Team#32879Page6of35
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.
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！
6
Team#32879Page7of35
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
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！
7
Team#32879Page8of35
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)
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！
8
Team#32879Page9of35
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.
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！
9
Team#32879Page10of35
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！