2015MathematicalContestinModeling(MCM)SummarySheet
IntotheVoid:AProbabalisticApproachtotheSearchforMissingAircraft
Inrecentyears,thedisappearanceofmajorcommercialaircraftoveropenoceanhasledtoexpensiveinternationalsearche?orts.Thesesearchesrequirethee?cientallocationofresourcesandtimeinorderto?ndsurvivorsofthecrashandtheairplaneitself.
Wedevelopagenericprobabilisticmodeltonotonlypredictthelocationofthedownedaircraft,buttoalsoaidinoptimizingthesearchinatimee?ectivemanner.Thismodelassumesthatatthemomentoflostsignal,theplaneexperiencesafailureandisnolongerpowered.Speci?cally,weaccomplishthefollowing:
?InitialProbabilityDistribution:Wecreateapriorprobabilitydensityfunctiontomodelthepotentiallocationsofthemissingplane.Thisdistributionisbasedsolelyontheknowledgethatwehaveabouttheplaneatthetimeoflostcontact:itslocation,bearing,cruisealtitude,andlift-to-dragratio.
?SearchPatterns:Weimplementfourindependentsearchpatternsanddevelopamethodtomeasuretheire?ectivenessbasedonthetotalprobabilityof?ndingtheaircraft.Weconstructanoptimizationalgorithmtodeterminethemoste?ectivemeansofconductingeachsearch.
?DynamicProbabilityModel:WeemployBayesianInferencetocontinuouslyadjusttheprobabilitiesofourdistributionasinformationfromthesearchiscollectedandprocessed.Thisallowsustocreateaposteriorprobabilitydistributionwhichre?nesthedatautilizedinsubsequentsearches.
?Versatility:Weexplorevariationsofcrashandsearchscenariosthatrealisticallysimulateactualincidents.Thisadaptabilityisachievedthroughtheincorporationofadjustableinputparametersthatre?ectuniquecircumstances.
Ultimately,ourmodeldemonstratesthataneasily-packedandspacially-e?cientpattern,suchastherectangularparallel-sweep,moste?cientlymaximizestheprobabilityof?ndingalostaircraftovertime.
也许大学四年，我们会一直在迷茫中度过，因为生活总是难以言说。赛氪APP与您相伴！
1
Team#32879Page2of35
Contents
1Introduction
1.1Overview......................................1.2Nomenclature...................................1.3SimplifyingAssumptions.............................2ModelTheory
2.1PriorProbability:RandomDescent.......................2.2Bayes’Theorem..................................2.3PosteriorProbability:SearchPaths.......................2.4OptimizationCriteria........
.......................3ModelImplementationandResults
3.1PriorProbabilityModel:ADiscreteGrid....................3.2GeneralSearchModelMethods.........................3.3SimpleSquareSearchModel...........................3.4OptimizedRectangleSearchModel.......................3.5SpiralSquareSearchModel...........................3.6OctagonalSectorSearchModel.........................3.7ModelVariationandComparison.......
.................3.7.1SinglePlaneSearchModelComparison.................3.7.2FivePlaneModelComparison......................3.7.3HighLikelihoodofStall.........................3.7.4ShortRangeSearchAircraft.......................3.7.5ComparisonofVariedSearchPatterns.................3.7.6TheE?ectivenessParameter......
.................
4FinalRemarks
4.1StrengthsandWeaknesses............................4.2FutureModelDevelopment...........................4.3Conclusions....................................
也许大学四年，我们会一直在迷茫中度过，因为生活总是难以言说。赛氪APP与您相伴！
2
3334558911131314151718212324252527282932323233
Team#32879Page3of35
1
1.1
Introduction
Overview
Sincethedawnoftheaviationage,crashesandotherincidentshavehelpedshapedthetechnologyintheaviationindustry.Notonlyhavetheyguidedthedesignofairframes,propulsionsystems,wings,andmanyothercomponentsofanaircraft,theyhavealsoheavilyin?uencedthetechnologyoftheentireindustry,includingthatofsearchandrescue.Inthelastdecade,worldwideattentionhasbeendirectedtowardsthesearchformissingaircraft,especiallyoverwater,duetotwomajorincidents:
?June,2009:AirFranceFlightFlight447(AirbusA-330)wentmissingovertheAtlanticOceanwithoutatrace.Ittook?vedaysforsearchersto?ndanysignsofwreckageandnearlytwoyearsforthe?ightdatarecorderstoberecovered[1].
?March,2014:MalaysianAirlinesFlight370(Boeing777)disappearedovertheSouthChinaSea.Anearlyyear-longinternationalsearche?orthasstillfoundnotraceoftheaircraft[2].
Inbothoftheabovecases,allcrewandpassengerswerelost(orareassumedtobelost),andmanymillionsofdollarswerespentinthesearche?ort.Thee?cientsearchandrecoveryofaircraftinthefuturecouldsavelivesandmoney.Inthisreport,wewilldetailaseriesofgenericmathematicalmodelsthatdescribeandoptimizethesearchforamissingaircraft.
1.2Nomenclature
Description
Areaofthesearchgrid
CruisealtitudeformissingaircraftLift-to-dragratioformissingaircraft
Numberofsearchpassesinasearchregion
ProbabilityofagridpointcontainingtheaircraftPosteriorprobabilityforasearchedlocation
ProbabilityofdetectingtheaircraftgivenitiswithinthesearchregionProbabilityofagridpointcontainingtheaircraft,wherenosearchisconducted
r??Posteriorprobabilityforanun-searchedlocationREntiresearchregion
sSquareregionsidelengthWLateralsearchrange
xEast-WestdistancefromthepointoflosingcontactyNorth-SouthdistancefromthepointoflosingcontactzDistancetraveledbythesearchplanewithinasearcharea
Wewillbeginbyde?ningalistofthenomenclatureusedinthisreport:Abbreviation
AhL/Dnpp??qr
也许大学四年，我们会一直在迷茫中度过，因为生活总是难以言说。赛氪APP与您相伴！
3
Team#32879
AbbreviationDescription
αParameterdescribingtheeaseof?ndingthemissingplaneβParameterdescribingthee?ectivenessofasearchplaneλE?ectivenessparameterφAngleofchangeinbearingσStandarddeviationθGlideangle
Page4of35
1.3SimplifyingAssumptions
Theaccuracyofourmodelsrelyoncertainkey,simplifyingassumptions.Theseassumptionsarelistedbelow:
?Theplaneispreciselytrackeduntilthemomentthatcontactislost.Atthatpoint,itisnolongerpowered(i.e.theenginesprovidenothrustandnosignalsaretransmitted).?Theplanelostcontactduringthecruisephaseofits?ight.
?Thepilotcanmakeasingleturnofnomorethan180?ineitherdirectionimmediatelyafterlosingcontact.Thisturnisassumedtooccurinstantaneously,asthetheoreticalrangelostduringthisturnisinsigni?cant.?Thereisnowind.
?Therearenooceancurrents.
?Theplane/debriswill?oatinde?nitely.
?Theentiresearchareaiswater(i.e.thesearchareadoesnotextendontoland).?Thesearchplanesonlysearchintheirsearcharea.Althoughtheir?ightfromtherunwaytotheirspeci?edsearchareamaybeoverothersearchareas,theplaneisassumedtonotbesearchingduringthistime.?Thesearchplanescanmakeinstantaneousturns.
?Thereisnolocalcurvatureoftheearth–thesearchareaisaperfectly?at,two-dimensionalsurface.
?Onagivensearchday,thereare12hoursofdaylightduringwhichasearchaircraftcanbe?ying.
Therearealsoseveralparametersofthesearchthatwerede?nedarbitrarilyinordertopresentconsistentresultsinthisreport.Theseparameters,however,canbeeasilyvariedtoaccommodateaspeci?ccaseofamissingaircraft:?Theplaneis?yingdueNorthwhenitlosescontact.
?Arunway,fromwhichsearchandrescuee?ortscanbebased,liesexactly400milesSouthofthepointoflostsignal.
也许大学四年，我们会一直在迷茫中度过，因为生活总是难以言说。赛氪APP与您相伴！
4
Team#32879Page5of35
2
2.1
ModelTheory
PriorProbability:RandomDescent
First,wemodelthepotentiallocationsforthemissingaircraftwiththeirassociatedproba-bilities.Weassumethattheplaneisnotpoweredafterthelossofsignal,sotheinformationonwhichtobasethesearchislimitedtothelastknownpositionoftheaircraft,thedirectiontheaircraftwastraveling,andthetypeofaircraftthatismissing.Thislastknownpositionwillbeusedtode?netheoriginofaregionRinwhichtheplanecouldbelocated.RegionRwillbede?nedintheCartesiancoordinatesystem,withNorthinthepositiveydirectionandEastinthepositivexdirection.Therearetwopropertiesofthelostaircraftthatareusefulindeterminingwheretheaircraftmaybelocated:lift-to-dragratio(L/D)andaltitude(h).Wewillde?neθastheglideangleoftheplanebelowthehorizontalandφastheaircraft’spossiblechangeinbearingwithrespecttoitsinitialbearing.ThesequantitiesareillustratedinFigures1a&1bbelow:
(a)Visualde?nitionofθ.
(b)Visualde?nitionofφ.
Theminimumvaluefortheunpoweredglideslopeangleθisde?nedbyL/Doftheaircraft[3]:
????
1
θmin=tan?1(1)
L/DAcommonplaneusedfortrans-ocean,longdistance?ightistheBoeing747,having?ownmorethan42billionnauticalmilesinitslifetime[4].ForaBoeing747-400,themostcommonvarietyofthe747,thelift-to-dragratiois17andthecruisealtitudeis35000ft[5].Duetoitsfrequencyoftraveloveroceans,the747-400willbethe?rstaircraftconsideredasthemissingaircraftinthisreport.Itisimportanttonotethattheprobabilisticmodeldescribedlaterisgeneralandcanthereforebeappliedtoanymissingaircraft;thevaluesofL/Dandcruisealtitudesimplyneedtobechangedinthemodel.FromEquation1,theminimumglideslopeanglefora747isabout3.37?.Bygeometricallyanalyzingtheglideangle,asshowninFigure2below,anexpressionde?ningthemaximumrange,rmax,ofthe
也许大学四年，我们会一直在迷茫中度过，因为生活总是难以言说。赛氪APP与您相伴！
5