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Figure9:ProbabilityDensityDistributionaftera5-daysearch.
Theoptimizationisclear;?vedistinctsquareregionswereremovedfromtheprobabilitydistributionfunctionwheretheprobabilitieswithineachassociatedareaweremaximized.Thecumulativeprobabilityofsuccessisalsoshownbelow:
0.8Single Plane Simple Square Search ModelCumulative Probability of SuccessDaily Probability of Success0.70.60.5Probability0.40.30.20.1002468Search Day101214161820Figure10:CumulativeProbabilityofSuccessaftera20-daysearch.
Thisstrengthofthismodelliesinitssimplicityandcomputationale?ciency;however,asthesearchareaisconstrainedtoasquareshapeitdoesnotnecessarilyoptimizetheparallel-sweepmethod.
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3.4OptimizedRectangleSearchModel
Arectangularparallelsweepsearchpatternissimplyageneralizationofthesquarepatterndetailedabove.Theonlychangeistherelationshipbetweenthesearchdimensionsandtheusablerange.Thearea,whichwaspreviouslycon?nedtoasquare,cannowbemadeupbyanyorderedpairof(length,width)forA≥length?width.Themoregeneralizedrelationshipbetweensearchsizeandusablerangeisshownhere:
A=Wz
(22)
Fromthisarea,theoptimizationcodealsotestedthetotalprobabilityofsuccessforallpossiblecombinationsof(length,width)ateachgridpoint.Aconstraintinthisprocesswastoensurethatthecombinations(length,width)wereintegermultiplesofthegriddimensions.After?vedaysofsingleplanesearching,theprobabilitydistributionfunctionisshownbelow:
Figure11:ProbabilityDensityDistributionaftera5-daysearch.
Thisdistributionresemblesthatofthe?vedaysquaresearch,butitleaveslessgaps,clearlyshowingthatarectangularsearchpathisgenerallymoree?ectivethanasquaresearchpath.Thecumulativeprobabilityofsuccessisalsoshownbelow:
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Single Plane Rectangular Search ModelCumulative Probability of SuccessDaily Probability of SuccessPage18of35
0.80.70.60.5Probability0.40.30.20.1002468Search Day101214161820Figure12:CumulativeProbabilityofSuccessaftera20-daysearch.
Thissearchpathmodeldoesanexcellentjobofoptimization.Thisoptimizationcomesatacosthowever,asthemodelismuchmorecomputationallyintensivethanthesimplesquaremodel.Onetime-stepofoptimizationtakesabout20-30secondstorun,comparedwithlessthanasecondforthesquaremodel.
3.5SpiralSquareSearchModel
Thenextsearchmodelisanexpandingsquarespiral.Thefollowing?guredescribesthe“spiral”shapedsearchpath:
Figure13:“SpiralSweep”throughasearchsquare[12].
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Thederivationoftherelationshipbetweenthesearchpatternareaandthearclengthtraveledinsideismoreinvolvedthanthatofeitherthesimplesquareoroptimizedrectangle.Considernowthefollowing?gure:
Figure14:Symmetryofthe“SpiralSweep”.
Thesymmetryofthepathaboutthediagonalsimpli?estheapproachtothisderivation.Eachsetoftwoturnsinthesearchis2Wlongerthantheprevious.ThesesetsoftwoturnsbeginfromthecenterasthearrowsshowinFigure14.Additionally,thespiralsearchpatternhasbeendesignedtoconsistentlyendatthebottomrightcornerofthesearchpattern.Extraallowabledistanceintheoptimizationofthesquareregiontobesearchedmaybeinterpretedasasafetyfactorforthefuelconsumptionofthesearchplane.To?ndthetotallengthofthesearch,thesesegmentscanbesummedasnecessary:
z=(W+W)+(2W+2W)+···+((l?1)W+(l?1)W)+(l?1)W
l?1??
=2W(i)+(l?1)W
i=1
(23)
l=(l+1)(Wl?1)+√2??1√
→l=(?2+18?16z))(24)
4
Thesimpli?cationofW=1hasbeenmadeinEquation24tore?ectourmodel.Theterm(l?1)Waccountsfortheeventofthepathendingatthebottomrightofthesquarewithoutextendingthesidelengthofthesquare.Asthesesidelengthsareadded,apatterndevelopswithinthesumthatcanbereducedtotheaboveequation,withthepreviouslyexplained(l?1)Wtermfactoredin.The?nalexpressionofpathlengthzasafunctionofsquaresidelengthlwastheninvertedsuchthatanoptimallforamaximumpossiblezcouldbeobtained.
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Usingthisrelationshipandthesameoptimizationprocessdetailedearlier,theoptimalspiralsearchover?vedaysandthecumulativeprobabilityofsuccessareshownbelow:
Figure15:ProbabilityDensityDistributionaftera5-daysearch.
Single Plane Square Spriral Search ModelCumulative Probability of SuccessDaily Probability of Success0.80.70.60.5Probability0.40.30.20.1002468Search Day101214161820Figure16:CumulativeProbabilityofSuccessaftera20-daysearch.
Thismodel’sstrengthisinitse?ciency.However,asitislessspatiallye?cientthantheparallel-sweepsquarepath,itwillneverperformbetterthantheparallel-sweep.
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