[SOLVED] AF 447

Case Study: Searching for the
wreckage of Air France AF 447

Copyright By Assignmentchef assignmentchef

L. Stone, C. Keller, T. Kratzke, J. Strumpfer
Search for the Wreckage of Air France Flight AF 447,

to appear in Statistical Science
[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.370.2913&rep=r

ep1&type=pdf] and
L. Stone, C. Keller, T. Kratzke, J. Strumpfer

Search Analysis for the Underwater Wreckage of Air France Flight 447
Fusion 2011 July 7, Chicago, USA

[http://www.sarapp.com/docs/AF447 Slides for INFORMS Jun 2011.pdf]

1 June 2009 AF 447 disappeared in the Atlantic Ocean
with the loss of 228 passengers and crew

2 June 2009 Wreckage sighted by search aircraft
May 2010 Black boxes still not found
July 2010 U.S. company Metron engaged to use

Bayesian analysis of evidence to redirect search
20 January 2011 Metron deliver their report on

probability map to guide search
Late March 2011 Search resumed based on prob. map
3 April 2011 Wreckage found on ocean floor

Air France Flight AF 447

Well use their work to motivate a very simplified
example of this type of analysis

At the time of writing, a similar search is underway for
Malaysian Airlines Flight MH 370

Based on the locations of where pings were heard
from the black box flight recorders,
a grid search is being undertaken
in southern Indian Ocean by
Australia, China, Japan, Malaysia,
, South Korea,
United Kingdom and United States

How can we use Bayesian analysis?

Image source: Wikicommons

Consider a grid search is the region of the ocean around the
planes last known position

Bij = true iff grid [ij] contains black box flight recorder
Pij = true iff a ping is heard

in grid [ij]
Assume a ping can only be caused by

a black box in the same or
adjacent grid cell

Assume only [11, 12, 21]
have been searched
i.e., include only
P11 , P12 , P21

in the probability model

Grid Search for

Observations and Query

We have the following observations:
Based on active sonar search so far we know

known = b11 b12 b21
Based on passive acoustic search so far

we know where pings have been heard
p= p11 p12 p21

Suppose we want to evaluate the query:
P( B13 | known, p )

Inference by enumeration

Unknown = Bijs other than

Query B13 and Known
For inference by enumeration:
P( B13 | known, p )
= Sunknown P(B13, unknown, known, p)

If|unknown| = 12
then 212 possible combinations of values to enumerate!

Full joint distribution: P(B11, , B44, P11, P12, P21)
We can rewrite this in terms of causes (Bs)

and effects (Ps) i.e., P( effect | cause )
P(B11, , B44, P11, P12, P21)

= P(P11, P12, P21 | B11, , B44) P(B11, , B44)

Simplify the probability model

Can be simplified:
Black box can only be in one square
All squares equally likely (idpt)

Can be simplified:
Pijs are independent of each other
Can exploit conditional independence betweenPs and Bs

Idea: observations (pings in [11, 12, 21]) are
conditionally independent of more distant squares
given neighbouring unknown squares

Define Unknown = Fringe Other

P( p | B13, Known, Unknown)
= P( p | B13, Known, Fringe)

Using conditional independence

22 combinations to enumerate
rather than 212 combinations!
Can manipulate query into a form
that uses this expression.

Further details

Add other types of prior knowledge into
model, such as flight path, ocean currents

Take account of false positives and
false negatives in detection of pings

For a detailed derivation of these types of
calculations, see RN Section 12.7

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