Search for a Moving Target: Upper Bound on Detection Probability

NPS55-78-029

NAVAL POSTGRADUATE SCHOOL Monterey, California i

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111

!F SEARCH FOR A MOVING TARGET: UPPER BOUND ON DETECTION PROBABILITY by Alan R.

Washburn

October 1978 Approved for public release;

distribution unlimited.

79 01 09 030

I NAVAL POSTGRADUATE SCHOOL MONTEREY, CALIFORNIA

Rear Admiral T. F. Dedman Superintendent

J.

R. Borsting Provost

Reproduction of all or part of this report is authorized. This report was prepared by: A$(na'

R. WasBurn,

Associate Professor

Department of Operations Research

Reviewed by:

Released by;

Michael G. Sovereiqn, Department of Operations Research

William M. Tolles Dean of Research

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SExtsting algorithms for computing the optimal distribution of effort in search for a moving target operate by producing a sequence of progressively bptte. distributions. This report shows how to compute an upper bound on the detection probability for each of those effort distributions in the case where the detection function is concave.

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I SEARCH FOR A MOVING TARGET: UPPER BOUND ON DETECTION PROBABILITY

by Alan R. Washburn Naval Postgraduate School Monterey, CA 93940

ABSTRACT Existing algorithms for computing the optimal distribution of effort in

search for a moving target operate by producing

a sequence of progressively better distributions.

This report

shows how to compute an upper bound on the detection prob-

dbility for each of those effort distributions in the case where the detection function is concave.

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TABLE OF CONTENTS PAGE i.

INTRODUCTION

2.

THE GENERAL

3.

THE CASE OF MARKOV MOTION AND EXPONENTIAL DETECTION FUNCTION

6

AN EXAMPLE

...........................

9

REFERENCES

...........................

11

4.

.......................... CASE

......................

2

3 3

I

1.

INTRODUCTION

The object is to detect a randomly moving target at one of the discrete times

0,1,...,T.

negative effort distribution

The searcher determines a non-

V(x,t)

applied at time t does not exceed

such that the total effort

m(t).

Our purpose here is to

establish an upper bound on the detection probability for every effort distribution.

This is intended to supplement existing

iterative procedures that develop sequences of affort distributions that improve monotonically.

2.

THE GENERAL CASE Let

w(x,t)

be an "effectiveness coefficient" for search

effort applied at position of the target at time t,

x

at time t.

If

X

is the position

then the probability of detection depends,

we assume, only on the total effective search effort T

z =

I w(xt,t) V(Xt,t) two

Specifically the probability of detection is

P(V) - E(b(Z)),

where the expectation operator is needed because stochastic process. for some function

We assume that s(z)

and for all

3

Xt

Is a

b(z') - b(z) 0.

when

Dr (0,x,t) > A_(t)

(x,t) Idx•

C'(x,t) -

t) 10 D¶Jx,

< 1(t)

for all x, and

In the discrete case,

from (4), ~T ) (5) E(s(Z) (Z'-z))

t=0

II T(t) P' (x,t) X

S~T - ~~I

-

x

t)

X[t (t)) mlt)

t=O

and (1),

Combining (5) P(

(6)

)

:- (*) 1.

for

and it

problems is relatively simple, increases with

and

n > 1,

Each of the minimization can be shown that p(*n

n [1,2,3).

For each of the minimization problems,

Xn(t) n such that

a function

for

Dn(x,t)
0

*n(x,t) - 0, where

Dn(xt) - w(x,t) P(*jn ,x,t) exp(-w(x,t) iJn (x,t))

n-l(x,t).

These are simply the Kuhn-Tucker conditions; in practice, sort of a search is made until xt)

-

m(t).

Xn(t)

is found such that

But

(

" )n(t) n

Qn (xt)/Qn-l (xt)