradian random error.

O P T I M A L T R E A T M E N T O F B I A S W eareinterestedin redesigningafilter fo rthelinearsystem5

z-z= H(x-x)+ v+ b 4.1)wherexisasix-dimensionalvectorandthebiastermisassumedconstantbutunknown.nourpreviousonstruction,iaswastreatedasinherentlyunobservable,withit seffectsinsomesensemodeledas an afterthought.n practice,however,itmaybepossibletolearnaboutbiasthroughthemeasurementsequence,whichal-lowsthefiltertoadjustthegainoptimally,withconsiderationgiventobothrandom andbiaserrors.T otreatbiasasanobservable,w efirstincorporateitintothestatevector.6orasinglesensor,define5For simplicity,the index i i s suppressed in this section.6SeeB .Friedland,1 9 6 9 ,p . 3 6 0 .

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T h eEffectof Bias Errors 51

M r x 0 x 2 x 2*3 x 3x 4 _ x 4x 5 x 5x 6 x 6x 7 b i

VX8 j W(4.24)

wherex1throughx6ar edefinedas before[seeEq.(3.16)],andbj= biasonmeasurementzx b2= biasonmeasurementz2.4.25)Concurrentwiththisstatevector redefinition,redefinetheH-matrixas H = 10z jz jzlz1zxzx8xjx 2x 3x4x 5x63z 2 3z 2z2z2z2z2V3xi 3x, 3x o 3x , 3x ^ 3x fi (4.26)

Sincethebiasenterstheproblemasanexacdy linearterm,thereisnoneedto generateabiasestimatefo r the purpose ofTaylorexpan-sion[seeEq.(2.18)]. hus,Eq.(4.1)is rewritten as

z -z=H(H-E)+ v,so thattheform ofthefilterwithoutbias isregained.

(4.27)

A saresult,theframeworkofChapter ThreemaynowbeappliedtothebiasproblembysimplyaugmentingthetatevectorandH-matrixwithadditionalterms.fw ewishtomodeltime-dependenterrorsassociatedwiththechoice((z1;z2)=((av,Q),[seeEq.(4.18)],w emayincorporate appropriatefactorsintoH:

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52 EstimationandPredictionof BallisticMissile Trajectories

H =3 z jzxztztz1 dzldxl 3x 2x 3x4x 5x 6dz2 3z 2z2z :dx1 3x2x 3 3x , 3z 23x 5 9Z2 n3g3x 6 (4.28)

Finally,accountingfo rthebiaserrorsof multiplesensorsrequiresadditional dimensions.orthe caseofstereoscopicprocessing(with time-dependenterrorsinQ),Eqs.(4.24)and(4.28)generalizeas

r.

where

x2x 3x4x5x6bn b2ib12

vb22y

bn= biasonmeasurementzvsensor1b21 = biasonmeasurementz2,sensor1 b12= biasonmeasurementzvsensor 2b22= biasonmeasurementz2,sensor2,

\

a n d

H =3z x 3zj 3zj 3z x 3zj 3z 23x x 3x ? 3x 3 3x 4 3x5 3x fi3z2 3z 2 3z2 3z 2 3z2 3z 23x xx2x 3x4x5x 6

(4.29)

(4.30)

10 100 g! 0 g2

(4.31)

withthesubscripts1and2onthetime-dependentfactorsreferringto sensors1and2,respectively.

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54 Estimationan dPredictionof Ballistic Missile Trajectories

MNOMR737-4.7

4 00Time(sec) Biaserror(microradians)10 0Figure4.7Sensitivity ofL P U (t-2 )to Bias Error(Two Sensors;OptimalFilter)

BMiOMFl737-4.B

EA: o3_i2

10 0 2 00 30 000 Time(sec)

50 0

Biaserror(microradians)10 030 10

600

Figure 4.8Sensitivity ofM L U (t=2 )to BiasError(Two Sensors;OptimalFilter)

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The Effect of Bias Errors 55

shouldbe roughlyindependent of bias.hisisnotth ecasefortw osensorsusingth esuboptimalfilter(Figure4.3),buton ecanverifythat withonlyonesensorthisfilteralsoyieldscurvesthatcoalescelatein th etrajectory.A ssensorrevisit t imesarevaried,th eLPU calculatedoptimally(seeFigure4.9)againshowsbehaviorsimilartothatcalculatedsubopti-mally,in th ecasewhenrandom an dbiaserrorsarecomparable. nth e other hand,MLUsdemonstrate consistentlysmallervaluesin th eoptimal treatment(Figure4.10),an dalsoexhibita greatersensitivityto th erevisittime.Insummary,w eshow LPUsandMLUsatapogeeinTables4.3and4.4,respectively,forvaryingrevisittimesan dbiaserrors(treatedop -timally).nallcases,errorsarereducedasth enumber of measure-mentsincreases.naddition,valuesarecomparabletothoseob-tainedusingth esuboptimalapproach w hen thebiasissmallerthanor comparable to th erandom error(seeTables4.1an d4.2).

RANDMH737-f.9 10030-microradianrandom an dbiaserrors

2 0 4 00Time(sec)

80 100

Figure 4.9Sensitivity ofL P U (I= 2 )to Revisit Tim e(Two Sensors; OptimalFilter)

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56 Estimation andPrediction of Ballistic MissileTrajectories

1E+07 RMtDMR737-4.lt)11E+06 30-m icroradianrandom an dbiaserrors '1E+05 -1 E + 0 41E+03 S r1E+02 1E+011E+00 110 0 2 00 30 000 Time(sec) 500 Seconds402010 52.560 0Figure4.10Sensitivity ofM L U (i= 2 )to Revisit T i m e (Two Sensors;Optimal Filter)

Table4. 3L P U s(1=2) fo r T w o SensorsProcessedin Stereo(OptimalFil ter 1 ,inkm 2)

NumberofBiasError(urad)

RevisitRate(sec) Measurements 0 10 30 1 0040 5 10.9 12.4 23.3 141.520 10 4. 8 6. 2 16.8 134.310 20 2. 4 3. 8 14.5 132.05 40 1 .2 2. 7 13.4 130.82. 5 80 0.6 2. 1 12.8 130.2a30-microradianrandom error.

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The Effectof Bias Errors 57

Table4. 4M L U s(I=2)atApogeefor T w o SensorsProcessedinStereo

(OptimalFilter3;Equivalent SphericalRadiiinkm )Numberof

BiasError(nrad)RevisitRate(sec) Measurements 0 10 30 1 00 40 5 30.5 30.9 32.8 35.420 1 0 22.4 23.3 25.6 27.710 20 15.8 17.0 19 .3 21.1 5 40 11.2 12.8 15.0 16.72. 5 80 8. 0 9 .8 11 .8 13.5a30-microradian random error.Keep in mind,however,thatthe suboptimal treatmentof bias can goawry evenin instances when the randomandbiaserrorsar ecompa-rable. smentionedpreviously,if thefilter applieslargegains duringthe estimationsequence,resultsobtainedtreatingbiassuboptimallymaydiffermarkedlyfromthoseobtainedwithanoptimalformula-tion.nal lcases,though,theformerapproachwouldoverestimatethe error. Inconclusion,treatingbiasasanobservablequantityconsistentiyyieldssmalleruncertaintiesthanthoseobtainedtreating biasasanafterthought.ithintheconstraintsofthislimitedanalysis,then,ouruse oftheterms"optimal"and"suboptimal"isclearly justified.

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hapterFiveCONCLUDINGR E M A R K S A stheatermissiledefensesarefieldedat thedecade'send,satellitesensorswilllikelysupportvitalT MDbattlemanagement functions.Waging "informationwarfare"willrequireincreasinglysophisticatedC3Inetworksthatarecapableofpiecingtogetherthemultifariouspacketsofinformation required toeffectbattlespacedominance.nthisregard,timelytransmissionthroughoutthetheateriscentral.B u tsuccessfulbattlemanagementrequiresmorethanconnectivityalone:hequalityoftheinformationbeingtransmittedispara-mount. s w ehaveshown,Kaiman filteringofsensormeasurementscaninprincipleprovidesuchinformationintheT MDoperationalenvironment.hus ,ourprimary focushasbeenondescribing howto estimate theoperationalimplicationsof thistechnique.Table5.1summarizessomeoftheresultsfromChaptersThreeandF o u rfo ranotionalmissilelaunchagainstIsrael. sing thistable,it ispossibleto derive order-of-magnitude estimatesfo rlaunch pointun-certaintiesinavarietyof situationsnotdescribedexplicitlyinthisreport.orexample,halvingtheT B Mburntimeto50se cwouldhalvethenumberofmeasurementsobtained,roughly equivalentto doublingthesensor revisittime.ncludinganopaqueclouddeckat10km wouldeliminatemeasurementsobtainedduringthefirst42 secofflight,againroughlyequivalenttodoublingtherevisittime.Thus ,a sensorwitha20-secrevisittimeand 30-microradian randomerrorswouldyieldanL PU ofabout1 1km 2( . 2 ,nobias)inboththesesituations.In short,Table5.1isausefulguideto theclass ofnumbers onewouldobtaininmanysituationsofinteresttoTMD.roperanalysisofa

59

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Table5. 1 LPUs(t-2)forTwo Sensors ProcessedinStereo3

(inkm2)

Number ofMeasurements

BiasError(urad)RevisitRate(seconds)

RandomError(lirads) 0 1 0 30 1 00

40 5 1 030

1 00 1 .5

10 .9 102.7

3.012 .4104.2

13.723.3

115.9131 . 2141 . 5241.3

20 10 1030 1 00 0. 64.8

47.22.06. 2

48.512.716.859.2

130 .1 134.3177.4

1 0 20 1030 1 00 0. 32.4

24.21 .7 3. 8

25.512.414.536.0

129 . 6132.0153.6

5 40 1030

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