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U n c l a s s i fi ed

U n c l a s s i f i

. .

OEG

Study 626

P r o b a b i l i t y- of - Damage P r o bl e msof Frequen tO c c u r r e n c e(U)

Prepared by

Operat i onsEval uat i on Group .

Of f i ce of t he Chi ef of Naval Operat i ons

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Copyri ght CNA Corporati on/ Scanned Sept ember 2002

THE OPERATI ONS EVALUATI ON GROUPh as s i n c e1942 had t he r e s p o n s i b i l i t yf ordoi ng o p er a t i o nsr e s e a r c hf o r t heChi ef ofNavalOp e r a t i o n sand Naval o p er at i ngf o r c e s. Thi s group of c i v i l i a nsc ien t i s t s ,managed undera Massachuse t t sI n s t i t u t eof Technol ogy con t r ac tand s har i n gs common manpower poo land r e s e a r c hmanagement wi t ht he Naval Warf are An al y s i sGr oup,a dv i s e st he Ch i e fofNaval Op e r a t i o nsand c e r t a i nF l e etand F o r c ecommander s i n o p er a t i o na lprobl em sus-c e p t i b l et o q ua nt i t a t i v e a na l y s i s. These i nc l ud eprobl em i n t he e v al ua t i o nof newweapons, o p er a t i o na l t e c hn i q u es ,t ac t ics , f o r mul a t i o nof new r e q ui r e me n t s ,t echni ca la spec t so f s t r a t eg i cp l a nn i n g,and t he c or r el at i o nof r e sea rchand devel opment w t hNaval needs.

OEG pub l i shes th r ee typesof f or malr e p o r t sof i t sresearch ,i n a ddi t i o nt o 'many memor andaof l i m t e dd i s t r i b u t i o n -

STUD ES present i ngt he compl e ter e s ul t sof t he ana lys i sof some s i n gl eNaval probl em s

REPORTS c o nt a i n i n gt he ex tens ivei n v e s t i g a t i o nof an e n t i r ewa r f a r ef i e l d ,or of somes i m l a rb r o a da r e aof r e sea rch.

SUMMARY REPORTS whi ch ar e r esumes of r e sea rcho r i g i n a l l yp ub l i s he d i nano therf o r m These presen to nl y r e s u l t swi t h o utsubs t ant i a t i ngana lys i s ,and ar e desi gnedf orgenerali n f o r mat i o n. -

A d i scuss i ono f e a c hy e a r s f o r ma l l y r e po r t e dr e sea rchi s a v a i l a b l ei s t he cur ren t0EGAnnual Repor tt ot he Chi e fof Naval Opera t i ons. Copi esof t h i sand otherpub l i cat i onsar ea v a i l a b l et o thosecommands and ' agenci esauthor i zedf or suchmater i a lf r o m -

The D i r e c t o rOPERATI ONS EVALUATI ON GROUPOnce of t he Chi e fof NavalOpera t i ons( OP- 03EG)Washi ngt on 25, D C

For Of f i ci alUse Onl y

Copyright CNA Corporation /Scanned October 2003

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DEPART MENT OF THE NAVYI I I / ~ OFFI CE OF THE CHEF OF NAVA L OPERA T ONS

WSH NGTON 25, D C

From Chi ef of Naval Operati ons- ' To : D stri buti on Li st

11 DEC 7959

4 . I f addi ti onal copi es are requi red, the wl l be f orwardedupon request .

D STRI BUTION Attached l i st

ew nl ~R MASONBy di r e ct i o n

I N REPLY REF ER TO

Op03EG si bSer 70P03

Subs ; Operati ons Eval uati on Group Study No, 626 f orwardi ng of

Encl ; (1) OEG Study No . 626, Uncl assi f i ed, enti tl ed "Probabi l i ty-of-Damage Probl em of Frequent Occurrence"

1 . OEG Study No . 626 ( encl osure ( l ) ) , prepared by the Operati onsEval uati on Group, i s f orwarded herewth f or retenti on

2 Thi s study summari zes some of the sal i ent methodol ogy whi chhas been devel oped for the sol uti on of probl em i nvol vi ng thecomputati on of the probabi l i ty of damage to ml i tary targets .

3 The Chi ef of Naval Operati ons r ecommends that thi s studybe revi ewed by those responsi bl e f or anal yses whi ch i ncl ude thederi vati on of the probabi l i ty of damage .

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D STR BUTI ON LI ST FOR OPNAV SER AL 70P03

CNOUP72

93313334505206C07T

SNDL(6) = C NCPACFLT, LANTFLT, NELM

22 ( FLTCDRS)24A ( NAVAI RPAC, LANT)

26F ( OPTEVFORC OUmS ( i ncl udi ng K4VTEDNLDEVDET : MnWar-Eval Det ; G1VEU-1)

OTHERDI RI I !SEG (2)SENNAV14i BRyVSEGDODR&ECOMASDEFORLANTPRESNAVWARCOL (5)SUPTNAVPGSCOL (5)D RNRLCOScD RNEL

coMxoLsCO&D R DTIdBCOl `dI I J OTS (2)COI uI NPGC yZCI NSORD APL/ J HUDept of AF ( Attn : , AFC. AS) (7Dept of Army ( Attn Ad Gen) (6)D R, Long Range Studi es Proj ect, NWCThe BAND Corporati on ( vi a USAF Li ai son

ORO J HU (3)O f i ce) (3)

o

4004304324364914 92493

BUSI i I PS04 5)

Chi ef , BUWEPS ( 10)

BUAER

AER (10)RS

BII ORDr-( 0)ReRe qRee

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UNCLASSI FI ED

UNCLASSI FI ED

OPERATI ONS EVALUATI ON GROUP

STUDY NO 626

PROBAB LI TY-OF- DAMAGE PROBLEMS OF FREQUENT OCCURRENCE ( U)

11 December 1959

Thi s Study represents t he vi ew of t he Operati ons Eval uati on Group at the t i meof i ssue I t does not necessari l y ref l ect t he of f i ci al opi ni on of t he Chi ef of NavalOperati ons except t o t he extent i ndi cated i n the f orwardi ng l et t er . I t i ncl udesi nf ormti on of an operati onal rather than a techni cal nature, and shoul d be madeavai l ab e onl y t o those authori zed t o recei ve such i nf ormti on .

Encl osure Z- - CNO Ser 70P03

Dat ed 11 Decemer 1959

Prepared by t heOPERATI ONS EVALUATI ON GROUP

O f i ce of t he Chi ef of Naval Operati ons

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These are some of the pri nci pal resul t s :

SI NGLE PONT TARGET

Consi der the t arget at the ori gi n of a rect angul ar coordi nat e syst em i n asui tabl e pl ane Thi s pl ane mght be the hori zont al pl ane or a pl ane norml t o thet raj ectory of a weapon whi ch woul d pass t hrough the t arget . Let the cond t i onalprobab l i t y of damge t o the target, i f the weapon i s fi red or rel eased on a t raj ectoryt hrough the poi nt ( x , y) , be pd(x, y) . Let the j oi nt probab l i t y densi t y f uncti on of

x and y be f (x , y) . The j oi nt densi t y f uncti on f ( x , y) descr i bes the densi t y of shot sor bomb-drops i n the t arget pl ane I t i s non-negat i ve and i s so chosen t hat the

ABSTRACT AND I NTRODUCTI ON

- Probl em i nvol vi ng comut at i on of t he probab l i t y of damge, or t he expect ed. f ract i on of the t arget damged, occur f requent l y i n ml i tary operat i ons research

These probl em have been wor ked on by many peopl e over a l ong peri od of t i me - Thi s study i s i nt ended t o summr i ze some of the mor e i mort ant resul t s and t o

. i nd cat e sources of i nf ormt i on on ot her probl em . No at t emt has been mde t oi ncl ude an ext ensi ve b b i ography or to t rack down the f i r s t man who deri ved any

- part i cul ar z . s u l t .

A t hough mny of the resul t s were known l ong ago - some of t hem duri ng t he19t h cent ury or earl i er - i t was not unt i l Wrl d Wr I I that systemt i c efforts weremde t o obt ai n answers f or a wde vari et y of si tuat i ons . Recentl y several revi ewshave been mde of t hi s mt eri al . One of t hese i s ref erence ( a ) , whi ch does noti ncl ude some of the resul t s gi ven here On t he other hand, ref erence ( a) gi vesresul t s on t he i nverse probl em that of det ermni ng t he probab e l ocat i on of t heai mng poi nt f rom the l ocat i ons of the burst poi nt s or f rom the damge produced The i nverse probl em i s of l ess i nt erest than the d rect probl em the resul t s are

of l i t t l e usef ul ness i n most damge probl em.

SI NGLE-SHOT PROBABI LI TI ES

I n t hi s secti on we w l l consi der the probl em of comut i ng t he probab l i ty ofki l l i ng the target , or of i nf l i ct i ng a stat ed degree of damge, by a si ngl e weapon( proj ect i l e, bomb, et c . 1 Because of i t s f requent occurrence t he t wo-di mensi onalcase w l l be used t hroughout f or the i l l ustrat i on of met hods and st atement s ofresul t s . I n most cases the mnner i n whi ch the resul t s woul d be adapted t o onedi mensi on or t o mor e t han t wo di mensi ons shoul d be evi dent . The deri vat i on mustbe examned, however, i n every i nstance i n whi ch onl y the resul t s are gi ven .

W w l l consi der bot h poi nt and area t argets, a poi nt t arget bei ng one whosedi mensi ons are sml l comared wt h t he "damge radi us " of the weapon re ati vet o the t arget . W w l l al so consi der t wo t ypes of "cond t i onal damge f uncti on",that i s, the f uncti on t hat descr i bes t he probab l i t y of damgi ng the t arget i f theweapon det onat es at a gi ven poi nt re ati ve to t he t arget . These t wo f uncti ons areusual l y ref erred t o as the cooki e-cutt er f uncti on and the Gaussi an f uncti on . I nthe f ormer case the probab l i t y of damge i s 1 wt h n some area about t he poi ntof det onat i on and i s zero out si de thi s area I n t he l at ter t he probab l i t y of damgevari es gradual l y f rom a val ue of 1 at the poi nt of detonat i on, decr easi ng t owardzero as the di stance f rom t he poi nt of det onat i on i ncreases .

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2

i ntegral of i t over any area i s the probabi l i ty that the weapon w l l f al l i n t hi s area Then the probabi l i ty of damage t o the t arget f rom a si ngl e weapon chosen at randomi s 00

P = SS Pd(X, Y) f (X, Y)dx

dY

- 00

Some cases are gi ven bel ow

C rcul ar Cooki e- Cut t er Damage Functi on

Y) _ ~1 i f x2 + y2 ' - - R2

0 otherwse

For thi s f uncti on the probabi l i ty of damage i s

P = SS f ( X, Y) dx dY

ci rcl e

wher e the i ntegra i s t o be t aken over the ci rcl e of radi us R wt h cent er at the ori gi n

" Probabi l i ty of dama g i ng si n g l e poi nt tar ge t wth a si n g e shot when the con-di t i ona damage f unct i on i s the ci rcul ar cooki e- cut t er f unct i on and when thedi str i but i on of shot s i s ci rcul ar normal and cent ered on the t arget :

More preci sel y, we assume t hat x and y are i ndependent vari abl es havi ng normaldi stri but i ons wt h means zero and common st andard deri vat i on, Q . Then f (x , y)has the f orm

f (X, Y) _ 2 eXP [ ' ( X2 + Y2)/ 2 a 2 ~ .27rQ

By el emnt ary i ntegrat i on we fi nd that

P = I - exp( - R2/ 2 Q 2)

t Probabi l i ty of damagi ng si ngl e poi nt target wt h a si ngl e shot when the con-di t i onal dama ge f unct i on i s t he ci rcul ar cooki e- cut t er f unct i on and when t hedi str i but i on of shots i s el l i pt i cal . and cent ered on the target :

Thi s case i s the same as the one precedi ng, except t hat the t wo standarddevi at i ons are unequal . Thi s s l i ght change makes the i ntegrat i on di f f i cul t . I f wechange t o pol ar coordi nat es ( r , A) and i ntegrat e wt h respect to A we obtai n t hef ol l owng resul t :

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3

27r R

. - 2 7TQ aS~ exp ~r2 cos 2A/ 2 oX2 - r2 si n

29/ 2 Qy2 J rdrdAx y 0 0

- . R 27r2 2

2 ~r Q o~ e- ar r dr S e r cos ~ do, 28)

y 0 0R

Q 1Q S e-ar I o~2) r dr1 2

R2

2 6 1 QS e- au I o( bu) du

1 2

Here Q1 i s the smal l er of vX a Y and Q2 i s the l arger of these . A so,

a = 4 ~~ +~

, b = 4 ( Q12 - Q22 )

and I o(x) i s t he mod f i ed Bessel f uncti on of zero order . I f we repl ace t he Bessel

f uncti on by i t s seri es expansi on and i ntegrate t erm by term we get the f ol l owngresul t :

P = 61 ~ ~_1 ~k(R2/ 2

- Q12)k+1 k C- 1/ 4> ( k> ( 2i )~1 - C I 0 2 )1 ,(k+ 1 ) ! 1 2

k=0 i =0

For cases i n whi ch R i s smal l compared wt h 011

thi s seri es converges rapi dl y

The i ntegral of the el l i pt i cal nor mal d stri buti on over a ci r c l e, as deri vedabove, i s equal t o the i ntegral of the ci rcul ar nor mal d st ri buti on wth uni t st andarddevi at i on over the el l i pse that has sem- axes R QX and R r y

The probab l i ty of damage i n the case of unequal st andard devi ati ons can beapproxi mated by

P = 1- exp (-R2/ 2 Q 2)

where o i s an appropri at e f uncti on of Q1 and Q2 Funct i ons f requent l y used are

the geometri c mean f f g , the ari thmet i c mean Qa, and the root - mean- square 6s .

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These are def i ned as f ol l ows ; , -

.Qg

2= Q Q2

as = ( Q 1 + Q 2)/ 2. , ,

QS2 = ( Q 12 + Q 22) / 2

I t i s evi dent t hat Qa l i es bet ween g nd QS i n fact , Qa2 i s t he mean ofQ92

and Ors2

.Let P , a, and PS be t he probab l i t i es of damge gi ven by

P = 1-exp ( -R2/ 2 a2) when Q has t he val ues or91

v a, and os respecti ve y O

these 3 approxi mat i ons t he best seem t o be

Pg when 0 : s R2 < 0 . 5 as

Pa when 0 5 vat

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5

where cp i s the normal error i ntegral :. ' x

' ( P( x) = 1 ~ e- t2

/ 2 dt

r2 ? r 00

Probabi l i t y of damag ng si n g l e po nt t ar g et wt h a si n g l e shot when the con-di t i onal damage f uncti on i s the ci rcul ar cooki e- cut t er f uncti on and when t Edi stri but i on of shot s i s e l l i p t i c a l normal and x and v are correl at ed

W assum that the mans are zero but x and y are not i ndependent . Let thecovari ance be QXy Th s i s equal to the expect ed val ue * of xy and i s equal to

pxy a Xory where pXy i s the correl at i on coef f i ci ent . The probabi l i t y of damage i n

thi s case can be obta ned f rom the above resul t s by repl aci ng a12 and v22 by X1and A2 respect i vel y, where X and X 2 are the root s of the equat i on

ax2 - X) ( oy2 - X) - QXy2 = 0

wth X1 ~ X2 . The root s of thi s equat i on are of t en cal l ed the e gen- val ues of the

covari ance mat ri x The equat i on i s obta ned by equat i ng t o zero the det ermnanti n wh ch the el ement s axe the correspondi ng el ement s of the covari ance mat ri xwt h the el ement s of the mai n di agonal reduced by X

Th s resul t can be obta ned easi l y f rom the j oi nt densi t y f unct i on, wh ch i s ;

f ( X, Y) _ (1/ 2 i r a ayV - - P2)

exp C( Qy 2x2 - 2p QX Qy xy + QX2y2) / 2 ax 2 ay ( 1 - P2) Jwhere Qxy = p QX c r y. W now rot at e the axes t hrough one of the angl es that

el i mnat es the cross- product term i n the quadrat i c f orm The t ransf ormat i on i sl i near wt h det ermnant equal t o 1 . I f u and v are the new rect angular coordi nat es,

_ , the f uncti on f (x , y) reduces to

' - . ( 1 / 2 7 r a exp ( - u2/ 2 X, - v2/ 2 X2)

. . where X and X2 are t he root s of the equat i on gi ven above Si nce t he t ransf ormat i on

does not al ter t he condi t i onal damge f unct i on, the resul t s can be obta ned f rom the

previ ous case by repl aci ng Q 12 and Q22 by X and A2

* Because the mans are zero More pneral l y, the covari ance i s equal to theexpect ed val ue o f ( x - x) ( y - y) , where x and y are t he mans of x and y respecti vel y .

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Agai n we can expand the Bessel f unct i on i nt o a seri es and i nt egrate t er m by term Thi s can be wri t ten i n a number of f orm of whi ch two are gi ven bel ow

-2 2 00 2Z1 / 2 0 ~ 2P = e

E (h / 2 Q 2~ (R ~2 O i - f - 1 , i ).

1=O1

1 2 2(R / 2 a

j1 -e- (h2 + R2)/ 2 Q2 ~ ( h2/ 2~2) i

i .i =0

" Probabi l i ty of damagi ng si ngl e poi nt target w th a si ngl e shot when the con-di t i onal damage f unct i on i s t he ci rcul ar cooki e- cut t er f unct i on and when t hedi str i but i on of shots i s ci rcul ar normal and i s cent ered at a poi nt h uni t sf rom t he target :

Here we assume agai n that x and y are i ndependent and are norml l y di stri but edw th equal st andard deviat i ons, but t he means of x and y are not both zero . Becauseof the ci rcul ar condi t i onal damage f uncti on, we need be concer ned onl y w th thedi st ance h of the center of the shot di str i but i on f rom the t arget . The i ntegral i nthi s case i s approxi mt el y as d i f f i c u l t as that of the el l i pt i cal di str i buti on cent eredat the t arget . I f we t ransf orm to pol ar coordi nat es and i ntegrate w th respect t o8 we obtai n the f ol l ow ng resul t :

R 2 2 2P =

or 2( ' e-

(r + t r ) / 2 Q I o (r h/ Q 2)r drJ0

R / 2 Q

2 2= e - h / 2 0 ~ e' X I ~ ( h r2i ~Q ) dx

0

Here, Pi s t he i ncoml et e gamma f unct i on t abul at ed i n ref erence ( b) . I t i s evi dentthat P can be wri t ten as a f unct i on P( R/ Q h/ Q ) of the two rat i os l i s t e d . Tabl esof thi s f unct i on are gi ven i n ref erence ( c) and i n numrous other pl aces . Anapproxi mt i on that i s usef ul f or sml l val ues of R i s the f ol l ow ng

2R2 e- 2h2/ ( R2 + 4a 2) R

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002' '

P=1 S e- x / 2

dx, k = ( h _ VR2 _, a 2)/ v R > 5 0V2 7r k

1 _ ~ k)

where 45 i s the normal error f uncti on

Probabi l i ty of damag ng si ngl e po nt target wt h a si ng e shot when t he con-di t i onal damage f uncti on i s the ci rcul ar cooki e-cut t er f uncti on and when thed stri buti on of shots i s el l i pt i cal normal not cent ered on t he t arget :

Th s case i s d scussed i n ref erence ( d) .

t C rcul ar Gaussi an Damage Funct i on

Pa(X' Y) = exP [ - ( x2 + Y2) / a2 ] = exp [ ( -r2/ a2) ]Th s f uncti on was f i r s t i nt roduced as f ar as i s known, by von Neumann wh l eworki ng as a consul t ant f or the Bal l i s t i cs Research Laborat ory . I n some casesi t i s a more real i sti c descri pt i on of the cond t i onal damage f uncti on t han i s thecooki e-cut t er f uncti on However, i t s bi ggest advant age i s the ease wt h wh chthe i ntegral s can be obta ned and the si ml i ci ty of the resul t i ng f orml as . Thef orml a i s somet i mes wi tt en wt h a f actor 2 i n the denomnat or of the exponent .Th s i s done to si ml i fy some of the f orml as s l i ght l y . I t appears pref erabl e thatthe f orml a be wi tt en i n the f orm g ven above, si nce i n thi s f orm the quant i ty ai s the equi val ent damage rad us, i n the sense that the i ntegral of the f uncti on overthe ent i re pl ane i s the area of a ci rcl e of rad us a . That i s ,

00

Pd ( X' Y)~ y = 7 r a2S

00

" Probabi l i ty of damag ng si ngl e po nt t arget wth~a si ng e shot when the con- di t i onal damages f uncti on i s the ci rcul ar Gaussi an f uncti on and when the di s-t rt bunon or snots i s ctrcui ar normal ann cent erea on the t arget :

I n t hi s case the damage f uncti on and the densi t y f uncti on depend onl y on theradi al vari abl e r . When we change to pol ar coord nat es, we f i nd t he probabi l i t yread l y

1 ( ' e- r2/ a2 - r2/ 2 Q2 r dr( F 2

= a2/ ( a2 + 2 Q 2}

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Probabi l i t y of dama gi n gsin 1e poi nt t ar ge t w t h a si n g e shot when the con-di t i ona l damge f unct i on i s the ci rcul ar Gaussi an f unct i on and when the ' str i buti on of shots i s el l i pt i cal nor ma and cent er ed on the t arget :

8

Agai n the i ntegral i s f ound i n a st r ai ght f orward mnner by separat i ng i t i nt o ,a product of 2 i ntegral s by the separat i on of vari abl es . The probabi l i ty i s . '

P = a2/ V (a2 + 2or

X2) ( a2 + 2 ory2)

" Probabi l i t y of damgi ng si ngl e poi nt t arget w t h a si ngl e shot when the con-di t i onal damge f unct i on i s the ci rcul ar Gaussi an f unct i on and when the di s-tr i buti on of shots i s ci r cul ar nor ml and i s cent ered at a poi nt h uni ts f romthe t arget :

Agai n the i ntegral can be comuted readi l y by the separat i on of var i abl es .The probabi l i ty i s gi ven by

P = Cat / (a2 + 2 Q2) 1 exp C - h2/ (a2 + 2 02) JJExact l y the sam expressi on i s obtai ned f or the case i n whi ch t he di stri but i on i s t heCarl t on di st ri buti on w th densi t y f unct i on

f(X, Y) _ (1/ 2 ? r Q2) expC-(x2 +y2 + h2) / 2 Q 2 J I o ( h x2 + y2/ Q 2 J

" Probabi l i t y of damgi ng si ngl e poi nt t arget w t h a si ngl e shot when the con-di t i onal damge f unct i on i s the ci rcul ar Gaussi an f unct i on and when the di s

shots i s el l i pti cal nor mal not cent ered on t he t ar

Assum that the cent er of t he di stri but i on i s at ( hX, hy) . Agai n the i ntegralcan be separated, and the probabi l i ty of damge i s

P = [a2/ ( a2 + 2 QXZ) (a2 + 2 ory2)

2) 1 .xp I - hX2/ (a2 + 2QX2) - hy2/ ( a2 + 2

Qy J

El l i pt i cal Gaussi an Damge Functi on

Pd (X' Y) = exp( - x2/ ax 2 - y2/ ay2)

I n t hi s case the i ntegral of the condi t i onal damge f unct i on over the ent i re pl anei s 7 r ax

aYthe area of an el l i pse w t h sem-axes aX and ay . W can det ermne

t he probabi l i t i es i n t hi s case f rom the cor r espondi ng probabi l i t i es f or the ci rcul ar 'Gaussi an damge f unct i on by mki ng the f ol l ow ng substi tut i ons :

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Repl ace i n the denomnators :

a2 + 2 aX2 by aX2 + 2 QX2

a + 2 oy2 by ay2 + 2Qy

2

a2 + 2 a 2 by ( aX2 +2a 2 (ay2 + 2 Q 2

Other Condi ti onal Damage Functi ons :

For some weapons and weapons ef f ects the condi ti onal damage f uncti on appearsto l i e somewhere between the ci rcul ar cooki e- cutter f uncti on and the ci rcul arGaussi an f unct i on, I n appendi x A, a sequence of f uncti ons of t hi s type i s proposed The f i r s t member of t hi s sequence i s the Gaussi an f uncti on, and the l i mti ngmember i s the cooki e- cutter f uncti on These f uncti ons have the property that theprobabi l i ty i ntegral s f or the usual cases can be computed i n f i n i t e f orm i n termof known f uncti ons . The second member of the sequence yi el ds probabi l i ti es thatare approxi matel y hal f way between the correspondi ng probabi l i ti es f or t he Gaussi anand : _noki e- utter f uncti ons . Thi s property coul d be used as a means of approxi -mati ng the di f f i cul t cases f or the cooki e- cutter f uncti on However, si nce thesecond member of the sequence probabl y i s cl oser t o the true condi ti onal damagef uncti on than i s the cooki e- cutter f unct i on, i t i s mre reasonabl e to use the prob-abi l i t i es gi ven by t hi s second member of the sequence,

~r SEVERAL PONT TARGETS

When several poi nt targets exi s t , there i s general l y a need t o answer questi onsof the f ol l owng type What i s the probabi l i ty of damagi ng al l the target s, or atl east a speci f i ed number of them wth a si ngl e weapon? (The case of severalweapons w l l be di scussed l ater . ) To answer, we must f i r s t compute the desi redprobabi l i ty under the condi ti on t hat the weapon detonates at a parti cul ar poi nt( x , y) , Thi s quanti ty i s then treated as the condi ti onal probabi l i ty i n the previ ouswork and we must average t hi s f uncti on over the shot di stri buti on

Far the cooki e- cutter damage f uncti on the i ntegral s i nvol ved are i nvari abl ydi f f i cul t . The i ntegral s are f a i r l y easy to compute, however, i n the case of theGaussi an damage f uncti on W gi ve one exampl e bel ow

Assume that the di stri buti on of shots i s ci rcul ar normal of standard devi ati ono and i s centered at a poi nt mdway between two targets . Assume that the targetsare l ocated at the poi nts (-h, 0) and ( h, 0) . I f the weapon detonates at t he poi nt( x , y)

the probabi l i ty of damagi ng both targets i s

+ y2 + ( x + h) 2 + y2 ] / a2

+ h2)/ a2 J ,(

Pd(x~Y) = exp j -L~x-h)

2l

= expL

- 2 ( x2 + y2

9

Repl ace a2 i n the numerator by ax ay

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10

To obt ai n the desi red probabi l i ty we must now mu t i pl y by the densi t y f uncti on andi ntegrate over the enti re pl ane . The resu t :

P (both) = Cat / ( a2 + 402) J exp ( - 2h2/ a2)AREA TARGETI n t hi s case we assume that the target i s spread over a l arge area, the maxi mum

di mnsi on of t hi s area bei ng at l east as l arge as the damage radi us of the weapon I t i s assumed that, for a l arge f racti on of t he detonati ons, some parts of the t argetwou d be damged c r i t i c a l l y wh l e other part s wou d be l e f t undamged . I n t hi scase i t mkes l i t t l e sense to tal k about probabi l i ty of damges . I nstead we mustconsi der the probl em of det ermni ng t he expect ed f ract i on of t he target that w l lbe damged, or the probabi l i ty of damgi ng at l east a gi ven f ract i on of the t arget .

Assume that the burst occurs at the poi nt ( x , y) . Then i f ( X, Y) i s any poi nt ofthe target, t he condi ti onal probabi l i t y of i n f l i c t i n g damage t o t hi s poi nt i s

pd ( x - X, y - Y)

Then the condi t i onal f racti on p(x, y) damaged i f the burst occurs at ( x , y) i s

P( X~ Y) _ (1/ A) SS Pd ( x- X, y - Y) dX dY

where the i ntegral i s t aken over the t arget of area A The expect ed f ract i on of thet arget damaged i s

00

F = SS p(x, y) f ( x, y) dx dy

- 00

where, as bef ore, f (x , y) i s the j o i n t densi ty f uncti on .

Let So be the set of poi nt s

So Poi nts ( x, y) such that p ( x, y) ' Fo

Then the probabi l i t y of damgi ng at l east t he f racti on Fo i s equal t o

P(~ Fo) = SS ( x , Y) dx dY

S0where t he i ntegral i s t aken over the set So Sandi a Corporati on l i t erature descr i bes

an anal ogue comut er wh ch was desi gned t o obtai n t hi s f uncti on for a wde vari ety oft arget coml exes and densi ty f uncti ons .

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To obtai n the probabi l i ty of damg ng at l east t he f racti on Fo we woul d have t o

sol ve the equati on obtai ned by equati ng p ( x, y) t o Fo . Thi s w l l be a ci rcl e whoserad us coul d be obtai ned by t r i a l and error . Let the rad us of t h i s ci rcl e be Ro Then the probabi l i ty of damg ng at l east t h i s f racti on woul d be equal to

2

P ( ] o) = 1 - exp ( - R02 / 2 Q2) ~ 0 < o < T2 C1 ` e~ - T2/ a2~ J0 , Fo 7 (at / T2) [ 1' exp ( - T2/ a2~

I f we repl ace the ci rcl e target by an equ val ent Gaussi an t arget of densi t y

exp[ _ (X2 + Y2) / T2 ]

at the poi nt (X, Y), the f ormul as become si ml er . Thus we f i n d

]-(x2 + y2)/ (a2 +T2)P ( X~Y) = J a2/ (a2 + T2)

Jexp.

LF = a 2/ (a2 +T2 + 2 v2)

11

W i l l u s t r a t e t hese i deas wt h one examl e . Let the target be a ci rcl e of rad usT and l et the cond t i onal damage f uncti on be the ci rcul ar Gaussi an f uncti on of equ val entrad us a . Assume that t he d stri buti on i s ci rcul ar nor ml of st andard devi at i on aand cent ered at the cent er of the target . Then t he cond t i onal f racti on of the t argetdamaged i s

p(x, y) _ {1/ 7r T2)j

exp - C ( r cos A - x) 2 + ( r si n 8 - y) 2J

I / a21

rdrd9l

0 0 ?/ a 2' ~

- (a2/ ,T,2) exP C- ( X2 + Y2)/ a2 ~ S e- u I o ( 2 u(x2 + Y2 ) / a ) du0

The expect ed f ract i on of t he target damged i s00

F = (1/ 2 ?ro2) S S p ( x, Y) exP

I -

( X2 +Y )/ 2 Q 2 ] dx dy

00

_ ( a 2 / ' i ' ) j 1 - exp C - T2 / ( a2 + 2 62) 1_ (

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00

Y~F [ P u, v)1 g u, v) du dv

J00

For i nstance we mght want t o f i nd the probabi l i ty of scori ng at l east one damagi nghi t w t h a sal vo I f the probabi l i t i es of damage by weapons i n a sal vo are i ndependent ,except f or t he common MPI , and n weapons are used i n a sal vo, t hen the quanti tyof i nterest i s

nF [P] = 1 - ( 1 - P)n = ~ - 1 ) i - 1 ( i PL ( u, v)

7 =1

12

1 - F (a2 + T2)/ a2] (a2 + T2)/ Z ~2, 1 30 ~F ~ a2/ a2 +T~p~~ Fob - o

Fo > a2/ (a2 + T2)

SALVOS

So f ar we have consi dered the probabi l i ty of damage wth a si ngl e weapon onl y .I f several weapons are dropped separatel y i n such a way t hat the probabi l i t i es arei ndependent , we can combi ne the si ng e- shot probabi l i t i es by wel l known met hods .An exampl e of part i al correl at i on i s t hat of a sal vo of weapons, that i s, a group ofweapons rel eased si mul t aneousl y or nearl y si mul t aneousl y . I n t h i s case the weaponsi n the sal vo have a part i cul ar di stri but i on whi ch i s not the same i n al l respects asthe di stri buti on that woul d be obtai ned on another sal vo A characteri sti c of the

di stri but i on t hat mght vary f rom sal vo to sal vo i s the po nt about whi ch the i ndi v dualweapons i n the sal vo are grouped, such as the mean po nt of i mpact - ( MPI ). Theweapons of a part i cul ar sal vo w l l have a part i cul ar MPI but t h i s MPI may vary f romsal vo t o sal vo I n f ocusi ng at tent i on on the MPI we do not i ntend t o i mpl y that t h i si s the onl y characteri sti c of the di stri but i on of weapons i n a sal vo t hat mght varyf rom sal vo t o sal vo I n the f ormul ati on bel ow however, we shal l consi der onl y theMPI .

Let f ( x, y I u, v) be the j oi nt densi t y f uncti on of the po nt ( x, y) of detonati on ofa weapon i n a sal vo that has i t s MPI at the po nt ( u, v) . By the met hods di scussedprevi ousl y we can f i nd the si ng e- shot probabi l i ty, g ven that the MPI i s ( u, v) .Let t h i s be P = P(u, v) and l et g(u, v) be t he j o in t densi ty f unct i on of u and v LetF [P] be any f uncti onal of P of i nterest . Then the average val ue of t hi s quanti ty i s

From t h i s we can readi l y f i n d the probabi l i ty of scori ng at l east one damagi ng hi tby i ntegrat i ng the product of t h i s f uncti on of u and v and g(u, v) over the ( u, v)pl ane .

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13

We w l l t reat i n f u l l one part i cul ar exampl e, perhaps t he si mpl est exampl e' - possi bl e . Assume that the condi t i onal damge f uncti on i s ci rcul ar Gaussi an w th

. equi val ent radi us a Assume t hat the di st ri but i on of shots i n a sal vo i s ci rcul ar- nor mal of standard devi at i on s and i s cent ered at ( u, v) . Al so, as, - ' ume that the

di st r i but i on of the MPI i s ci rcul ar nor mal w t h st andard devi at i on Q and i s cent eredat the t arget . Then f rom the sol ut i on t o the pr obl em concerni ng t he ci rcul ar nor maldi st r i but i on of shots cent ered at a poi nt h uni ts f rom t he target, we have

P(u, v) = A exp C - ( u2 + v2)A/ a2 1 , A = a2/ (a2+2s2)

J

Al so00

Y SPi ( u, v) g( u, v) du dv = A / ( 1 + i / B ) , B =( a2 + 2s2)/ 2 v2

The l at t er resul t i s obtai ned by st rai ght f orward i nt egrat i on Fi nal l y, t he prob-abi l i t y of scor i ng at l east one damagi ng hi t can be wi t ten as

n~- , ( - I ) i - 1(n ) Ai / ( I + i /B>L

where

A = a2/ ( a~ + 2s2) , B =(a2 + 2s2) / 2 Q 2

The probabi l i t y of at l east 1 damagi ng hi t i n thi s case i s appr oxi mat ed byassumng that P i s so sml l that we can est i mt e the probabi l i t y of at l east 1damagi ng hi t when the MPI i s ( u, v) by

- F P) ~ - exp (-nP), P = P u, v)_ Usi ng t hi s approxi mti on we can expr ess t he aver age val ue i n the f orm

00

r2/ 2 02 - nA exp ( - r2 A/ a2) 1 r drZ) ~ exp[_

J_ =~' ) =1 - ( 1/ 0

Lett i ng t =nA

exp ( -r2A/ a2) we obt ai nnA

( ~ 1) - 1 - B( nA) - B S e- t t 8 -1 dt = 1 - ( nA) B P B + 1) P nA/ V_B, B- 1)o

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14

where P B + 1 ) i s the compl ete gamm f uncti on andP

( u, p) i s the i ncompl etegamm f uncti on tabul ated i n ref erence ( b) .

The above exampl e i ndi cates the general procedure i n probl em of thi s ki nd . 'The reader my wsh to compare the probabi l i ty obtai ned i n the above exampl ewth that obtai ned by f i r s t f i ndi ng the over- al l di stri buti on of weapons, then deter-mni ng the probabi l i ty of damge wth a si ngl e weapon, and f rom t hi s deri vi ng theprobabi l i ty of at l east one damagi ng hi t wth n weapons wh ch are consi deredi ndependent . The probabi l i ty of scori ng at l east one damagi ng hi t by the l att ermet hod shoul d be greater than the probabi l i ty f ound by the f ormer (and correct)met hod Wapons f i red i n sal vos consti tute one of the si mpl est exampl es ofcorrel ati on

SEQUENCE OF SHOTS

I n mny probl em two or mre weapons are f i red on a si ngl e run or event i nsuch a way t hat the i ndi vi dual probabi l i ti es are not i ndependent and there are mnycharacteri sti cs of the di stri buti on t hat change f rom shot t o shot . Exampl es are surf ace- to- ai r anti ai rcraf t f i r e by guns, rockets and gui ded mssi l es ; ai r- to- ai rengagement s wth guns, rockets, and mssi l es ; sti ck bombi ng and i nterval ometerbombi ng by ai rc raf t , etc . I n fact, most of the real i sti c si tuat i ons i nvol vi ng mrethan one weapon are of t hi s type Usual l y the correl ati on ef f ects, both auto- cor-re ati on and Lexi an, are too l arge t o be i gnored . Numrous efforts have beenmde t o sol ve probl em of thi s sort . Most of the sol uti ons requi re that the processbe stati onary . But t he process i s usual l y not stati onary i n the probl em encountered

A sequence of shots that f orm a si mpl e Markof f chai n probabl y i s the si mpl esttype Wth a s uf f i ci ent l y l ong chai n t hi s met hod i s adequate t o treat the auto- cor-re ati on effects . The probl em of the Markaf f chai n, i ncl udi ng the l i mti ng l aw ofsmal l numers ( the anal ogue of the Poi sson di stri buti on, i s treated i n ref erences

( e) and ( f ).

There seem to be no easy met hod of treati ng Lexi an ef f ects, exceptby taki ng the we ghted average of the probabi l i ti es f or t he separate popul ati ons .

Submtted by :.

_v&

J . M DOBB EOperati ons Eval uati on Group

Approved by :i

1 , x , 1

J . STE NHARDTD rectorOperati ons Eval uati on Group

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Ref : ( a) Andre G Laurent, Bombi ng Probl em - A Stat i s t i cal Approach,Operat i ons Research 5, 75 (1957)

( b) K Pearson, Tabl es of the I ncoml ete Ga mm Functi on, B omtri kaOf f i ce, Uni versi ty Co l ege, London 1934

( c) RAND RM330 Uncl 26 J an 1950

( d) RAND P-94 Uncl 28 J ul 1949

( e) B O Koopmn, A general i zati on of Poi sson s di stri buti on for Markof fchai ns, Proc . Nat . Acad Sci . , U S . A Vol , 36 (1950) pages 202-207

( f ) B O Koopmn, A l aw of sml l numers i n Markof f chai ns, Trans .Am Mat h Soc . Vol , 70 no 2 (1951) pages 277-290

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A- 1

APPENDX A

A SEQUENCE OF COND TI ONAL- DAMAGE FUNCTI ONS

The condi t i onal damge f uncti on i s a f uncti on whi ch g ves the probabi l i t y that

a t arget wh ch i s l ocat ed at~- a part i cul ar poi nt and bear i ng f rom the burst poi nt w l lsuf f er at l east the stat ed degree of damage Th s probabi l i ty i s a f uncti on of di s-t ance r of the center of the t arget f rom ground zero, he ght (depth) of burst,bear i ng of ground zero re ati ve t o the t arget headi ng damge cat egory, t arget vul -nerabi l i t y, and weapon yi el d .

I t i s evi dent t hat a f uncti on whi ch accurat el y descr i bes the ef f ects of al l t hesevari abl es w l l be coml i cat ed . Fi rst, l et the l ast three vari abl es - - damage category,t arget vul nerabi l i ty, and weapon yi e d - - have as"si gned val ues . Then the condi t i onaldamage probabi l i ty i s a f uncti on of t he t hree coordi nat e var i abl es - - hori zontaldi stance r, he ght ( dept h) , and beari ng Fi rst average over t he di st ri but i on oft hese vari abl es f or a se ected ai mng poi nt (somt i ms cal l ed t he i ntended groundzero, I GZ) and i ntended he ght of bur- st .

The mst i mor t ant coordi nat e vari abl e i s hori zontal di stance r . For si mp l i c i t y the condi t i onal probabi l i t y w l l be wi tt en expl i ci t l y as a f uncti on of t hi svari abl e onl y I t i s under st ood that a part i cul ar hei ght , based on weapon yi el d,ef f ect desi red, dud probabi l i t i es, etc . , has been set i n the f uze I t i s assumedthat errors i n he ght produce a neg i g bl e ef f ect on the probabi l i t y of damge, orthat an average- over the correspondi ng er ror di st ri but i on w l l be t aken Such anaverage can be t aken at t hi s poi nt i f the he ght error i s i ndependent of er ror s i n thehori zont al pl ane, whi ch i s true f or mst del i very and f uzi ng syst em i n use However, f or ai r -burst f uzes i t i s f ound t o be conveni ent t o def er t hi s average unt i ll ater .

I t i s assumed that changes i n bear i ng produce negl i gi bl e changes i n damageprobabi l i ty . For some t arget s, not abl y sh ps and a i r c r a f t , t hi s i s not true How

ever, i n t aki ng t he average over the di st ri but i on of weapons, t hese changes wt hbeari ng can usual l y be i gnored wt h negl i gi bl e er r or .

When t he condi t i onal damage probabi l i ty i s wi tt en as a f uncti on of r, i t w l lbe assumed that the ef f ects of changes i n damage category, t arget vul nerabi l i ty,and weapon yi e d can be descr i bed adequatel y by changes i n par amet er s i n t hi sf unct i on, wt hout chang ng the f orm of the f uncti on .

Let p ( r ) be the average probabi l i ty of damge at hori zontal di stance r f or ag ven damage cat egory, t arget t ype, and weapon yi e d Several f uncti ons whi ch have

. ' been used are the f ol l owng

( a) Def i ni t e- range or cooki e-cut t er f uncti on

1 i f x : 5 RPOW (A-1)0 i f r >R

where R, the damage radi us, i s a paramt er wh ch depends upondamage cat egory, t arget t ype, and weapon yi el d .

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A-2

( b) Gaussi an f uncti on

Pi( r ) = exP ( - r2/ a12)

where "al " al so i s a parameter whi ch depends on the damagevari abl es . A f actor of 2 i s somet i mes used i n the denomnator of theexponent t o si mp i f y l ater f ormul as ; t hi s shoul d be kept i n mndwhen compari ng val ues of paramet ers .

The def i ni te-range f uncti on appears t o be reasonabl e i f al l the vari abl es exceptr have f i xed val ues . Thi s requi res that the damage category be def i ned preci se y,a part i cul ar t arget of known vu nerabi l i t y be consi dered, the weapon yi e d be knownexactl y, and the he ght and beari ng of the burst posi t i on be f i xed . As exp ai nedabove, p(r) shoul d be an average over some of t hese vari abl es - - part i cu arl yvari at i ons i n damage whi ch woul d be i ncl uded i n a gi ven damage category, vari at i onsi n target vu nerabi l i ty among targets of a gi ven t ype, and vari at i ons i n he ght ofburst un ess the average over the he ght di st ri buti on i s t o be t aken l ater .

For any condi t i onal damage f uncti on p ( r ) , l et g(R) be the cor respondi ng densi tyf uncti on of the damage radi us R Wi te po( r) i n the f orm po(r, R) t o di sp ay the

r ol e of R Then00 00

= S Po(r, R) g (R) dR = S g (R) dR (A-2)

o r

= Probf

R ' r

i s the di st ri but i on f uncti on of the damage radi us . Hence, the probl em of f i ndi ngthe probabi l i ty f uncti on p(r) i s equi val ent t o that of f i ndi ng the di stri buti on f uncti onof the damage radi us R

For p ( r ) equal t o the Gaussi an f uncti on p ( r) the cor respondi ng densi ty f uncti onof R s

g, ( R) _ - Pi (R) = 2R eXp ( - R2/ ai )al

An obj ecti on t o t hi s densi ty f uncti on i s t hat i t g ves too much wei ght t o verysml l and very l arge val ues of R As w l l be shown l at er, onl y 65 percent of thedi st ri but i on l i es bet ween 0 . 5R and 1 . 5R, where R i s the average damage radi us .

To get a h gher concent rat i on about the average we can ml t i p y the exponent i alf actor by a power of R greater t han one For examp e, i f

3g2 (R) _ ~ xp ( - 2R2/ a2) ,

a2

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( a) the average R( b) the mdi an Rm

( c) t he equi val ent radi us Re ( def i ned bel ow)

and f or pn ( r ) i t i s

1 . 3 . 5 . . . ( 2n - 1) anR = Y

2n (n - 1) ; n

A- 3

the correspondi ng probabi l i ty f unct i on i s2

p2 ( r ) _ ( 1 +~ exp ( - 2r2/ a2)a2

Thi s f unct i on and the Gaussi an f unct i on may be consi dered as the f i r s t t wo f uncti onsof a sequence obtai ned f rom the densi ty f uncti on

g(R) = 2 -y

nR

2n- Iexp ( - nR

2/ a

2n ( n - 1) an n

n

The correspondi ng probabi l i ty f unct i on i s

k 2 k n-1pn ( r ) _ ( 1 + kn + 2 + . . . + n - 1) : ~ exP ( - kn)

(A- 3)

where

kn = nr2/ an

For any of t hese f uncti ons i t i s necessary t o determne the val ue of the para-mt er f rom a . masured or est i mt ed val ue of the damge radi us . Three val ueswhi ch have been used are :

The average i s gi ven by

00 00

R g (R) dR = - S R dp (R)= S

0 0

For the Gaussi an f unct i on p1 ( r ) the average i sa

R Z 7r = 0. 89 al

Forp2

( r ) the average i s

R=~ ~ 0 . 94 a2

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The medi an Rm

i s def i ned by

A- 4

As n i ncreases, an approaches R

y0 5

For the Gaussi an f uncti on the medi an i s

Rm al I og 2 = 0 . 83 al

For p2 ( r ) i t i s 0 . 92 a2 approxi mat el y . For pn ( r ) the paramet er an i s greater

t han t he medi an, whi ch i t approaches i n the l i m t .

The equi val ent damage radi usRe

i s def i ned by the equat i on00

2 7r S r p ( r ) dr = ~r Re0

The quant i ty on the l e f t i s the expect ed number of t arget s damaged i f t hey areuni f ormy di stri buted wt h uni t densi ty over a l arge regi on about the burst poi nt .The quant i ty on the ri ght i s the correspondi ng number f or a def i ni te-range f uncti onof radi us Re The par amet ers an have been i nt roduced above i n the f orm f or whi ch

t he paramet er i s al ways equal to Re that i s , an = Re f or al l n .

To compare these f unct i ons we assume that the average R s known and t hepar ameters have been determned i n t er m of R The densi t y f unct i ons g1 (R),

g2 (R), and g3 (R) are pl otted i n f i gure A-1 . The correspondi ng probabi l i ty f unct i ons

and the def i ni te-range f uncti on are pl otted i n f i gure A-2 As n - - > 00 , pn ( r )

approaches the def i ni t e range f uncti on A measur e of how cl osly R i s di stri butedabout the mean s the probabi l i ty t hat R l i e s between 0 . 5R and 1 . 5R_ . Thi s val uei s 65 percent f or p1 ( r ) , 83 percent f or p2 (r j and 91 percent f or p3 ( r ) .

Anot her measur e of di spersi on i s t he st andard devi at i on f rom the mean Whave

00 00

S R 2 g ( R) dR = - S R2 dP(R)

0 0

I nt egrat i ng by parts, and not i ng that, f or al l the f unct i ons consi dered, p (R) - ~ of aster t han R- 2 as R - ~ oo , t hi s becomes

2 ~ R P ( R) dR = R ae n

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Rg(R)

1. 4

1. 2

1. 0

. 8

. 6

. 4

. 2

Jr

0 . 2 . 4 . 6 . 8 1. 0 1. 2 1. 4 1. 6 1. 8 2 . 0

R/ R

FI G A-1 : DENSI TY FUNCTI ONS OF DAMAGE RADUS R

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Probabi l i ty of Damage p(r)

. 2 . 4 . 6 . 8 1. 0 1. 2 1. 4 1. 6 1. 8 2 . 0

r1R

0

FI G A-2 PROBAB LI TY OF DAMAGE AS FUNCTI ONOF HOR ZONTAL RANGE r

. ,

1. 0

S

. 6

4

2

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A-7

Hence, the vari ance f rom the mean i s Re - R2 The st andard devi ati on f rom

the mean i s 0 . 52R, Q 36R and 0 . 29R f or pl ( r ) , p2 ( r ) , and p3 ( r ) respecti vel y .

. ~ I f we know R and the st andard devi ati on of R from R the appropri atefuncti on can be chosen from the sequence of functi ons . I f onl y R s known, howsensi ti ve i s the probabi l i ty of damage to the choi ce of condi ti onal damage functi on?

Let ( x, y) be the rectangul ar coordi nates of ground zero rel ati ve to the centerof the target . Assume t hat x and y are i ndependent and norml l y di str i buted w th

the same st andard devi ati on o and means hX and by respecti vel y (Unequal

st andard devi ati ons make the f orml as more compl i cated but do not i ncrease thed i f f i c u l t i e s of i ntegrati on ) The densi ty functi on of x and y i s

1 2 [(x- hX)2 + (Y - hy) 2

Jf Y)

2 ~r Q

Then the average probabi l i ty of damage f or the n th functi on i s

00

Pn = ~ Y Pn ( X' Y) f ( ' x ' Y) dx dY

- 00

where p ( x , y) i s the functi on gi ven i n equati on (A-3) when r2 i s repl aced by

(X2 } Y2

Expandi ng the powers of (x2 + y2) and compl eti ng the squares i n the exponent,i t i s evi dent that the i ntegral i n Pn can be comput ed for any n W omt detai l s

and gi ve the resul t after the i ntegrati on i s compl eted . At t hi s stage we have

-HPn = e n ( 1 - an) Sn (A-4)

wherej-1 n-1 a

S = L L + --~ S j k i nk (A- 5)k=o j =k 2J " j

1 = 1 + = 1 +22n-1

[ (n-1) ~ 2 Co ~ 2 ( ~' - 6)oe 2n Q 7r [1 . 3 . . . (2n-1)]

n

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h2H = ---~ an (A- 7)2 Q

f l n ~ 1 - 1) Hna

and S j k i s asum whi ch does not depend on n . The quant i t y h i s the hori zontal

separat i on between i nt ended ground zero and the target .

The f actorSA

reduces t o

S _ ~ 2jj k k . (k~Putt i ng t h i s i nt o (A-5) we have t hat

n-1 n-1

( i ) a~ ~nk- f -

k=o j -k k k .

L ( ~n0

n) k U~k~ {~ )k=o ( k : ) n n

where U(n) ( a n) i s the k ~ deri vat i ve of

A- 8

Un(a n) = 1

+an + acn 2 +

. . . . . .+ a

n n- 1

Fi nal l y, the expressi on f or Pn becoms

n-1 k

Pn =e _Hn ( 1 - a c n) L ( 1 - a n2

)U(n) (a n) Hnk

k=o ( k .

wth n n andHn

as i n (A- 6) (A- 7) , The f i r s t f ew of these are ;

Pl = e- H ( 1 - a 1)

H ' ~

P2 =e- 2 ( 1 - '2) C ( 1 + a 2) + ( 1 - a 2) H2

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The di f f erence between Pl and P~s as h gh as 20 percent i n extr eme cases .

Hence, i f we are i nterested i n accuraci es better than t hi s , i t i s i mor tant t odeter mne t he approxi mte shape of t he condi ti ona damage curve .

However, t he val ue of P2 sel dom di f f ers f rom e ther extr eme by more than

10 percent . Theref ore, P2 i s a good val ue t o use i n those cases i n wh ch l i t t l e

i s known about t he shape of t he curve except t hat i t i s not as gradual as t heGaussi an curve or as abrupt as the def i ni te- range ( cooki e- cutter) curve

TABLE A- 1

COMPAR SON OF PROBABI LI TI ES(percent)

ho' 0 1 2

R o Pl P2 P0 0 Pl P2 Poo Pl P2 P0 0

. 5 14 13 12 04 08 07 03 02 021 . 0 39 39 39 29 28 27 11 10 081 . 5 59 63 68 48 49 51 26 24 21

2 72 78 87 ~ 62 67 73 41 41 403 85 92 99 79 86 96 62 69 794 91 97 100 87 94 100 76 85 97

A- 9

The extr eme member s of t hi s sequence are obtai ned wth n = 1 ( Gaussi an) and- n = co (def i ni te- range) . The val ues of Pl , P2, and

P00are l i s t e d f or some val ues of R Q

and h/ Q i n tab e A- 1 . For sml l val ues of R v the val ue of Pl i s greater than

t hat of P , wh l e the opposi te i s true f or l arge val ues of R Q . The val ue of R 600at wh ch they are equal i s 1 . 0 f or h/ r = 0, 1, 2 f or h/ R = 1, and about 2 . 0 f orh/ Q = 2 or more

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1 . 0

00 1. 0 2 . 0 3, 0 4 . 0 5 . 0 6 0 7, 0 8 . 0

h

FI G A-3 CONSTANT PROBAB LI TY CURVES FOR P2

h-10

R

8 . 0

7 . 0

6, 0

5 . 0

4 . 0

3 . 0

2, 0

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Unc l a s s i f i ed

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