1、2015 Mathematical Contest in Modeling (MCM) Summary SheetInto the Void: A Probabalistic Approachto the Search for Missing AircraftIn recent years, the disappearance of major commercial aircraft over open ocean has ledto expensive international search e orts. These searches require the e cient alloca
2、tion ofresources and time in order to nd survivors of the crash and the airplane itself.We develop a generic probabilistic model to not only predict the location of the downedaircraft, but to also aid in optimizing the search in a time e ective manner. This modelassumes that at the moment of lost si
3、gnal, the plane experiences a failure and is no longerpowered. Speci cally, we accomplish the following:Initial Probability Distribution: We create a prior probability density function tomodel the potential locations of the missing plane. This distribution is based solely onthe knowledge that we hav
4、e about the plane at the time of lost contact: its location,bearing, cruise altitude, and lift-to-drag ratio.Search Patterns: We implement four independent search patterns and develop amethod to measure their e ectiveness based on the total probability of nding theaircraft. We construct an optimizat
5、ion algorithm to determine the most e ective meansof conducting each search.Dynamic Probability Model: We employ Bayesian Inference to continuously adjustthe probabilities of our distribution as information from the search is collected andprocessed. This allows us to create a posterior probability d
6、istribution which re nesthe data utilized in subsequent searches.Versatility: We explore variations of crash and search scenarios that realisticallysimulate actual incidents. This adaptability is achieved through the incorporation ofadjustable input parameters that re ect unique circumstances.Ultima
7、tely, our model demonstrates that an easily-packed and spacially-e cient pattern,such as the rectangular parallel-sweep, most e ciently maximizes the probability of ndinga lost aircraft over time.1Team # 32879 Page 2 of 35Contents1 Introduction 31.1 Overview . . . . . . . . . . . . . . . . . . . . .
8、 . . . . . . . . . . . . . . . . . 31.2 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Simplifying Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Model Theory 52.1 Prior Probability: Random Descent . . . . . . . . . . . . . . . . . .
9、 . . . . . 52.2 Bayes Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Posterior Probability: Search Paths . . . . . . . . . . . . . . . . . . . . . . . 92.4 Optimization Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Model Implementation a
10、nd Results 133.1 Prior Probability Model: A Discrete Grid . . . . . . . . . . . . . . . . . . . . 133.2 General Search Model Methods . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Simple Square Search Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 Optimized Rectangle Sear
11、ch Model . . . . . . . . . . . . . . . . . . . . . . . 173.5 Spiral Square Search Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.6 Octagonal Sector Search Model . . . . . . . . . . . . . . . . . . . . . . . . . 213.7 Model Variation and Comparison . . . . . . . . . . . . . . . . . .
12、 . . . . . . 233.7.1 Single Plane Search Model Comparison . . . . . . . . . . . . . . . . . 243.7.2 Five Plane Model Comparison . . . . . . . . . . . . . . . . . . . . . . 253.7.3 High Likelihood of Stall . . . . . . . . . . . . . . . . . . . . . . . . . 253.7.4 Short Range Search Aircraft . . . . .
13、 . . . . . . . . . . . . . . . . . . 273.7.5 Comparison of Varied Search Patterns . . . . . . . . . . . . . . . . . 283.7.6 The E ectiveness Parameter . . . . . . . . . . . . . . . . . . . . . . . 294 Final Remarks 324.1 Strengths and Weaknesses . . . . . . . . . . . . . . . . . . . . . . . . . . .
14、. 324.2 Future Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332Team # 32879 Page 3 of 351 Introduction1.1 OverviewSince the dawn of the aviation age, crashes and other incidents have
15、helped shaped thetechnology in the aviation industry. Not only have they guided the design of airframes,propulsion systems, wings, and many other components of an aircraft, they have also heavilyin uenced the technology of the entire industry, including that of search and rescue. In thelast decade,
16、worldwide attention has been directed towards the search for missing aircraft,especially over water, due to two major incidents:June, 2009: Air France Flight Flight 447 (Airbus A-330) went missing over the AtlanticOcean without a trace. It took ve days for searchers to nd any signs of wreckageand ne
17、arly two years for the ight data recorders to be recovered1.March, 2014: Malaysian Airlines Flight 370 (Boeing 777) disappeared over the SouthChina Sea. A nearly year-long international search e ort has still found no trace ofthe aircraft2.In both of the above cases, all crew and passengers were los
18、t (or are assumed to be lost), andmany millions of dollars were spent in the search e ort. The e cient search and recoveryof aircraft in the future could save lives and money. In this report, we will detail a series ofgeneric mathematical models that describe and optimize the search for a missing ai
19、rcraft.1.2 NomenclatureWe will begin by de ning a list of the nomenclature used in this report:Abbreviation DescriptionA Area of the search gridh Cruise altitude for missing aircraftL=D Lift-to-drag ratio for missing aircraftn Number of search passes in a search regionp Probability of a grid point c
20、ontaining the aircraftp0 Posterior probability for a searched locationq Probability of detecting the aircraft given it is within the search regionr Probability of a grid point containing the aircraft, where no searchis conductedr0 Posterior probability for an un-searched locationR Entire search regi
21、ons Square region side lengthW Lateral search rangex East-West distance from the point of losing contacty North-South distance from the point of losing contactz Distance traveled by the search plane within a search area3Team # 32879 Page 4 of 35Abbreviation DescriptionParameter describing the ease o
22、f nding the missing planeParameter describing the e ectiveness of a search planeE ectiveness parameterAngle of change in bearingStandard deviationGlide angle1.3 Simplifying AssumptionsThe accuracy of our models rely on certain key, simplifying assumptions. These assumptionsare listed below:The plane
23、 is precisely tracked until the moment that contact is lost. At that point, it isno longer powered (i.e. the engines provide no thrust and no signals are transmitted).The plane lost contact during the cruise phase of its ight.The pilot can make a single turn of no more than 180 in either direction i
24、mmediatelyafter losing contact. This turn is assumed to occur instantaneously, as the theoreticalrange lost during this turn is insigni cant.There is no wind.There are no ocean currents.The plane/debris will oat inde nitely.The entire search area is water (i.e. the search area does not extend on to
25、land).The search planes only search in their search area. Although their ight from therunway to their speci ed search area may be over other search areas, the plane isassumed to not be searching during this time.The search planes can make instantaneous turns.There is no local curvature of the earth
26、the search area is a perfectly at, two-dimensional surface.On a given search day, there are 12 hours of daylight during which a search aircraftcan be ying.There are also several parameters of the search that were de ned arbitrarily in order topresent consistent results in this report. These paramete
27、rs, however, can be easily varied toaccommodate a speci c case of a missing aircraft:The plane is ying due North when it loses contact.A runway, from which search and rescue e orts can be based, lies exactly 400 milesSouth of the point of lost signal.4Team # 32879 Page 5 of 352 Model Theory2.1 Prior
28、 Probability: Random DescentFirst, we model the potential locations for the missing aircraft with their associated proba-bilities. We assume that the plane is not powered after the loss of signal, so the informationon which to base the search is limited to the last known position of the aircraft, th
29、e directionthe aircraft was traveling, and the type of aircraft that is missing. This last known positionwill be used to de ne the origin of a region R in which the plane could be located. Region Rwill be de ned in the Cartesian coordinate system, with North in the positive y direction andEast in th
30、e positive x direction. There are two properties of the lost aircraft that are usefulin determining where the aircraft may be located: lift-to-drag ratio (L=D) and altitude (h).We will de ne as the glide angle of the plane below the horizontal and as the aircraftspossible change in bearing with resp
31、ect to its initial bearing. These quantities are illustratedin Figures 1a the valuesof L=D and cruise altitude simply need to be changed in the model. From Equation 1, theminimum glide slope angle for a 747 is about 3:37 . By geometrically analyzing the glideangle, as shown in Figure 2 below, an exp
32、ression de ning the maximum range, rmax, of the5Team # 32879 Page 6 of 35plane without power can be de ned, seen in Equation 2. Since the plane is assumed tolose signal during cruise, h will be equal to the cruise altitude. A 747 has an impressivetheoretical maximum glide range of nearly 113 miles.F
33、igure 2: Derivation of Maximum Glide Range.rglide = hcot( ) (2)Our initial probability density function is based on the likelihood of every possible crashtrajectory. Each crash trajectory is de ned by a value of and . Both of these variables arerandom and independent, as and characterize di erent as
34、pects of the random ight pathexperienced upon the loss of signal. The region R of the probability function is bounded bythe calculated maximum unpowered range of the aircraft, swept in all directions from thepoint of lost signal. Because they are independent, and are assigned their own probabilityfu
35、nctions.A value very close to the minimum glide angle min represents a power outage in theplane and the pilots decision to glide at that set angle, maximizing distance and minimizingchance of damage upon impact with the water. values closer to 90 signify catastrophicfailures, such as sudden loss of
36、lift due to a stall or an explosion, both of which would causerapid descent. The probability distribution is modeled as a bimodal normal distribution,with weighting toward the extreme cases of an optimal glide and a catastrophic failure. Thedistribution itself is the sum of two mutually exclusive no
37、rmal distributions. The weightingof each is shown below as a ratio of the probability of a sudden crash, pcrash, to that of aglide, pglide = 1 pcrash.f( ) = pcrash f( crash) +pglide f( glide) = pcrashp2 1e 12( max 1 )2 + pglidep2 2e 12( min 2 )2 (3)The quantities 1 and 2 correspond to the standard d
38、eviations of the crash and glidedistributions, respectively. Similarly, the mean of the crash angle distribution is max = 90 ,while the mean of the glide angle distribution is min.is either a random value, if the loss of power causes a loss of control, or a pilot-dependent value that describes the d
39、egree of turning deemed suitable for accident mitigation.However, since nothing is known about the decisions of the pilot or the controllability of theaircraft at this time, will be varied according to a normal distribution with a mean of zero,corresponding to the most likely scenario that the plane
40、 does not alter its course.6Team # 32879 Page 7 of 35f( ) = 1 3p2 e 12( 3 )2 (4)The standard deviation of the probability distribution is denoted as 3, with a meanof zero explained previously. The standard deviations of both the and distributions werechosen arbitrarily to approximate a likely scenar
41、io. These may be adjusted based on crashstatistics and standards of the missing aircraft. We let 1 = 15 , 2 = 20 and 3 = 30 .Since these two variables are independent and random, their probability functions canbe multiplied to obtain a probability function that spans the entire possible search space
42、, in( ; ) coordinates.p( ; ) = f( )f( ) (5)This probability in ( ; ) space does represent the probabilities we are interested in;however, it is more meaningful to searchers to map these probabilities to a Cartesian (x, y)coordinate system. These transformations are given by the following equations,
43、where isthe characteristic turning radius of the lost plane in a -radian turn. From these equations,the probability distribution can now be transformed from p( ; )p(x;y).x =p2 2(1 cos( ) cos( 2 ) + (hcot( ) j j) sin( ) (6)y =p2 2(1 cos( ) sin( 2 ) + (hcot( ) j j) cos( ) (7)Equations 6 and 7 apply to
44、 a precise conversion of and to Cartesian coordinates,accounting for elevation lost both during a straight glide and during any initial turn thatthe plane may have made. However, these equations may be simpli ed such that the turn ismade instantaneously with no elevation loss, and the plane is then
45、free to glide at any anglewithin the possible values. This simpli cation of the aircraft trajectory is allowable dueto the insigni cant loss of altitude during the planes turn with respect to the full cruisingaltitude. Using this simpli cation and substituting = 0 into the above equations, the initi
46、alprobability distribution of the planes landing location is mapped into Cartesian space.Figure 3 below displays the initial probability distribution for our example aircraft, theBoeing 747-400, in (x;y) space. There is a spike at the origin, corresponding to the prob-ability of the sudden crash cas
47、e, and a more gradual increase in the initial direction of theplane until the maximum glide range is reached, at which point the probabilities decrease tozero.7Team # 32879 Page 8 of 35Figure 3: Initial Probability Distribution of Plane Location.2.2 Bayes TheoremAfter the initial probability has bee
48、n determined, we must model how this probability dis-tribution is a ected by the search. To accomplish this, we separate the information knownbefore the search is conducted from information gathered during the search. Bayes Theoremwill be used to derive a general expression for the probability of nd
49、ing the wreckage. Bayestheorem stated mathematically isP(AjB) = P(BjA)P(A)P(B) (8)for events A and B. P(A) and P(B) are the probabilities of A and B, while P(AjB) is theprobability of A given that B is true. In our case, event A is nding the plane and event Bis the plane being in the location we are searching. The variable q will represent P(AjB) infurther equations6.This theorem can be rewritten to model the manner in which search information a ectsthe probability distribution. If a location is searched and the plane is not found withinthat region, the new probabilit
