15年美赛B题一等奖论文

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Mathematical Contest in Modeling (MCM/ICM) Summary Sheet

(Attach a copy of this page to your solution paper.)

Type a summary of your results on this page. Do not include

the name of your school, advisor, or team members on this page.

Searching a crashed plane in the sea is a hard job, while searching a lost plane presumed crashed in the open sea is much harder. To help find a lost plane, we restore the whole process, divide it into three periods and construct models respectively.

The first model is a Stochastic Particle Simulation Model(SPSM), which describes the process that the plane loses contact with the ground and falls into the sea. Then we treat debris of the plane as separate particles and build a Drift Model based on Stochastic Particle Migration Model, which helps us to describe the motion of the debris of the plane and find the possible area containing the lost plane. Finally, we use BP-Artificial Neural Network Algorithm to choose the most suitable type of search planes and try to plan the optimal routine based on Time Homogeneous Markov Chain Model.

In the first period, we break it down into two models: Fly Model and Fall Model. In the Fly Model, considering the great uncertainty on the plane crash, we use SPSM to find the distributions of the position where the plane lost power. Also, we get the distributions of flight speed, fight course and flight duration in that position. Then we divided the crashed plane into two types: with gliding function and without gliding function. Each type of the plane falls down in different way.

In the second period, our goal is to simulate the motion of the debris in the water. We assume that the debris of the plane float on the surface, and it is small enough to ignore the affection of wave force. Based on the Leeway Model, we analyze its acceleration while considering the disturbance of environment at the same time. Then we check the model with data from National Oceanographic Data Center (NOAA) and get a good result.

In the third period, we should choose the most suitable type of search plane and plan the optimal search routine. Using BP-ANNs Model, we determine the input layer as some factors on sea states and the output layer as several factors on the performance of a search plane. The outputs we get are the criterion by which we choose the most suitable plane. Then we try to find the optimal routine based on Time Homogeneous Markov Chain Model.

We conduct Sensitive Analysis on the BP-ANNs and find that the model is robust. We also analyze our strengths and weaknesses and give a brief conclusion.

Contents

1Introduction6

1.1Restatement of the Problem (6)

1.2Literature Review (6)

1.3Our Work (7)

2Assumptions and Justi?cations7 3Notations8 4Model Overview9

5Models in Different Periods9

5.1Fly&Fall Period (9)

5.1.1Model for Fly Period (10)

5.1.2Model for Fall Period (11)

5.2Drift Period (14)

5.2.1Analysis and Assumptions (14)

5.2.2Build Drift Model (16)

5.2.3Numerical Calculation (19)

5.2.4Results (19)

5.3Choose&PlanPeriod (19)

5.3.1Model for Choose Period (20)

5.3.2Model for Plan Period (23)

6Sensitive Analysis25

6.1BP-ANNs (25)

7Strengths and Weaknesses26 8Conclusions26 9Non-technical Paper27

1Introduction

Plane Crash is the most severe type of aviation accidence and incidence.As forced landing is easier on the land than in the sea,most of the plane crash has happened in the sea.When a plane is lostand presumed crashed,we oftenuse satellites to help locate it?rstly.While the conditions in the sea are always extremely complex,satellites cannot work as effectively as work on a land problem.Also,rough location will lead to a wider search and available search facilities are limited.Thereby,the searching operationbecomes a reallybig challenge,let alone rescuing the survivors.In this paper, we will try to help tackle this problem by mathematic models.

1.1Restatement of the Problem

We are required to build a genetic mathematic model to help?nd a crashed plane in the open seathat can apply to different kinds of crashed planes and search planes.We discompose the problem into three sub-questions:

?Build a model that can predict all possible search areas and its probabilities;

?Build a model to plan the optimal search operation using different kinds of search planes;

?Try to apply models to several speci?c situations.

1.2Literature Review

Search theory originated in World War II.It is the study of how to most effectively employ limited resources when trying to?nd an object whose location is not precisely known.The goal is to deploy search assets to maximize the probability of locating the search object with the resources available.And maritime search is the most dangerous, and the most complicated part in the search and rescue(SAR)system,which involves many computational problems.

Since the U.S.Coast Guard?rst articulated its search planning doctrine in the form of a search and rescue manual helping for the SAR problem,there was a great de-velopment in the?eld.International Aeronautical and Maritime Search and Rescue Manual[11]published in1999uni?ed and standardized the national aviation and mar-itime search and planning method.And with the rapid development of the computer technology and automation technology,people began to use computers to help SAR.

There are two main factors that in?uenced the success of Maritime search.Firstly, the searchers have to search in the correct area;secondly,the searchers in the correct area must have the ability to detect the https://www.360docs.net/doc/9f10768156.html,ually the probability of success(POS) depends on the probability of containing(POC)and the probability of detecting(POD). Then we have a formula to calculate:

P OS=P OC×P OD(1)

According to the different types of conditions,there are several methods to?nd the correct area,like Analytical method and so on.For detecting,there are a lot of models to analyze the best searching path and the best resource allocation,like the"Inverse Cube"Model.However,it is still extremely hard to search and rescue in the actual life.

1.3Our Work

In this paper,we try to solve these sub-questions stated above by the following steps:In this paper,we try to solve these sub-questions stated above by the following steps: In the?rst step,we seek to a model to simulate the process from the time the plane sent its last signal to the time it fell into the sea.We can use the information in last signal from the plane to output the possible areas where the plane fell into the sea. Most importantly,our model should consider some uncertain scenarios in this process, for example,the type of the crashed plane.

In the second step,we try to approach the motion of the”crashed plane”in the water in order to?nd the possible search https://www.360docs.net/doc/9f10768156.html,ing the outputs from the?rst step and data about sea state,we can obtain outputs containing possible search areas of the crashed plane and their possibilities.

In the third step,we design a plan model to choose the most suitable type of search plane for the operation.We will consider sea stateto output an optimal choice of the search plane type.Then,we explore a model to optimize the search routine of the plane in the possible area.We inputthe locations and probabilities and get outputs of the optimal search routines.

We start the presentation with the section of assumptions and justi?cations,where we state the basic and well-justi?ed assumptions through this paper.We then list all notations that will be used in the paper andgive a brief overview of our models.Fol-lowing these,we present the models in detailone by one and apply them to a real data set to get some results.We then discuss the strengths and weaknesses in these models and further work remained to be done.Our paper is?nished by a clear conclusion for the whole process.After these,we attach a non-technical paper for the airlines to use in their press conferences concerning their plan for future searches.

2Assumptions and Justi?cations

To simplify the problem,we make the following basic assumptions,each of which is properly justi?ed.

?We need to?nd only one plane at the same https://www.360docs.net/doc/9f10768156.html,ually,two or more planes get crashed in the same sea area is a small probability event.

?There is no land or island near the area where the plane got crashed.From the requisition we learn that the plane may have crashed in open water,thus this is

a natural assumption.

?We assume that no signal would be sent out from any part of the plane.This means that all communication facilities are out of order,even the Flight Data Recorder(Black Box)cannot send signals.

?We regard the surface of the earth as a plane when we do the calculations.It is reasonable to ignore the radian of the earth’s surface since the radius of the earth is far larger than the displacement that might happen in thisproblem.

?last signal we have received from the plane is a ping signal.This kind of signal is sent automatically once an hour.

================================

3Notations

All variables and constants used in this paper are listed in1

Table1:Notations

Symbol De?nition Units m Mass of the plane kg

g Acceleration of gravity kg·m/s t pi Time interval of the plane contact with the base station s

t rl Time of the plane lost power s

x pl Abscissa of the last known position which the plane reported on km

y pl Ordinate of the last known position which the plane reported on km

x rl Abscissa of the actually last known position km

y rl Ordinate of the actually last known position km

P nd Probability of a normal distribution unitless P rd Probability of a uniform distribution unitless x nd Abscissa of the position where the plane lost power km

y nd Ordinate of the position where the plane lost power km

v pl Speed of the plane the last time it reported on km/s v rl Speed of the plane in fact when it lost contact km/s v nd Speed of the plane when it lost power km/s θpl Angle between the?ying direction and due east in the last report rad

θrl Angle between the?ying direction and due east in fact rad

h rl Height of the plane when it lost contact km

h nd Height of the plane when it lost power km

f and Force from wind when the plane lost power N

f gnd Force from gravity when the plane lost power N

x io Abscissa of the position where the plane crashed into the sea km

y io Ordinate of the position where the plane crashed into the sea km

t glide Time the plane glided in the air s

v p Horizontal speed of the plane when it fall down km/h f ow Wind force in the air when it fall down

ρGlide ratio of the plane unitless C Current force to the drift N

L Leeway(wind force)to the drift N

W Wind speed above the sea level N

v d Drift speed of the debris m/s

a Wind slope unitless

b Wind intercept unitless τRand disturbance of leeway coe?cient unitless σL Variance ofτunitless u Rand disturbance of the estimate of wind speed unitless σw Variance of u unitless w Rand disturbance of current speed coe?cient unitless σC Variance of w unitless L Estimate of L N

C Estimate of C N

W Estimate of W N

X Displacement of the drift m

4Model Overview

In the introduction section,we divided the solution into threesteps.We attempt to establish three models to?nish thethree steps in order.Now,wefocus on the three stepsand the models.

The?rst model allows us to get thelocations of possible fallen areas.The core point in this model is that we need to notice three kinds of uncertainty in our simulation: bias in the information from the last signal,plane’s motion after the signal and ways of falling after losing power.We will treat the plane’s possible conditions afterlosing power as stochastic particles and apply the Stochastic Particle Simulation model to to ?nd the most possible position.Then in the Fall Period,we use the output from Fly Period to?nd the point where the plane dropped into the water with the consideration on different types of planes.As the model deals with the falling process of a lost plane,we call it a"Fly&FallPeriod".

The second modelhelps us to simulate the motion of the plane’s debris in the sea. We will design the model based on hydrodynamics,considering the wind force,wave force and current force.We then apply the Stochastic Particle Migration-based Drift-Modelto get an overall drift displacement formulation.As the model deals with the drifting process of a lost plane,we call it a"Drift Period".

One part of the third steps views the problem of choosing the most suitable type of search planes in different sea state.We will select some important factors to describe the sea state of the possible areas and some other factorsto represent types of the search plane.We treat sea statefactors and factors on plane performance as input layer and output layer respectively to build an Arti?cial Neural Networks model.The model will output the optimal plane factorsand we can use them to choose the most suitable plane.The other part of the step deal with the optimize the search routines.Here we work out the problem by Time Homogeneous Markov Chain Model.As this part solves the choosingand planning problem,we can name it as"Choose&Plan Model". 5Models in Different Periods

In this section,we will present the several model for different period.We will start by the Fly&Fall Model to learn about the?rst step.Then we discuss the Drift Period when the plane’s debris drifts in the sea.We?nish this section investigating the Choose&Plan Period to choose a suitable type of search plane and make a routine plan.

5.1Fly&Fall Period

In this model,we attempt to simulate how the plane moved from the time it sent the last signal to the time it fell into the open water.To make the problem clearer,we divide it into two parts.

In the?rst part,we use Stochastic Particle Simulation to simulate the process be-fore losing power;in the second part,we will use basic physics model to simulate the

process considering different kinds of crashed planes after losing power.

In the second part,we predict the way of falling down for different kinds of planes and then get the possible areas where the plane fell into the water.

5.1.1Model for Fly Period

Stochastic Particle Simulation is largely used to generate a probability graph in multi-scenarios given the initial probability distributions.In this problem,before the plane lost its power,there were two random events withsome probability distributions and what we try to obtain is the location probability graph at the time the plane lost power. Thereby,Stochastic Particle Simulation can be applied here well.

Scenarios Analysis and Assumptions Since we only get hold of a small amount of information about the plane crashed.There is often more than one scenario about the plane disasters.These scenarios could happen in the following two phrases.

?The last signal would contain some information of the plane’s conditions at that time,such as the?ght height,velocity,position and?ght course,but bias often exists in the information.

?After sending the last signal,the movement of the plane is not determined.The plane may keep?ying and drop down at any time or lose power immediately.

Usually,all of the four?ight conditions reported in the last signal are treated biased in practical application and the time losing power is not certain.Thus,we make some assume that?ght height,velocity,position and?ight course from the last signal follow some known distributions respectively and the?ight duration after the last signal also follows a certain distribution.What’s more,we treat the signal sent by the plane as Ping Signal,a kind of signal sends by plane automatically at a certain time interval unless the plane has crashed.

Build theStochastic Particle Simulation Model In this section,we will apply the S-tochastic Particle Simulation to the problem by take some certain probability distributions into the model.We will employ several steps to?nish the simulation.

Step1:establish coordinate system Using the Last Known Position(LKP)as the origin,we establish coordinate system.

Step2:determine probability distributions In view of actual deviations,the last location from the plane can follow a three-dimensional Normal Distribution in the area.Mathematically, the probability density function is

f(r)=

1

√

2πσ

exp

?

r2

2σ2

.(2)

The?ying durationin LKPtrl should in a range of

0≤t rl

and follow aRandom Uniform Distribution.Similarly,the real?ght course thetarl and the ve-locity vrl in LKP followRandom Uniform Distribution in range of

θpl?0.5rad≤θrl≤θpl+0.5rad.

and

v pl?60nm/h≤v rl≤v pl+60nm/h.

respectively.

Step3:sample randomly The model will select sample randomly in the circle with a2sigma error radius generated by the above Normal Distribution to determine the initial position of the plane,(xrl,yrl).Then,select real?ight theta,velocity vrl and?ying durationtrl randomly in the Random Uniform Distribution stated above.(jiayigexiangzhishangyi yang de tushi) Step4:get one possible position where the plane lost power Using the selected initial posi-tion in the step3(xrl,yrl),the selected?ight course theta,velocity vrl and?ying duration trl, we can calculate one possible position where the plane have lost its power.

Step5:repeat Step2-4describe a way of locating one position where the plane lost its power. It is a location process only for one particle.If we repeat step2-4for N times,then the model can output a probability graph composed by N random particles.

Results We choose some reasonable and feasibledata to assign the distributions above and simulate the process for more enough times.Then,we can get an intuition on this model and this Fly&Fall period.

The assignments are as follows:

As for the normal distribution1.we assign sigma=80.And we repeat step2-4for1000 times.That is N=1000.

After step5,we use matlab to generate a stochastic particle scatter graph as the2shows. The original point is the LKP of the plane and its reported?ight course is vertical downward in this graph.Each spot represents a possible position where the plane lost power.We notice that the densest area is like a downward sector with the origin point as the center of the circle.

Then,we divide the whole area into an8*8gridding as the?gure3shows and count the spots in each grid.This makes it more obvious where the the probability is higher.The grid with42 spots owns the highest probability.Thus,we will choose the central point(-25,-150)in this grid as the most possible location where the the plane lost its power.We will use the information in this point as input in other parts below.

5.1.2Model for Fall Period

After the plane lost its power,it would de?nitely fall down from the sky to the sea in our problem. The methods of falling largely depend on the plane types.In our paper,we classify the crashed plane and predict the possible falling areas in the sea for different kinds of planes.

Figure1:Distribution of Stochastic Particle in a huge amount

Analysis and Assumptions We classify all crashed planes into two main types.One type is planes with gliding function and the other type is planes without that function.Gliding function helps the plane glide for some distance without power.Generally,an ordinary gilder can as long as the pilot want to survive.

Based on this classi?cation,we make some assumptions for each type to simplify our analy-sis.

?Planes without gliding function cannot reduce horizontal velocity on the condition that air resistance is neglected.

?Planes with gliding function will glide to reduce horizontal velocity,even ifair resistance is neglected.

The two assumptions are all reasonable.For the plane without gliding,its motion type can be treated as Parabolic Motion.The plane will drop into the water quickly almost without horizontal displacement and thus,the air resistance will have little effect on horizontal velocity. That’s the the justi?cation of the?rst assumption.As gliding function is designed to make the plane land softly,the second will be not hard to understand.

We also assume that:

?we treat the plane’s movement in vertical direction as a typical free falling processwith a stable acceleration of gravity,and

?air resistance need not to be considered in vertical direction,and

?the?ght course will remain stable after losing power

Figure2:Number distribution statistical chart of Stochastic Particle

Planes with Gliding Function Planes with gliding function are one type of plane that our model will recognize.In this paper,we will use the data about the gliding to predict the location where the plane has dropped into the water.

According to basic physics formulation,we can describe the gliding function using rou.And

the rou can be calculate by[24,p,25-2].Thus,if we the?ght height and rou are known,we can

get the length of the gliding and then the location problem looks much more easy.

After losing the power,the plane can keep gliding for a long way until it lands or drops

into the water.The gliding length is usually very big.According to plenty of empirical data,a typical glider owns a glide ratio about8～11andwe assume thatrou follows a Random Uniform Distribution with a range of[8,11].Then we select a random number in this range.

Take the selectedρand known?ght height into

L x

≈ρ

h

,we can calculate the gliding length.We assumed the?ght course will not change before.As

we already know the start location,we can get the position where the glider fell into the sea.

Here,our random and known values are as following:[table-szl-2]Using by above values,

we know in this situation,the gliding length is98.4km and the fall position(-19.8510km,173.3450km).

Planes without Gliding Function In this situation,we can abstract the motion from the

time the plane lost power to the time it drop into the water into a?at parabolic motion.

According to basic physics principle,we can get an equation group to calculate the length of this motion.Here are the equations:

Using the values of the start point of this stage stated in[table-szl-2],we can calculate the coordinate of the point where the particle dropped into the water.The length of this motion is 13.0095km and the coordinate of the last point is(-24.3191km,87.9911km).

5.2Drift Period

We will simulate the motion of the debris in the water and try to?nd the possible search areas with probabilities.To solve this problem,we will start with analysis of the problem and essential assumptions.Then we apply Stochastic Particle Migration model to our drift model.We?nish this section by using some real data to get some visible results.

5.2.1Analysis and Assumptions

The motion of debris in the sea is really complex in reality.In this paper,we will try to simplify the analysis with some reasonable assumptions.The followings are assumptions we make for this drift period.

?We assume that the debris is https://www.360docs.net/doc/9f10768156.html,ually,losing power is a main reason for air crash.And we assumed the plane haslost its power in the sky in previous discussions.

?We assume that the plane crashed at the moment it fell into the sea.Because of the pressure difference,a whole plane must crashed when it drop into the sea.Even if the plane becomes disintegrated in the air,the situation will be similar to this one.

?We assume the debris will?oat on the sea level.Lots of the plane components can ?oat in the sea,while some may sink.Floating debris is easier to be found by a search plane and the?nd can help us search for sunken ones.So we can assume that we should search the?oating debris?rst.

?We assume that the amount of the debris is n and it will not change.Though some of the debris may break up again,the situation will not change a lot.So we only discuss the simpler one here.

?We assume length of all debris is less than50m.It’s reasonable to make this as-sumption since it had been broken down into many pieces of debris

Force Analysis To understand the motion of debris,we should?gure out its force situation ?rst.Generally speaking,three main sources of force will affect the motion of debris:wind,wave and current.Now,we choose one debris as the object and try to analyze the the force situation of the object.

Based on the assumptions stated above,we can get the object’s acceleration equation,

(m+km )dV/dt=F a+F w+F c.(3)

from the Newton Second Law.We?nd that the wave force can be neglected when the length of the object is less than the wave length in previous studies.And the wave length is usually about50m.Thus,we can neglect the the wave force[1].The other forces satisfy

F a=1

2

C dρA a W2

a

and

F c=1

2

C cdρw A w L2

b

respectively[2].

At the beginning,the object has a high acceleration generate by external force.After a while, these forces reach a balance and the velocity can remain stable.Thereby,we can reach

F a+F c=0

.Then,we substitute

F a=1

2

C dρA a W2

a

and

F c=1

2

C cdρw A w L2

b

into

F a+F c=0

to get[3.7].It is not hard to?nd that the left part in the equation totally depends on the wave velocity vector Waand the right part depends on the relative velocity vector between the object and the water surrounding,lb.So as long as the information of wind?eld and attribute parameters are known,we can calculate lb.

Drift Velocity Analysis After the former analysis,we now try to obtain the drift velocity vector of the object.According the relative velocity formulation

V=V current+V relative

,we can calculate the drift velocity vector of the object vdrift as long as we know the current ve-locity vector vcurrent and the relative velocity between the object and surface current,vrelative. As we neglect the wave effect,we can presume vrelative and ld are equal[2].

Then,we can simplify the drift velocity vector expression of the object as

V=C+L.

.The drift model satisfy

X(t)?X(0)=

t

0V(s)ds=

t

[C(s)+L(s)]ds.(4)

Figure3:Resolution of leeway vector

Leeway Analysis Leeway is the amount of drift motion to leeward of an object?oating in the water caused by the component of the wind vector that is perpendicular to the object’s forward motion[3].Leeway here is a vector and its size is equal to the relative velocity between the object and surface current.

6depicts the decomposition of leeway.The downwind leeway componentld can be estimated by

L d=a d W+b d

and the cross leeway componentlc can be estimate by

L c=a c W+b c

.All coef?cients in the two regressions can obtain from Leeway Table from Allen’s work in 2005.We will use this data in our following calculation.

5.2.2Build Drift Model

In this section,we will build a Stochastic Particle Migration-based Drift Model(SPM-based DM)to simulate the drift process and?nd search areas.

Stochastic Particle Migration model treats each object as independent stochastic particle with all attributions of the objects and we employ this method to locate the object.Then we establish a Drift Model to describe the drift process.Thus we name the combined model as Stochastic Particle Migration-based Drift Model(SPM-based DM).

The drifting process of the object can be understood as a Markov Process.Mathematically, the process satis?es

P(X t+1|X t,X t?1,...,X1)=P(X i+1|X t)

.Then we can use the following differential equations to specify the process:

dX=V(X,t)dt+dX ,

(5)

dX =K1/2dw.[3.16]

DX’is a random disturbance and represents the possible errors in the data about sea state. Dwt is a random increment,following a normal distribution with known mean and variance. And K is a turbulence diffusion coef?cientwith a following expression:

T

K=σ2

v

.Whereσ2v is variance of the Perturbation Velocity Fields and

σ2

=<(dX /dt)2>

v

;T is the disturbance time,usually equaling to

T=dt/2

.

More assumptions We give some more assumptions that are only essential in the Stochastic Particle Migration-based Drift Model(SPM-based DM).

?Each object isindependent and they cannot affect others’motion.

?Two components of object’s velocity are independent.

?All disturbances here obey normal distribution.

?The disturbances in wind?eld and current?eld are not related.

These assumptions may not agree with the real situation,but they can help us simplify our model.That can make the model easy understanding,and what’s more,simple models can always present the most basic part of the motion.

Leeway Coef?cient The leeway coef?cients come from the linear regression of real experi-mental data and re?ects a character that the object drifts effected by wind.In order to measure the uncertainty in the experiments,we give a disturbance to the coef?cient.

If there are n particles,one of the particle j has following leeway coef?cient disturbance formulations:

?a j=a+τj/20.(6)

?b

=b+τj/2.(7)

j

τj∈N(0,σL)j=1,2,···,n.(8)τj is the disturbance of the coef?cient following a known normal distribution.The mean is zero and the deviance sigmaL is accessible in Allen’s work in2005.In the simulation process, we will select ajianandbjian randomly.Thus,we can get the estimate leeway of object j with the following expression:

?L

j

=(a+τj/20)W+(b+τj/2),j=1,2,···,n.[3.23](9)

Wind Field andCurrent Field Disturbance The data about wind?eld and current?eld is available in external database,including their standard errors.We select the10m-wind velocity over the sea and0.5m-current velocity to re?ect the sea state.According to earlier study,we get

W10=W z(10/z)1/7

.With conditions that wind and current?eld disturbances are independent and obey a known normal distribution,we can get10m-wind disturbance as

?W j =||W+u

j

||

,

u

j

∈N(0,σW)j=1,2,···,n .Take

?W j =||W+u

j

||

into

?L

j

=(a+τj/20)W+(b+τj/2),j=1,2,···,n

respectively,we can obtain the estimate value of leeway:

?L

j

=(a+τj/20)?W j+(b+τj/2),j=1,2,···,n.(10) The estimate current velocity is

?C j =||C+w

j

||

and

w

j

∈N(0,σC)j=1,2,···,n

.If we take

?L

j

=(a+τj/20)?W j+(b+τj/2),j=1,2,···,n and

?C j =||C+w

j

||

into

X(t)?X(0)=

t

0V(s)ds=

t

[C(s)+L(s)]ds

,we can reach to a Stochastic Particle Migration-based Drift Model(SPM-based DM).Math-ematically,we can us the following formulation to describe it:

X j(t)?X j0=

t

0V j(s)ds=

t

[?C j(s)+?L j(s)]ds.(11)

5.2.3Numerical Calculation

Here,we conduct a numerical calculation as the?gure shows below.

Figure4:Steps of predicting random particles drift trajectories

5.2.4Results

In this section,we try togive some numerical results of the model,using real or estimate data sets from previous studies or opendatabases.After substituting data,the distribution of the objects is showed in the following graph.Particles in the dark area are more that these in the white area.According to the distribution,we can?nd the area with high probability to?nd the debris.

This graph represents a400km2open sea area,so we divide this area into a2*2griding. Thus the darker grid is more possible to?nd the debris of the plane.

5.3Choose&PlanPeriod

After we have the possible search areas and their probabilities,we change to focus on how to choose suitable types of search planes and?nd the optimal search routine.In this section,we will divided this problem into two optimization situations and establish models to solve them.

Figure5:Distribution of Stochastic Particle in the sea

5.3.1Model for Choose Period

What we try to solve in this period is mainly an optimization problem.Now,we will employ theBP-Arti?cialNeural Networksalgorithm to?nd the most suitable type of search plane.

BP-Arti?cialNeural Networks are a family of statistical learning algorithms inspired by biological neural networksand are used to estimate or approximate functions that can depend on a large number of inputs and are generally unknown[4].Typical BP-ANNs contain three layers: input layer,hidden layer and output layer.Each of them connects with the upper layer and next layer,just like the synapses connect the layers of neurons.The synapses store parameters called "weights"that manipulate the data in the calculations.In the learning process,we give both the inputs and outputs of the model to the system and it will learn to get reasonable weights. Then,the model can be used in new input data without output.

The BP-Arti?cial Neural Networks algorithm is suitable in this problem.As both the inputs and outputs are combined by several factors,this is clearly a combinatorial optimization prob-lem.We should give each input parameter speci?c and feasible weight.Thus the BP-Arti?cial Neural Networks algorithm is applicable in this situation and can satisfy our requirement. Build BP-Arti?cial Neural Networks According to the analysis above,we now try to build BP-Arti?cial Neural Networks for the choosing problem.We determine the input layer has?ve neurons,the hidden layer has three neurons and the output layer has six neurons.

Each input layer neurons represents one piece of information of sea state.The meanings and value rangesof these neurons are as follows.

?Wind Force measures the wind level over the sea.Nine is the highest level,while?ve is the lowest one.

?Wave Height measures the height level of wave on the sea.Ten is the highest level,while three is the lowest one.

?Visibility measures level of eyeshot over the sea.Three is highest level,while zero is the lowest one.

?Temperaturerepresents the temperature level over the sea.Zero means low temperature, while one means high temperature.

?Water Area means the distance level of the search areas.Zero means that the search area is an offshore area,one means that the search area is far away from offshore.

These factors can describe the sea state in almost all essential aspects that might affect the search operation.So these factors are necessary enough and simple in some degree.

Eachoutput layer neurons represents performance of the search plane in one?eld.The mean-ingsof these neurons are as follows

?Speed means the highest speed of a search plane.

?Endurance means how long the?ght can last.The time will include the outward and return process.

?Anti-wind Ability measures the search plane’s ability of resisting the wind over the sea.

?Detecting Capacityrepresents different electronics and sensors with different detect abili-ty that the search planes are equipped.

?Main Powermeasures the main engine power of the search plane.

?Communication Function measures the communication ability with satellites,ground platforms or other other planes.

All these factors are essential in our model.Search planes with high level performance seem better than planes with poor performance.However,we don’t need such high level performance in some situations and the plane’s performance will be prevented by bad sea state.So these output factors can help us?nd the most suitable type of plane instead of the best one.

Search Ability,Search Dif?culty and Search Effect are factors that we need consider in the search operation.Thus we set these three factors as the hidden layer.

The?gurex show our networkstructure.Each circle represents a neuron and the lines among the neurons represent their connections.

Learning Process Learning process is a crucial part of the BP-ANNs.In this progress,we give the networks a set of data containing both the input values and the output values.We run the networks to a huge number of times to let the network"learn"weights between different parameters.

To solve this problem,we use a known data set from Weihong Yu’s previous work as train sample and let the network learn for?times to make it satisfy the required accuracy.Part of the

Figure6:schematic diagram of BP-ANNs

Figure7:compare the outputs from neutal network with real values sample is recorded in the following table.After the learning process,the networks will memory this experience as connection weights and use these weights in decision process.

The graph1-3describe the similarity between the real values and estimated value from our BP-ANNs.We can?nd that our neutral networks have good learning effect.Thus,we can say that this model can be used in this problem.

Results Here we input a set of value of factors on weather state of ocean(8,10,1,0,1)andthe networks output a ser of values of factors on the search plane(0.8002,-1.1842,-0.2814,1.0149, 0.8604,0.3605).The factors range and meaning has been stated above.This input(8,10,1,