essential search mathematics for sar managers & planners presented by dan o’connor newsar

Essential Search Mathematicsfor

SAR Managers & Planners

Presented by

Dan O’ConnorNEWSAR

“Windows” CASIE

Computer-Aided Search Information Exchange

FREE at http://www.wcasie.com

“Closed”System

“Defective”Probability

“Open”System

No ROW100% POC

ROW + Segments= 100% POA

IPP

“Background”3 Types of Search Systems

30%

50%

SA Less Than100% POA or POC

PhysicalLimits

Physical &Psychological

Limits

1. Theoretical vs. Statistical Search Area (SA)

What’s the difference?

THEORETICAL Search AreaThe Straight-Line Distance that a Lost Person could have traveled “in theory” over the Elapsed Time since reported Missing

Rate x Time = Distance (as est. of radius)

2 mph x 12hrs = 24 miles

A radius of 24 miles means a Circular Search Area of 1,810 Square Miles!

Equivalent to a 40 mi by 45 mi Area!

STATISTICAL Search Area

An AREA based on Distances that other Lost Persons have traveled in the PAST.

Ideally, these distances traveled are compiled byLost Person Category (child, elderly, hiker, etc.)

Search Managers typically draw Statistical Search Areas based on the MEDIAN (50th Percentile) & 75th & 90th & 95th Percentiles

Maybe should be called “Potential Search Area”

Q. Why are Potential Search Areas Drawn as Circles?

A. Because in the Absence of CLUES, we have no idea about the Lost Subject’s Direction of Travel

Sources for STATISTICALDistances Traveled . . .

1. Ken Hill (Nova Scotia data) published inthe NASAR MLPI Text & CASIE

2. “Lost Person Behavior,” Robert Koester

3. ISRID Koester & Twardy et al

4. SARSTATISTICS.org (under development)

5. Your OWN or other Local Agency Data

CASIE Source Distances Traveled

2. The MEDIAN: the value which divides the Data in Equal Halves.

50% is At or Above the MedianAnd

50% is At or Below the Median

“The Median home price in the area is $300,000.” Half sold at or above, half sold at or below.

The POSITION of the MEDIAN

Is NOT the VALUE of the MEDIAN!

IMPORTANT!

To find the POSITION of the MEDIAN in a SORTED Dataset use:

MEDpos = 0.5 * (n+1)

For 99 data points, the POSITIONOf the Median = 0.5 * (99+1) = 50

17 SORTED Lost PersonDistance Traveled Data Pts

POSITION DATA Percentile (P)

1 0.5

2 1.2

3 1.8

4 2.3

5 2.7 Position of

6 3.4 Median

7 4.1 formula

8 4.4 0.5 * (N+1)

9 4.8 50th 4.8Median 9

10 4.9

11 5.5

12 6.2 Average: 67th Percentile (12/18)

13 7.1

14 8.6

15 9.9

16 16.9

17 20.3

104.6 Sum

6.2 Mean or "Average"

The MEDIAN is More Stable, The MEAN is More Variable

1. Consider our 17 Data Points , from 0.5mi to 20.3 miwith Mean=6.2 mi and MEDIAN=4.8 mi . . .

2. If we ADD 2 more data points at 1 mi and 30 mi, the Mean goes to 7.1 mi, but the MEDIAN=4.8!

3. The Mean is sensitive to Outliers – the Median is NOT!

The MEDIAN also defines the position of the 50th Percentile

Data: 0.5mi 4.8mi 20.3miPercentile: 0 10 20 30 40 50 60 70 80 90 100

The MEDIAN lives here at the 50th Percentile OR end of the 5th Decile

Why Use the MEDIAN?When to Use the MEDIAN?Why not 75th or 90th Pctile?What should “r” radius be?

Questions on Radius “r”

IPP

20.3 mi

4.8 mi

50%

50%

AREA= ?

AREA= ?

Which Area is easiest to search? Both represent 50% of cases . . .

AREA of a Circle = pi * r^2

For r = 4.8, Area = 3.14 * (4.8 * 4.8) = 72 sq units

For r = 20.3, Area = 3.14 * (20.3 * 20.3) = 1294 sq units

Area of Outer Circle (annulus) = 1294 – 72 = 1222 sq units

Area of an Annulus in CASIE

IPP

20.3 mi

4.8 mi50%

AREA= 72 sq mi

AREA= 1222 sq mi

Which Area is easiest to search? Both represent 50% of cases . . .

50%

IPP

20.3 mi

4.8 mi

50%

50%

pDen= 50% / 72 sq mi = 0.69% per sq mi

pDen= 50% / 1222 sq mi = 0.041% per sq mi

Another way to look at it . . . pDENProbability Density: % Statistical POA per Unit Area

CONSENSUS POA is different from

Statistical Probability. The Area with

the top 50% of cases might be assigned

only 10% POA initially as a Region

NOTE!

• RESOURCES Are LIMITED

• TIME Is Limited

• HIGH Coverage is Required

• Increased Urgency for Good Confinement

• It’s a 50-50 Tradeoff for a smaller SA

WHEN to Search Within the Median

Statistical Circles are

NOT Limits to the Search Area . . .

Go wherever the CLUES Lead!

“Closed”System

“Defective”Probability

“Open”System

No ROW100% POC

ROW + Segments= 100% POA

IPP

“Background”3 Types of Search Systems

30%

50%

SA Less Than100% POA or POC

PhysicalLimits

Physical &Psychological

Limits

3. Analyzing OWN Agency Data

A. Sort and Compute Percentiles

B. Compute the “75% Plus” Range of Finds

Advantage to “75% Plus” . . .

• Uses STANDARD DEVIATION in Data to estimate Variability in LPDT values

• Very Robust for SMALL Datasets

• “Conservative” way to proceed

222 34 7 8 9 1011 26

MED = 7

Sorted Data LP Distance Traveled11 Data Points in Miles

75th Percentile = 10 (9th Position)

MEDpos 0.5 * (11+1) = 0.5 * 12 = 6

The Data Value “7” is at the 6th Position in the Dataset

For “75% Plus” Compute SampleSTANDARD DEVIATION in Excel by using:

+STDEV(data range)

then for “75% Plus” range calculate:

Mean – (2 * SD) = lower bound Mean + (2 * SD) = upper bound

222 34 7 8 9 1011 26

Lower = 0.0

Sorted Data LP Distance Traveled

For MEAN=7.63 & SD = 6.975

Upper = 21.59

75% Plus Range = [Mean – 2*SD to Mean + 2*SD]Reflects VARIABILITY Within the Data;

When Lower Bound is NEGATIVE, Use Zero

4. Methods for Creating a Consensus

In CASIE there are 3 Methods available:

1. MATTSON (numeric POA’s = 100%)2. O’CONNOR (use Verbal Cues)3. PROPORTIONAL (rate relative to Baseline #)

MATTSON

O’CONNOR

PROPORTIONAL

Initial POA’s from Proportional Consensus

5. 2-Methods for Updating a Search

• Bayes Formula, With ROW

• OPOS Summation, Without ROW

Bayes Formula, With ROW

Based on P(A|B) or “the Probability of A, Given B”

The fact that I have searched in B affects theprobability of finding the subject in A. Once

B is searched, the POA of A goes UP.

B A

Bayes Formula, With ROW

BIG SCARY Formula . . . Hard toDo by Hand, especially multiple updates

Do It In CASIE or a Spreadsheet!

Bayes Formula, With ROWUpdate in CASIE

Seg# POA-0 POD POA-1

ROW 27.50% -- 38.63%

1 33.50% 86% 6.59%

2 24.17% -- 33.95%

3 14.83% -- 20.84%

Overall Probability of Success, Without ROW

Seg# POA-0 POD POS POA-1

1 33.33% -- -- 33.33%

2 33.33% 86% 28.66% 4.67%

3 33.33% 86% 28.66% 4.67%

OPOS 0% -- 57.32% --

6. Optimizing Resources

• Brute Force, Calculate to Exhaustion (David Lovelock, Retired Math Prof, U of AZ)

• Washburn Algorithm (Alan Washburn, Naval Post-Graduate School)

• Both require estimating Resource POD

Optimizing Resourcesin CASIE go to . . .

1. top menu “What If” then “Resource Allocation Advice

2. Create a New Table

Resource Allocation Table:Estimated POD for Each Resourcein Each Segment of Interest

WHY BRUTE FORCE?

BRUTE FORCE ADVICE – 3 Scenarios

Washburn Algorithm – 1 “Optimal” Scenario

7. The Mathematical Importance of CONFINEMENT

At a 1 Mile Radius (5,280 feet), Step ONE FOOT farther and the AREA increases by 33,179 sq ft. About 3/4ths of a Football Field (210’ x 150’) to the 74 Yard Line!

8. COVERAGE & POD

Use the Exponential Detection Function (EDF) to find POD from COVERAGE

At COVERAGE = 1, POD = 63% “Efficient”

At COVERAGE =2, POD = 86% “Thorough”

Note: It takes TWICE as much Effort (Resources)to get a Coverage=2 as it does to get Coverage=1.

The “Expanded” EDF

Exponential Detection Function POD v. Coverage

0%10%20%

30%40%50%60%70%

80%90%

100%

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Coverage

PO

D

Too Efficient, Not Thorough

Too Thorough, Not Efficient

Optimalregion

63%

86%

Determining Grid Spacing fromCritical Separation

Chart 1: Convert CS to est. ESWChart 2: Select Desired CoverageChart 3: Obtain Spacing

Example: For a CS of 80 @ 0.6 est ESW=48;for 86% POD Coverage=2, & Spacing = 24.

(Note: for AMDR, skip Chart 1; multiply AMDR by 1.5 to calculate est ESW, then use Charts 2 & 3)

86%

63%

Version 1.2 Source: [email protected]

9. Estimating EFFECTIVE SWEEP WIDTH (ESW)

In the Absence of an Appropriate Detection Table,Sample the Terrain to be Searched using . . .

CRITICAL SEPARATION, or

Avg. Max. Detection Range (AMDR)

and Adjust for an Estimate of ESW

Mt. Greylock base trail,Berkshires, MA – Various Seasons.

Source: Rick Toman, MSP

The Complexity of the Ever-ChangingLandSAR Environment

Determining Critical Separation - 1 CSUnder Prevailing Conditions

If the Object changes, or the Conditions change,a new CS value must be computed!

½ CS

½ CS





86%

63%


10. K9 POD for SAR Managers

Major Environmental Factors that Affect K9 POD

1. Sun Angle (High is Bad)

2. Wind (Still is Bad)

3. Cloud Cover (Clear is Bad)

10. K9 POD for SAR Managers

You debrief a K9 team on a hot August day in Arkansas . . .

They have been out for 4 hours between 10am and 2pm.

The sky is clear and the wind is still.

The Handler says that their POD=95% for 40 acres.

Q. What is your Response to that POD?

BALONEY!

Many factors go into estimatingK9 POD . . . Best bet . . .

BUY The MLPI Text at the NASAR Bookstore and refer tothe Table on p.225!

11. Calculating Cumulative POD

1. Table in MLPI & Field Guide

2. Exp Detection Function (EDF)

3. CASIE (different vs. same teams)





86%

63%


12. GRID SEARCH PLANNING Formulas

Assume Ground Searcher SPEEDOf 3.5 Hours Per Mile . . .

How Fast is that in mph?

1 Mile / 3.5 Hours/Mile = 0.286 mph

12. Find Required # of Searchers

13. Find Searchable Area

14. Find Hours needed to search

15. Find required Spacing

Bonus! Coverage & Track Spacing from #15 Inputs

MLPI Planning Exercise (p.223)

1. High Pressure! Congressman’s Relative Lost!

2. IC wants 80% POD over 1 sq. mile

3. Gives you 100 Ground Searchers

4. ESW estimated to be 60 feet

5. How long will this take? You have 2 minutes!

MLPI Planning Exercise (p.223)

1. Solution: Use CASIE!

2. Find Coverage at 80% POD

3. Find Spacing at Coverage = 1.6 with ESW=60

4. Use HOURS Planning Formulas for Time

5. Answer: 5hrs (4.9 rounded up)

THANKS!

[email protected]

ENCORE?T-CARDS!

essential search mathematics for sar managers & planners presented by dan o’connor newsar

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