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Optimal Pinging Frequencies in the Search for an Immobile Beacon - - PowerPoint PPT Presentation

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Optimal Pinging Frequencies in the Search for an Immobile Beacon David Eckman Lisa Maillart Andrew Schaefer Cornell University, ORIE University of Pittsburgh, IE Rice University, CAAM r

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Optimal Pinging Frequencies in the Search for an Immobile Beacon

David Eckman Lisa Maillart Andrew Schaefer

Cornell University, ORIE University of Pittsburgh, IE Rice University, CAAM ❞❥❡✽✽❅❝♦r♥❡❧❧✳❡❞✉ ♠❛✐❧❧❛rt❅♣✐tt✳❡❞✉ ❛♥❞r❡✇✳s❝❤❛❡❢❡r❅r✐❝❡✳❡❞✉

May 22, 2017

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OPTIMAL PINGING FREQUENCIES IN THE SEARCH FOR AN IMMOBILE BEACON ECKMAN, MAILLART AND SCHAEFER

Motivation

Deep sea searches for missing aircraft

  • Air France Flight 447 (2009)
  • Malaysia Airlines Flight 370 (2014)

Flight data recorder (FDR)

  • AKA “black box”
  • Keeps electronic record of aircraft operations
  • Extremely important to follow-up investigations

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 2/28

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OPTIMAL PINGING FREQUENCIES IN THE SEARCH FOR AN IMMOBILE BEACON ECKMAN, MAILLART AND SCHAEFER

Finding the Flight Data Recorder

Each FDR is equipped with an underwater locator beacon (ULB).

Figure: Flight recorder (orange box) with ULB (silver cannister).

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 3/28

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OPTIMAL PINGING FREQUENCIES IN THE SEARCH FOR AN IMMOBILE BEACON ECKMAN, MAILLART AND SCHAEFER

Underwater Locator Beacon

How the ULB works:

  • Activates when submerged in water
  • Produces ultrasonic pings (roughly once per second)
  • Battery life of ≈30 days once activated

Finding the FDR before the ULB’s battery dies is critical

  • Other search methods are less effective and/or slower
  • E.g., high altitude fly-overs, side-scan sonar searches

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 4/28

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OPTIMAL PINGING FREQUENCIES IN THE SEARCH FOR AN IMMOBILE BEACON ECKMAN, MAILLART AND SCHAEFER

Finding the Underwater Locator Beacon

Ocean-surface search vessels

  • Pass over search area on parallel runs
  • Drag a towed pinger locator (TPL) ≈1000 ft above ocean floor
  • Move slowly (2-3 mph) and turn slowly (3-8 hrs)

Figure: Search path over a rectangular section of the search area.

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 5/28

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OPTIMAL PINGING FREQUENCIES IN THE SEARCH FOR AN IMMOBILE BEACON ECKMAN, MAILLART AND SCHAEFER

Search with Towed Pinger Locator

Figure: Ocean-surface search using towed pinger locator.

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 6/28

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OPTIMAL PINGING FREQUENCIES IN THE SEARCH FOR AN IMMOBILE BEACON ECKMAN, MAILLART AND SCHAEFER

After Detection

Once pings are detected

  • Perpendicular runs to box in FDR position
  • Triangulation
  • Recovery by diver, submersible, or remote-operated vehicle

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 7/28

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OPTIMAL PINGING FREQUENCIES IN THE SEARCH FOR AN IMMOBILE BEACON ECKMAN, MAILLART AND SCHAEFER

Beacon Design

Proposed recommendations by governing bodies:

  • 1. Add a second beacon with
  • lower frequency of sound
  • increased range of detection (8 miles vs 2.5 miles)
  • 2. Extend battery life from 30 days to 90 days

A less costly alternative: Modify the beacon’s pinging period—time between successive pings

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 8/28

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OPTIMAL PINGING FREQUENCIES IN THE SEARCH FOR AN IMMOBILE BEACON ECKMAN, MAILLART AND SCHAEFER

Beacon Design Question

Need to determine the pinging period before other search parameters (e.g., search speed) are known.

Is the industry-standard pinging period...

  • ...too short?
  • ...too long?
  • ...just right?

Method: develop a simplified search model to get a first-order answer.

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 9/28

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OPTIMAL PINGING FREQUENCIES IN THE SEARCH FOR AN IMMOBILE BEACON ECKMAN, MAILLART AND SCHAEFER

Existing Search Models

We adapt the linear search problem (LSP) for an immobile object.

Figure: LSP with distribution of hidden object.

Novelties

  • 1. Object is intermittently detectable
  • 2. No switching/turning
  • 3. Search vessel speed is selected from known distribution

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 10/28

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OPTIMAL PINGING FREQUENCIES IN THE SEARCH FOR AN IMMOBILE BEACON ECKMAN, MAILLART AND SCHAEFER

Our Model

Assumptions

  • Linear search problem (by unfolding parallel runs)
  • Definite range law
  • P(Detect ping) = 1 if within range during ping
  • A single search vessel
  • The search terminates once a ping is detected

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 11/28

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OPTIMAL PINGING FREQUENCIES IN THE SEARCH FOR AN IMMOBILE BEACON ECKMAN, MAILLART AND SCHAEFER

Notation

  • [0, L]: search space represented as an interval
  • B: location of beacon
  • r: radius of detection
  • n + 1: number of pings (including at time 0)
  • ν: search speed
  • τ: pinging period

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 12/28

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OPTIMAL PINGING FREQUENCIES IN THE SEARCH FOR AN IMMOBILE BEACON ECKMAN, MAILLART AND SCHAEFER

Fixed Search Speed ν

Figure: Search on [0, L] under three settings of the pinging period τ.

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 13/28

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OPTIMAL PINGING FREQUENCIES IN THE SEARCH FOR AN IMMOBILE BEACON ECKMAN, MAILLART AND SCHAEFER

Probability of Detection

Assume B uniformly distributed on [0, L]. Let θ(ν, τ) denote the probability of detection for a search speed ν and pinging period τ.

Objective

Find τ ∗ that maximizes θ(ν, τ), assuming ν is fixed and known.

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 14/28

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OPTIMAL PINGING FREQUENCIES IN THE SEARCH FOR AN IMMOBILE BEACON ECKMAN, MAILLART AND SCHAEFER

Two Cases

  • 1. Sufficient pings: n ≥ L−r

2r

  • Enough pinging instances to search entire interval [0, L].
  • 2. Insufficient pings: n < L−r

2r

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 15/28

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Sufficient Pings

Figure: The probability of detection as a function of the pinging period.

τ ∗ = L−r

nν , 2r ν

  • DEEP SEA SEARCH

MODEL ANALYSIS END MATTER 16/28

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OPTIMAL PINGING FREQUENCIES IN THE SEARCH FOR AN IMMOBILE BEACON ECKMAN, MAILLART AND SCHAEFER

Insufficient Pings

Figure: The probability of detection as a function of the pinging period.

τ ∗ = 2r

ν , L−r nν

  • DEEP SEA SEARCH

MODEL ANALYSIS END MATTER 17/28

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OPTIMAL PINGING FREQUENCIES IN THE SEARCH FOR AN IMMOBILE BEACON ECKMAN, MAILLART AND SCHAEFER

Two Search Speeds

Suppose search speed has distribution ν =

  • ν1

w.p. λ ν2 w.p. 1 − λ for λ ∈ (0, 1) and ν1 < ν2. Eν[θ(ν, τ)] = λθ(ν1, τ) + (1 − λ)θ(ν2, τ).

Objective

Find τ ∗ that maximizes Eν[θ(ν, τ)], assuming ν1, ν2, and λ are fixed and known.

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 18/28

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OPTIMAL PINGING FREQUENCIES IN THE SEARCH FOR AN IMMOBILE BEACON ECKMAN, MAILLART AND SCHAEFER

Sufficient Pings

Assume ν1 < L−r

2rn ν2, otherwise τ ∗ = 2r/ν is optimal.

Figure: Expected probability of detection (red line) as a function of the pinging period for two speeds with λ = 1/2. From picture, τ ∗ ∈

  • 2r

ν2 , L−r nν1

  • .

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 19/28

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OPTIMAL PINGING FREQUENCIES IN THE SEARCH FOR AN IMMOBILE BEACON ECKMAN, MAILLART AND SCHAEFER

Solution to 2-Speed Problem

Theorem

There exists a threshold ˜ λ such that τ ∗ =

  • 2r

ν2

(left endpoint) for λ ≤ ˜ λ,

L−r nν1

(right endpoint) for λ ≥ ˜ λ. An explicit expression for ˜ λ in terms of the search speeds ν1 and ν2 can be easily solved.

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 20/28

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Sketch Proof

Figure: Linear upper bounds on probability of detection (dashed blue line) and expected probability of detection (solid blue line). Tight at endpoints.

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 21/28

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More about ˜ λ

Corollary

When the faster search speed ν2 is as or more likely than the slower search speed ν1, the optimal pinging period is the longest period that ensures no intervals are left undetected between pings; i.e., τ ∗ = 2r/ν2. That is, ˜ λ ≥ 1/2.

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 22/28

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OPTIMAL PINGING FREQUENCIES IN THE SEARCH FOR AN IMMOBILE BEACON ECKMAN, MAILLART AND SCHAEFER

Insufficient Pings

Harder case because the linear upper bound may not exist.

Proposition

If the faster search speed ν2 is more likely than the slower search speed ν1, i.e., λ < 1/2, then τ ∗ = L−r

nν2 .

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 23/28

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Three or more search speeds

Closed-form solutions are harder to come by, but using a

  • ne-dimensional optimization algorithm is always an option.

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 24/28

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Case Study: MH370

  • Disappeared over Indian Ocean
  • Preliminary air and sea searches found no matching debris
  • Revised search area of 85,000 sq-miles wherein ocean depth is

between 10,000–15,000 ft

Figure: Search area for MH370.

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 25/28

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OPTIMAL PINGING FREQUENCIES IN THE SEARCH FOR AN IMMOBILE BEACON ECKMAN, MAILLART AND SCHAEFER

Case Study: MH370

Conservative estimates for variables

  • Shortest range of detection: r = 0.6 miles
  • Fastest search speed: ν = 5.8 mph

Calculation

τ ∗ = 2r ν = 2(0.6 miles) 5.8 mph = 12.4 minutes. Industry standard pinging period: τ = 1.1 seconds. Difference in magnitude of ≈ 700!

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 26/28

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OPTIMAL PINGING FREQUENCIES IN THE SEARCH FOR AN IMMOBILE BEACON ECKMAN, MAILLART AND SCHAEFER

Acknowledgments

This material is based upon work supported by the National Science Foundation under grants CMMI–1100082, CMMI–1100421, CMMI–1333758, and CMMI–1400009 as well as Research Experience for Undergraduates supplements to National Science Foundation grants CMMI–1131172 and CMMI–1333758. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 27/28

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OPTIMAL PINGING FREQUENCIES IN THE SEARCH FOR AN IMMOBILE BEACON ECKMAN, MAILLART AND SCHAEFER

Sources

Images

  • Black Box and ULB: ❤tt♣✿✴✴✇✇✇✳r❛❞✐❛♥t♣♦✇❡r❝♦r♣✳❝♦♠✴

✇♣✲❝♦♥t❡♥t✴✉♣❧♦❛❞s✴✷✵✶✺✴✵✼✴♠♦✉♥t❡❞❴❜❡❛❝♦♥✳❥♣❣

  • TPL: ❤tt♣s✿✴✴st❛t✐❝✳❣✉✐♠✳❝♦✳✉❦✴♥✐✴✶✸✾✺✽✺✶✷✹✷✻✽✷✴

❇❧❛❝❦✲❜♦①✲❢✐♥❞❡r❴❲❊❇✳s✈❣

  • MH370 Search Area: ❤tt♣s✿✴✴✇✇✇✳❡①tr❡♠❡t❡❝❤✳❝♦♠✴

✇♣✲❝♦♥t❡♥t✴✉♣❧♦❛❞s✴✷✵✶✻✴✵✼✴❙❡❛r❝❤❆r❡❛✳♣♥❣ References

  • Optimal pinging frequencies in the search for an immobile
  • beacon. Eckman, D., Maillart, L., and Schaefer, A. IIE

Transactions, 48(6): 489–500. 2016.

DEEP SEA SEARCH MODEL ANALYSIS END MATTER 28/28