Essential Search Mathematics for SAR Managers & Planners Presented by Dan O’Connor NEWSAR
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Essential Search Mathematics for SAR Managers & Planners Presented by Dan O’Connor
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Essential Search Mathematics for SAR Managers & Planners Presented by Dan O’Connor NEWSAR. “Windows” CASIE Computer-Aided Search Information Exchange FREE at http:// www.wcasie.com. “Background” 3 Types of Search Systems. “Defective” Probability. “Closed” System. “Open” System. 50%. - PowerPoint PPT Presentation
Transcript of Essential Search Mathematics for SAR Managers & Planners Presented by Dan O’Connor NEWSAR
Essential Search Mathematicsfor
SAR Managers & PlannersPresented 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 Median7 4.1 formula8 4.4 0.5 * (N+1)9 4.8 50th 4.8Median 9
10 4.9
11 5.5
12 6.2 Average: 67thPercentile (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
POD
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
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]
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)
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]
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)
ENCORE?T-CARDS!
