2015年美赛O奖论文B题Problem_B_32879(2)

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planewithoutpowercanbede?ned,seeninEquation2.Sincetheplaneisassumedtolosesignalduringcruise,hwillbeequaltothecruisealtitude.A747hasanimpressivetheoreticalmaximumgliderangeofnearly113miles.

Figure2:DerivationofMaximumGlideRange.

rglide=hcot(θ)

(2)

Ourinitialprobabilitydensityfunctionisbasedonthelikelihoodofeverypossiblecrashtrajectory.Eachcrashtrajectoryisde?nedbyavalueofθandφ.Bothofthesevariablesarerandomandindependent,asθandφcharacterizedi?erentaspectsoftherandom?ightpathexperienceduponthelossofsignal.TheregionRoftheprobabilityfunctionisboundedbythecalculatedmaximumunpoweredrangeoftheaircraft,sweptinalldirectionsfromthepointoflostsignal.Becausetheyareindependent,θandφareassignedtheirownprobabilityfunctions.

Aθvalueveryclosetotheminimumglideangleθminrepresentsapoweroutageintheplaneandthepilot’sdecisiontoglideatthatsetangle,maximizingdistanceandminimizingchanceofdamageuponimpactwiththewater.θvaluescloserto90?signifycatastrophicfailures,suchassuddenlossofliftduetoastalloranexplosion,bothofwhichwouldcauserapiddescent.Theθprobabilitydistributionismodeledasabimodalnormaldistribution,withweightingtowardtheextremecasesofanoptimalglideandacatastrophicfailure.Thedistributionitselfisthesumoftwomutuallyexclusivenormaldistributions.Theweightingofeachisshownbelowasaratiooftheprobabilityofasuddencrash,pcrash,tothatofaglide,pglide=1?pcrash.

θ?θmax)2pglide?1pcrash?1(θ?θ(σmin)2

2σ212e+√e(3)f(θ)=pcrash?f(θcrash)+pglide?f(θglide)=√2πσ12πσ2

Thequantitiesσ1andσ2correspondtothestandarddeviationsofthecrashandglide

distributions,respectively.Similarly,themeanofthecrashangledistributionisθmax=90?,whilethemeanoftheglideangledistributionisθmin.

φiseitherarandomvalue,ifthelossofpowercausesalossofcontrol,orapilot-dependentvaluethatdescribesthedegreeofturningdeemedsuitableforaccidentmitigation.However,sincenothingisknownaboutthedecisionsofthepilotorthecontrollabilityoftheaircraftatthistime,φwillbevariedaccordingtoanormaldistributionwithameanofzero,correspondingtothemostlikelyscenariothattheplanedoesnotalteritscourse.

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1φ21

√e?2(σ3)(4)σ32πThestandarddeviationoftheφprobabilitydistributionisdenotedasσ3,withameanofzeroexplainedpreviously.Thestandarddeviationsofboththeφandθdistributionswerechosenarbitrarilytoapproximatealikelyscenario.Thesemaybeadjustedbasedoncrashstatisticsandstandardsofthemissingaircraft.Weletσ1=15?,σ2=20?andσ3=30?.Sincethesetwovariablesareindependentandrandom,theirprobabilityfunctionscanbemultipliedtoobtainaprobabilityfunctionthatspanstheentirepossiblesearchspace,in(θ,φ)coordinates.

f(φ)=

p(θ,φ)=f(θ)f(φ)(5)

Thisprobabilityin(θ,φ)spacedoesrepresenttheprobabilitiesweareinterestedin;however,itismoremeaningfultosearcherstomaptheseprobabilitiestoaCartesian(x,y)coordinatesystem.Thesetransformationsaregivenbythefollowingequations,whereρisthecharacteristicturningradiusofthelostplaneinaφ-radianturn.Fromtheseequations,theprobabilitydistributioncannowbetransformedfromp(θ,φ)?p(x,y).

x=

????

2ρ2(1?cos(φ))cos(

π?φ

))+(hcot(θ)?|ρφ|)sin(φ)2

(6)

π?φ

y=?cos(φ))sin())+(hcot(θ)?|ρφ|)cos(φ)(7)

2

Equations6and7applytoapreciseconversionofθandφtoCartesiancoordinates,accountingforelevationlostbothduringastraightglideandduringanyinitialturnthattheplanemayhavemade.However,theseequationsmaybesimpli?edsuchthattheturnismadeinstantaneouslywithnoelevationloss,andtheplaneisthenfreetoglideatanyanglewithinthepossiblevalues.Thissimpli?cationoftheaircrafttrajectoryisallowableduetotheinsigni?cantlossofaltitudeduringtheplane’sturnwithrespecttothefullcruisingaltitude.Usingthissimpli?cationandsubstitutingρ=0intotheaboveequations,theinitialprobabilitydistributionoftheplane’slandinglocationismappedintoCartesianspace.Figure3belowdisplaystheinitialprobabilitydistributionforourexampleaircraft,theBoeing747-400,in(x,y)space.Thereisaspikeattheorigin,correspondingtotheprob-abilityofthesuddencrashcase,andamoregradualincreaseintheinitialdirectionoftheplaneuntilthemaximumgliderangeisreached,atwhichpointtheprobabilitiesdecreasetozero.

2ρ2(1

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Figure3:InitialProbabilityDistributionofPlaneLocation.

2.2Bayes’Theorem

Aftertheinitialprobabilityhasbeendetermined,wemustmodelhowthisprobabilitydis-tributionisa?ectedbythesearch.Toaccomplishthis,weseparatetheinformationknownbeforethesearchisconductedfrominformationgatheredduringthesearch.Bayes’Theoremwillbeusedtoderiveageneralexpressionfortheprobabilityof?ndingthewreckage.Bayes’theoremstatedmathematicallyis

P(A|B)=

P(B|A)P(A)

P(B)

(8)

foreventsAandB.P(A)andP(B)aretheprobabilitiesofAandB,whileP(A|B)istheprobabilityofAgiventhatBistrue.Inourcase,eventAis?ndingtheplaneandeventBistheplanebeinginthelocationwearesearching.ThevariableqwillrepresentP(A|B)infurtherequations[6].

Thistheoremcanberewrittentomodelthemannerinwhichsearchinformationa?ectstheprobabilitydistribution.Ifalocationissearchedandtheplaneisnotfoundwithinthatregion,thenewprobabilityof?ndingtheplaneinthatsearchareashouldnotequalzero.Thisisduetothefactthatthereremainsthepossibilityoftheplanebeingtherewhilenotbeingseen.Bayes’theoremcanberewrittentomodelexactlyhowtheprobabilitydistributionshouldchangeafteralocationissearchedandnothingisfound.Forthegridpointthatwassearched,

p??=

p(1?q)

(1?p)+p(1?q)

(9)

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wherep??istheposteriorprobability;inourcase,thisisthelikelihoodof?ndingtheplaneinthefutureafterhavingnotfounditpreviouslyinthesamelocation.Similarly,pisthepriorprobabilityof?ndingtheplaneatanygivenlocation.Giventhispoint,thequantityp(1?q)representsthelikelihoodofthelocationcontainingtheaircraftandtheaircraftnotbeingfound,and(1?p)isthelikelihoodthattheaircraftisnotatthelocation.Thisformulaappliestolocationsthataresearched,reducingtheposteriorprobability,p??[7].

Afteralocationischeckedandtheaircraftisnotfound,thelikelihoodoftheaircraftbeinginadi?erentlocationmustbeincreased.Torevisetheseprobabilities,weusethefollowingformula:

r

(10)

1?pq

r??andraretheposteriorandpriorprobabilitiesrespectively,andareanalogoustop??andpinlocationsthatweresearched[7].

Theseformulaeallowustocontinuallymodelnewprobabilitydistributionsbasedonnewdataaboutsearchesthathavealreadytakenplace.Theyalsorenormalizetheprobabilitydistributionsothatateverytime,theplanehasa100%chanceofbeinginthefullsearcharea.

r??=

2.3PosteriorProbability:SearchPaths

Havingalreadysetupthetheoryformodelinganinitialprobabilitydistributionandhowthatdistributionwillchangeovertime,wenowaddressthelikelihoodthattheplaneisdetectedgiventhatitisatalocationthatisbeingsearched.Thisprobability,q,willbede?nedasafunctionofthreevariables:thelateralsearchrange,thetotaldistancetraveledbythesearchvehicle,andtheareathatboundsagivenlocation.LetthelateralsearchrangebeW,whichcanvarywiththeelectronicsbeingusedandthevisibilityintheregionofinterest.Thetotaldistancethatthesearchaircrafttravelswhilesearching,z,cangreatlya?ectthesuccessofthesearch.Inthesamevein,A,theareacontainingthegridsquaresalsoa?ectsthee?ciencywithwhichthesearchisconducted.

Tovisualizethesearchpath,thefollowing?guredepictsasamplepath:

Figure4:Visualde?nitionofW.

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Toderiveanexpressionfortheprobabilityof?ndingtheaircraftinthisregionA,we?rstneedtomakemoresimplifyingassumptionsaboutthesearchpath.First,thetargetdistributionmustbeuniformovertherectangle.Second,disjointsectionsoftrackmustbeuniformlyandindependentlydistributedintherectangle.Lastly,noe?ortwillfalloutsidethissearchregion.Compliancewiththesethreeassumptionsconstitutesarandomsearch[8].

Nowconsidersomeincrementalstep,h,inthistrack.Letg(h)beafunctionthatdescribestheprobabilityof?ndingthetargetintheincrementhgiventhefailureto?nditpreviously.Therectanglesweptoutinthisnewincrementalstepisshownbelow:

Figure5:Visualde?nitionofh.

Itfollowsthat

Wh

(11)

A

becausethegreaterthewidthanddistanceofthesearch,thehighertheprobabilityofsuccess,butwhentheareaisincreased,thisprobabilitydecreases.

To?ndtheexpressionwewantforq,wewillfollowaderivationoutlinedbyStone[8].Letb(z)betheprobabilitythatthetargetisdetectedaftertravelingsomelengthz.Then,byBayes’Theorem,[1?b(z)][g(h)]isthechanceoffailingtodetecttheobjectafterlengthzbutofsucceedinginthenextsteph.So,

g(h)=

b(z+h)=b(z)+[1?b(z)]

with

b(z+h)?b(z)W

=[1?b(z)](13)

h→∞hA

Theabovedi?erentialequationissimplyanon-homogeneous,lineardi?erentialequationwithconstantcoe?cientsforb(0)=0.Itssolutionisthen

b??(z)=lim

b(z)=1?e?

10

zWAWA

(12)

(14)

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！

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