Case Study: Searching for the
wreckage of Air France AF 447
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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
• We’ll 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
plane’s 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 Ù ¬b12 Ù ¬b21
• Based on passive acoustic search so far
we know where pings have been heard
p= ¬p11 Ù p12 Ù p21
Suppose we want to evaluate the query:
P( B13 | known, p )
Inference by enumeration
Unknown = Bij’s other than
Query B13 and Known
For inference by enumeration:
P( B13 | known, p )
= α 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 (B’s)
and effects (P’s) 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:
– Pij’s are independent of each other
– Can exploit conditional independence between P’s and B’s
Idea: observations (pings in [11, 12, 21]) are
conditionally independent of more distant squares
given neighbouring unknown squares
Define Unknown = Fringe È 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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