Hit Probabilities

7/27/2019 Hit Probabilities

http://slidepdf.com/reader/full/hit-probabilities 1/51

AD/A-OOl 533

MULTIPLE ROUND HIT PROBABILITIES ASSOCIATEDWITH COMBAT VEHICLE FIRE CONTROL GUN

SYSTEMS

Louie R. Cerrato, et al

Frankford Arsenal

Philadelphia, PennsylvaniaMay 1974

DISTRIBUTED BY :

Natioua Toical liofsrri eU.S. DEPARTMENT OF COMMERCE

I1 . . . . .IIIAl



UNCLASSIFIED

SECURITY CLASSIFICATION OF THIS PAGE MengaDaD bf*4 __________________

*REPORT DO)CUMENTATION PAGE 33 FORE COMPLILTIG FORK

2. O-VT ACCMSAON NO. t. N NTS CATLGNME

4. TITLE (and SW6,1dU.1)S.TP M - O T & E R D C V R D

MULTIPLE ROUND HIT PROBABILITIES ASSOCIATEDRE

WITH COMBffAT VEHICLE FIRY, CONTROL GU N SYSTEM4S ___________

* S. PERtFORMING ORG. REPORT MUMUER

7 AUTHOR(S) GIN imcc6

LOUIS R. CEHEATO AMCMS Code: A U

KENNETH R. PFLEOER 662603.12.22500.03DA Project: 1W626603'A005

S. PERFORMING ORGANIZATION HANOI AND ADDRESS PO CT ?AIX

______ __________ ______ ______ _____ ___________ __________

Fran~tford ArsenalA~TTN: SARFA-FCA-WFhileadeluhia, PA 193

11. CONTROLLING OFFICE NAME AND A0,0R11SS1 12. REPORT WATE

ARMCOM May 197418, SlU1111ER0F PAGES

52

14, MONITORING AG9NCV NAME A ASURNIII(H 101fe~mntb CoMb,01*IA'1 0940.) 1S. SECURI1TY CLASSI (ofthi tou.l

UNCLASSIF:I.ID

I~a.~jC~SIICATI0NI/OW01GRADING

16. DISTRI@UTION 111AYTEMEN7 (at Mud. o~vel

Approved for public release; distribution unlimited.

Ill SUPPLEMENTARY NOTES

NA .0N '''*N1:A

II VA I

1S, I(EY WORDS (Cmntinue onl .v.,ee aide Itnecessary end IdesDf.lip black numb.,.)

Multiple Round Hit Probability Correlated Bursts

Automatic Cannon

Average Number of Rounds to hit

10, ASTRaAcT (cetame onMaEvores side if nOesoWIP -' idenDimpy block number)

This report describes a method fo r evaluating the hit probabilityassociated with multiple round or bu~rst fire wieapons when used on

combat vehicles against ground targets. The problem of correlation

between consecutive bursts is also addressed and some hi t probability

expressions are derived. Table. and curmcs for representative calcula-

tions are presented in an Appendix.

DI CAIid31473 EDITION OF I NOV565 IS08SOLETE UNCLASSIFIED -

4 SECURITY CLASSIFICATION OP THIS PAG W~Wn lk na



one 6giQUu.-d I

J ....IC ..lo .. . DISPOSITION INSTRUCTIONS. . .b . . ,.... .. ........ .. . . . . I

IV ... ..... ... ...... ...

eIsuIau'i;:," Destroy this report when it is no longer needed. Do not return it

i, ;..'• to the originator.

DISCLAIMER

The findings in this report are not to be construed as an official

Department of the Army position unless so designated by other author-

ized documents.



ITABLE OF CONTENTS

INTRODUCTION * . . .................

DITSQUSSION . . . . . ... . . . ....... . . . . . . . . . . 4

Classification of Errors ... .... . .. : 4

Assumptions d. . . . . . . . . . . . . . . . . . 5~Single Shot Hi robabilities 5.....S~Multiple Round . .. ..... '.... 6

Probability of at Least LN Rounds Fired . . . . . .. . .. . . . . . . . 10Expected Number of Hits . . .. . . .... 11

Variance of th e Number of Hits. . . . . . . . . . 12

Average Number of Rounds to Achieve a Hit . . . . 13

Correlation Bewteen Bursts. .... ........... 14

"APPENDIX . . . . . . . . . . . . . . . . . . . . . . 1

GLOSSARY ..

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

4DISTRIBUTION............... . . . . . . . . .49

List of Tables

Tables

SI. Probability of Achieving at Least One Hit

of a Tw o Round Burst. . . . . . . . . . . 19

II. Probability of Achieving at Least One Hit

of a Three Round Burst. . . . . . . . . . 20

III. Probability of Achieving at Least One Hitof a Five Round Burst .. ...... . 21

IV . Probability of Achieving at Least Four Hits

of a Five Round Burst . . . . . . . . . . 22

V. Probability of Achieving at Least One Hit ofa Ten Round Burst . . . . . . . . . . . . 23

*" " "". * ,~ i • r • " I " '•: ' '' -'•-'• '• ¶ :: "'• - . ...... ......... ... ............. '



List of Tables

VI. Probability of Achieving at Least

Five Hits of a Ten Round Burst. . . .... 24

VII. Probability of Achieving at Least

Eight Hits of a Ten Round Burst . . ....... 25

VIII. Probability of Achieving at Least One

H it of a Twenty Round Burst . ..... P6

IX. Probability of Achieving at Least Five

Hits of a Twenty Round Burst ......... 27

X. Probability of Achieving at Least Ten

Hits of a Twenty Pound Burst . , . , . 1 2

XT . Probability of Achieving at Least Sixteen

Hits of a Twenty Hound Burst ......... 29

XII. Single Shot H it Probability, P(H} ..... 30

XIII. Probability of Achieving Exactly Two Hitsof Tw o Rounds Fired, P{HH}. ........... .. 31

XIV. Average Humber of Hounds to Achieve a Hit. 32

List of Illustrations

Figure1. Probability of Achieving at Least One Hit

of a Two Round Burst .... ........... 33

2. Probability of Achieving at Least One Hit

of a Three Round Burst . . . ....... 34

3. Probability of Achieving at Least One Hitof a Five Round Burst. . . . . . . . 35

Probability of Achieving at Least Four Hits

of a Five Round Burst. .. ........... 36

2



List of Illustrations

7

Figure5. Probability of Achieving at Least One Hit

of a Ten Round Burst . . . . . . . . . . . 37

6. Probability of Achieving at Least Five Hitsof a Ten Round Burst ..... ..... 38

7. Probability of Achieving at Least Eight Hits

of a Ten Round Burst . . . . . . .. .. 39

8. Probability of Achieving at Least One Hit of

a Twenty Round Burst .... ....... 40

9. Probability of Achieving at Least Five Hitsp •of a Twenty Round Burst .... . . . . .

10. Probability of Achieving at Least Ten Hitsof a Twenty Round Burst ......... 42

11 . Probability of Achieving at Least Sixteen

Hi+s of a Twenty Round Burst . . . . . . . 43

12. Single Shot Hit Probability, P{H} .... .

13. Probability of Achieving Exactly Two Hits ofTwo Rounds Fired, ({H}l). . . . . . . . 45

14. Average Number of Rounds to Achieve a Hit 46

3



INTRODUCTION

This report describes analytical methoOs which have

been formulated and used in the evaluation of fire controlsystems fo r both large caliber and automatic cannon weaponsused on combat vehicles against ground target,;. Currentrequirements fo r hig'h performance automatic cannon systemshave made this type of analysis necessary.

DISCUSSION

Classification of Errors

Combat vehicle fire control errors 1 can generally bedivided into 4 clasnifications: fixed biases, occasion-to-

occasion biases, burst-to-burst biases, and round-to-round

errors. An occasion is one tactical engagement betweenvehicles. A burst is a series of roundr fired in rapidsuccession with the same point of aim.

Fixed biases, as their name implies, are constant ata given range. Gravity drop-off, drift of spin stabilized

projectiles, and parallax error (due to sight loffset fromweapon) are examples of fixed biases. Another source

which may contribute to this type of error is fire control

equipment which is damaged or out of adjustment. In a mod-erately sophisticated, well maintained fire co:itrol system,correction is made for all fixed biases.

Occasion-to-occasion biases are errors which changefrom time-to-time, but change so slowly that they can beconsidered constant over the period of an engagement.Errors due to vehicle cant, wind velocity, air density andtemperature changes are considered to be occasion-to-occasinn

biases. These biases vary from engagment to engagement ina :aneom fashion. In general, these errors are functions

of weapon system parameters (ammunition type, muzzlevelocity, etc.) the geometry of the tactical situation(range) and the firing conditions.

iFor a more complete discussion of combat vehicle error sources refer

to: K. Pfleger amd R. Bibbero, "The Evaluation of Combat Vehicle FireControl/ Gunnery Systems," Frankford Arsenal Report R-1937, Sep 1969.



Burst-to-burst biases are random and generally have

different values fo r each burst fired during an engagement.If a gunner lays the reticle onto th e target before each

burst of fire from an automatic cannon, then the laying

error is a burst-to-burst type of error.

Round-to-round errors are random taking on differentvalues for each round, so that no correction can be intro-

duced by th e fire control system. Ammunition dispersion,

(due to number and size of projectile propellent grains,fit of projectile to case, orientation of th e round in th e

chamber, etc.) must be treated as a round-to-round error.

Assumptions

The following assumptions are made in analyzing theproblem of multi-round hit probabilit ies:

1. V is a normally distributed random variable

which represents th e component of fire control error that

remains constant during the period of time (several minutes)

of a single engagement. The mean of V is the fixed bias ofthe system; and th e standard deviation of V is th e standard

deviation of the occasion-to-occasion biases.

2. B is a normally distributed random variable whichrepresents th e component of error that remains constant

during a burst but has a different realization fo r eachburst. B has zero mean.

3. R is a normally distributed random variable which

represents th e round-to-round error. R has a zero mean.

4. V, B, and R are mutually independent.

5. The horizontal and vert ical componenits of th e

errors, represented by (VxV y) , (B x,B y) and (R x ,R y) are

uncorrelated.

Single Shot Hit Probabilit ies

The following expressions give th e single shot h it prob-

ability in terms of th e variables defined above (the round

fired is considered to be a single burst of one round):

Throughout this report random variables are represented by upper

case letters (e.g., V, B and R) and realizations of these random

variables bv the corresnonding lower case letters (e.g., v, b and r).



The rounds are not, in general , independent since part

of th e total error is shared by all rounds fired during an

engagement; another part is shared by all rounds within th e

burst; an d yet another part changes from round-to-round.

If 'ze measure the cartesian coordinates of each round

as it passes through a vertical plane containing th e target

(with th e center of th e target as th e origin) an d if the co-

ordinates are designated by (S,T), then

S V + B + R (2)x x x

T Y + B + R

y y yEach round in a particular burst will have th e same

realizations of th e bias random variables, V an d B, but

different realizations of th e round- to- round random variable,

R. Letting the horizontal bias, V + Bx, be represented

by X and th e vertical bias, Vy + , be Y; Equations 2 become

S -X + Rx

T= Y + H (3)

If realizations of X an d Y are mpecified fo r a particu-

lar burst, that is, if X = x an d Y = y, then S and T have

means x and y an d variances u2

an d v2 respectively. The

conditional probability of getting ayhi t on th e target with

an y round of th e burst given th e realizations of X and Y is

2I

"S" Ixx f e- ! V2 do2 21 2 IN (J4)

y 22 j

2

To simp lify th e notation,the above expression is re -

written as

P H Ix,y P(HxiX) P{H yly



Furthermore, if realizations of X an d Y are specified,

each round within the burst is independent of the others.

The probability of hitting th e target with two particular

rounds given x an d y, is P(jlIx,y}

2

In general , th e probability of getting exactly K hitsof an N round burst, given x and y, th e components of th e totai

bias fo r that burst is

N{,NXY1jP(JI XY)] [l-P(itIY)jI N-K

M ultiplying the above expression by the probability densityfunction for the variables X an d Y, p(x,y), an d integratinp,over all possible values of bias yields th e required results,

P/Il. (CN)[ xI] [l-P(HiY1XY) -Kp(xy) dx dy (6)

The expression for p(x,y) ca n be obtained since X an d Y are

normally distributed independent random variables with

E{X= E (V + Bx }=

Var {X} Var [V + B )ar12 , y~ 2 +x x x

and similar expressions fo r E {Y}and Var'{Y). The joint

density function for X and Y is then given by

PNY-2yny" 7

The multiple integral (6 ) with th e integrand made up of

terms from Equation 4 an d Equation 7 can be evaluated numer-

icrlly using either Simpson's rule or a Gaussian quadrature

formula. The tw o single integrals which make up P {Hlx,y}are usually available as library subrout ines in a digitalcom;uter system.

8



I

An alternate method of evaluating P{K/Nlis to expand{N ~ i`. by ueof' Uhc lb riomriaA theur.'m.

Y, N- !

Substi tuting this into (6) interchanging summation and inte-

g ation, and letting n=J+K yields.

where

n 2;

oft .. (9),

p{fn/n} represents th e probabil i ty of Achieving exactly n hits

of an n round burst. P{1/1} is th e single shot hit prob-

ability, P{H}. Then the following matrix equation can be

written fo r P(K/N) in terms of P{n/n). Note that P{ 0/O}) 1.

9



1o 0 .o...... a 0 )N)/M r{10 I

* ' * (ao)Pin/n)

The problem thus reduces to an evaluation of the

quantities P{.n/n),n = O,N, defined by Equation 9 rather

than the computationally more complex expression of Equation6. The difficulty with this method is that th e elements

of th e matrices become large as N increases and, since th e

signs alternate in one matrix, small errors in th e evalua-

tion of P{n/n} may cause large errors in I(K/N} . N< 20 hasbeen tried with excellent results.

Probability of at Least L Hits of N Rounds Fired

The performance specification of an automatic cannonsystem may be stated in terms of achieving a certain prob-

ability of getting at least L hits of an N round burst. Ifth e quantities P{K/N) are known, this probability is easily:

calculated. N

P {at least L hits of N rounds} = •P{K/N} (11)

Tables and plots of these probabilit ies for several valuesof L and N can be found in th e Appendix.

N

Since L P{K/N} =1, Equation 11 can be rewritten as

K-20

L-1

P(at least L hits of N rounds } 1 ZP{K/N1 (lla)

K=O

so if L-ON Equation lla reduces th e number of terms that

have to be summed.

Specifically if L = 1 we get

Probability{at least 1 hit of N roundi}-l - P{O/N} (a2)

10



Expected Number of Hits

The expected namber of rounds that hit a target, EH,of

a burst of N rounds is given by

EE K P (K/N)

using the expression fo r P{K/NI of Equation 6, E becoresH

EE.ZMff [(H jx'y)K [,_p{(Hlx,y KK p(x,y)dxdy

Interchanging integration and summation yields

N Ki

H a[pIp{Hlx,y p(x,y)dxdy

N

Using the binomial theorem,(l+r)N= rK

K=O

differentiating this expression with respect to r, and multi-plying by r yields

N-1Nr(l+r) - =r

K=0

P{HIx,y)

letting r= l-P{Hlx,y} ,EH can be written as

EH= NffP Hlx,ylp(x,y)dxdy

-) -0

=NP fl/l}

NP{H} (13)

11



The expected number of hits on a target of a burst of

N rounds is equal to the single shot hit probability mult i -

plied by N. Values of P{H} are t;abulated in the Appendix.

Variance of th e Number of Hits

The variance of the number of rounds that hit a target

in a burst of N rounds is

N

2Z(K-E )2 P{ K/N}

K=O

which can be written asN

°2=-E2 + K2P{K/N}

KaO

Substituting the expression of Equation 6 fo r P{K/NI,o 2 be-

comes N'

;02 -2 K2 j[NJ H Ixy K[1PH x,y, N-K p(x,y)dxdy

K=0

In terchanging integration and summation

a2 ~~ - ol-p{Hjx,y},,'HIxyH

K= p{Hp,y p(x,y)dxdy

N

Again using the binomial theorem, (l+r) jr

differentiating twice with respect to r and multiplying by

r2 yields

N

Nr(l+Nr)( l+r)N-2

= ZK2fVK 0

K=O

12



Pý(H X,y}Letting r a - o• becomes

I-P {HjKa,y}

.u~~dd --

Letting P0{Ifl}P(2/2) ,that iathe probability of getting exactly two

hits of two rounds fired,

U2O[ H - P M]+ -1p M) 01) 2

Values of P{H) and P{FH } are tabulated in th e Appendix.

Av-rage Number of Round-,.to Achieve a Hit

In this section an expression is derived for th e aver-

ag e number of rounds which must be fired at a target before

th e first hit in achieved. It is assumed that no at tempt

is made to sense miss distance or "Lo make corrections be-

tween rounds.

When a series of rounds is fired at a target,the prob-

ability that th e first hit occurs on th e Kth round is

ff Hx.,] [1-P(HI~y} p~xov)dxdy

The average number of roundo which must be fired be-

fore achieving a hit is

EEt C "PR~l -PtHi x,V1K-l p(x.y) dxdyr

In terchanging summation and integration an d making use of'

th e relationship

K-1

13



yields the required expression for the expected number of

rounds to hit the target fo r the first time when

r= [-PHIXY)]

p(x,y)dlxd

24 X , 11 (1 )

The integrals in (15) diverage fo r n)v, that ii, E? is

infinite. For n~v th e integrals converge and tha yalues ' of

E are tabulated and plotted in the Appendix.

Correlation Betweon Bursts

When two burst are fired in succession a t th e same

target, the events are not generally independent. There is

a component of bias error, which is common to both bursts,and another component which is, in general, different foreach burst .

The horizontal and vertical components of the total

bias for th e first burst are

XiMVx + Bxl

YIuVy + By1

For th e wecond burst th e components of 'bias are

X2=V x + Bx2

Y2 = Vy + By2

Since the components BxBx2 s ByI and By2 are mutually inde-

pendent burst-to-burst random variables, it fol lows that

(XI-Vx), (X-Vx) -v) an d Yi V,) are mutally inde-

penden wh the folloing co n itio a l probability density

function f(xl y.i.,x.,,yL2IVx=Vx,Vy=Vv)

1le.(Xl-VX)2 (y1-Vy)2 (x 2 -vx )2 (Y+ ("(

Z1 + +x

72¼w*-'Px;y Lx y j

14



and the probability density function for the occasion-to-occasion biases is given by

g(vx.vy)u Wxy+(7

The-probability of achieving ezactly Kj hits of M, roundsfired in the first burst and K2 hits of1 rounds firel in

the second burst is

f(x1 ~lx,y, tYy1vy. y )g(vx,vy) dxidyldx~dy2dv~dv ~ (18)

Inertngo h variable. v andov~ he 46bove cx-

[~Ký Kp 19

where

h(xlyl,x29y2)u w4?:' 20l-p~x) 2~ X

2ffn-p2 xl

2 V2

and15'



That is, th e biases are Jointly normally distr ibutedwith a correlation coeff icient which depends on th e relative

size of th e variance of th e burst- to-burst error and th evariance of th e occasion-to-occasion error. If x-spy W 0

th e bursts are independent.

Equation 19 can be simplified by making use of th e

following relationships:

1. From Equation 20 h(xl,yl,x,,y 2 ) can be written

in the form

h(Xlyl,X 2 ,y2) o hx((X,x 2 ).h(y{ 1 ,y'2 )

2. From Equation 4

P({Hx 1 ,Yl) . PFH x 1)#.P(KyIly.)

P{HJx 2 ,y 2 ) - P(Hx Ix 2 )'P(H IY2)

3. By use of th e binomial theorem

NJ-K 1

i-O

N2 -Kn

[1Px 2 yjUp-K -2

J-O

By substituting th e above expressions into Equation 19,

interchanging summation and integration operations and let-ting nui+Ki and mr'J+Ki, th e probability of achieving exactly

K hits of N rounds ?ired in the first burst and K2 hits

oi N2 rounds fired in th e second burst becomes

N2 NJ

P(K1N1,K2/N2}( (N2)Z, (_)n~m-KlK 2 (Nlj'I) (N2- 2)PnnrI(1

mwK2 n=Kl

a6



where

P(n/n,m/m}*/ [P(HxIxl']nP(H~xjx2] m hx(xi-x2)dxl,dx2

-m ,5

xJfII~i2[P%,I 2 .) (22)

This procedure reduces the complexity of the cslduls-

tions. However, as noted previouslysthe appearance of th e

binomial coefficients in the expression puts a practicall imit on the size of N1 and N2 .

IT



APPENDIX

Tabulation of Numerical Calculations of Hit Probabilities

Tables I through XI give the probability of achieving

at least L hits of N rounds fired at a target, fo r vbrious

values of L and N, as a function of the total bias standarddeviationn, and the round-to-round standard deviation

Table XII gives the single shot hit probability, P{H) tor

P{1/1}), and Table XIII gives the probability of achieving twohits of two rounds fired at a target, POWH} (or 0{2/2)). Table

XIV gives the average number of rounds to achieve a hit, ER.In all of the tables n varies from 0 to 20 mils and representsboth the vertical and horizontal components of bias (i.e.,nx- i *n);v varies from 0 to 6 mils, also equal in both dirnvn-siege (vx*vy). These tabulated data have been plotted inFigures 1 through 14. The figures have been numbered tocorrespond with the tables.

All calculations were made fo r a target size o! 7.5' x7.5' at 1000 meters. To use the tables and figurus fo r other

size targets at other ranges the following equations should beused:

T 1000

T 1000

where n~ and Y are tabulated valies and in(RT and Y H,¶) arethe errors at a

range R (in meters) against a target measur-ing T x T (in feet).

As an example, suppose we wish to know the probabilityof achieving at least 1 hit of 2 rounds fired at a 15 ' x 15'target at a range of 1500 meters, given that the 'bias stand-ard'deviation is 0.8 mils and the round-to-round standarddeviation 0.4 mils. Applying the above equations we get:

7. y i0a. x o.8 - o.6

1510

S ... x 1500 . 0.4 0.315 1000

Consulting Table I fo r n a 0.6 and v 00.3 the probability ofat least 1 hit of 2 rounds fired is 0.9187.



T -D z .z D %0 %V 04 c- r1 0'0 - 04 k0 0%

o~~~ ~0'0'000'0'00 N r-4' r4 0 0

1.0 '.0 '. '.0 '0 3~ el' T4 ( n r- 0% -l 00 P'-%D 0-.0 ýa'D 0 D. 'D0 '.0 4. m..0 m'z

u0 0 o000c,0,0000 "A

A 4~ N-H 000 0000

- ;4 & R n W- - -

4-4 - Z -: -- - M~Z4 W. .r .iH, 40 0t02 9 . * * * 9 * S 9

('e400eln %'. % 1 I .. J~2f' LM m m z C4 C-4 V-4 0% a, N. 4No 9-40 00O LPn e

I- M N D 00 00 Lm 0 m 'D.00f.

m qW4 0 .0 vi LM

%W m m m~v o~ s-I-4 P4 0 iroo * 0 r-v n o r-- o0 vi 0p

9

-4 -t e 0, Mi N M' fnv 00 wul'%iz

P- ý0 0%00 11 ~ i0o %DL1 0%:t9 C"~ 04 "4 00

0 0 * S 9 9 *0 * 9

00 M r- V4CAJ Lfn 0 0 0 00o N o in L r-.r-4 -4 0 ,.0 en m W,C

00o r- M N- %

r-4 N 4 r- 0 C-4 -n p. ý M ;10 N in00 00N '

.0 00 0 '-0 in 4- r 00%c. F. 4M P-I " -d 0T0 0.1C-4 P.4 M % % 0 0 'iv i -4 0 0 0 0 0O L M V -

mJ 0 *ýt m~ t,4 r- 0- u' (n 000 N 00-I0CA %0~ 00 00 Mt " 0 N ' %D0V co40,L

to0%0 co 00 t, Lr V-49. t N C O O Sý4 P

- S - - - - N- -4 r-- - P-

00 Ný4t I,-- Ni O '.

0ý0% cl fH @ 0 N 011'. . sIO 0"

% 01%0% LM. '0 P.4 &M o4 m-0 - V- 0 N , m im- - C - C-

m c 00 0'0 (7A M ON 0 N -v Ln e N v-4 -00 0 0% -

00t- ko 04N' vi C4~

clO 04 8 8 o0%0 r-0 9.. :t% CiN.0~ 0- 0. m- in 8

00 00 0%. Np 'D N% t0* )mI .v0 gc

oil I -4* - * * . u u *

50 ka V4 N 0

.0, C- 2 00o0 N0

ON cOv N0Ct 0%041 14j '0 0 NS

I-I - 4 0 . 0 00 0 a a 6

-~~~' V T---- --

00 0 0 00a s4~ I44 "4. N0 m N

TSIN NI 1ANOLIJVIma amrvNyis SVIR Iv1loi

L9



Ci O 0% 0 0~.01co c, eC4l %'I - aD -i '0o C14 CN N Nl 04 Nq - 0 co NO C'J- P4 ON 'r-. r-4 0 '0 ON

*~ 1 -4 H44 -I 14 "4 I H 0 0 ON ir- '.0 i .4 cn "4 0

0 4 M -n 0 m 4 N0o - 0 t- r% ,4 -.m .t '. 00VI, U, 1-U~ 4 C 4 0% 00 '.0 9 m r- 0

U,-1 "4) L"24 -- U' L4 -4 V4-4 0 0 0 000 0

-4 0) %.D 0 H- O %a WA0 00 "4 N ON. 0 t M, kO' 0 ON LO0 L k.0 I0 , * (N H- 00 0~0 C l 'n.0 t I'- r' 0 V) %D0 ON

(N O P4. 0 0N ~~ 00 ON'. r- 9-4. 9o "4 0%r 00 0.

c 0 CA 0011(((Clwcfl(11 1 - c rD H 0 0n000N

go - I- h- -- It M "q-.C n 40 n m "

y-4 P) c l in cn co c0 i n N N0 V-4 V-4 LI 0 -' 0 0 0

0h iný-. P4 te) GoU aN so in, so G

Vo in 4.' :,. 4clN Onr-. en 00 r,-.P4

0 0

in 1so so0 so.0 'p0 ko'. 'o L -4 r.q 0'p~ 0 0

Hl -1 6.~- -

(y)C U) C)10 '.(n O "H , 00 -4 (N-4 0 .N 4 O 4"Im) m0 m, ill V-4 CNI N W.0 CT__ '.Cl-. t-.~~C "4 'D H ,n " M 0 .0 t-.4CN ~ c "U . -4 r-4 0 0 0 0 0

4-4 - - -Nn C4 O 0- -y C r-v- Y %. t4r) 0 00 110 ~ N 0 0.r-) 1, 1" " s ON P4 ,U PH N%. 1

Lm ON 00.0 N P4 0- 0 inc-0 Cl 4Ut. (e0-4M% m4.4 W~000 0%-4 ' ON -4NU Z,- H0 0-i r s

V-4 ol as ON N -" ~ .NNm 0 0r. koC4 r0% 0 0 01-4 1 #0, a 0 0 0 0 0. 6. 0. 0

so 0H ko 1-- as 1%% g%0001 a% m.4 o 00 0

"a, as ON -\ - -4 -1 -0 - - 4

C) N~.% OH ON0 o N 0 .o i N 00 t10 ON m%0 0 m% m% a0 in F- r4 N00 000 0G

bn3 m ON m 0N 0 C. ON -. GO "4 ' LM N

0 ON ON0 U, O -0 C74, %'.O "4 N 0 0 ~ 0

Cf0.0 ~0%0 0 0M0I 4&,on 0 HrUl 00 0DIr4

0,00 ZZ %D ~ 0-, 8-0 6N~0 1-4H C)C'1 i -k4 C) C)00 0%0N .

a H .N 4..01O

0 s0 0 *,'. 60 -0'0 ,o-4 0 0 00SI"',C"

0000 N CIN go(00I~l NHN 0.000 00 C.4 a, 00

C) 0) ON0% m40 wI- 04, 4.ON N ON 0

04 . 0 .0% 00 0% N '000 W 0 000000f)"47 olo .) a099 5 5 5 5 9 5

C000 C' 000 "4o0 '.0NU,,N.lN so w"r4 0 00000% C7l0V)%.o!2 % C 4 lN s"(00 C

.0 a00000 0' 00 ('4 0 U, 0. "4,C %~400 * 0 s 0% as0 I0n en. NHOOO 0 000

.0 "4 H0 k-. "4Aa a% N9 5 S 5

N0 00 -- 00 Q30( MN "00 Lf ~d M 0 U.00' a% M00U L0 L. N C

0 * m 11 1 Ci% 91 1J% .ý op.4l "4ý C9ý *i *ý * * . 5 5

"4 "4 "4 Nn P5 P4 q.

000 0(0."4 0 l N NN 4 %4% .4'.00

0SUI 0 0% c 0 CO ,-' .0 '.0 CINI VI l (1 L 00

0 00 % % '00% 1h-.j 4 00% U Cl'J 400

9 . 0' 0 0%O'-,.4 l HO 000 I00d



0 00 00 0 0 0C00 0 0 0 0

4 0 00 00 000) 0c C00 O 000 0 000D() O

4000 0 0O.0 0C)a0 ((D0 0000D 0 00 00

o00000 C) 000

008000000000000 0

000 000 0W *1 *1 0 a a1 6 0 9 a$

No~ Zn~ (nC) 0 Cu ON LN "4 H 0M~o ~ MN 04 04003

Lk ~ ~ ~ ~N N V v r -7 'V) 0 ON r- ONf P.4**a a

0 (. 1 - - M 0 % L '

o- P-4 P- 0 0 8 800 0 CM--r0j

goW oin r - 0Ci go "4 as No C I m N 0 a

* q 1 4 1-1 o a a C)0

tm~r' uln tN-- 0 0 0) 8 00 0

-PV' OD NDcq ONO gou 0CAw OaaN C)H 0 0 00 0-o A a a a a a a No m a P4

00 G0 ID LncqC

0N ON C14 "4 "4 0N 4~

04 0, 0m 0' CN 0O ~~N L M N0 ON C'.M -~ - ~ O i~ M 0 aNN .N ON r H 0t0 0 0

-H ml FNu ma~ cp-- -wO"4 NM ON r c ON Lr 0I000 ON ko "

*a CD ea 0 c 0 e q c a Sn Va o a 00 0% 0% - 0 0 0H C lM c H~' m.N

N- - - - - - - 1"S-IIWN iNIVAK ~v~~ssii 'v

cq ,n



01 Vcni N 0% rI V 0 cn c0 0lo en~ ca 00

(7 N a% cou co 0 0 -r - In N~ H O HO%4 w In (

V1 N I ( 4 Ni (N c - v-4 -4 0 0 0

in en ci t- H C1 it) N if 0n cc %0n N - 0%Q ~ 0 V:n 4 m H Nt I4 ON In Hýn c n "

*0% -4 "4 P- -4 P " 0% CI Gotr-.-I 00 4 co %t 10 '

-1~ 1 Nt Nn Nn m cNN N ,-4 P-4 H- c0

-J . . . . .1 .6 . . OR 61 0

4* H0 H pI 1- 4 U4 00aN 0' %V

In m * *0 * 4 *

"4 ý4 C"NNj 0 I MYN 0 n eq r, tm ~ %D 0%% rm %D el v-

0. . . 1 9. .. * *. . * * * OR *

P-14 crM 1. 4 - %0 = a% N- N, co 0% D en 5 Q0 ON '.o In

m Nu C'4 0o Ny-I% 0z "% " m2 v' H.cc2 00 00( " r- %0 en V NNI ON N- 0 m v-

N- en H-0 h i .tN NT ('qcv ccn~V NNn N0' UC 7 N 0N P- %0 "H N

* N NN N VU1 C Ij'%4 00 4 V14 (H 0 %D % 0% VII Go 0% V Non M' N H 0 01

E.4 (', Nr V' 0- CM %DII~ N r.'4 "d0,4N_ ON Ns VN ON CA m O~ 4 HN 00- .0 40 N 0-

4 OH en 04 to 0% cn C4 I- V-% In ON f', '4 MH 0 0L% 00 -

-N Il r- %o ýo H00 (5' %1 r4 " -0%f '.0 4a Vs ra'

04N ON ON CA 9 V C ,N %D n4 N O HD V N . P1.W ON ON ON a% ON O as% C% tr- "1 Ha' -4 ', (,

Ha' 0%0 %0-- In0 co ch CON u'1' Hn,'I-O4% 0 0 0 0.

0 0 0ON 00 C4

'M)C a4) N if'V0 W C t I 000000N% N N l ON 11 C4 0 NH)4 N-" c ) m r

- o .-i , e4 N M 0 r I, Is c % ocoV 0 0 a% f4n u N r 4 ý , 0 4 t

m0 0 0) 0 ItN Nn ID e5'n M VON (n' N H

a C * .0% 01 In V- 44. IN0 M4 M 00%0 000 "

Zf' Ný Ný c%4 Lr0) Nt'

qu i 0 00 0 VOVr c qC N 0r- M dIn 00000a 0 o0% CAN4 00I N 0~ Mo 0ON I

0~ ~ ON, 4%aI N H00000( NG0

m0 ON % 0% Ln c; -%% H 0V '

00 % N n0 M'. Mc'% 00 V41M

00a0 00 ONV ON N 04 t0 0 N0-1C)0 0 0 C7% 0(7% CN 00 N- In M,4 H0 0

-4- - - - - - - n ON - 7 1- (D

Hn 000 Vr MF.4 004 0 Na 4 n N0%

m 0 0 0 0 0 a, 0% % 4 C1 iO 'N 4 V Nr-0000 "%V 0- r! 'I0 % c:4 N %I

u 04 Co * 0 .0 % t%01 0 N '.0tH 0 0 0 0 0 3 004%.c

0 on -

N 0 0 0 0 0%H N N In' 0tN H q D w% 0 1 i'

.00000 w C) N0% 00 C':4 ND 4M Nq H- 00

0 o *4 . 00 en0NL' NH 000 %000

O 0 0) 0 V. (D, w 0% 4 %D' %0 N "4 04 V 10% 0m 00 'NHO 0 0N (N (J N l 4 m s n en"0 N ) o0 C3 aN a, w0. N4 C Cf(Y000 C 00C

00 "t V IC N Nn 0 NZ

510% O"Vc in.4 n H 0 0 0H; H. *: C45* *

- - In.-iD, --- I -

00000.....



a~ o 0 010 0 010 0 0 0 0 0 00 00 000 0 00 00Cc0 00 00 0 0 0 a0

a000 0 000 o 010 0100 0 0 00 00000 00 00 00 00 0 0 00

000 000~0 000 0 0 0 0 000 0 0 0 0 0 0 0 0 0 0 0 0 0 0

-4 v-f P- 1- -4 4 r-4 -4 0 0 0 0 0

0 00 00 00) 00 0 0 000 0 0 0o oo ooooooooOOOO

C-4 .-. 0 0 0 0' i o 018 Q ' 00 000 ,4.I 0- p. 40 I. 10 0 0 Is0 0 0 0

-4 r-000 00 000000 1-0000000 m o

" "4 V-1I12'N %D c W 00 C-4In 00(n00 Ln Cn NP "- 0 C:

r 0 000 4 -. 0- -4 V- V-4 r0 0 c0j0 ' o a, - ),00 0

o1 t-N % I N4 p ,0 "0 VIQoo 00011% 000 0 000 0000- tc ýo' ýN 00 000o V40

In4 MNO 00 ~0'4 v-4 0V 0 NO -0 0 0 (01-4 0 i- - 0'a . ' O O1 . . . . . 0 .' a d- 0 0!.

cn en aYD -d' m - ýo ý 00 ýOI - ONNU M- C-4 t-I r-. V-4A 0 0r- 0 0 0 0 0 0 00)cn' C ý

*. Aý- En A A A * p

0 N ,l N 0'i 1. N C4 uIt 0$ ' N (r) Lf) 00 uI ý0do "- 0 I~ 0l4~ rM %0 .0~ r- V) 0D w P04

w CISC7 M0 M WY M N in 0' u ýu- f uI uI0 0 0 0 01-1

- A M4 1% a'4 00r, 0 000000"

N P--4- -4 -4 r-

Ch) g 4 ' r- 0 0 0 N N C4 co N0 ~ ~ c-r-NL~C' - 0 Go 00 u-I in0' C N . -

P-I %t I. %D " C)0l (cluI ul0 000

m~ I=. v.4 00 (y, ON N i 4 "4 C) 0 0

00 u- 0 N0 %0I r 0 (N a%~ ki 0 0w 000)'A m,-4~~~.T n4 O' - h r 0 t N (n m V) 0 0(

0 0 0 .A c VI 4 uN u 0 (3N ult) "ri C'

in 0 0' Nl W (IN 0 P-4 ka I'. Ch m~0 c* 000 1 '0 ) V4 00i N -40 00

0000 NO (n mu) ON1 00 U q'

0 0 N) -o0 0. I-. 04 l (N C'l ' ) co Noa0 C0 0 IM0 Ifq m NO .4- -0' V) 1-1 N ~ O en

-4 0 0 0 0 m Mu40 N 00 -40 CC)'I)

.0 0 t u'I 0' :ý n 1-I' 1' 4f 1-4

C) 0 0 NL 0 0 -4 -1 40 N m'4- C 00 "4 N0 0' Ll0NLP0 0' F A10 N1 I

0U N*OVA,( NTII SV'0'' 0 l0 0 0 0



0 0100 CI 00 0 000~ 00 0 00 0 000 00

0 0 0 0 0

0 000 00 a0 0 0 a 000a 0 0 0

0 0 00 00 000 00co0 000 0 00 0

ci 00 ,00C 00 000000000

$4 .i to 9 a 0 a 8 0 9 9 9 . .

9 0 9 0 81 8 8 a* 9 9 9 0 0

0 00008 0

N 0 00

Li fn r4 H 0C ,0 4"

C44 v- r--4"4

000 8 0 a000 800004~ 8 09 01 8 0 0 01§ 0 a 0 0

i-4 00 00 m %' 00C0 m~ 0% - V8C' 1 N v-r4 0 0eq 10 Ln N as WD0 04 -4 00 im A. 4 0 OO OO

N P-4000 0 0 0 0 0 0 0 0 00v

U,4 r- "4D Nn * 0a kn M- "a a'.4- -n in - - -

0 &o4 C~ HH M N 0.-4 r- 0 ' N ' IA~ -1-4~~ 90 0 0N-r0-U ,0 ' ' - 0 0 0 0 0

a~~~ .1 of or $1 eAM0 0 0

44~ - - m N N -r- 0 0 9 0 0 9 0

r%001 N 00, 00 %0 r, -- 4 P1 V1 "N -0 0 0 N1. %D C-4 r-l0r4- H 0 0 0

0 N N 0 - 0 n4 n "04 H 0 a 0 0 0

% N . na . *

*A I 00 - 0 'ON Ný ID HM 0N 1-4 r-40 0cm m0 a 00 00 1- .4 cn q 1 0 0 0 0 0 0 0 0

4 06 r- go %0 04 64 P- t- N

'H -4 - y " 4 r1-I NUý I'- W~' Hn 's NO4609 c'0 9

4j , Lm CM H 00 ON W) %.a'00 cN r- 4-40-, cn ~ Lm A- 9'N ;, NH 0 0

,4 W0N N 0 .. 00 0 N CA N n0 . N -

.44c 0' C' -Nr 4 00 1- ,-4 0' 0 A N4-

0'C-4 H -0 0 0 N0 N. en en~ U )N 4- g- 0 0ý 0 0104g 'D (M P' H' t-'0 C n 0.44-400000000

a%. r* "0N e0 %D n t- ;g C1 -49 0 * 9 9fri a ý ci o k- t M N 0 -

r-00kc LM L00 P,4~ M,4. . 10%0 'C -u 11

C~ S 9.0' a 0% m 0 -. 0in CAN H 00 0 0 0~v-4 r-1 w * * l e * . * el . a l of 61 1 a

.0n (I00 1

a00 * '' HtIAA0h O -u 0 4 00 0 0P.44 "* 6 4 4 6 4 a of a C Ao of 6 q 4 1 *

4.4 1 1i 19 9 * 9 199 c; S, 1

0 0 0 10 C3 0 r00% -. -4 - N 111 10 .'- HO 0'.4 N

H~ ~ ~ U1 4o vm'vi svi iAvio-0l0C 0 N4

* 0 ' 0 4 0 0 % 0~ '.00 .4A25H O



r= Lno "' OsN 040 cr. I~ c. 00 in O4 oo ico o oo f-- r-- r-. r-~ r- r- .a % D~ r4 - N - 0 IIm -4 U%

M. (n~ m'P C"i cVn M C4n Cn fn M4~ Cr N N N r -4 4 0 .~10 44i . o. 44 0. "00 "00 . . 4 . . 4 4

cn N 0'3 wd P-4 ON .4 I-% (n CO 4 .t i-I en ~I- I-.4o 0 w r'- r- '0 -- ON~ w%o C . N m ~ r0,4 P-4 '.0 Wo .* 0% CI" O em C% a% 0% 00 Z9 I- mO en r- (n %c P-4 kA~C'

UL .4 .4 . -T It II ýt~.4 . .4 mC c'n N N' a4 44 a . 4 4 . 1 44 1 0. a4 4 . 44 .4. all

%fC C ' o '0 '.0 LF% ,-4 r- IOw 0m C 4 0o % 0o . . Co 9-.4 0 ON N 0 '.0 U0 '0 N

w o % %%o0.o 'D tn.("m0

o4 04 NO -C-4p 0' P-4 '00C40 M L M M r D 0&MO COC CO V-1 0 M 0t un a in

(NO' %0 %d go0 00 00 cof0% n cn 4P4 "

cr 4-4- ON 4-N m 4 IC4 P - S M " C 1V

0 -lo G a 0 O fý I o f-- - - - - - 5 - -

&rz 0% GO W0-'O% M V% M 00 r,% N Cu4 9-0% 0

ON4 ON ON0% 0 0' Ch I'l w%C N N aI 0 N

t 4ONO 0N 0%ON 3a' L )N Cl0 M- 0- .4 0 I V

- 0 0 0% aC %J ,-0 C0 0 I%

m ~ 0 I 0%0 0 %N M 4 F %v- r- CO)-4 tn M~ 4

LW ~ ~ 0 0 0ý 0% C)4 pf Co t- 0% LM0 v-o CO) v.0 N7 P

~0000 04O %N oN -ý2 ON C o 44 '.00 44- PP4 a0 0 00 (7% ON ON ON N N 4 '0 - O4- 4 -

j OH- 4 -.4 1 -4 I -.4 a-4 4 4 4 S S 4 4 4 4

4-4.ý4 0 0D 0 0 ON m4 Nf %. r- -d 00 Y 0tP O '0 ') CT'i00000 0 %D0 N ON N 00C N CO t00(Y) L UDM r% F

.- . 1 ON ON ON CON co 11 f) CC) 4- v-.4 0- 0 0 0 0>~~~~ ~ ~ ~ 4P-4 P- 44ý

.0 0 00 l-%

o O 0 0 oONN w~ '- 14 A )0

2~~~ m. V)**a'0.0 % o'0 N ý-4 OD V0 0

0 0 0 0 ca r- C4 C4 0 C U Co . N C .4N

Ct40 0 0 t%' 1100 C-4-4 0.C4 0' . 0 0

0l 44 (D 0 000% 0%4 0%o NO. 00 -

Im a4 0 0 10v-4 v41 004 P-4 N - pO WO N. U80N 000% th .i. IO)% 0 O -4C 0Oi a-0 a, .440%Co LN L)t N )0

0.44 00 0% 17, a' Co0 LM )N -4 i 0w2 m N00

01

140000N0" O N enn N- NO OL) CiN4 -P 4 0000., 0c 01 ON m w4%0%Q mCC' 4 00 0 000

04.~. rCO LI4 NY' H CO .40 . -

04

900 CO 41) N .40%(:4 00O c. C) H -4 0

0.r 4 C%ON %CO - 1.0 P'40 0 0 0 0"4- "-4 -4 P4 J-4 4 44 4 4 44 4 4 44 4 4 44 4 4 4

00 r%%'00 i-I 0%0 IIC'M P-4Q0*~~O f" P0%% 0 0()O O ~ O

"0 m-I "4J CY) If)'0 C 0 Ný ) =0 0 0 0 0 0

0 0 01 00 0 0. 0 H i-I N icq )'0 CO r- -

sr'lw RI tL UNNI~IAaal UalVaVIS SVIG 'IV;1O.L

26



0 0C a00a000 0 0 00 0 0 0 00 0

0 )0cC 0000000 0 o0 0 0 0 a 0'.00 00000000 00 00 0010 0

at e rn~4 4 4 N4 N' N N '~4 H H O 0

000 00 00000 0 00 0 0000 00 0

u~00 0 0 a 0 0 0ý 0 0 0 0 0 0 0 0 0 0 .0 a

NCO Ln 4 Nt w0 r% N LC- q 0o c7 n mP4 r-.. n N

OC4~ V ' ' ' N N N N- 0j 0 0 0

- - aM 4 In Go %t N N - m r- a 1- W- - - a m

0 ~ ~L0 j N Wf H N -I' CY C% 44 % .0O N -I N1-"4 H4 r- 4 H 'C N H--4 0 0 0

1%% N I %NJ NN NI 00 0WO00 1

in ')4 N A WO4 O H - Pu 0

0 0 0 mO m44 wO mO m% t-O -0 en %0 mNHChr I-N I%O N N- m~ w' In mP-4 0 0 Q

V) V-1 C'% N - C n N H 00 0 0 0 0 0 0

a 1 a %a0 a a -. ON a o-4 " a r- w Z m l N

C N ON UI r- H- %0 r 1f .0 -O .0 = t)hw r- ýD N N CO m

Cý mO 3N t" N .ý N C, 1.0 C,~ tf 004 L,)r4 0 00 0 0

CNC H 0%ON tU HD %D N O N-4ý Ch 00OCOC41 C4 P-4 HC) 0

0 Ch Ot CM M 00 r- ýn ýt I-i -w '0 0

o aN O0% N 0 V- P4 N11 CIA O V-r4 N H OCV) WH 0 0 0% 0%0m r-I CO Le '.c ' , It NHO 00 0

al a aN aN C-0 -- a0 a 4 a . 0 a a a a

NH '.00 ()00% .40 HO-) -41 0 r- CO) r CIS 00 INod 0 0% C.) N 0h 0% '.0l 0- LI ! 00% LriC N 4 n en

C 0% 0 0% 0% m % u4 r-4 0m t n e n C4 "N0-4 Me N O 0 0 0

to COO N N cN cn ~ m0 C4'0 N - 0 r44JI -

CO 0 0 0%C Or 0 cr% 00 en r * 00%nO ~ r

. 0 0 0 a %ON '0 CO %0 -.0 % Cq I- r 14 H O 0

H 4 H . 4 .m * * . rS Cý *ý * * *

en P4aM a - - a a -4 aa4 a a -4 IM

M ilh M0 0 00% HcI t. 41 o~ 8 (D4 0t 040U4 '. 4.-4 0 - 0 0 0% U 0 v of .1 0% H e~ N 6 O

0. *n%0 P% r0

0 O ' 0 0 000 000 0' 0 '.0 00 .C CO P-4 M' N H 00

sf4 00 0 0O 4 C f1i 10* - 4 v .0 % . 101"0 "0 m4 % N if40

0 5 S 5 0% 0% 00 N fi4 N H 0 C

on an a4 a0 r-- a - a "a 0 ea

V- 0000 0% c NN as M % Ul N N OH00 Hz ON ;88 0 mC m N N 0 4 00% It" 0% m4O 0%N C)%0

.0 NN4 M1 NNc04P 0

.0 0000%D "OC 0 MCO H 4 CO 4 r-% HtD0

-00 k0c40 H 0 Lr i 4 Nr "%L4 H) %CI~~ toOONOCOC%0 0~1 HD a,

I00n CO 0 O I i i C4 I.~ý N L4H1010 0 .0 0% 0% CO CO "4 " N Hn4OO X 00 000V-

-~~ -ýn - - a aa-aa- -- Al-

T~l NI tZNU COIT NH(NJV9r~~



-~~ ~- - - 0-1- - a1 a-

0 00 000 00000 OQ0 0 QQ 0 Q i

0 8O0 8 O00 8

0000000000~00 00 0 I 0.n0 0 00 0 0 0 0 0 8I 00 0

Z 0 C30 000000N0000000

8~ 00000a0, 0 00 0 00 00 0

%D- -% 04 " -

" 0 0 000888 200 CO880000ý

44~0 0D8 . o rý e 4 V4

,,4'O co %a f0 en -4 G~N % 00 "%%0.0 0MD o C

Sm0 0 r- Nýuoo i) h e N s-I

0'0' tN Uý- Nn 1n(4P4"0 ( N 48 0 0'% 0 0% O r-C N 00 0 0O01C G i.

00 C4' -4 0 M0e ,00%I C4& N n- 94 v-4a,00a, %r.v 0N LnP4 " N M4M0 40 %a nO s w%m04%4 0% S

N000 no as M' %D cn C4u Cq wg~-

%o U,%D% ON goIPS0 V4IMOSO O00 ý 0%- N 8%

00 d, 0 %h M 'I 0 0to."

4

o ~ Sl Sý 9 CS7,4 - 0- 0% m I,- -n o r-- %D U) m- -4 1

wko% Ln N. "404

in0- V00 Ns a ON %D % 4P4 00 8 ON40%% 0% P0-4L'4 N-N00 0 00 M0CI

r 44 ~ON o 9-4U 00 c'. 00 0 0, C0 00 U-. ~nO% f81 (%1O

V4I P-4

P-4 t . . .... . .9

0 C) (N*r v000 m' P4 CIO (a-IS-4 - . ON S0. Snn P4 S

00 01 0 0% r-I ZI ON ON 4 U~. 0 -

288



00 0,000 0 0 0 0 0 0000000 00 00c 00 0 0 0 04 000 0 0 0 CD0 0 01000000000 00 00 00 0 00 0

t000( 0 0 0 0

* 0000 0000 0 0 0 0 0 0D140 00 0 0 0 0 0 0 0F0; 0 0 0

0.0000000 0 00

0 0

C.,0000 0CD 0

C -0 0 01 1 0 I

PQP- 0 0 0 0 0 0 0 0 0 01 01 0 0

.48 10 0 08 0 0 0 00 00 c00200

0000 00 0 a '00P-4 H 0 0 0' 0r"00 000

W4. * o N "* vC ON 0 *c 1S f. a, -.I N m *% (q * n C4

co - 00 C - P4 t - g C -4 t--& 4 1-4 t- " '.o N , .0 It4 m 0-I 0 00 00 0 0 0

o -n M N N P-00a 0 00 00a0 0 00C0 0

- ý -00 ON 0- , l O P -

cn -o N~l 0A0 C n ( u 1 0CLMo 0.4- f- 0)n, r 4 0 1

Cc N~ IN4 P-4*-000 n Z 0 Q 0 40 0

o -~- --in Le 4 -(c" 0 0 )4 0

.,4r- Ll h O I- P4 M .- ý4LM " -4:1 M r ,L N M. tr) M0 v-40

>0 m O -%o fn 0%%0 co 1N 0%- (n4 a~ u0 N CI (3, p.4 C~%~ l 11 9-4"4 -0 0- 0 0) 0 a, c 0

(D 0~N N41 0t,01 ý100 0 0 0 0 0 0 0 00Cr04

C) 0N~ o '-0 - " ND 1-4 0-0 0 ON~ w. m. - r-. r- 0% N '- 4 00 0

tON~ 4 Nv4- 0 0 0 0 a c

0. i 0 001 '- ". 0: r- cIJ- 0 -'9- . 1, I III ar .Y 6 0

>1 0%~ P.4 m~ ,o. 1 ~ ' '-400 0 e0

6.4 CD CD5 I

LA- -

"0 r ýN r-''. % %/) ~-t0 9i4%

4. 4 0% 0.

".4 0 0%% 0 "-4 to cn k,j -1~ 0. 0 u,,0 4- 4 '0 .4-4% r -4.0 0 CO Wo0 IN CC . I-~rO. 0 N "4 P-4 0 0 0

C;i0 cr0 M% Go %D Lli P-4- 0 0 0 0 0 0 0 a

M N0 v-4M0 ".4 -10n (n4 Mr 00 e',- . .&0 0 0g 0 Z 40 U, 0 9 r4-0 0 0 0 Cj0** o0% 0%t m N Ln4Cm N- .0 0000000 C

000,-4 If4-1 00N Otin 0 .4 ~o C;0 c 0 - - 0 001 '

1.4U I'1( IN NISSM IV~



0 h ON 00 OQCh'b r,00 ~ q

Ln~ ~ Nlý8( N "4 V.4

enoco0000000

L,0 0 000 0 00 0 0

C-4 .t 0%;- cm, 0[c m C 4 NWD . 0 II N ton P.4 N. N - - 0, 0.. %0 :n0 0 I-m No 04 '00 0 %0 IC0 cc,N00 0

m % O 00 r- %D6n

-I~~ ~~ P- F- *-F- , 0

t-- -4 r- 60 ý -n* "I -, " -fgo "4wl %C lliO %0 o 0 0 00- r- N, P -4 N eq

en N- N r,oh- .o o .cc N N- "4 O %q, 0 r- m. %D ý' N:, a, co ey 40% H l"4 - - 0r- C% %T Nl ,- 0 0

W~4 0 oO8 00

0 U 00 M C,4 00y, .zt C4 0N V.m LIM LC W) 4 ~ 11 hh %D % 0 a

00 c I a- L, ..4 N 4 M r. %0% 0% mI N~~, Un uNr 0 enn LI'.0 '~sI N N 0 N 4, 0P. P L, LI, II 4 -n' NC. P4 0

00 0 0

CA % "4 m % N 0 .qN r- M 00 II, "I, MM LIo cM O 0 LI H N0 00, L.v 'C 00 u M0; 00. N 0 '00 L, 4 M N O OOc

w . 00 M P% %0 wl , %0CN'

0 cis 0 ON 001

00ai r

* * Ic *D S *0*

O0 1.0 NN "DO to,r "4 -m Go~ 0 ON, C% 0 aN65 % Pq 19 N'C~04 'N -4 0 00 0

0N P* N

:g 'n :, s ,0 9 9 N ." A ID 0h ONI), cc0 %0 P-0 coL, 000 u 0 i M '0

'4 O r-4N-0 '.h0 P-4 0% ,-4 e 000. Ný '.0 0 "I, ON Qý W)

0 %% 0% 47 0, 0%cs0 0

"I "NU.l 2- m4I 0 0 0 0 04I I ONS m 0" 4 0' 0

CI 4

ýNHNM(~O 0 00%W 1 01~ .- 0- 0%n CI, r4 ehr " 0% 0% WO LI.q 0 V4C4 00 0 04

§n fn V-4 0. c)0

0 30



NoON LM m . 0o J' % 0

C, 4 0.o ' ~.-a - C ~ ~ a ena a CCO _ I

w~ ON 0 N IN. 1 a,

L , h 0'. 0, m % 0%, 0N -4 (4 W~h~ 0%

0 .ý00 c,4 . 0O m f

0% O% 0% 0; a, 0, 0% O% 0 0 N ý CN4"P-44 " 4 "4l P4 N N N N4 GoIn0'- o mr I.% %.00 aM a% '00% m (% 00 m4 clh 0%

*c! ý In . 6 1(ý -4 "a- 4 r.4 r-4 N N 4. G af-4 1 P4 1 -4 --. a - - 4 F-4-4 -4A P - -m

0 gN N 00 '. 00 04 ml %a0 m .4

-4 v! . .- 9 -ý - - - -ý-1N WL V0% M in ý% 1. F-4 40

%f0 I C. %0 LMa M N N4 '

a. H4 "4;f"44**

(.44 N- N, in m0 Lm 0 - - - -~ ot 4 m .0 N LI NC14N N N4 N % N ( gaN - -I

'.4 '.04 4 C 4 i A 0%

N koro. W t4 4 V

kn NO YA2 % N OP-4 S S I(AQ -4 P- P- P.4 P-4 N. N m iM

c 00 P.-4'.W ND 0 N4 CI I -4 -4 an 04- -4- - - ~~- 4 - -- - a

v-I4 - I

bO 4 "~~,4 4 v4- -

- - - r- 4 - -- - - --

" "4o%

SI NI NOT-- a mliVI - a - - -

32i a



0

00

0.00

Jr- -c [. *

04-

AI j flquqoad

33



C-4

p-W4

A~~C 41~q.1

314 in



T'..I I

C4.

l1eit~LJ'



r174'A ;:

IL I7t] ~ I

1.0 1 0

144

c36

4,4



40

00

t IL1

3A1



14.

il~ it w'' do~~

.4A

AIrIqiuqo~zt d

39



tLO

IIn

040

a'. 40



CAI

04

tl;

ILLIN

A3001 flpq.

A4'



II

~t

1 09

fII

IV-

to

~~14

L.L4



Ic.

I Hill I

W In

144

C14 J

Ln

.0.40

-4-

Xi111,qvqozaa

'43



r0

It -or41

CCA

'-4

IN

1 771147



1.LTill'!

0J

i I iIJ1

2o k N

~4 5

1000



600 ~-1*~500

- ~ .27

400 ~ - * .4 : .-

300 3--

60

p4.

7/r Pj~ g S tuanda rij 1e vl ,4

fl m i l~.

7o24 6 8 1

Ro~und To Roundc StAndard Deviation (mils

Figure 14. Avera~ge Number of Rounds to Achieve a Hit

46



GLOSSARY OF SYMBOLS

The symbols used in this report were chosen to permit

reference to other texts on th e subject and to maintain

consistency throughout th e report itself.

GRE4K LETTERS

I, G~~ E The fixed bias in th e horizontal and vertical

dimensions rbspectively.

2y,y The var iances of the occasion-to-occasionbiases in th e hor.• 'ontal an d vertical d.i-mensions respect ively.

ip •2

The varianceo of the burst-to-burtt bisesx y in th e horizontal an d vertical dimensions

respectively.

n! fl,2 The var iances of th e total bias proseAt 'dur-X y ing a burst in the horizontal &nd vertical

dimiensione respectively.

2 '12The var iances of the round-to-round errors

xv y in th e horizoý:tal an6. vertical dimensions

respect ively.

PROBABILITIlES

PNH First round hit probability.

PP/N}_ The probability of obtaining exact ly K hits

on a target, from a sequence or N rounds fired.

p{HIxys The conditional probability of a hit given

the values x an d y,horizontal an d verti.%al

oomponents of bias respectively.

P{K/Nlx,y) The conditional probability of obtaining ex-actly K hits of N rounds fired given th e

values x an d y, th e horizontal and vertical

components of bias respectivejy.

P{A../N,K 2 /N 2 } The probability of obtaining exact ly K hitsfrom an N, round first burst an d exaci ly K2

hits from an N2 round second burst.

47



OTHER TERMS

ER The expected or average number of rounds

which must be fired to obtain a first hit.

EH The expected or average number of hits ob-

tained when firing a burst.

02 The variance of th e number of hits obtained

when firing a burst .

48

Hit Probabilities

Documents

Transcript of Hit Probabilities