*** Multi-layer searches - calculating Cumulative POD ***

Date: Wed, 28 Jan 1998 09:45:47 -0400 (AST)
From: "Patterson, Tony" <PattersonT@DFO-MPO.GC.CA>
To: Multiple recipients of list <rcc-l@rcc-net.org>
Subject: [RCC-L:37] RE: Multi-layer searches - calculating POD

In reply to Rod Johnson's question of January 26, 98, here are two
methods to calculate cumulative POD with multiple SRUs
(Search Resource Units).

Readers must bear in mind that the solutions given below are valid only
under certain conditions. These conditions are:

1.  Probability of detection is being used in its purest form. That is,
it is the conditional probability that a target will be detected and
recognized given that the target is within the area swept. The method
below cannot be extended to probability of success which takes into
consideration that the target may not be within the area swept.

2.  Each SRU in the search is operating independently of all other SRUs.
That is, the ships and aircraft are not exchanging information during
the search which will alter the way in which look-outs scan their
sectors. (or radar screen as the case may be).

3.  The SRUs are using a parallel track method, and that sufficient
tracks have been assigned to validate the sweep width assumptions.

In both methods, the probability of detection for SRU "A" is given as
P(A); for SRU "B" as P(B); and so on. The probability of missing a
target (e.g.: 1-P(A)) for SRU "A" is P(-A); for SRU "B" is P(-B), and so
on. [I have not figured out how to use the standard A bar notation in
my e-mail package]

Method 1:

Directly calculate the cumulative probability of all the search units.
This method works ok for two units but gets VERY messy for 3 or more:

Cumulative POD = P(A) + P(B) - P(A)*P(B)  (two SRUs)

Cumulative POD = P(A) + P(B) + P(C) - P(A)*P(B) - P(A)*P(C) - P(B)*P(C)
- P(A)*P(B)*P(C)  (three SRUs)

Method 2:

Calculate the cumulative probability of NOT detecting the target by any
of the search units, and subtract from 1. This method works well for
any number of SRUs, and is the method I prefer:

Cumulative POD = 1-[P(-A)*P(-B)*P(-C)...]  (for any number of SRUs)

Examples:

Probability of Detection for SRU "A"    = P(A)  = 0.4  ie (40%POD)
Probability of No Detection for SRU "A" = P(-A) = 0.6
Probability of Detection for SRU "B"    = P(B)  = 0.7  ie (70% POD)
Probability of No Detection for SRU "A" = P(-B) = 0.3

Method 1 Cumulative POD:

C POD = 0.4 + 0.7 - (0.4)(0.7)
C POD = 1.1 - 0.28
C POD = 0.82
C POD = 82%

Method 2 Cumulative POD:

C POD = 1 - [0.6*0.3]
C POD = 1 - 0.18
C POD = 0.82
C POD = 82%


Note from Martin Colwell ~
Method 2 is the method taught in Ground SAR,
where Resource A has a POD of 40%, Resource B has a POD of 70%:

C POD = 1 - (1 - P(A) X (1 - P(B)...)
C POD = 1 - (1 - 0.4) X (1 - 0.7)...)
C POD = 1 - (0.6 X 0.3)
C POD = 1 - 0.18)
C POD = 0.82
C POD = 82%


Hope this helps.

Anthony Patterson