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LLVEj') IDA PAPER P-1373

ANALYTICAL REPRESENTATIONS OF BLAST DAMAGEFOR SEVERAL TYPES OF TAR(CETS

Leo A. Schmidt, Jr.

.2 October 1978 D D C

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The wwk reperted hoi th~s declaneitt wis conducted under Conenit No.OCl'AOI1-7-C-O21,' for the Defense ClIA P'aaoness Agneny. Thepnbko-~ ",; this MA Pspwr 14 rist Ind.-ate andrsement by theDefense CvIW Prmparuness .4gency, For should the ci'tvts be con-strued as refltting the e*tWl oosftln of that ageny.

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ANALYTICAL REPRESENTATIONS OF BLAST ItrmRprDAMAGE FOR SEVERAL TYPES OF TARGETS / Inei Rpr

AUWOU~a IA' Paeer P-l373~7. A040s . QQ TRC onN qnsrWTVj

Leo A. Schmidt, Jr. DCPA0l..77-C-0215

9. P~P~,4@ORA..mZATIOe eMAN AMC 30 LOONSI U000a £LSU1M?.07

0J1C? IIA

INSTITUTE FOR DEFENSE AALYSES . 6in

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I I co r"0 61012 FFIE 1AIAE 440 CON81It REPORT GAM ,..

Defense Civil Preparedness Agency ,October 19181; biumo irO0&0,4S

Washington, D.C. 20301 72V 4,09*C 14A09~aN &@M~ MAUI R apmugsefv 1"3M C.SMI*OZ18,eu 3#0.) 64CURITy C&S.ra$Ste #hg. m0*0

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blast dam'age, target complexes. nuclear attack, damage laws,damage functions

is. gat .CtC~M. .um'. ,of RSiod aid0 40W 4moso :=9e 614*a"40

~' Irhispaper is concerned with the ýSquare Root Damage Law's-an analyticalmethod of computing blast damage on target complexes. The Law assumesthat target value is a continuous function of position, and replaces a setof inJividual weapons attacking a targe~t by a weapon density function thatis optimized to give maximuim damage. rhis basic approach is applied toseve~ral types of target damage functions and methods of weapon targeting. 4

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UNCLSSIFIED$CC:-1gV CL..A. IFICATIC1l11 OOP ?Ia S PAUMP14, 1111 DIMS 1n0

20. (continued)

"-:Attention is restricted to cases where the results can be presented inrelatively simple analytical form. The derivations are presented in somedetail to illustrate the nature of the results obtained. Charts of damageare presented for sets of dimensionless parameters that govern the damagesensitivity to weapon usage. As a practical application, they can formdetailed nimerical calculations of the damage for a particular target from a

specifik set of weapon aimpoints.

"b-

ACCESSION for

NMIS White SectaiDOC BuffSedop 13UNA N CUPIEDJUISTI'. 1-,',IIN ,

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H0

IDA PAPER P-1373

ANALYTICAL REPRESENTATIONS OF BLAST DAMAGEFOR SEVERAL TYPES OF TARGETS

by

Leo A. Schmidt, Jr.

t

forDefense Civil Preparedness Agency

Washington, D.C. 20301

"October 1978

I U~~DA Kiwliw NMInMs mwd WW OW pmaw• Am dM adm dN *A

I DAINSTITUTE FOR DEFENSE ANALYSES

PROGRAM ANALYSIS DIVISION400 Army-Navy Drive, Arlington, Virginia 22202

Contract No. DCPAO1-77-C-0215Work Unit 4114C;

-- - -~.-... ~ -~ '71_

CONTENTS

ABSTRACT . . . . . . . . .. .. .. ..... . . . . . . . vii

I . INTRODUCTION . ... .. .. .. .. .. .... ....... 1

II. ANALYSES .. ................................... . . . 3

A. Square Root Damage Law Derivation . . .. .. ... 3

B. Survivors From Population in a Uniform DiskTargeted as a Gaussian Population .. .............. 6

C. Survivors From a Gausaian Population Targetedas a Uniform Disk. ............................... 10

D. Survivors From a Gaussian Population Targetedas a Gaussian Population of a DifferentDispersion. ....................................... 13

E. Square Root Injury Law .. ......................... 17

F. Survivors From a Gaussian Population Targetedas a Unifop'm Population in an Annulus . . . . . 23

[i ¼G. Survivors From a Population in an AnnulusTargeted as a Gaussian Population..........29

H. An Alternative Square Root Law Derivation . . . 33

I. A Second Alternative Square Root Law Derivation 37

J. An Application of the Second AlternativeDerivation to Value a Linear Function of Area . 41IK. An Application of the Second AlternativeDerivation to Value an Exponential Function ofArea to aPower .. ................................ 43

L. Factors9 Influencing the Probability of Kill asa Function of Weapon Density .. ................... 53

:,-• .. ~~ ~ ~ - --- 9--"~-~-- = .. .. - .... --:---'" •:-•- - ---• T .-•-• % "_ _-_--- _ pm •• •-•: ••: • .

IiF

FIGURES

1 Fraction Survivors for Population in Disk of Radius[• R With Gaussian Targeting for Various Values of

"With Weap ns Normalized by Targeting--0- R2/2B2 ,X - kW/nO . .. . . . . . . . . . . 9

2 Fraction Survivors for Population in Disk forGaussian Targeting for Various Values o5 / WithWeapons Normalized by Population--p R/22 a2/02I Y = W/T°2 . • k•/,•a. ..•.• . .•. . •. •. . • .• • . • . . 11

3 Fraction Survivors for Gaussian Population forTargeting of Uniform Disks qf Radius R With WeaponsV Normalizeo by Target v = 2a4/2R - a2/021

V., X = kW/ 82 . . ...... . . . . . . . . . . . . . 14L4 Fraction Survivors for Gaussian Population for

Targeting of Univorw Disk With Weanons Normalizedby Population = aL/s Y - kW/r8e . . . . . . . . 15

5 Fraction Survivors of Gaussian Population WithStandard Deviation 0 With Targeting Against GaussianPopulation With Standard Deviation a With WeaponsNormalized for Population 0 a2/0 2 , X a kW/wr 2 . . . 18

6 Fraction Survivors of Gaussian Population WithStandard Deviation 0 With Targeting Against GaussianPopulation With Standard Deviation a With WeaponNormalized for Population-- a 2/82, Y - kW/vr . 19

7 Fraction Survivors as a Fraction of Weapon Usagefor Vulnerability Index k and Target k with WeaponsNormalized for Attack--t a kc/k, X -kW/wra2 . . . . . 22

8 Injuries as a Function of Weapon Usage WhenAttacking for Fatalities--� = k /k, X a !,W/wa 2 . . 24

a9 Injuries as a Function of Weapons Usage When U

Attacking for Total Casualties--E - k /k, X - kW/W0 2 . 25

10 Mean and Standard Deviation of Annulus as a Functionof Inner and Outer Radius. . . . . . . . . . . . .. 27

11 Survivors From a Gaussian Population Targeted as anAnnulus With Center a Constant Distance--n a m/o 2,P= 8/. .............. ......... . . .. 30

iv

Iia nuu ihCn.n.it..... 0.25.

12 Survivors From a Gaussian Population Targeted asan Annulus With Cons-cant Width .... p - /0 0.25,4 M / 0 - / . . . . . . . . . . . . . 3 1

13 Survivors Prom a Rim Population Targeted as if aGaus~ian Population for H/L - 2.9 ........... 34

14 Fraction Survivors as a Function of Normalized WeaponsWith Value a Linear Function of Area .......... 44

15 Variation of Value, Accumulated Value, AccumulatedLogarithn of Value, Survivors and Attack Sign WithArea Covered by Attack . .. . .4........... 45

16 Value as a Function of Distance for N Greater Than 1. . 49

17 Value as a Function of Distance for N less Than 1 . . . 50

18 Integrated Value as a Function of Distance. . . . . . . 5119 Fraction Survivors as a Function of Distance for

Various Values of m 52

20 Linear Plot of Survivors as a Function of FractionCoverage for Several Types of Attack. . . . . . . . . . 58

21 Semi-Logarithmic Plot of Survivors as a Function ofFractfon Coverage for Several Attack Types. 59

22 Probability of Kill as a Function of NormalizedDistance for Several Kill Functions .......... 61

v

-----

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

'I, - - -I

. * K*

ti

ABSTRACT

This paper is concerned with the "Square Root Damage

[ Law"--an analytical method of computing blast damage on targetcomplexes. The Law assumes that target value is a continuous

function of position, and replaces a set of individual weapons

attacking a target by a weapon density function that is optimized

to give maximum damage. This basic approach is applied to

several types of target damage functions and methods of weapon

targeting.

7 Attentton is restricted to cases where the results can be

presented in relatively simple analytical form. The derivations

are presented in some dctail to illustrate tne nature of the

• •results obtained. Charts of danage are presented for sets ofdimensionless parameters that govern the damage sensitivity to

~I' weapon usage. As a practical application, they can form analyt-

S~ical bases for explaining the results obtained from detailed

numerical calculations of the damage for a particular target

from a specific set of weapon aimpoints.

vii

•.-U.

Chapter I

INTRODUCT ION

71 In assessing the effects of nuclear attack upon target

complexes, attention often isa restricted to blast effects, or

more precisely to effects where the probability of damage is

taken only as a function of the distance from a weapon, inde-pendent of direction. In many cases the target complex is

Lt assumed to consist of a set of discrete target elements; eachelement has ain overall probability of damage obtained by com-

bining the probabilities of damage from separate individually

located weapons; and overall damage is calculated by summingI the effects on each target'element. While this type of calcula-K tion is the natural way of assessing blast damage from a

particular attack, the results are strictly numerical and no

analytical itnsight is gained from such a calculation.

Iii Analagous to the dichotomy between particulate and continuum imechanics, an approach to casualty estimation can be developed

where target elements are replaced by a target value as a con-

tinuous function of position, and weapon locations are replaced

by a weapon density function. When the target value function

i~s a circular normal function, the local fraction of damageis an exponential function of weapon density, and weapons are

optimally targeted against value, a simple expression can be

obtained for overall damage to the target complex as a f'Unctionof weapon usage. This expression has become known as the"tSquare RootZ Damage Law." A derivation of this law, using a

somewhat more general expression for probability of damage as

17

a function of weapon usage, is given by Galiano and Everett.'

A comparison of the results of this formula with the resultsof calculations against individual target elements is given bySchmidt' for a variety of cases. A surprising degree of agree-

ment of the two methods is evident.

Despite the venerability of the Square Root Damage Law, noefforts have been-made to apply this general approach to other

situations. That is precisely the intent of the present paper.In this attempt, the algebraic structure is presented in some

detail to assist in understanding the nature of the results.

Chapter II presents a derivation of the damage laws whenthe population is in a circular normal distribution, or uni-

formly distributed in a disk or in a ring. Results are obtained

from situations where the targeting is optimal against one typeof target value distribution or hardness, but the actual hard-ness is of a different type. Such results are of interest when

two separate types of target values are collocated, e.g., where4the targeting is optimized against the industry in a target

complex and the evaluation is against the population.

Following. this, a more generalized derivation of the damage

law allows damage calculations for value distributions that area function only of radial distance, but are arbitrary in shape.Several applications of th~s derivation are presented. Finally,

some factors influencing the shape of the damage functions are

presented.

'Robert F. Galiano and Hugh Everett, III, Defense ModZe IV, Famiy of Damage

Fwotion. for Z•tipZe Weapon Attaoks, Lamda Corp., Paper 6. March 1967.

'e A. Schmeidt, A Sensaitivity AnalisA of Urban Blast Fatality Calculations,Institute for Defense Analyses,, IDA P-762, Januaryj 1971.

2

44,_ At

NI

Chapter II

ANALYSES

A. SQUARE ROOT DAMAGE LAW DERIVATION

This section will rederive the Square Root Damage Law in

some detail as a basic algebraic theme from which variationswill be taken. In so doing, the algebraic structare rather

than the logical basis is of prime interest.

Assume a population, V., is a circular normal distributionso that the population density at any distance r from the centerof the distribution is

V(r) - Vo/27rt 2 exp(-r 2 12a 2 ) . (i)

In subsequent discuscions, the distribution of the targeted

va.ue, VT, will usually be different than the actual one; indescribing the former, a will be replaced by 8; in the latter,

a will be replaced by a.

Assume a weapon density w causes a fraction of fatalitiesF(w) and that the functional form is'

F(w) -1 e (2)

The damage at akuy point is

h(r,w) = V(r) F(w) . (3)

Through the introduction of a Lagrange Multiplier X, one can

'For a discussion of the relation of a weapon density to a discrete set oftargeted weapons, and how to determine values of the constant k, seeSchmidt, op. cit.

3

fti

see that for an allocation that maximizes total return

H = fh(r,w)dA (4)

as a function of total weapon usage

W = fw(r)dA (5)

we have

ah(r,w)/3w U

whenever w > 0. Since

Bh(r,•)/Bw Vke-kw

we have

/kV(6)

or

0, kV/X < 1

W II

i/k in(kV/X), kV/X 1>I,(),

and

F = 1- X/kV (8)

when w # 0. At this point it is convenient to define I

y 2r 2 /kV • (9)

y is proportional to the Lagrange Multiplier X. Now we can

write

0 exp(-r 2 /2& 2 ) < y

and/ .• 2 22

F 1 - y exp(r /2a ) . (11)

A• ~..

r .-

The weapon usage W is now found from

W 1/k ln(l/Y exp(-r 2/2a2)) 2wr dr

R 0

where r. solves

exp(-r2 /2 - y • (12)

This is readily found to beit.

W 2v/k [In(l/y) r 2C/2 r r 2/8a

W 2,W- 2w/k [ln(l/y) • r /14] ,

SW (•Ir/k)a 2 2 (l/y) , (13)

using the definition of re twice. A normalized weapon usage isintroduced by

X; kW/Ta , (14)

from which

I X X in2 (1/y)

and

y - exp(- /- ) . (15)

A critical point in achieving a final expression for survivors

directly in X, rather than two expressions in the parameter y,

is solving for y in terms of X. *1

The total fatalities are found from

icVo/2'rc ep-2/c 2

H Y exp(r2/2a e2/2) 22a. 2nr dr

0

5

= ~ --- ~ ,=t . .LŽ .t1t-~.4= rnl. -A - "•:' - ' "

True i~s readily integrated to

H -V 0 (1 - exp(-r2/2m2)) -yV 0 r,2/2ct 2

The fatalities are expressed as a fraction and the definition

of r. is used to give

H/V0 - 1- y - y ln(1/y)

4H/V 0 = 1 y (.1 + ln(1/y)) .(16)

Now using the expression for y found from integratingthe weapon usage,

HV 1 - exp(- ,'r )(1 + AF-) .(17)

If S is the fraction of survivors, then

S exp(- V )(1 +v7 ).()

B. SURVIVORS FROM POPULATION IN A UNIFORM DISK TARGETEDAS A GAUSSIAN POPULATION

TARGETED

Assumne a population V., for which damage is to be

assessed, is uniformly distributed in a disk, of radiua R with ;pOPulation density ý

6

V(r))

r R.()

The targetir1~g is assumged to be against a Gaussian distri-

r buted pooulation given by

- V -VT/ I2wrB 2 exp(-r 2/20 2 (203)

with a damage function as d~efined in the squar'e root law deriva-tion.

To relate the assessed population to the tar~etings csall

2 2e R/? (21)

Since the targeting Is identical to the previous section, then

X ln 2(l/Y),

using the derivation of the previous section with a replaced by

A To compute damage, two cases muuit be corm4idered, n~amely

Case I: r R

Case II: r c > R

where r0 as before, is the limiting radius for w > 0. For CaseI. using Equation (11)

H uf V/WH2 (1 -Y exp(r /202) 2ir drJ

t~iu22 [r2/2 0~~( 2 28 )

H/V0 2/R -r 12- & , -0)

7

Vflow using the definition of r.,

H/Vo 2 2 2 /R 2 (in(l/y)- y/y + y] 1 I/0 [in(1/y)+ y - 1]

In te'rms of X, using Equation ý15)

,•. ~~H/V'0 1/o [exp(- .'" +• 1 22)

Poir Case II

H f V0 /R 2 1 - y exp(r 2 /20 2wr dr

'4 0o--2

Thus

WHNO 21R2 [R2 12 -_2 y exp(R2 /20 2 ) + B2y1

SoH/V0 -ii,," [, - y exp(•,) + y] , (23)

or

'H/V0 1/,w [, + exp(- T )(1 - exp(,))] . (24)

To distinguish between Case I and Case II, we notice that

(, rc - R at the ýborder between the two cases; then, by definitionS• of rc

exp(-f 212/ 2 ) = y (25)

from which, using Equation (15), at a critical X

Xc

Summarizing, the fraction fatalities are given by

Si/i. [exp( - P ) + "--- 1], x _

H/V01/40 [0 + exp(- f-X )(! - exp(,))], X > 2 * (26)

8

In terTms of survivors,• Ii+ 1/10 - 1/o lexp(-,- + OT-), X < 1_

S ) ex(- e))(exp(O) - l)/X, x >e2• (27)

The fraction of survivors as a function of X is presented

in Figure 1 for various values of'P. The dashed line separatesthe two regions where the attack radius is less than or greaterthan the target radius of the assessed population. Large val-

ues of# have more survivors at constant X, in part simply

because the attack is the same and the target is larger.

To express survivors in terms of weapon usage normalizedto the assessed rather than the targeted population, leto bethe standard deviation of the targeted population about the

targeted population disk's center; i.e.,

R2 2 2wr dr

0

so =R/IT 2 (28)

SIf o is expressed in terms of a, we have 0 a /0 To normalizein terms of the actual target, let

Y = kW/ra (29)

thenY " "

Survivors can be expressed by replacing X in the previousformula by OY, i.e.,

1 + 1/v - 1/v (exp(- A'Y + PY ), Y < '

•1 • ~exp(- /#P )(exrD(- 1)/0 - > (30)

_ A

S.. . .... ,. ,.. • . .,,....• • .,.• •..,:..••;_ •. • •;; • •.:•:

4 I4

- 9-Iw

ad

Fi ca

J C 4~

I -hi

U. LL I.

IV-ID Laio-C co'

F.~# I p.- cmaa

44 =2 cc

- 0 0* 0

SHWAtAvls Noinm~

10

I

Survivors as a function of Y are presented in Figure 2.

P Since the normalization is by assessed population, the lower

curves occur with more efficient targeting. It is interestingto notice that for #a 8, where the assessed population is much

more dispersed than target population, the targeting is more

efficient for heavy attacks (Y * 10) than when 0- 1 with the

two populations matched. This occurs for heavy attacks, since

many weapons are expended outside the assessed dopulation disk

where = 1. An optimal attack against a disk has a uniform

weapon density. The survivors are readily seen to be (see(35)):

S= e-Y 2 (31)

The curve labeled 'optimal targeting' in Figure 2 exhibits

this formula and is a lower bound for any of the other curves.

C. SURVIVORS FROM A GAUSSIAN POPULATION TARGETED AS AUIIFORM DISK

ASSESSED TARGETED

Assume a population, V0 , is Gaussian distributed with

standard deviation a2 2 2

V(r) - V0 /2ro exp(-r /2a2) . (32)

11

-. 4 i~-. - v------ 7

04__ -A

C4-

ItIItQ

I- C

IJ

40

-Woo

- -~--~-- ~ ~U-, I--CI

The targeting is assumed to be against a disk of radius R with

r TIER2 R

V(r)

0 r > f. (33)Let B be the standard deviation of the disk about the center asin the last section, so

S- R/ /,7.

As before, efine 0 a to describe the relative size of

the two populations. It is clear that the optimum targeting

against a uniform value, is a constant weapon density, thus

W 7 rR2W

then

F(tw) 1 i - exp(-kW/!R 2 )

Define X by

X = kWi'w 2 , (34)

then

F 1 1 - exp(-X/2) . (35)

The damage H is given by

p R( exp(-X/2)) Vo/2fro exp(-r /2a2) 2wr dr

0

This is readily seen to give

H/V0 = (1 - exp(-X/2)) (I. - exp(-P 2 /2a 2 ))

or'

H/Vo - [1 - exp(-X/2)] [1 - exp(-l/t)] . (36)

.1 13

j .,~-.

For survivors

S - exp(-X/2) [1 - axp(-lM)] + exp(-l/#) , (37)

To express survivors in terms of the assessed population,

I we define 02

Y-kW/wo (38)

Then

Y - X/O (39)

Lj soS - exp(-YO/2) El - exp(-1/%)] + exp(-l/,) . (40)

The fraction of survivors in Figure 3 is for weapons nor-

malized by targeting and in Figure 4 for weapons normalized by

Il population; also Figure 4 shows the Square Root Damage Lawwhich represents the optimal attack. As such, it must be alower bound for other attacks. As can be seen for 0 - 1/2, the

actual survivors ara close to this lower bound for almost all Y.

D. SURVIVORS FROM A GAUSSIAN POPULATION TARGETED AS AGAUSSIAN POPULATION OF A DIFFERENT DISPERSION

Ii 1TARGETED

Assume a population, V0 , is Gaussian distributed with

standard deviation a so

14S€o4

lo ow44

N 'N 0

0.4

CL1.

IN I-

It -

- - - - -" w

':IN (A

II~~~1 ItB U

a cc;c 11

coUCc I*

L6

C4~

SSOAIAsnS N0113"A

15

SCQ

r4a

-. _o.-U-k/ L1 W,ItI It

Fiii

CL

I a

CA

- ll uCI C 3'_ A9

S•IOAI^IIINO NPikYI:It

16-1

H I. 1 "-

- -. ,. . , ...

C;L

SV0AiAvnS N0I1i:YdAI

4 r16

I 2w 2 2 2V(r) 0 Vo/2to exp(-r /202) . (*41)

The targeting in assumed to be against a Gaussian distributed

population given by

V(r) - VT/2,w 2 exp(-r 2 /28 2 ) , (42)

As before, let the ratio of the two dispersions be specified

by • * 02/02 The targeted population requires a weapon usage

es computed for the Square Root Damage Law, so, by Equation (15),

X a Tn 2 (I/y)

Moreover, for P -we again haveF - y ecxp(r2/282) , r • rc

where r0 solves (H1

2 2exn(-r /2S) Y .(44)

I. The damage H is given by

H f [l - y exp(r/282 )] V/2a exp(-r /lo 2 ) 2wr dr

Integrating gives

H/V0 - [l - exp(-r'/2a2 ))

S- y/(1 - o)[i - exp(z%2/2 2 exp(-r 1/22) .

Now using

K * exp(-r•!2c 2 ) a (exp(-r /28 2 ))$1/

17

- -

and recalling the definition of r. gives

H/V0 y/(1 - A1 -).

Now defining X as kW/wO2 and substituting for y using

Equation (15),

H/V 0 - 1 - exp(- vT/')

-[exp(- V•XT-)/(l -i)]CI - exp(- vT-/P)/exp(- V"T)]

(46)

For survivors,

S - (•/•-l) exp(- v'r-/7/) - exp(- /X-(•-l)) . (47)

To normalize by the actual population, let

Y-kW/ra2. (48)

So

Y = X/0 . (49)

Then 4

S ; (o/l-l) exp(-vY--) - 1/(-l) exp(-. f-- ) . (50)

It is interesting to notice that ifPis replaced by i/t,

the same formula is obtained. This implies that if the target

population is either too small by a factor of X or too large

by a factor of X, relative to the actual population, the same

number of survivors is obtained. The fraction of survivors is

presented in Figure 5 with weapons normalized for targeting and

in Figure 6 with weapons normalized for assessed population.

E. SQUARE ROOT INJURY LAW

Assume a population is Gaussian distributed so thatV(r) = V0/272 exp(-r2 12a2 (51)

Assume the population is targeted so that

FM= 1- e"k. (52)

18

--------------------- ,

-~~o -- 1 - - - -

= m

N N za

m ('4

Ot-._____49 wn --

to

LL. 4A f.- ~ C 2c~

_jb4Q-c lilt

CDwE

CA 0a

Lo 92cc *x

uia

3-

-' C C4

SHOAMIArS NOIIDS4

F 19

Pj

I: 0

b in

4BA~

K 0K0

ac Q

400CL.~ Q

____C m.-J

ac Vi

16(Ao

.2 .

uLuI,4Czi

-~~ 94L. f

0=-

swoA iAvnS NOILNVA

20

NQw consider anotherx.uation function,

-_ k w

F (w)1-e -

C

This function will: beuslfor two caess'

Case I: The actum., vulnerability is'.k and the effect

'of the. attacker using an i•correct vulnerabilityis~p~ int. jrestrest

Case II: T c.ie k reresents a casualtl,, rate andwe wish

to determ•n.rle injuries (casualties minus fatal-

ities).

In either case, define

"= kc/k * (54)

For fatalities, when', F(w) is used directly with k, the

square root law derivation is used, leading to

H/V0 = 1 -* exp(- -X- )(1 + v ),

where X is defined as kW/ao2 .

Call Hc the Ni-a7.tie obtained using kc.. /C

• Fc(w) V(r) dA .(55)

r H0 f0

As before, the Integration could be directly performed using

Fc -k- Y exp(g r 2 /20 2 )

However, the following change of varS.ables leads to less algebra.

Let

dV = -V dA/27r 2 , (56)

21

I''

so

V0 /2,o 2

2f 0 0 w"H 27"a i F<, (w) dV .(5TS)

From the definition of k., and substituting for w,

-k-wFc * 1 - e 1 - .(58)

2 f rl- (x/kV)] dV.H 2n~

X/k

Now let y = kV/X to give the simple expression

H-/V = 1/dy' ~1 }

From which

H /v- -(59)

For Case I, we wish values of Hc directly. Recalling

"y = exp(- -"T )

and calling

Sc =1 -H /V0 , (60)

Sc - exp(- ) - exp(-X/- ) (61)

Here the weapon usage is normalized by the original attack. iValues of S are given in Figure 7. Normalizing by the actual

+• vulnerability gives

:.Y 2-Tr P, = X (6,2)

22

- •- • -• -- .. ,- '• ,, . .= -- -.:....,..,: . .. t , . ."•/ i .,• " .• '"""-• • ''

nd(N '-I

baIIIt * 6III 9I,4', X

IN_ '0I.

ILL.

WLa

LiJ-

W- o- I-w

V I.-I-3:PC

nIV

16..

Then

S a exp(- V-7J- exp(- M (63)

Replacing [ by i/C in this formula leaves survivors unaf-

fected, which implies that either an underestimation or an

overestimation of vulnerability will give the same result.

L!i Moreover, this formula is identical to that for survivors of

-4 two Gaussian populations of different dispersions in the pre-vious section with v replaced by E. The curve of survivors in

that case, Figure 6, also applies here.

For CaseII, call injuries the difference betwe~n fatal-

ities and casualties, so

I - Hc/V - H/V0. (64)cO0 0

Then since H/V0 = 1 - y - y in(1/y), we hte, from Equation (59),

0I

I = + y ln(I/y). (65)

Thus I

S= exp(- f ) -. exp(- T-X ) + • exp(--X )

or

I=exp( -x.)i-~p(l-• (66)

Injuries are presented in Figure 8 for values of & greater

than 1. In this Figure, the attacker is attempting to maximize

fatalities. If toe attackei' is attempting to maximize total

casualties--injuries plus fatalities--then values~of t are lessthan 1; i.e., the vulnerability of the evaluated population is

less than assumed by the attacker. The injuries in this case

are presented in Figure 9.

24

! •1

C43

C.4 - D

U-INN~~U It_ _

IIV)

1-1

Ciz

s~vn~iN0-:VH

25-

IcIU.I.

400

Cii

L~sin~ ____ ________ -

26U

F. SURVIVORS FROM A GAUSSIAN POPULATION TARGETED AS A UNIFORMPOPULATION IN AN ANNULUS

I ASSESSED

TARGETEDpim

LI I I , I

II

Assume a population, V0 , is Gaussian distributed with

standard deviation a so• 2•2 -2 22

V(r) = V0 /2ra exp(-r/ ) .

The targeting is assumed to be against a uniform ring with

inner radius L and outer radius H. In order to normalize the

evaluation in a way comparable to other situations, a "center

of gravity" and "standard deviation" of the ring is defined

by

H• m 1=/A r " 2nr dr , (67)

where A is the ring area = Tr(H2 - L2), and

HB - 1 'A (r 22r dr (6n)

L

27 2

-rn areawher A i theW(H anHI

This yields

2 H 3 - L 3 (69)3L

2 H4. L4 m2

B f (70)

The expression for B is a measure of dispersion about the"center strip" of an annulus. Thus, as the inner radius of

the annulus approaches 0, B does not approach the value com-

puted for a disk which is about the center. In fact, with

L - 0, we have

Bring 1/3 Bdisk

There is no simple solution for H and L in terms of m and

B, but a simple Newton's method type of procedure will allow

for quickly determining numerical values. Figure 10 shows m

and B as a function of H and L. As the thickness of the ring

becomes small compared to a radius, the ring curvature is

insignificant in determining m and B, so that we have approxi-

mately

iH Pjm + 3B ,B/m << 1,

•L ;3 m - /3B- ,B/m << 1

The departures of the contours of constant m and B in,•..Figure 10 from straight lines where L/H is small indicate '

departures from this approximation.

The ratio of target to actual standard deviation, B/a,we shall call p, and the ratio of ring center to standard

deviation m/a, we shall call n. At small L, the 0 used previ-

ously is related to p by 0 = 1/9P 2 if 0 is substituted for

3B.

The total weapon usage W is given by weapon density times

ring area, or

28

4-00000'e\

ac

C4C

_~'AC

00,,

CC

00I-

1~~a SfIOoNIN

Hw 29

-~~~c 71 77-' a.~.a 4½. ,~..I--~

W a r(H 2 - L2 )w

Call X the weapon usage normalized to the target, or

X , kW/wB 2 a kW(H 2 - L2 )/B 2 (71)

Call Y the weapon usage normalized to the assessed population,

Y -kW/wB 2 - kw(H 2 L2 )/a 2 . (72)

The succeeding calculations will be in terms of Y but could

be readily converted to X by changing a to B. The calculationswill be done explicitly in terms of H and L, which are functionsof m and B. As the expression for Y shows, these functions can

be considered as normalized by a; I.e., the expressions are in I'

terms of H/a, L/as p and n.

The fraction survivors k is given by

F 1 1 - exp(-kw) 1 i - exp(- ) (73)H2/a 2 L2 /02 14

The value destroyed is given by

H

H *f F *V dA;

LI

fY 22 2(1- L2 ) V,/2wa exp(-r 2 /2a 2 )2wr drL H2a2 - L2/aL

Integrating, and letting the fraction of survivorz, S -

I -H/V 01 gives

S" 1- Il -expY [exp (-- -exp -.S(H/a2 _ (./o)2 .c 2!

(74)

(A similar formula based on S is attained by substituting X for

Y and B for a.)

30

Fraction survivors are indicated in Figure 11 for the annu-lus center, m, at a constant value with varying annulus width,

B; and in Figure 12 for the annulus width, B, cor.itant and vary-

in& distance of the center m. In these figures a curve for

L - 0 is given which reduces to the disk calculations presented

earlier.

When L/H is near 1 and the approximate formulas for H and

L can be used, we have

S f 1 - [1- exp(-Y/(4 /vnp))][exp(-(n- /3P)2)

- exp(-(n + /1P) 2 /2)] . (75)

G. SURVIVORS FROM A POPULATION IN AN ANNULUS TARGETED AS A

GAUSSIAN POPULATION

TARGETED

- ASSESSED

Assume a population is of constant density in a ring ofinner diameter L and outer diameter H. Let m and a be themean ring distance and a the standard deviation. The relations

between H and L and m and a are the same as in the previous

section, with c replaced by B. The targeting is assumed againsta Gaussian target distributed

2 2 2U ~V - V /27wo exp(-r /20

As in the square root law derivation, the weapon usage is

31

-:

7 M-J

Sn -

00

oc

I-Lýo iJII

IIL

C-i

0~

ALL

LIl

I " INC

h-

-~ II I. -

a •i

- I AiAvnS NOII:)"I

33........ ..........

. ". -I, i _-. - ,. .: .

x - kWwO2 - 1n2(l/y) y

with

22wO A/kV 0

The fraction damage F isS2 2

F w 1 - y exp(r /201

and extends to a distance rc where rc solves

y a exp(-r6/2B' 2 )

For r less than L there is no damage and-the fraction

survivors is 1. Otherwise, for damage

J

H a J El - y exp(r2/?2S)] IV 0 /w(H 2 - L2 )] 2wr dr , (76)L

where

J " min(r ,H) • (77)

This readily givesj2 2 ( 12 epj/12

S I H/V0 - - 2 exp 2 X 2 exp(J2 /202

H L H L

- exp(L 2 /28 2 )]

If rc > H this simplifies to

-r 2/ a2

(-exp 2 2 [exp(H2/282) - exp(L /282)] . (78)H L

For r. < H this can be written

S =1 2- 22 V - L 2 + 2 2 ep Y_ x( 22/ XL 2 22 2 2

Sa -+ H _L2- 2 exp(- IT ) •82 exp(L2/282).''

H -L H -LH2 L2

(79)

• 34

In terms of the assessed population, the normalization becomes

SI kW/wBs I (80)

where, * B2 /B 2 . The fraotion of survivors as a function of Y

is shown in Figure 13 for an annulus of constant ratio o.f outerto inner diameter of 2.9. When the targeted population is notdispersed enough, the weapons are concentrated in the hole in

the annulus. As the standard deviation of the targeted popu-lation is increased, the tarpe.-ing becomes closer to optimal

targeting until finally the ,apons are too dispersed and th3

ý4 targeting effectiveness drops off.

H. AN ALTERNATIVE SQUARE ROOT LAW DERIVATIOIO

In this derivation, the geometric context of the square

root law will be emphasized. As before, let

V(r). V0/2Trcta exp(-r /2ai)

Let

•;2 2

I' .S1i!•'! = • (81).

Then (8

V(r) Y Vo/T6 2 exp(-r 2 /6 2 ) .

Now calli'2!•, nS AT (82)

so AT is the circle which contains 1 - l/e, or 63 percent of the

population.

Let

A r•

•..• 35

I 12i __ __ -

IN

CI 2Ln

-~~______ __ __a-

-Go NOV0

SHO~lmnS N III)"

~ ____ _36

Then

V(r) - Vo/AT exp(-A/AT) . (83)

As before, let

F(w) - e-k

The optimal allocation yields the requirement

Ee-kw - X/kV, for X/kV < 1

Now let s,(w) denote the fraction of survivors, locally, so

s(W) - e-kw - X/kV

Call G = sV the survivors per unit area. Then

G - V • AIkV = X/k . (84)

Thus the survivors per unit area are a constant for the area

attacked.

The attack continues to a radius where X/kV = 1. Call the

area enclosed Ac. At this radius the target value equals the

survivors per unit area. Thus,

V/G =1,sooVO/GAT exp(-Ac/AT) = l,

or

ln(V0/GAT) = AC/AT (85)

Define, as before, a dimensionless weapon usage by

AcX =kW/n2c 2 /AT kw dA. (86)

0

Now, since

e-kw = G/V

37

---------------------------, .------.. -- .. •,

kw - ln(V/G) - ln(V0/OAT) - A/AT

ThusAc

X - 2 /AT f (ln(V0/GAT) - A/AT) dA0

- 2/AT ln(Vo/GAT)AC - 2/AT2 Ac2 /2

- 2 AC/AT (ln(VO/GAT) - 1/2 Ac/AT)

Since Ac/AT - ln(VO/GAT) '

X (Ac/AT) 2 (87)

or alternatively,

2X - in (VO/GAT)

Divide survivors-into the area in the circle Ac and therest of the plane

S- S1 + 3

Si - GA•

S Ac V dA Vo/AT exp(-A/AT) dA

Ad Af

= V0 exp(_Ac/AT).

S - GAC + V0 exp(-Ac/AT).

Normalizing,

S/V 0 - GA /V 0 + exp(-Ac/AT) (88)

Now S will be computed both in terms of G and Ac.

38

First, geometrically:

S/V0 - AT/V 0 • AC/AT + exp(-Ac/AT)

Now,

- ln(GAT/V) -A/A

so

,AT/V 0 - exp(-AC/AT)

SS/V 0 , exp( Ac/A T) AC/AT + exp (-AC/AT )

Sexp(-Ac/AT)(AC/AT + 1) . (89)

And since

Ac/AT -

S/V0 , exp(-4 7 )(X f-+ 1)

Now in terms of survivors

S GAT inV 0 + GAT

0 V0 GA 0

" GAT/V (In(V0/GAT) + 1 (90)

Now

GAT/V0 - exp(-vrT ),

so S/Vl exp(-Vf-)(vT-+ 1)

1. A GENE(.,i,•ZATION OF THE SQUARE ROOT DAMAGE LAW

Suppose value, as a function of area covered, is an arbi-

trary decreasing function. Then the use of weapcn density can

still be used for a more general form of the square root law.

39

"7. 7

Let

0 A 11A)(1AT AT

j be the value per unit area where value density is a decreasingfunction of area; i.e.,

df) 0 (92)i dxLet

g(x) - f(t)dt . (93)

'f X

wJV(A~dA V (0 r A. dA-Vo f (AlAJ (0 T AT

'V f(A)dA

Thus we require

g( 1. (94)

Moreover, to normalize distance in a fashion comparable to the

Gaussian case, set

1 - 0.63 (95)e

The damage, as a function of weapon density, is still

F(w) =i-e-

U The optimization of an attack yields

a V(A)F(w) V k-•

~4o •o ~Ii

- . - - - -- - -.---.-• ... . . . . . . . . . . ... . . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . . . , . . . .- . " , , • .: . •

50

e-kw A f -x

Call a the local fraction of survivors and 0 the local numberof survivors per unit area. Then

G - Vs - V(l-f) - Ve- a - Av- x.

Thus 0 Is a con3tant in the area attacked. The attack is made

out to a radius where

kV

or

V-

Call A the area attacked- ThenC

(A A T (96)A "-V_

The weapon usage W is given by

W " wdA

Let X be the normalized weapon usage, as before, so

A C

2kW 2A - ;T- kwdA ( 97)

Now

e-kwn G_V

so kw -n(i) l(in '(0))

41

Then

2 fA f 1

Ac0 0

Ac

Ac 1 P' I T inf.)d (98)

IT

20

Now define

K he h~x) = in (t~t))dt . (99) iA2 V 2hI l( f~x) (100)

S) T he n

X -2- in +2h T (100)[ The total survivors is the sum S1 of the survivors in the areaattacked and S2 in the unattacked area with

S2 -V -

242

S A - G

The fraction of survivors is then

S AC OAT A

IHJ. AN APPLICATION OF THE GENERALIZED DAMAGE LAW TO VALUE

A LINEAR FUNCTION OF AREA

Let value be a linear function of area, say of the form

2 A

V(A) (102)

0 A > AF

For a circularly symmetric value function, value woulddecrease quadratically with distance. After normalizing withthe area AT) we have Ts;Vo(

T- - -' AT

v A (103)0 , ,:/A 1.

'IT e

where 6 - 1 -v'/7e m 0,39346934. (10J4)Then

25x e 2x2 , x <g(x)

, X >

and it is readily seen that as desired for

normalization. Moreover, in terms of t•he original triangle,

AT - 5AF

"43

For h(x) we have

-h(x) x - Olog(2- 2i 2x) x + log(25) (105)

Now suppose a value of 0 (survivors/unit area) is given.

Then with AC defined as the area under attack, call x AC/AT.

Then

i W AT 1x° """ "0 24-• '

Now from

X 2x in ( + 2h(xc)

we have

2 In 0 1 T + 2(log 26- 1) (106)T - .and from

S_ !AT• 1 g(x,)

we have

s 1iGA T 1 GA T2 17

Numerically, this gives

~~ifGA\-5 -0 8 2 9 8 8 1 i n .+6 .4 5 9 1 9 2 6 .3 0 0 8 9 7 2

V/ = 2.5444 \ .34900

V VolvolIn this case no explicit function -I(x) is presented.0

GAT

Solutions must be obtained in terms of the parameter -T--"

The valid range of area attacked xc is 0

I0 < X Xc _e,

4414

____ ___

which gives

SOAT24 " 0.79693868 > V-A 0

SThe survivors, as a function of attack intensity, is pre-sented in Figure 14. The square root law, with Gaussian tar-gets, is included for comparlson. As can be seen, practicallyno difference occurs between the two curves until the survi-vors are under 50 percent. Then the larger amounts of area at

low value density in the Gaussian targets begin to signifi-cantly affect the attack effectiveness. In Figure 15, thesurvivors, attack size, and several other parameters are pre-

sented as a function of the area attacked. As is clear in the

Figure, not until almost all the area is attacked does the

weapons usage become relatively large.

K. AN APPLICATION OF THE GENERALIZED DAMAGE LAW TOVALUE AN EXPONENTIAL FUNCTION OF AREA TO A POWER

Let the value as a function of area be given by

V(A) - V0i exp{-(BA)n} , 0 < n < . (108)

The area A is wr 2 , so this can be written

V(A) V V0i exp(-{Bir21n)

For n = 1 the standard Gaussian shaped curve is obtained.

For n- an exponential decay is obtained. 3 and 8 areparameters whose values will be determined by normalizing the

value function. The damage will be calculated using the secondalternative derivation. To normalize, call

f(x) = • exp({SAP) , (109)

and X

g(x) f(t) dt , (110)

45

Rum

F

A

0Z

CD,,

in•,.

,

UJ x

t

o,-A ,,'

IN • '

-

f

0

* ,_

" II

-I lip It

0.'t

'-'1o) coC

SVOA;AvnS NWvlý)"

C4C

X 4m

M ~ N C ~ '

-I~NO

L" UVCn7.c 4Ku X

--I_ J -

AtU

ZIP

s40

INI-

It

UK)

(*1 C4

47a

Now for normalization we need

g(ca) =1i.

From the integral

xne-(rx)m dx mrn r , n+l, r, m > 0]

we have

from which

f(x) = n exp((x)n) .. :12

rI~and

g(x) exp dt exp( y y . (113) ""*

V. fThr rfzl

For normalization we wish

g(l) -1 01 - .'63212055509

Thus 8 can be obtained numeric'ally a P afi'lnction of n.. n..

An approximate equatIon for 8 i . . *.t

I0(-0. 2 2 5 + 1.3 2 (iOg,04 - iogi)'), n < 4 .

0.57 + 0.036! g N , n .. N.,",.)

calling a an. approximate equatidn fora is (115)

n

0.8.+ 5.81og1 n , m <,.6

1/a, ((116)0..4i453(l m-I'.09) m > M. ,.,

I'. 1 14

Thef f(x) a ex~p(-(Bx))

w. mxe now a and:, 0 cani be expressed as-'functions of n. Now for,

val-w,-, ie' carvi a1so write

_,\n)A (117)

and pr'eaerve-norma-aizatioh. -Here AT is the Arlea 'Within which

I,- of t)ie Population resides., Ther, ,x A/ T~cms ra1Trexpressed in un is of -AT.,

wihNow let a num~bet, of' 6uYrvivor5 per unit area,' G be giv~en. We,wihto. f i.nd %nf, x 0 Ac/A) s0 t hat

f~x GAT . 18

h~)f C) i) dT(1)

0

Theus x ~ x solves.,p((~~n )t

a x ,,xIy 19

c

7 '" 7 -7 -- -' ---

ln particular,n-+1

h(x =x` in W ) - a(122)

Now

X A2 n(1X + 2h'(x ).(13C C

Finally, for survivorp~ . . . .

IC C.

The followinK table compsires, exact-values uf 0 and a' withvalues f romm the f'itted c,.urves.

n 0)KIC't 1i F() 1 -In "e xact. Oi.

0.125 g7.100.0,00 197,840 5.040 Y0.0014878 -6. 51044 672.1230 84.600.25 357.473 2.01 O .0M714 -2.701 14.89 14.760.5 . 4.6224 5.0O 1 0.43268 -0.0~777 2.3-112 2.580.75 1.55201 I.Eý 6.89296 0. 7672 i -4.6506 1.3035 1.18'1.0 1 .40231 0.9;6 1.00 0.9977 -0.00h3 1.0623 1.001.5 0.716634 064 1.3640 .1.25$6 0.230?9' 0.7A39 0.85292.0 0.63778 C .623 1.1;24 1.349A 0.3289 0.7197 0.7897,4.0 0.58720 0ý691 3-AI46 1.5436 0.31 0.647A 0.70668.0 0,497143' 0.602 17.534 4. V76 0.4554 0.b341 0.6?#0S

54.0 0.24_{ 0. 61P 63'.44 V.b99 0.4674 0.6329 0.6437

Figitres .76 and 17 pre' 6e nt values 'r, a f'unction of distancef or n > I r..ad n < 1. that iQ, the .ý.o'ves preý,-nt f (x) as a funn-

tion of /iE. The I'ntegr&l of' the value F,(x) 'is pr,ý-ented in

Figure 18 as a function.Qf distance rx 'for the extreme valuesof n considered%. In!P'i;ure 1.9 the fraction survivors as AA

'Ttme ,,exacttl values9 of arnd a ame f'cun thrOV&i solving the equation

e f

This wa&s acccmUp1:Ahed. n~anrica11li by evaL~uating the integral by Sinpson'sfrule tjlijg 1,000 s4Peps .tur dif.06ring, values of 8 urntJ.l the desired value war,obtai~n~d. Sincee at nwl 1 0 ir~mld be 1 ex'a-tly', VtIe quoted vp&&Ie givea~ an,

i ýic noi of the error In this ruinerica1 prccedure.

1,J 50

Iii

000

Nt-

x

V 1-4

F 51

7 ._ _ _ _

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ___ _a

CD

Lk.1

tj

10 0 L

C. 0*~ 0

100

52*

_ _ _ _ ._ _ ._ _ ._ _ ._ _ .q ...

I - ..

LLI-

CCc

LA-

53K

- . - ~ a ~ n r w w a c

II _ _ _ _ _ _0_

_ _ _ _ _ _ _ _ f

* Cp

NOI1DV~e~. 'c

function of normalized weapon usage is presented. For n *1,

the curve is the usual square root damage law. For n -64,

the value is almost uniform and then rapidly drops so the dam-

age is close to that of. a uniform disk optimally targeted; i.e.,

the fraction survivors is an exponential function of weapon

usage. For n a 1/2, the value is an exponential function of

distance. For values of n less than 1/4, no curves are pre-

sented since the decay of value with distance is extremely

rapid and numerical integration using Simpson's rule over inter-

vals (the algorithm implemented here for numerical evaluations)

becomes tedious.

1. FACTORS INFLUENCING THE PROBABILITY OF KILL AS A FUNCTIONOF WEAPON DENSITY

In this section some rational for the use of a weapon den-

sity is presented. This is done first by studying the survival

probabilities when a larger number of weapons arrive at random

in a large area. Next the probability of Survival from weapons

delivered in regular patterns in an area are considered. Thisis done analytically for weapons where the probability of kill.[ for the weapon is one inside a weapon radius and zero outside,

and numerically for several typical types of assumed weapon

kill probabilities as a function of distance.

Suppose weapons are delivered In an infinite plane so that

any point in the plane is as likely to receive a weapon as any

other. Then In any specified area the number of weapons deliv-

ered is a Poisson process with the distribution of number of

weapons given by

P(ni) -e~wA (wA)n (125)

where A is the size of 'the area. Since the expected number of

weapons in a unit area is w, we will call w the weapon density.

55

, ! -

Consider any monitor point in the plane and let this moni-tor point be surrounded by a set of N non-overlapping but touch-

ing annuli, where the jth annulua is centered a distance rafrom the monitor point and has a thickness dr .

The area of the jth annulus is given by

A - w(2r dr + dr 2 )

The probability of n weapons arriving in this annulus is

-WAj (wA )n

P(n) = • n( )

The overall probability of kill of the monitor point fromthweapons in the j annulus is the sum of probabilities of kill

for one weapon arriving, two weapons arriving, etc., or

PJ" " Pk (rj )caAaej + .. +(- (i - Pk (ra ))f)(wAa -- a•n.

The rings form a mutually exclusive set of areas covering

te plane, except for a distant final outer section surroundinge rings. Then the probability of survival of the monitor

point is given by the product of the probabilities of surviving

the weapon arrival in each ring.

Now let N become indefinitely large so,(l) for all j for

j 1,2,...N, lim A - 0 and (2) lim rN ,. It is clear that

this can be done with a countable set of rings; for example,let rj j a 1,2,...N. Then the area of each ring is

A + < '72 i37-2)

and lmA = 0.

56

Then

N -wASS Up l n P k k(r )wA e J 0(

Nj

[~. - O~r2)11 [M Pk(rj)w2wrrjdrj + (drN*o J~1

Now taking logarithms of both sides,

in S - in limn [ - Pk(rJ)w 2 yrjdrj + 0(dr 2]

N

- irn 1 in(1 - Pk(rj 2Q rj drj + O(drj 2))

-jI

Now2 3

in (1-x) - - (x + + i + "'" ) Ixl < AiI

Thus,

inS rn0 (dr+0a k (j )

Now, assuming the Pk(rj) are such that the sum converges,

which is, ertainly true for all practical damage functions,

pA

in S lf _Pk(rj )w 2 wrj dr

-W Pk(A) dA (126)

57

::AM

I2

Now Pk(A) dA is often equated to wWR 2 where WR is0

called the weapon radius. The circular area is the area of a

cookie cutter damage function which does the same damage.Thus, finally,

2S e-W(wR) (127)and2

P 1 1 -

This is the form of the damage function of Equation 2 which

underlies the square root law formulation.

A suggestion by John Donelson1 gives the same result in amuch shorter fashion and in a way which makes the answer

intuitively clear. The expected fraction of the plane covered

by lethal effects is wr(WR) 2 . A small change in this fraction,

i.e., Aww(WR)' gives a proportional fraction change in tha

survivors. So

_ -• .* Aw r(WR) (128)

or

-AILn S - AwI(WR) 2

2(WR)2S J:- "

The random pattern of Poisson arrivals Just discussed A

might be appropriate for an attack against certain types of

valuable targets in an area where population happens to be cal-

culated, and where the location of the population is not sig-

nificantly influenced by the location of the other valuable

targets. On the other hand, if an attacker were interested in

1Formerly with Institute for Defense Analyses, currently associated ItthScience Applications, Incorporated. 2

58

attacking an enti:.e urban area, then he might place his weapons

in a regular pattern. Auy target in the urban area is some-if where within the pattern, but unless some reason ia given tofix the location of the weapon pattern, the target may be any-

where within it.

Suppose weapons are placed at the intersections of a squaregrid with mesh length 1. Then the weapon density w is simply

W • 1/ 2 . If the probability of kill is I inside some weaponradius WR, and 0 outside, then the expected kill is simplydetermined by the fraction of the plane covered by weapon circles.Call p - W./,/. the fraction coverage, and expected kill is

readily seen to be

•ir I 2

w22

S2cos-'() 1< p <

1 < p . (129)

The fatalities can be given as a function of the fractioncoverage of the plane, C, to be consistent with the previous

section. Then I'C •(WR)2 W it 2 (130)

If the weapons are dropped in a more closely packed patternof equilateral triangles of side length 1, then w 2//3k2 .

Again, calling p WR/A/2 for fraction fatalities,

,- 0 < < I

V- 2cosl(-) 1<0P< 2

2

The coverage is given by c = n 2

59

LLA

-' 1J. I-- I%tAj LM0

VIa <II B CD

z

I-C LU.

LL..

, -4 I iL~

C4-* . I I_ _ _ I.

SS*~n j a * iV -

I 60

_ ---- AIR BURST 12 PSI I=

0 -AIR;BURST 12 PSI 6 0.5

. . . . . . . . ...... 6..... SURFACE PMJRST 10 PSI 6=0.5

':4,,,-- SURFCE BRST 0 PS 0.

OF FRACTIONSCOVRAE BORS SEVTERALATCNYE

0.061

.161

The f rac't ion survtvojras f`Udqtup.1on Pj ..qovet~age v pre-

sent ed as a. l1inear~plot 4:Vgr-2Oapd asa. Oemli-'1of~arit hznic

plot in Yigure'21, 4.,Also 'presented -are: suwrvivorIs' f or,tbe rand~lmdelivery, con'side'red earjier. ,Thý,*bottom .atraizhtiline' n.

Figure 20' repreaei'eits, th4e 3U~rvivQra-:if? weapon Jrot Tect' co, ld, be

perf~ectly. rlaced,, e0g'., -kqa~re I e ýha tk, eectz' a~e~as covering

the plane foir weapons pla.qe#i in' t"he 6enter'o~f. each square. As

can,.be 3aeen* th*pre is ''no, oveisJla for' the eq~u' ater~l, trlantle,

-~tenuntil 0.9 _Vý& th' lah6. is cov~ez, b~ pots ,effec ts'.

Even then there, i-a ?ess- than "four,.,percent differi~nce betw~een

thla coverage and t~hie no0 overl'ap val'iation,

rTT t~he fra~cticnf damage as a function of distanc~e is a

gradually decrea-ing fun~ction of the distance, t~e n e~ff eats of

.Over'lap m~ust be, considere4. Figures 20-i:nd -2.1. calqulat±i~ns

for Tol.r casefe,.of Fizure 2.2 with wepn(,tot--.n surs

In Figures 19 ad20'j$t anbsenhat the e frome

t~he equivalent of almost perfect plac~ement to al~most. random.[ ~placement.

Some typicul, .fr-actlon damage as a function of distance

example3 are shown -in Figu~re 22. They rep~rese~nt a ran~ge from

very steep cutoffs in weapon lethality to ,quite gradual ones.The labels give- an indicatiun of the type of weapons target corn-

bijnations considered. In obtaining these curves fraction kill

as a function of distance is obtained by combining a function of

fraction damage as a function of overpressure with a function

or overpressure as a f'unction of distance. The fraction of

damage as a function of overpressure is assumed to be a -.uru-

lative log normal curve and the value a in Figure 22 is the

ratio of the standard deviation of the cumulative log normal

curve to the maean lethal overpressure.

62

S- 4 4

3,;3

- 40

) 3..

... . ... .............. ......__ ...... _ .... ...........

LAU

K .. z

00

co 4-4

00

U-

Id.-

oS

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