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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 “Open” System “Defective” Probability “Closed” System 50% IPP Physical & Psychological Limits ROW + Segments = 100% POA 30% Physical Limits SA Less Than 100% POA or POC No ROW 100% POC

1. Theoretical vs. Statistical Search Area (SA) What’s the difference?

THEORETICAL Search Area The 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 12 hrs = 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 by Lost Person Category (child, elderly, hiker, etc. ) Search Managers typically draw Statistical Search Areas based on the MEDIAN (50 th Percentile) & 75 th & 90 th & 95 th 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 STATISTICAL Distances Traveled. . . 1. Ken Hill (Nova Scotia data) published in the 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 Median And 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.

IMPORTANT! The POSITION of the MEDIAN Is NOT the VALUE of the MEDIAN!

To find the POSITION of the MEDIAN in a SORTED Dataset use: MEDpos = 0. 5 * (n+1) For 99 data points, the POSITION Of the Median = 0. 5 * (99+1) = 50

17 SORTED Lost Person Distance Traveled Data Pts ON POSITI 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 10 4. 9 11 5. 5 12 6. 2 13 7. 1 14 8. 6 15 9. 9 16 16. 9 17 20. 3 104. 6 50 th 4. 8 Median Average: 67 th Percentile (12/18) Sum 6. 2 Mean or "Average" 9

The MEDIAN is More Stable, The MEAN is More Variable 1. Consider our 17 Data Points , from 0. 5 mi to 20. 3 mi with 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 50 th Percentile 0. 5 mi 4. 8 mi 20. 3 mi Percentile: 0 10 20 30 40 50 60 70 80 90 100 Data: The MEDIAN lives here at the 50 th Percentile OR end of the 5 th Decile

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

Which Area is easiest to search? Both represent 50% of cases. . . 50% IPP AREA= ? 4. 8 mi 20. 3 mi

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

Which Area is easiest to search? Both represent 50% of cases. . . 50% 20. 3 mi 4. 8 mi IPP AREA= 72 sq mi AREA= 1222 sq mi

Another way to look at it. . . p. DEN Probability Density: % Statistical POA per Unit Area 20. 3 mi 50% 4. 8 mi IPP p. Den= 50% / 72 sq mi = 0. 69% per sq mi p. Den= 50% / 1222 sq mi = 0. 041% per sq mi

NOTE! 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

WHEN to Search Within the Median • 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

Statistical Circles are NOT Limits to the Search Area. . . Go wherever the CLUES Lead!

“Background” 3 Types of Search Systems “Open” System “Defective” Probability “Closed” System 50% IPP Physical & Psychological Limits ROW + Segments = 100% POA 30% Physical Limits SA Less Than 100% POA or POC No ROW 100% POC

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

Sorted Data LP Distance Traveled 11 Data Points in Miles 2 2 2 34 7 8 9 1011 26 MED = 7 75 th Percentile = 10 (9 th Position) MEDpos 0. 5 * (11+1) = 0. 5 * 12 = 6 The Data Value “ 7” is at the 6 th Position in the Dataset

For “ 75% Plus” Compute Sample STANDARD DEVIATION in Excel by using: +STDEV(data range) then for “ 75% Plus” range calculate: Mean – (2 * SD) = lower bound Mean + (2 * SD) = upper bound

Sorted Data LP Distance Traveled 2 2 2 34 Lower = 0. 0 For MEAN=7. 63 & SD = 6. 975 7 8 9 1011 26 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” B A The fact that I have searched in B affects the probability of finding the subject in A. Once B is searched, the POA of A goes UP.

Bayes Formula, With ROW BIG SCARY Formula. . . Hard to Do by Hand, especially multiple updates Do It In CASIE or a Spreadsheet!

Bayes Formula, With ROW Update 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 Resources in CASIE go to. . . 1. top menu “What If” then “Resource Allocation Advice 2. Create a New Table

Resource Allocation Table: Estimated POD for Each Resource in 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 3/4 ths 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 Too Thorough, Not Efficient 86% 63% Optimal region Too Efficient, Not Thorough

Determining Grid Spacing from Critical Separation Chart 1: Convert CS to est. ESW Chart 2: Select Desired Coverage Chart 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) CHART 3: Searcher Spacing from est ESW & Coverage 63% Searcher Spacing 86% 110 105 100 95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0 SPC @ Cov=1. 0 SPC @ Cov=2. 0 10 20 30 40 50 60 70 80 est Effective Sweep Width (ESW) Version 1. 2 Source: [email protected] org 90 100

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

The Complexity of the Ever-Changing Land. SAR Environment Mt. Greylock base trail, Berkshires, MA – Various Seasons. Source: Rick Toman, MSP

Determining Critical Separation - 1 CS Under Prevailing Conditions ½ CS If the Object changes, or the Conditions change, a new CS value must be computed!

Determining Grid Spacing from Critical Separation Chart 1: Convert CS to est. ESW Chart 2: Select Desired Coverage Chart 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) CHART 3: Searcher Spacing from est ESW & Coverage 63% Searcher Spacing 86% 110 105 100 95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0 SPC @ Cov=1. 0 SPC @ Cov=2. 0 10 20 30 40 50 60 70 80 est Effective Sweep Width (ESW) Version 1. 2 Source: [email protected] org 90 100

10. K 9 POD for SAR Managers Major Environmental Factors that Affect K 9 POD 1. Sun Angle (High is Bad) 2. Wind (Still is Bad) 3. Cloud Cover (Clear is Bad)

10. K 9 POD for SAR Managers You debrief a K 9 team on a hot August day in Arkansas. . . They have been out for 4 hours between 10 am and 2 pm. 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 estimating K 9 POD. . . Best bet. . . BUY The MLPI Text at the NASAR Bookstore and refer to the 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 from Critical Separation Chart 1: Convert CS to est. ESW Chart 2: Select Desired Coverage Chart 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) CHART 3: Searcher Spacing from est ESW & Coverage 63% Searcher Spacing 86% 110 105 100 95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0 SPC @ Cov=1. 0 SPC @ Cov=2. 0 10 20 30 40 50 60 70 80 est Effective Sweep Width (ESW) Version 1. 2 Source: [email protected] org 90 100

12. GRID SEARCH PLANNING Formulas Assume Ground Searcher SPEED Of 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: 5 hrs (4. 9 rounded up)

THANKS! [email protected] org

ENCORE? T-CARDS!