HUMAN FACTORS IN FIELD EXPERIMENTATIONDESIGN AND ANALYSIS OF AN ANALYTICAL

SUPPRESSION MODEL

Michael Peter Mueller

NAVAL POSTGRADUATE SCHOOL

Monterey, California

THESISHuman Factors in Field Experimentation

Design and Analysis of an

Analytical Suppression Model

by

Michael Peter Muellerand

Karl-Heinz Pietsch

September 1978

Thesis Co-Advisors: D. P. GaverG. K. Poock

Approved for public release; distribution unlimited

Tl•

UnclassifiedSECURITY CLASSIFICATION OF THIS PAGE (Whan Dolo Entarad)

REPORT DOCUMENTATION PAGEI. REPORT NUMBER

READ INSTRUCTIONSBEFORE COMPLETING FORM

2. SOVT ACCESSION NO. 1. RECIPIENT'S CATALOG NUMBER

4. TITLE fond Subllllm)

Human Factors in Field Experimentation Designand Analysis. of an Analytical SuppressionModel

S. TYPE OF REPORT » PERIOO COVEREDMaster' s thesis:September 1978

» PERFORMING ORG. REPORT NUMBER

7. AuTHORfa)

Michael Peter MuellerKarl-Heinz Pietsch

S. CONTRACT OR GRANT NUMBER|"«;

». PERFORMING ORGANIZATION NAME AND AOORESS

Naval Postgraduate SchoolMonterey, California 93940

10. PROGRAM ELEMENT. PROJECT, TASKAREA 4 WORK UNIT NUMBERS

II. CONTROLLING OFFICE NAME AND AOORESS

Naval Postgraduate SchoolMonterey, California

12. REPORT DATE

September 1978II. NUMBER OF PAGES

140

TT MONITORING AGENCY NAME a AODRESSCI/ dllloronl tram Controlling Ottleo)

Naval Postgraduate SchoolMonterey, California 93940

IS. SECURITY CLASS, (at Ihlm raport)

UnclassifiedISa. DECLASSIFICATION/ DOWNGRADING

SCHEDULE

IS. DISTRIBUTION STATEMENT lot thlo Ropott)

Approved for public release; distribution unlimited

17. DISTRIBUTION STATEMENT (at lha mbmtrmet onlotod In Block 30, II dllloronl Iran Ropon)

18. SUPPLEMENTARY NOTES

1*. KEY WOROS (Conllnuo on roworoo mid* II nocooomrr ond Idontltr or Block nwmimr)

Suppression modelData analysis of suppression

20. ABSTRACT 'Conllnuo on rorotoo oldo II nocoooorr mod Idontltr or oloek numbor)

The primary objective of this study was to provide a contribution to thephenomenon "Suppression" as an aspect within the military environment.

Analystical models explaining these aspects were developed in order toidentify the influences to suppression. Techniques are examined for includingthe suppressive effects of weapon systems in Lanchester type combat models,which may be useful in wargame evaluations of military judgements, and in

DD ,

'«*, 1473(Page 1)

EDITION OF I MOV SS IS OBSOLETES/N 0103-014- «601

Unclassified

SECURITY CLASSIFICATION OF THIS PAGE (Whon Doto tntorod)

Unclassified

(t eu wty ei.tmnc'io'i o* tmis »»atf |»««

force level planning. The study also provides techniques to analyze andfit experimental data to the analytical models.

The data to verify the models were obtained from related experimentsperformed by Combat Development Experimentation Command (CDEC), Fort Ord,California.

The result for the modelling approach to suppression indicates- sourcedependences on quantitative as well as on qualitative features.

The functions are left quite general, although some functional formsare derived and discussed.

DD Form 14731 Jan 73

S/N 0102-ni4-6601Unclassified

jeeuoiT'' claudication of tmi* *»oer»*M< o«»« f»""<i

Approved for public release; distribution unlimited

HUMAN FACTORS IN FIELD EXPERIMENTATION DESIGN AND ANALYSISOF AN ANALYTICAL SUPPRESSION MODEL

by

Michael Peter MuellerCaptain Federal German Army

Ingenieur (graduiert) , 1974, Technical Military Academy,West-Germany

Karl-Heinz PietschMajor Federal German Army

Betriebswirt (graduiert), 1971, Technical Military Academy,West-Germany

Submitted in partial fulfillment cf therequirements for the degree cf

MASTER OF SCIENCE IN OPERATIONS RESEARCH

from the

NAVAL POSTGRADUATE SCHOOLSeptember 1978

ABSTRACT

The primary objective of this study was to provide

a contribution to the phenomenon "Suppression" as an

aspect within the military environment.

Analytical models explaining these aspects were

developed in order to identify the influences to

suppression. Techniques are examined for including the

suppressive effects of weapon systems in Lanchester

type combat models, which may be useful in wargame

evaluations of military judgements, and in force level

planning. The study also provides techniques to

analyze and fit experimental data to the analytical

models.

The data to verify the models were obtained from

related experiments performed by Combat Development

Experimentation Command (CDEC) , Fort Ord, California.

The result for the modelling approach to

suppression indicates source dependences on

guantitative as well as on qualitative features.

The functions are left quite general, although

some functional forms are derived and discussed.

TABLE OF CONTENTS

I. FORMUIATION OF THE PROBLEM 7

A. INTRODUCTION 7

E. BACKGROUND 8

C. MILITARY SUPPRESSION IN COM EAT-EN VIRONM EHT. .

.

9

1. Definitions 10

2. Structure Of Fire Suppression Process.... 11

a. Signal Process 13

b. Human Process 14

c. Reaction Process 15

d. Performance Effects Process 16

3. Suppression In Field Experimentations.... 1b

D. APPROPRIATE OBJECTIVES 17

E. POSSIBLE ALTERNATIVES 18

II. CONSTRUCTION OF AN ANAYTICAL MODEL 19

A. ASSUMPTIONS AND SIMPLIFICATIONS 19

B. RATES FOR THE MODEL 20

1. Rates Of Suppression 20

2. Rate Of Rise 39

C. SUPPRESSION MODEL 48

III. FITTING EXPERIMENTAL DATA DESCRIBING SUPPRESSION. 60

A. EXPERIMENTAL DESIGN 60

1. General Aspects 60

2. Setup And Realisation 60

3. Presentation Of The Data 63

B. DATA ANALYSIS 63

1. Parameters Of The Model 63

2. Suppression Time 80

C. VERIFICATION OF THE MODEL 86

IV. DISCUSSION 95

Appendix A: ORIGINAL DATA 81/105/155MM 97

Appendix B: DERIVATION OF CONSTANT A 102

Appendix C: DERIVATION OF VARIANCE ,. 104

Appendix D: ANALYSIS OF RESIDUALS 81 MM 106

Appendix E: ANALYSIS CF RESIDUALS 105 MM 112

Appendix E: ANALYSIS CF RESIDUALS 155 MM 118

Appendix G: ANALYSIS OF RESIDUALS (METHOD 8) 124

Appendix H: ANALYSIS CF PARAMETERS K, e . .. 125

Appendix I: APL REGRESSION FROGEAMS 128

Appendix J: ANALYSIS OF SUPPRESSION TIME 129

Appendix K: PROGRAMS FOR COMPUTING RATS3 OF RISE 132

LIST OF FIGUBES 133

BIBLIOGRAPHY 134

INITIAL CISTEIBUTION LIST 138

I. FORMULATION OF THE PROELEM

A. INTRODUCTION

In recent years considerations of Human Factors aspects

in the military has gained more and mora importance. The

study of psychological, physiological, and environmental

conditions and their influences on the performance of men in

"Man-Machine-Systems" supports the development of new

doctrines, design of weapon systems as well as training

programs for troops.

One phenomenon in military man-machine-systems is

suppression. Modelling this phenomenon has only recently

been given much attention.

This is partly because modellers did not understand

exactly neither the causes of suppression nor how it affects

the course of combats.

There is an intuitive feeling that when a soldier or a

combat unit is being fired upon, it will be less effective

than when it is not receiving fire. This is generally

referred to as suppression, but it can include much more. In

the broadest sense, suppression can effect individuals,

units, or weapon systems in different types cf combat.

This paper will limit itself to individuals or small

infantry units.

In order to be able to build up functional relationships

based on the suppression idea_, between different categories

of people and time, this paper will first examine different

rates as input to analytical Lanchester-type-models . The

rates will be based on stochastic aspects. The modelling is

done under different viewpoints; their results will be

compared and discussed.

In the second part of the thesis, the parameters cf the

models will te tested in a regression analysis against real

world data which were placed at the writers' disposal ay

CDEC.

The goal of this paper is not tc present a final

framework cf suppression; however it may contribute to

clarification of aspects of suppression, and help to embed

it as a component in future large size models.

B. BACKGROUND

The strains imposed on individuals in our society are

constantly increasing. Modern technologies and constant

efforts for improvements of standards of living lead to

growing difficulties in adjustments or sometimes to complete

failures to adjust. These facts create stress and ve may

observe that the degree of stress increases with the

difficulty of the adjustment-problem.

The term stress will be used as a substitute for what

might be called otherwise as anxiety, conflict, emotional

distress, extreme environmental conditions, ego-threat,

frustration, threat to security, tension or probably

arousal. Stress can be thought of as the result of almost

any environmental interference [ Applay , p. 2 ].

The stress-generating features of the civilian

environment are great, but the environment created by modern

warfare possesses additional features which result in an

increase of stress. The combat environment created fcy the

weapon power of the enemy causes a constant threat to life.

The soldier has to operate under this threat and naturally

he will respond with constantly recurring fear. This may

break down the soldier's psychological and physiological

resistance.

Fear and anxiety in battle is common, being experienced

by between 80 and 90 % of combatants. Pains in the stomach,

fatigue, dizziness, perspiration, and enhanced heart-beat

are some vegetative correlates to fear and anxiety. Of

course the moment when the individual soldier reaches his

breaking point varies and depends en individual

psychological and physiological resistance and the severity

of the battle.

One interesting observation from Agiell was that

auditory sensations convey the stress of battle most

strongly and most directly. The psychological effect of

weapons is directly related tc their sound level and the

freguency with which the sound occurs [ Agrell ,p. 2 1 5 ].

Each enemy grenade causes the soldier to react

constantly with fear. Stress and fear can have a

significant sensory operating characteristic, e. g. the

detection threshold and/or sensitivity may decrease as a

result of stress [Weltmann, G, Christianson, R.A. and

Egstrom, G.H., p. 423-430].

C. MILITARY. SUPPRESSION IN COMBAT-ENVIRON MENI

When weapons are used in combat, there are two types of

effects that they have. The first type of effect is physical

damage or injury to the target and the second type of effect

is psychological.

This second effect of a weapon has led to the term or

concept "Coabat Suppression". There exist a general belief

that fire suppression is important, but the importance of

suppression effect on combat outcomes as compared to the

effect cf other areas such as firepower, mobility,

intelligence, command/control has not bean quantified

adequately.

1. Definitions

Suppression can be generally defined asthe temporary degradation in the qualityof performance of an individual soldieror unit by an internal or externalstimulus.

CDEC, 1977

A more useful operational definition in terms of

performance capability changes is provided by the "Report of

the Army Scientific Adviscry panel Ad Hoc Group on Fire

Suppression"

.

Firecauscapa(supwhenknowcaustheresuto 1

sues tbillpres

fus toed otnreItesse

ppreempoti essednctioe

y siat ofromn ri

ssion israry chang

ofsystem) froning inpassive,gnals from£ deliverebehaviors

sk to the

a process whiches in performancethe supp lessee

om those expectedan environment he

These changes aredelivered fire or

d fire, and theythat are intendedsuppressee.

Ad Hoc Group, 1976

This definition emphasizes that suppcession is not a

10

single effect which can be measured totally on a single

quantitive scale. Suppression effects are multidimensional

and the "amount of suppression" varies among these

dimensions (e.g. fire impact points, soldier's

characterists, and reaction to the fire, his combat

experience etc. ) .

Suppressive fire in a combat environment can suppress a

number of combat activities; for example: firing, search for

and observation of targets, movements of units or command

and control.

2- Structure Of Fire Suj:j;ression Process

Fire suppression is a complicated process involving

many physical, environmental, physiological, behavioral, and

operational variables. The important point to emphasize is

that the behavior involved is in response to stimuli that

originate both externally (combat environment) and

internally (personal background, training and experience) to

the soldier suppressee. These aspects, however, are not

included in this paper.

The intensity and duration of suppression can not be

predicted from a knowledge of the combat environment alone.

It requires an analysis of the underlying motivational and

cultural factors and of the context of the combat

environment

.

Xha Fig 1 [Ad Hoc Group, p. 36 ] shous a schematic

description of a process when suppressive fire is delivered

and its affect on the combat result.

11

5

W > i~t

H'HOr4(;O U M M t! "<J

n. rl ^ i: i.i u\

O .u lu 1 1 <0. o 1 1

CI *-»

W •<-» rtJ

•H -^ <fl (/Jwo o

O *-. >. uiT* 01 >. *-* Ul

rtJ u o > u«-t rtj Ul -* Q.3 W C *-> CLn. .y <j u 3O .C t/l -C wl0.O I I I

15

»(b WO. rt

41

8d a Li

X tv. a.

•

aoH*J•0

VI O

y '3

Ul M

>>«h VI

c o cio o O*! £ OJ o MXblQi

0) O

> d

u u

O H VI

1. o o o1 J ^ -4t: u w mJ (, <Jt MU OH 4t-i 0* XfH

,

iA

M 14

"J <9

>5 UHZu a.

o (3 O

O a L» *J o «J <D

OH 3 o j Q DJ

13 c QJcoo•H u S«

<n < 0,

O••a

—

.5

go oSui ifl UPH -Hu^coomw *»a o o u« <y > 1'j <a nj

n i; r- *-» "->h > -m o

w w 3 ft. t* < I i l

1-4 "J 01

n ti a»coo0) t: oH o u

H iJ 3 £ Ul3-H r J-H j-

O" ^ n > O r-*

< u. u x: q >i i i i i i

uooouVI £2. H '«-• ^3^ e <o > c«to CJ E-* C O

fr.^>O OJ

> M1/1 o uc y «j

o urt -lOw-H^ Ul

o 3 "1'HtZ CMT3V 1 1 1

«

Cu oo r* yi

iS CT

c > U Va fl n up X a o3 CJ cc U— a — a.

4"̂3 (*

O O UlCT>-n4 01

c w o•3 u ox o «<

OHO.i

uco atH O

T)S1.v -3 a*s

>s en O U3 ^Jic u c B Q ol

o -^ H H ^ ^H > C U iamUlJi'H -j -Q 3 'On ft *-* <1 o r: o <n-.h T u U o o ^ y«r fr< < t-i i i i

a o o j

n. at

>*^

I (h O

m^ '

ye -^<a ^ *-»

W C H OIA (Q (0<H> C Utn <-< o cj t. -o I

O "H ^ o H .c <-* JCWt4U^<^^0O-C Q-JUL>r«Cu<3 a. o o i i i i

AT?^ I

(J JO.WO tfl

0- vsI

Figure 1 - SCHEMATIC STRUCTURE OF THE FIRE SUPPRESSION

PROCESS (Ad Hoc Group, p. 56)

12

The process is discribed by individual functions.

a. Signal Process

Ihe first process in fire suppression is the generation of

signals provided by suppressing weapons.

Inputs to this process are the

characteristics of the weapon systems (caliber,

amount of porpellant, warheadtypes etc) and

the environmental characteristics (trajectory,

platform, arrival points etc).

Characteristics of the weapon systems can vary in order to

increase or decrease suppressive effects.

Some parameters that are considered to be important to

suppression signals are:

Muzzle velocity (an increase in muzzle velocity is

associated with an increase in signal variables) ;

Caliber (as caliber increases, the firing signals

and projectile signals increase along with

lethality)

;

Projectile weight( penetration depends on weight

and velocity at impact and increases shock coupling

to ground) ;

Warhead charge weight ( the explosive charge weight

determines the energy in the pressure pulse) ;

Additional parameters like fire freguency and

proximity of shots could also be mentioned.

Environmental characteristics are also variaoles but thay

can not be determined completely.

Environment has an influence on the signal

13

generation -and transmission-process. For example, auditory

signals that result from the impact of projectiles depend

heavily on the nature of the object or material impacted.

A soft yielding material such as dusty ground or

sand receiving the impact of a projectile will produce a

different pulse and sound than will hard unyielding ground

under the sane impact.

Sound signals can be attenuated by the shadowing

effect of large obstacles or may be increased by echo or

reverberation. Visual signals are strongly modified ty the

condition of lighting. Haze, fog, rain, and snow act

similarly to smoke and dust. The visual field is also

reduced and interrupted by terrain and other obstacles.

The environment modifies the produced stimuli

when they are transmitted to the location of the suppressee.

Outputs of this process are the attenuated sensory signals

that become imputs to the hunan sensory receptors.

b. Human Process

Many of the determinants of the soldier's performance on

the battle field are unkown or at least uncertain - thought

of as influenced oy chance factors. This emphasizes the

difficulties of predicting human benavior in a combat

environment. Tne human process (sensory and perception)

converts the received signals into a perception of the risk.

Eattle field stimuli effecting the individual are

detected and converted into sensory data by a process such

as vision and audition, so the sensory process suggests that

the weapon systems stimuli relevant to suppression are the

-loudness, and

-visual impact.

There exist moderating factors that influence

the operating cnaracteristcs of the sensory process and that

determines which stimuli are effective.

14

Sensory modifiers (i.e. earplugs,

night-vision-devices) serve to change users' sensitivity

range. A major effect of these devices is to change the

salience of stimuli.

High concentration on an activity or a high

level of effort on an activity (e.g. missile-gunner is

tracking a target or reloading his system) may increa-se the

absolute threshold.

Ihe posture of a soldier (standing or sitting)

and the sequence of posture (observing, ducking, observing)

influences the sensory capabilities (e.g. observing for 10

seconds continuously is not equivalent to observing 5

seconds , ducking 10 seconds and observing 5 seconds).

Ihe perception process integrates sensory and

other information into a perception of the risk. Risk refers

tc the uncertainty of damage, injury, or loss. It

characterizes decision situations in which the consequences

of choosing an action are uncertain.

If there is no uncertainty in the possible

outcome, there is no risk.

Perceived risk is a function of uncertainty and

the subjective value the individual associates with each

outcome.

Eerceived risk represents the output of the

combined 'sensory and perception process. It depends en the

individual's experience and training in assessing risk from

sensory information. Also cover provided by the environment

and the individual's pesture may influence risk perception.

c. Reaction Process

Given the input perception of risk, this process causes

physical and mental reactions, which depend on the

-current mission

-task

-activity

15

-combat training doctrine and experience

-group dynamics and

-the quality of leadership.

It is conjectured that two individuals who perceive the same

degree of high risk , but who have different amounts of

ccmbat engagement experience, might be lively to react

differently tc the risks.

The soldier's reaction is also influenced hy his

current state. A soldier who has recently ducked may be more

likely to duck than one who has not , given the same

delivered fire.

Prior reaction or sequence of reactions may te a

good predictor of the coming reaction.

d. Performance Effects Process

Given the reactions of the human behavior process, it is

conjectured that these directly affect, the performance of

certain activities of the suppressee in a calculable way.

If for example the suppressee takes cover, he

may fire less often and less accurately and also might be

less vulnerable. The magnitude and duration of these changes

in performance are dependent on the characteristics of the

system employed by the suppressee and the target of his

activity

.

So the nature and duration cf change in

performance capabilities is determined by the performance

effects process.

3 . S ujD_p_ression in. Field Exp erimentations

A fire suppression research program reouires

significant experimentation on behavioral attitudes and

reactions tc risk. This necessity causes tremendous

difficulties in trying to induce actual behavior in soldiers

16

in field experiments. Former studies shewed, that the

soldiers felt true psychological stress only in situations

in which they believed that th-ey were in real danger. Such

situations are difficult to contrive and tc control. Social

and ethical limits and legal regulations preclude the

introduction of actual physical risk. Scldiers must be

taught the "rules and risk" defined in that context'. The

success of playing the role, being an individual

participating in a combat engagement, depends on the

soldiers' motivation and willingness.

Because of these reasons and the nultidimensional

shape of the fire suppression process as mentioned earlier,

suppression in field experimentations may be restricted only

to some variables involved in this process.

D. APPROPRIATE OBJECTIVES

The overall objective of a fire suppression research

should be to relate changes in performance capabilities

caused by fire suppression [Ad Hoc Group, p. 110].

Responding more directly the following objectives may be

determined:

Indicating the effects of suppression on combat

results, i. e. to develop rates cf suppression.

These values may be compared to other effected areas

and probably employed in computer sinulations. These

numbers represent two kinds of variables: Weapon

system variables and human suppression performance,

given operational and environmental conditions.

Determining characteristics cf suppressive fire

systems, characteristics which should be assigned to

such a weapon system. Results are developed

experimentally . Chapter III B. will display some

evaluated parameters and constants for the developed

17

model, which may also represent the suppressive

characteristics of the used weapon systems.

Heducing suppressive effects. Ways to reduce the

effect of suppressive fire may also be considered as

an appropriate objective of suppression research.

Special training or equipment can be assigned to tne

soldiers or new tactics can be developed. This

objective is beyond of the research of this paper.

E. POSSIBLi ALTERNATIVES

In ordei to get information anouc the fire suppression

process, previous investigations were based on interviews

and questionaires , because valuable information of the fire

suppression process is stored in the minds of combat

veterans. Studies on veterans of the Vietnam conflict and

the wars cf the Near East would be especially useful, since

newer weapons were employed and the combats were shorter and

mere intensive.

These studies may provide a good insight tc the

suppression process and/or may also deliver valuable inputs

for the modelling approach.

18

II. CONSTRUCTION OF AN ANALYTICAL MODEL

A. ASSUMPTIONS AND SIMPLIFICATIONS

Based on the foregoing discussion, the following model

is a detailed model [ Taylor , 1978, p . 12 ] which starts cut by-

considering the behavior aspect of a human being under the

influence of artillery fire power. It is assumed that such

simple uicdels, which represent a small part of a total

scenario, can be used profitably to investigate system

dynamics of lore complex models. The value of the model

derives from the fact that it forms intuitively plausible

and transparent subsets in a large composition of other

subsets which determine the basic structure of the complex

operational model. In other words, the whole is described

in terms of the sum of its parts.

The basic concern of the analytical model developed here

will be to model the behavior of an individual experiencing

artillery fire, considered as a function of time, where that

behavior depends upon ammunition types and location of

detonating rounds.

On the basis of particular assumptions and

simplifications it will be possible to apply the results

cbtained to a group of people (a force) en a battelfield.

The result of these considerations will provide a

relationship between time and tne actual number of people

affected by the fire power of the artillery . This last step

of the model is carried out by using Lanchester type

equations, so called after the pioneering work of F. W.

Lanchester. Finally, the models enable one to estimate the

total firepower of the force at any point in time; depletion

19

of total firepower is caused by attrition and by

suppression.

E. RATES FOE THE MODEL

1 • Rates Of Suppression

The tasic considerations in the preceeding paragraph

support the assumptions, that the behavior of a suppressee

can be expressed by a conditional probability of suppression

as a function of miss distance r and aspect angle 9 . This

function is represented by the family of surfaces of the

form:

P(S/6,r) = exp[- ~ r2(l - e cos 9)] (2.1)

where P(S/9,r) is the conditional probability of suppression

given that a particular round impacts under a certain aspect

angle 9 and a certain miss distance r away from the foxhole.

The line along the angle 9=0° is identical with the

line of sight. It is the main direction of ccservation.

lhe constant K and e are parameters, which are

determined by the experiment itself and by the environ mental

conditions.

Ihey can be influenced by factors as discussed in

chapter I which may be recalled here briefly.

-type of ammunition

-freguency of arrival of rounds

-perceptual damage

-total time spent in the foxnole (learning process)

20

-noise appearance of the rounds

-flash light intensity of the rounds

-performance

-personal factors like age, personal condition, etc.

-degree of stress

-motivation

The computational evaluation of the constant K and z will be

performed in chapter III B. and C. In particular it will be

important to determine K as it varies with different types

cf ammunition.

The mathematical conditions for the two parameters K and e

are

:

< e < 1

K >

If 6 is held fixed ,0°<8 <360°, and ?(S/9,r) varies

between 0<P (S/$,r ) < 1 we obtain a family of functions, which

is two-dimensional and shows an exponential relationship

between P(S/9,r) and r.

This is illustrated in the following figure, where

the angle 9 is held fixed at C'and 180*. The parameters X and

e are assumed as being 1500 and 0.7 respectively.

21

r/m = distance/meter

P(S/e ,r)

r/mfor G «1S0 foxholo

I 1 11 1 1 1 B»-

100,r/m

for 9 0°

Figure 2 -1 2

FUNCTION P(S/eQ,r) = cxp[- - r (1 - c cos 9)]

22

If en the other hand P(S/'0,r) is fixed,

0<P (S/ 0/ r) < 1 , and the equation is solved for r, we will

obtain an "egg-s haped" function with iso-levels of

probability of suppression P(S/8,r). The foxhole is located

in the middle of the coordinate system. Along 9=0°, the range

r is a max for a certain fixed probability cf suppression ,

while at 8=180°, r is a min for the same probability.

The function takes the form:

-K £n P(S/0 ,r)

1 - e cos 8(2.2)

Ihe following graph shows the family of functions for 4

representative selective probabilities of suppressions.

P (S/8 / r)=0.1«i i = 1,2,3,4 (2.3)

The parameters K and e are again assumed to be 1500 and 0.7

respectively.

23

r/ra - distanco/mater

j ur/me=270

c

r/n" .Cor 0-1 80

e

Figure 3 - ISOLINEb r =—

c Cos 'i = 1 '

2. 3 .

4 '

24

These "egg-shaped"-f unctions of iso-probabilities of

suppression simulate the reaction of human beings looking

along the main axis 6=0* in a very simple way. The functions

take into account the visual and accoustical perception

resulting from any given detonation of a round, where noise

and light are the major stimuli.

One could think of other functions wich simulate tae

behavior of an antitank gunner exposed tc artillery rounds

for instance:

1 2P(S/0,r) = L-exp[- - r (1 - e cos 8)] (2.4)

Ihis type of function allows the probability of suppression

E(S/0,r) to te smaller than one at its maximal value and has

the same general behavior as the function before. As a

third modification:

P(S/0,r) = exP [- ||r|(l - e cos 6)] (2.5)

these functions have the disadvantage that they have a

discontinuity at r=0. The integration which is necessary in

the following derivation is more difficult than the chosen

one.

However, the crosscut sections of these functions

are not "egg-shaped" iso-f unctions but rather simple conic

sections (ellipses) where the foxhole is located in the

center of one focus point.

The paper will continue with the function first

described in (2. 1) , because of its simplicity and variety

cf applicaticn.

In order to derive a rate of suppression for the

model, it is necessary to evaluate now, in a second step,

25

the unconditional probability of suppression P (S) for any

incoming round, no matter where it will impact around the

foxhole. This will be performed by matching the conditional

probability P(S/9,r) with the area-hit-probatility P(A).

Consequently, to the engagement procedures of the

artillery, the targets which shall be suppressed by the

artillery are categorized as small personal targets.' This

implies tnat the mean point of impact (tfPI) of a given set

of rounds lies on the target, which means also, that the

density has its max value at that point. According to the

U.S. Army Field Manual FM 6-161-1 page 2-2 it can be

ascertained that the MPI-error is destributed normally with

its mean at the aim-point. Thus it will be hypothesised that

in sucn a case the distribution of incoming rounds is

bivariate normal with parameters

v2

=

o . = a„ = a

(2.6)

In order to simplify the model, it is proposed that the

dispersion of rounds expressed in the standard deviation is

equal in all directions. With

p = (2.7)

no correlation is assumed between horizontal and vertical

deviation. The normality is also preserved if more than one

artillery gun is shooting. The essential change which has

to be made when a whole artillery unit will deliver the

rounds will be the value of the standard deviation a

In addition, a is determined by factors like:

dispersion

the fact that the location of the target is only

estimated and an artillery unit delivers rounds in a

fire area.

26

type of ammunition

conditions of the weapon systems

ballistic properties of the rounds

(precision)

distance to target

caliber

wind and other weather conditions.

It follows frcm the hivariate normal density for artillery

hits:

1f(x ,x

2)

=2 2

2ir(l-p )<J

(2.8)

(1

x exp -

( 2d-p2

) L\ai

Vri

Xl

rl\

/

X2~r2

+

x2-r

2

2-i

and by inserting the previous mentioned assumptions and

changing to pclar coordinates:

2 2 2r = x. + x

2

cos 9 = —r

(2.9)

that the unconditional probability of suppression P(S) can

be written in the form:

G=2-rr r=°°

P(S) =/ / P(S/0,r)«f (r,0) do r dr

e=o r=0

(2.10)

Eecall that an area element in polar coordinates can be

ex pressed:

r-dG-dr = d(-j-) d0

Ihe conditional probaoility of suppression is

1 2P(S/8,r) = expt- - r (1 - cos 9)]

27

(2.11)

and the density for artillery hits f(r,9) is

f(r,8) =

2tto2

exPt" I <f>]

(2.12)

inserting both into the expression anove , we can perform

the integration:

Q=2t7 j-=oo j 2

P(S) = / / exp[- £ r2(l - c cos 9) —^y exp[- j(~) ]-d(y-) d9

6=0 r=0 2,a( }

after a change of integration variable;

2

2 n =r=0

dOy-) = dz

= z

Z = oo

(2.14)

we find

PCS)

p(s)

9=2ir z=c°

= / J exp6=0 z=0

9=2tt

/

2 ,-, 1- (1 - £ cos 6) + —o 2ua'

dz d9

~ 2'

:'

jZ[\ 0- - e cos 6) + ~

a

• exp/2

- ( - (1 - £ cos a) + —z=C J

de

PCs) =9=2tt

j1

6=0 2tto2

/ |- Y cos 8 +^22Ka

d0

(2.15)

K0=2t7

PCS) =-±j /4tto 6=0 (1 + -~) + C-e) cos

28

• de (2.16)

by replacing

b = 1 +_K_

2a2

c = -e (2.17)

the integral reduces to a known form which can be solved.

P(s) . -JL.T *-/

2 Jn b + c * cos

2 2where b > c

kJ7 de

p(s) -ri/.bT., - ,, ,., b + ccos 6 „ 22tto 9=0 2ira

(b-c) tan9 -i

arctzan

/X7 vV^b -c

PCS) =-^ 2

2 *° yx?==> p(s)

2a v b -c

(2.18)

Inserting back the expressions for b and c, one obtains:

PCS) = (2.19)

2o \\ 1 + - c

2a'

After the data analysis in chapter III B. and C.

where the parameters K, e and will be determined, it

will be possible to evaluate the probability of suppression

29

E(S) according to the preceeding formula.

This probability adequately models those cases where

temporary suppression is the only possible response tc a

given detonation of a round. It may be useful to expand the

model by introducing a second paired outcome at any given

impacting round. Until now, we had one paired outcome:

The individual was either suppressed

cr not suppressed.

In addition to these we consider a second paired outcome:

The individual is permanently suppressed by being

wounded or killed

or net permanently suppressed.

If we focus the attention on modelling the effects under

this expansion, it is possible to evaluate a new probalility

of suppression ?(S) and a probability of kill P (K) by Baking

the following assumptions:

The conditional probability of kill unlike the

ccndtional probability of suppression does not

depend on the aspect angle 9.

It can be represented by a smooth curve of the

following form:

P(K/r) = exp(- ~ r2

) (2.20)

where H is a positive constant i.e. H> 0. This function was

selected on intuitive grounds rather then based on

real-world data. Taking a vertical cut through the surface

function above along any angle 9, one obtains the following

figure. The parameter H is arbitrarily chosen as being

H=50. Clearly this parameter is a function of different

factors like terrain and ammunition.

30

r/m = distance/meter

P(K/r)

r/mfor 9-180° foxhole for O °O

£

1 2.Figure -4 - FUNCTION P(K/r) = exp (- - r )

31

If on the other hand P (K/r) is fixed, 0<P|K/r)<1, and the

equation is solved for r, one will obtain concentric circles

with iso-levels of probability of kill P (K/r) around the

foxhole.

The function has the form:

r = - H-2,n P(K/r) < (2.21)

The following Fig 5 shows the family of functions for 4

representative selective probabilities of sup fressions.

P (K/r) =0. 1 -i i=1,2,3,4 (2.22)

The parameter H is again chosen to be 50.

32

r/m0-270°

foxhole

r/mfor 0-180°

r/m distance/meter

Figure 5 - ISOLINES r = -H-£n P(K/r)

33

The concentricity around the foxhole expresses the

fact that no matter in which direction the person in the

hole is looking the kill effect is just determined ty the

distance. It would be beyond the paper to verify this

assumption , and it would be extremely difficult to collect

data for it. The reader must be content for the moment with

the earlier presented intuitive argument.

This newly introduced function leads to a revision

cf the probability of suppression P(S) and the evaluation of

the unconditional probability of kill, P(K). The density

assumed earlier for artillery hits is used again:

f(9,r) = j exP2ira

2 V (2.12)

Probability cf suppression P (S) but not kill is:

9=2tt r=°°

P„(S) = / / P(S/0,r)[l - P(K/r)]-f (0,r) r dr dG (2.23)

6=0 r=0

Remark: The subscript K at p is used to indicate that this

suppression probability appears together with the

probability cf kill P(K).

The expression ? (S/9 ,r).(1 -P (K/r) ) means that a

suppressed but did not kill the individual at (r,9)

round

Erobability of kill P(K)

9=2tt r->-°°

P(K) =/ / P(K/r)-f(e,r) r dr d9

9=0 r=0(2.24)

Ihe computation for E(S) and P(K) runs along similar lines

as in the earlier evaluation of P(S). Fci the specific

functions we find:

3a

0=2-rr r=<»i ? l i 2 2

P(K) = / / exp(- i r ) • —4- exp[- j (J)] d(y-) dO

0=0 r=0 2tto

(2.25)

9=2tt r=°°, 2 12

P|/ s ) = / / exP[" 7 r (1 - g cos 0)] [1 - exp(- - r ) ]K

6=0 r=0* H

2ira

1 r 2 r2

exp[- ± £) ] d(-y) dG (2.26)

change again the variable of integration

= z

:

z' r -*£ •*- oo

zr=0

0; d(-y-) = dz (2.14)

9=2 z + »

P(K) = / /

P(K) =

exp

z=0 2ttcj

1

a

dz d0

(a /H) + 1)

(2.27)

(2.28)

0=2tt 2-*»oo

PK(S) = / / exp -[£ (1 _ e C0:3 9) + 1 j2 |

1 dz d (2.29)0=0 z=0 iva

" / / T exP Mv (1 - c cos 0) + - + "V]z' dz d6(2.30)0=0 z=0 2tto '

K H 2a

Recognizing that the first double integral is exactly the

same as fcefore we can simplify to

PK(S) -

2.2\/(l +

K

2a

- E

)=2tt

/ TfT3=0 2™ K

cos2 1

+ - + -^H 2

a

d6

(2.31)

35

PR(S) = P(S) -

K0=2tt

7 //

4tto 6=0

dO

1 + -\ + | )+ (-:) cos

2a

(2.32)

The integration can be done by using the same formula as for

the computation of P(S) , except that this tine b is:

b »( 1 + JL + in (2.33)

2a2 H

From this it follows:

PK(S) = P(S) -

V s) = 7T2a

2a

1 +

K

2a'

— z

(2.34)

(2.35)

After having derived the desired probaoilities as summarized

on the following figure, we are able tc evaluate the

different rates of temporary and permanent suppression.

Ey assuming a certain fire rate A with which the

artillery is firing in the area where tha anti-tan/: gunners

are located, the different rates will have the following

form.

iwo-event model (unsuppressed-suppressed)

Bate of suppression X = X^ P(s)r s f(2.36)

36

Four- event- mod el ( un sup pressed- sup pressed- sue vived-lcil led)

Rate of suppression X = \ •? (s) (2.37)S 1 t\

Hate cf killing Xk

= Xf

P(K) (2.38)

One important rate oust still be developed; it is the rate

cf rise from a suppressed state. He model this by stochastic

process metheds.

37

PCS) -

2o2 ,1*4

2a'

- e

(2.19)

P(K) =

H+ 1

(2.28)

PK(S) - P(S) -

-2

V(i+^r-* 2

(2.34)

Figure 6 - SUMMARY OP THE DERIVED FORMULAS

38

2. Bate Of Rise

The model so far represents a situation in which

there are three possitle outcomes. When any independent

round impacts, the person in the foxhole is assumed to be

either

suppressed,

net suppressed, or

is killed.

Naturally this is only true if we assume that he is always

hack up again when the next round impacts. This fact trings

up the necessity of considering the process along the time

axis.

Eecause of the complexity of a stochastic process

which reflects the behavior situation of an individual

exposed to arriving artillery rounds, it seems useful to

start with a very simple process, in which the time of

suppression fcy a particular round is considered to be fixed.

In a next step, this constant response time is randomized

over the numcer of people under consideration, i.e. each

individual has his own random time which however is assumed

constant for the process itself.

Finally, the response time of a single person say be

considered to be a random variable coming from a certain

distribution.

Eoth approaches are simple, and represent a first atteupt to

describe the actual situation in different ways. Certainly

the suppression time can also be considered as a function of

both above mentioned processes or of the miss distance r and

the aspect angle 9, or of a combination of all. But these

dependencies would rather complicate the mathematical

derivations and lead beyond this study.

Both processes consider rounds that arrive in the

39

neighborhood of the foxhole in accordance with a Poisson

Erocess having a rate ^f . The model consists of two process

states. One is the "up"-state which means that the person in

the hole is unsuppressed and able to act according to his

mission. The other state is termed the "down"-state which

results from an incomming round and a possible reaction to

it in the form of suppression. In this state the perscn is

not able to fullfill his mission, he is physically down.

The first process assumes that the person in the foxhole

changes froi the "down"-state to the "up"-state only if he

recognizes a gap of at least T time units before the

appearance of the next round. This means that the

"down"-'State has a duration of exactly time T if no rounds

arrive in the time interval (0,T) . The time T in connection

with such a process may fce called the critical gap.

[Gaver, p . 48 1 ]. The following figure shows the basic

relationship

:

40

FIRST CASS : t' > T

State "up"State 1 "down"

T = critical gap

A State

Impact of firstsuppressinground

Impact of second suppressinground (process starts allover again)

t/sec

5C0ITD CASS : t' ^ T

A State

Impact of firstsuppressinground

1 •

T

Impact of second suppressinground (process starts allover again)

1

^-J

t/sec

Figure 7 - SCHEMATIC DESCRIPTION OF IHE MODEL

U1

For the time being T is assumed fixed. However it is mere

reasonable to describe it by a random variable, since the

time gap T is determined by factors like learning,

accustoming to, cr overcomming of, fear, stress etc. The

assumption of a fixed T has to be changed in a later step,

where it will be defined as a random variable.

Furthermore, the change from the "up"-state to the

"down"-state is accomplished by arriving rounds to which the

person reacts through suppression ("down"-state) with tne

earlier computed probability of suppression P (S) .

Now let t be equal to the total time of being

continually suppressed. It is possible tc define x in the

following way:

!T if no suppressing rcund arrives in (0,T)

t' + x' if the next suppressing round falls (2.39)

before T.

In this case the random variable t' is an independent version

cf the random variable t.

According to this definition it is possible to compute the

expected value of x which represents the mean time spent in

the "down"-state, or the mean time of being continually

suppressed. The symbol used for it will be E[t].

In order to get the arrival rate cf suppressing

rounds which contribute to the time of being suppressed, we

have to multiply the fire rate A- with the probability of

suppression, P (S) , which was evaluated in chapter II.B.1. We

cbtainAs

= Af-P(S) (2.36)

First we have to state the following two expectations:

E[x|no round in T] = T

E[tIround In T] = t'

(2.40)

42

where t' = time of second suppressing round. Removing the

condition on t' leads to expected time of being suppressed

under the influence of incomming rounds.

-\ T T -X t' -XTE[ T ] - T«e

S+ J e

SX dt'ff + E[t]]+B[t] [1-e

S]

S

E[t] =-X T

T.eS

+ X

- -X ts

-

e

, 2(~ t " 1)

X— <5 _

T T

->0

-X T+ E[t] (1-e S

)

(2.41)

-XT. -XT -XTE[t] =Te S

+ f [1 - (1 + X T) eS ]+ E[t](1 - e

S)

A S

.-XTE[t] = T + _x T

[1 - (1 + XsT)e

S]

E[t] = r±- (eS

- 1) (2.42)

inserting X = X 'P(S) we obtains r

, X p(S)*TE[T]= i7(sy (e -« (2.43)

Extending the idea for a fixed time gap by applying

it to a group of people separately, we are able to

reformulate the process in the following way:

Assume a group of perscns in which each individual sticks to

a certain but randoi time gap T when responding to a

suppressive round, and assume that this random variable T

based on tne group is distributed with a density f (t) :

T ~ fp(t) (2.44)

43

The suoscript P indicates the origin of the density

(people) . We may express t which is defined as before as

follows:

t if no suppressing round

arrives in (0,t) or t>t

t = \ where t is the time of the

first suppressing round.

t'+ t' if t'<t

(2.45)

Remark,! 1 has the same distribution as T. A similar

derivation as before leads to the following result.

-A t's

-X t<

E[x|T=t] = t / X es

dt'+J [C1 +E[x |T=t]]X e

Sdt< (2.46)

aecall, the expression A e

suppressing rounds.

-A ts

is the density of incoming

-A t t -X t' ^X t

E[ T |T=tJ = teS +/ t'Xe s

dt » + E[t |T=tJ (1 - eS

)S

,i

-A t -X t

E[t|T=t] = ^- - ^ eS

+ E[x|T=tJ(l - eS

)

s s

(2.47)

Solving for E[ tI T=t], we find:

E[t |T=tJ = -^ (eS

~ 1)Ks

cr replacing A again by ^s

= A «P(s)

(2.48)

44

E[T|T=t] *x7Tsy (e

XfP(S)C

- X) (2.49)

So far this value represents the expected duration spent in

the "down state" given a particular time gap t for a certain

individual. Clearly the conditional expectation has the

same fcrtn as the unconditional expectation E[t] (see formula

(2.43)) for a fixed time gap T.

Removing the condition in equation (2. 49) we find

the expected duration time of suppression based on the

considered population.

E[t] = / E[T|T=t]-f (t)dc

, X P(S)t

EtT] / r^k (e -D-fP(Odt

(2.50)

(2.51)

E[t] =XfP

- XfP(S)t

<k{ e fp(t)dt -

XfP(S)

(2.52)

For an evaluation of this expectation, the density f (t)P

to be known.

has

A further variation of this process leads to the

second approach. Here the time gap T for a certain

individual varies randomly according to a particular

distribution. This approach is limited to one individual,

and will not be extended to a group.

45

Suppose an individual reacts with a random time gap t

T=t, X" I t *. -J / • • •

en any incoming suppressing round i, and assume that this

random variable T is distributed with

T ~ fT(t) (2.53)

The subscript I indicates the origin cf the density (time).

The duration x is defined as in formula (2.45) before. The

conditional expectation can be written as:

-X t ts

-X t's

E[t|T = t] = te5

+ / (t» + E[t]) X e ° dt 1

os

(2.54)

Ihis time T is conditioned on the individual's first chosen

time gap 1.

'

-X t -X t

EfxlT^t] = -±- (1 - es

) + E[t](1- es

s

) (2.55)

Removing the condition in equation (2.55) we can express the

expected duration time of suppression based on the

individual's time gap distribution.

E[t] = / E[t|T =t]-f (t)dtL

(2.56)

E[x] = /

. -X t -X t~ (1 - es

) + E[t](1 - es

)

s

fT(t)dt (2.57)

46

Solving tor the expected value E[ t ] and inserting

X e= X,«P(S)

s £

(2.36)

we find:

-X P(S)t

/ (1 - e ) f (t)dt

E[t](2.58)

XfP(S) [w°

L o

-X P(S)t(1 - e ) f

T(t)ck

]

cr expressing in Laplace transform with s as an argument:

E[t] =XfP(S)

f(s)

- 1 (2.59)

A further evaluation of this value requires the distribution

yt) of i.

comparing toth expectations,

Expected duration time of suppression based en the

population

E[t] =XfP(S)

« x p(s)ce f

p(t)dt - 1 (2.52)

and expected duration time of suppression based on

the individual's time gap distribution

E[t] =X-

fP(S)

f(s)

- 1 (2.59)

47

lie observe that they are different. However if we suppose

that both distributions f (t) and f(t) are concentrated atP .

TT = z (delta function) it is possible to reduce both

expressions to the very first derivation (2.48)

the time gap 1 was fixed.

where the

The reciprocals of these expectations approximate the rate

at which the foxhole occupant (e. g. member of a group of

antitank gunners returns from the suppressed state into the

unsuppressed state) returns back to continue his mission.

1Au

~ E[t]

where E( t ] represents any of the derived expectations.

In summary the three rate coefficients developed are:

(2.60)

Ak

= Xf-P(K)

Xs

=W S)

A =u E[t]

(2.38)

(2.37)

(2.60)

If killing as an additional event is considered, the rate of

rise is computed with the same formula (2.60) except that

this time P„(S) is used instead of P(S) .

C. SUPPRESSION MODEL

The model which will be developed in this section can be

set schematically in the following framework:

48

ANTI TANKGUNNERS

A,, . AS jK

MOVINGTANK UNIT

Y-FORCEFIRE

SUPPORT

ARTILLERY

figure 8 - SCHEMATIC REPRESENTATION OF THE SITUATION TO BE

KCDELED

49

This analysis is restricted to the effects en the anti-tank

gunners... in this case the X-forces. It is behavior of

members of the X-force that was uodeled previcisly.

The approach to formulate the situation of the anti-tank

gunners is done by using the ideas of Lanchester-type

eguations and their further development [ Tayler, 1978 ,p. 20 ].

The differential equations representing the model are

all deterministic in the sense that each of them will always

yield the same output for a given set of input data. Even

though combat between military forces is a complex random

process. These equations shed light on combat dynamics and

may oe useful in defense planning studies.

Ihe basic idea is that artillery forces use "area"-fire

tc suppress or eliminate forces like anti-tank gunners.

"Area" in this context means the fact that

the artillery unit "knows" tne area in which to

shoot, but does not know the location of each

anti-tank gunner,

the anti-tank gunners are "invisible" to the

artillery unit.

If we further assume homogeneous forces of X, it is possible

to set up differential equations which model the rate of

change of the X-forces:

dX (t)

—rf= -X X (t) - A.X (t) + X X (t)

at s a k. a us (2.61)

dX (t)

—f = X X (t) - X X (t)dt_ s a us (2.62)

Bemark: Clearly X here is taken from the 4 -event-model . Thes

50

variables X^t) and Xg(t) are respectively the number of the

X-forces which are either active in the foxhole (able to use

guns and anti-tank-weapons) , jyt) , or suppressed , Xs(t).

In this simplified structure, the two equations describe

the most essential factors of the assumed situation. They

are mathematically approximate , because the solution of the

two differential equations will furnish the number of

gunners active on the battlefield and the number of gunners

suppressed at any point in time. The^

derivation uses the

Laplace transformation:

£ {i£l) . „<„ - X(0>

^ {X(t)} = x(s)

(2.63)

ii€ perform Laplace transfor maticn upon (2.61) and (2.62)

sx (s) - X (0) = - (X + X ) x (s) + Xuxs(s) (2.64)

sx (s) - X (0) = Xsxa(s) - A

uxs(s) (2.65)

Solving these two equations for XJ-) and Xg(t) and

translating them aack to X&(t) and X^t) gives the desired

time dependent quantities:

-Xa(0) = - (X

s+ X

k+ s) x

a(s) + X

uxs(s) (2.66)

(2.67)-Xs(0) = X

sxa(s) - (X

u+ s) x

s(s)

-Xa(0)[X u

+ s] = -(Xg+ X

K+ s)(X + s) x

a(s) + X

u(Xu

+ s) xs(s)

-Xs(0) X u

= XsXuxa(s) - (X

u+ s)X

uxs(s)'

-Xa(0)(X

u+ s) - X

s(0)X

u- X

sXuXa(s) - (X

s+ X

k+ s)(X

u+ .) x

fl

(s)

xjo) xu

+ xs(Q) x

u+ sx

a(0)

(2

-

68)x (s)a (X + A, +s)(X +s) - XX

s k u s u

51

multiplying (2. 66) and (2.67) with Ag

and

(X + X, + s) respectively we receive:S K

X (0)X + X (0)(X + X, ) + sX (0)

xs(s) =

(x +x1t+ sH\ -fs) -x x

(2,69)

s k vi s u

Suppose that at t=0 the number of active gunners (able to

watch and shoot) is equal to X and the number of gunners

being suppressed (down in the foxhole) is equal to 0,

for i.e. t=0

X (0) = xa u

(2.70)

X (0) =s

the equations for X (s) and XJs) can be rewritten:3. S

Xn X Xn s

x (s) = -, °-^ + °- (2.71)

s + s(X + A, + X ) + X, A s + s(X + X. + X ) + X. Xsku ku sku ku

Vsx (s) = -5 ^ (2.72)S

s + s(X + X, + X ) + X, Xs k u k u

Using the Laplace correspondence:

f leat -ebt

l 1<*-

) a - b I~ 1 for a r b

v ' s + (-a-b)s + ab

(2.73)

£at , bt )ae - be (

a " b 12s + (-a-b)s + ab

for a ^ b

52

where ° k u

and -a - b = X + X. + Xs k u

(2.74)

hence a = ~[-(X + X. + X ) ] + /(X e + X. + X )

2- Ax. X* I s k u s k u ku

and tC-(a + x, + X )] _ J(x + x. + x )

2- AX. X2sku sku ku

(2.75)

Ihe equation for X (t) and X (t) then are:a s

X (t) = \ [(X + a)eat

- (X + b)ebt

]a a - b u u

(2.76)

S a - b s(2.77)

Since \ . A, and X have tc be always greater or equal tos k u

zero

(2.78)

Xs

>_

Xk

>

Xu

>

The constants a and b are always negative and real numbers.

a <_

b £

This leads to the basic shape of the function

(2.79)

X (t) = f(c)a

X (t) = f(t)s

(2.80)

shown in the following figure.

53

A xa(t)

x -t

A x (t)

Figure 9 - FUNCTIONS X (t) AND X (t)a 3

54

In discussing the functions X (t) and X (t) tha followinga s

properties can be seen.

for t -

V0) - a-^o" "*u+ *>*"° " <A

U+ ^'^ = X (2 - 81)

X (0) = X [e - e ] =s a - b s

(2.82)

for t =• °°

X («•) = lim ~ [(x + a>eat - (X,, + b)-e

bt] = (2.83)

a a - b u u

X (») = lim X [e - e =s a - b s

t •*<»

(2.84)

for t =t (i.e. max values for X (t) and X (t))max a s

X (t) has no max in (0,°°), i.e. max value is at C =a

V° = xo

(2.85)

X (t) has a max at

£n(n/b)

'max b - aw i t h (2.86)

X_ at bt

X ( t ) = °-x [e "iaX

- en,aX

] < X(2.87)s max a - b s U

Ihe foregoing model assumed a rate of killing in addition to

a suppression rate and a rising rate. It is possible to

simplify this model by leaving out the third rate. It will

ce shown in chapter III. C, that this rate

Ak

= Xf-P(K) (2.38)

55

for such a scenario may be very small in comparison to the

other two rates.

For this reason we can state that leaving cut the

killing rate will not drastically oversimplify the codel,

yet it will simplify the ccmputat icnal procedure in

obtaining the wanted dependency between time and the number

of gunners active or suppressed on the field.

This leads to the following set-up of differential

equations

:

dX.(t)

dt-A X (t) + A X (t)

s a us (2.88)

dX (t)

dt

= A X (t) - A X (t)s a us

(2.89)

where the total sum of people either suppressed or active on

the battle field is equal to a constant X (no killing)o

i.e. X =2 (t)+X (t)o a s

which enables us to rewrite the equations:

dX (t)

—J—- = -A X (t) + A (X_ - X (t))dt s a u a

(2.90)

dX (t)

f- = X (X. - X (t)) - A X (t)dt s s us (2.91)

Using again Laplace transformation as earlier.

A XsX (s) - X.(0) = -A X (s) + ~" - A X (s)

a s a s a a(2.92)

A X

sX (s) - X (0) = —S_Si - A x(s) - A X (s)s s s ss us

(2.93)

56

x xn X (0)

V b;' s[s + (X + A )] s + (A + X )

s u s u

x>;

x xa(0)

Xs(s) =

s[s + (A + X )]+

s + (A* + X )s u s u

using the Laplace transform

;£ii «-"">! -1

j

s(s + a)

.£<•-"> -ris + a

(2.94)

for both agnations we find X (t) and X (t) fee the simplified3. S

model.

The tatt As

has to be derived this time frcm the

probability cf suppression P (S) without Killing included as

a possible event. We recall;

P(S) = K, (2.19) A - X • P(S) (2.36)

s r

2-iTO 1 + - E

2a

Hence

X (t) = A Xn±— (1 - exp[-(A + A )t]) + X (0) exp[-.(\ + A, )t]

a uUA+A s u a s us u

(2.95)

Xg(t) = A_X

for t =

s"0 A^T- (1 ' eXp[ - (Xs+ V t]) + X

s(0) ^f-<^

s+V t]

X (0) = xn , X (0) =a s

I he equation for X (t) and X (t) then are:a s

(2.96)

(2.97)

x (O = -,

a A + As u

xs(t) = x

o T-TT

[A + A • exp[-(X + X )t]]us s u

(1 - cxp[-(As+ A

u)t])

(2.98)

(2.99)

57

Bemark: For the sua of both assymptotic values of

X(t) and X(t) is equal to X .asThe basic shape of these two functions X(t) and X (t) can be

a s

seen in the following Fig 10.

If we compare this result to the result on figure 9 we

can see that the addition of killing to the model has an

influence on the shape of the functions. So is e. g. the

value for X(t) and X(t) in the first model (figure 9) for

large times approximately zero, while in this case here

(figure 10), the total number of people on the field (X(t) +

X(t)) is always constant, i. e. both functions approaches tos

a limit value not equal to zero.

58

A *a( fc)

A x (t)

Figure 10 - FUNCTIONS X (t) AND X (t)a 3

59

III. FITTING EXPERIMENTAL DAIA DESCRIBING SUPPRESSION

A. EXPERIMENTAL DESIGN

1 General Aspects

The mathematical model developed in the chapter before

shall now be supported by an experiment which was conducted

by the US Army Combat Developments Experimentation Command

(CDSC) , fcrt Ord, California.

The experiment described here belongs to a series of

similar experiments, all dealing with the objective of

collecting and analyzing data of the suppression process.

The data of this particular part of the CDEC experiment were

based on the relationship between suppression and distance.

Ihis analysis tries to make use of the data by analyzing the

relationship among suppression, distance, and angle. In a

further experiment conducted by CDEC, tne objective was also

to include the angle as an additional variable for the

suppression effect.

2« Setup And Realisation

The data were taken from a part of the experiment which was

executed at Ecrt Hunter Liggett, California.

four foxhole-bunkers were constructed which

guaranteed the safety of the players as well as reproduced

the real scenario as close as possible. Their tops were

celow ground level, and covered with several layers of wire

mesh and steel plates. This provided overhead protection

from fragmentation. Each bunker was equipped with a mirror

system which allowed the player to view events in front of

60

his position.

A pop-up silhouette was installed forward of each

player position. The player was able tc control the

silhouette as well as the cover of the mirror system, i.e.

when the cover was opened (allowing the player to view

down-range), the silhouette was in exposed posture. He was

asked to respond as if he would be in the position of the

pop-up silhouette.

Each bunker was connected to controll bunkers by

communication and instrumentation wires and power bunkers

for data recording and supply. In the forefield of the

bunkers, different types of projectiles or equivalent

charges (31 mm, 105 mm, and 155 mm among others) were placed

and randomly detonated one at a time with the time between

detonations randomly selected from three possibilities of

ten, fifteen, and twenty seconds. The figure on the next

page shows the schematic setup of the rounds, the location

of the bunkers and the angle of sight for each foxhole. It

can be seen that the explosions were visible to some players

but not to others. Since each type of ammunition has a

different lethal radius, it was necessary to have different

range configurations for each type. The aspect angle and

the miss distances from the foxholes to the different points

of detonation are summarized in appendix A for each

ammunition separately. Since all of the projectiles used in

this part of the experiment were statically detonated, it

a as not possible to model the kinetic contribution to the

terminal effects. In order to keep the fragmentation pattern

as close as possible to those of incoming rounds, the

projectiles were supported with sandbags [ 5 uppex, p. A- 1 2 ].

61

V.

•s.

ao•H4->

oga>\fH-P•H.S03 bfl

•HP" CO

•H03 «Ha o

CD

oPaj

-PCO

•H-d

CD

H Ho d aj

a .s3-P

o co o

"t aj

<H oO <H •P X

O •H*H P 03CD CD O p-2 H H <U

S bO CO p.3 P £h p^P aj CD oj

!>II II P 0)

O CD

•H CD O CO

on

. WJN \ av h -Ho a<H O

•H C0 S

o3 -P 03H -HCD CO p•H O O'Hfth

\

N.

\

OO o

COo

Figure 11 - SCHEMATIC RANGE FOR 81/105/155 mmSUPPRESSION EXPERIMENT (SUPPEX II)CDEC

62

The players were divided into two four-man teams*

For each trial, members of one team occupied individual

foxholes and provided the performance data for that trial.

The mission of the players was to track moving target tanks

by operating the periscopes. This mission was interrupted by

the players responses to detonations in case he was

suppressed (change of his state to "down") . The state

change and the pericd of suppression were automatically

recorded

.

3. fiesentaticn Of The Data

Appendix A summarizes the data which are the basis for the

succeeding data analysis and serve tc evaluate the

parameters and rates for the model.

E. DATA ANALYSIS

1 • Parameters Of The Model

The data presented in the section above are separated into

three different sets:

Set one consists of data related to the conditional

freguency of suppression P(5/9,r) and the time of

suppression when rounds of caliber 81 mm were fired.

Specifically, if n trials were made under conditions

(r,9), the number s.cf suppression was tabulated.

Then s./n is an estimate of P (S/r. 8.,).

Sets two and three provide the same data except that

they are related to 105 mmm rounds and 155 mm

rounds.

The main part of this analysis is to make use of the data in

order to estimate the parameters of the model described in

63

chapter II, and to see how well the real world situation can

te described by the model developed earlier.

The model is fitted (numerical values of the

parameters are determined) by the method of least squares.

[De Groot, p. 527-538]. With the basic model being of the

form:

Yi

= 6lUi+ 6

2Vi+ Err

i(3.1)

And the data are analyzed on the following two different

assumptions; the second is certainly the most realistic:

Homoscedas ticity of the data, i. e. the

variance of Err is assumed to ba constant.

error

Heteroscedasticity of the data, i. e. we will assume

that each error term Err is distributed with

variance a 2, where the variance is not constant over

observations. Errors are also assumed tc be

independent. If convenient, error terms will be

assuned to be approximately normally distributed.

These assumptions imply different regression

methods. For the first assumption, the result of unweighted

single step and iterative regression methods will be

presented. On the assumption of heteroscedasticity, two

special iterative methods will be used. When we allcw for

heteroscedasticity, ordinary least-squares estimation places

the same weight on the observations which have small error

variances as on those with large error variances. By

applying a weighting regression, it is possible to adjust

for the heteroscedasticity. So the two announced methods for

the second assumption are iteratively weighted least sguares

regression methods.

64

In order to prepare the formula of the model fcr the

regression, we apply a log-transformation to the equation

(2.1) in chapter II B.1. This is a convenience, for it

transforms the problem to one of linear fitting. We obtain

the following result:

P(S/9,r) = exp[- - r (1 - e cos 9) 1 (2.1)

Jin P(S/8,r)1 2 ^ e 2

- - r + - r cos (3.2)

Using the notation

y = In P (S/9,r)

u = r (3.3)

v =r-cos9

the equation can be applied to each datapoint i and the

aquation can be rewritten -as:

yi

= " K Mi+

KVi

(3.4)

Ihe model car then oe expressed as follows;

Yi

= 8lUi+ B

2vi+ Err

i(3.1)

where the unknown parameters are: 8- = - — ^2 =K

( 3 - 5 )

65

and the random variable

Y = £n P(S/6i,r

jL) ( 3 - 6 >

Ihe "hat" on the letter P represents an estimate of the

probability of suppression at (r,9). The fitted regression

plane for this function has the form

:

{jU + B2v (3.7)

The log-transf crmation of the conditional frequency

cf suppression P(S/9,r) causes difficulties because of

experimental results, which lead to an observed frequency of

zero suppression. This fact influences the estimate

E(S/8,r), that is used.

Having specified the model it is possible to apply

the following different regression methods en the assumption

of homoscedasticity:

In other words, suppose ni

observations were made at

experimental condxtions (r., 9.) and of these s were successes.11 i

Then first consider the estimate

Y = Jin P(S/r ,8 ) = Zn(s./n.) ( 3 - 8 )

where Sj^ is the number of suppressions, and n^ is the number

of exposures to suppression, under condition i.

Consequently, the transformed response to conditions (r-, 84.)

may be In (9/ni

) = - °° formally, causing embarassment. We

will subsequently suggest alternative ways of dealing with

this problem.

66

Hethod 1: (zero-probabilities are omitted)

With this method , the datapoints which had a

frequency of suppression P(S/r,6)=s7n =0 are

deleted. Consequently the followinq data points of

the oriqinal sample of appendix A as shown in' figure

Fig 12 were net considered.

67

Caliber No. P(S/G,r) 9

81 mm 29

30

31

32

34

35

36

41

36

60.5

46

60.5

46

36

117

127

95

108

95

108

127

105 mm 15

19

23

24

28

29

31

17

36

46

46

56

56

71

104

158

135

135

108

108

97

155 mm 3

15

17

18

14

23

14

200

209

125

209

Figure 12 - DELETED DATA POINTS

68

Method 2 and 3 (Clustering of data points)

One or more zero data points i , i.e. P(S/e,r) =

i=1,2,...m will be clustered with one other data

pcint j which has a nonzero conditional frequency of

suppression, i.e. P(S/9,r) * and which possesses

the closest distance r and angle 9 tc the zerc dataJ J

pcint (s) .

Ey clustering (m+1) data points together and taking

the mean of the frequencies , 9 and r within each

cluster, we are able tc replace the data points of

the cluster sets. This procedure is inevitably

somewhat arbitrary.

P(s/e,r) = ^- I P^S/e.r)1=1

1m+1 j m+1 ( 3 -9)

=m+I I 9

ir =

m+1 Jx

ri

The bars above the symbols represent the cluster

average. With these averages two different

replacement procedures can be applied:

Each cluster will be replaced by its cluster

average. This reduces the data sample.

Each cluster point will be replaced by its cluster

average. Consequently the numoer of data pcints

stays unchanged.

figure 13 lists the replaced data points of the samples of

appendix A.

69

OX.4-1

oj

(0

4)

3t-l

CO

>

55

u

CD

CO

P_

r-~ r» f»»

CM CM CM

1/1 1/1 m inCM CM CM CM

o o o o m m u*i

CTi ON <r> o o co eno o oco co co

cm cm cm cmoooo o oo oCM CM

r~- f^ r-~ 10 mO O O CM CMCM CM CN i—I i—

I

VO vO v£>

CO CO COOOOO-3- <• -J <T

O O O\0 \0 \D

KO M3CO CO

r~ t^ t~-

<r <r <r<r -J- <r <fvO vO vO \o

O O CM CM CM CO COCM CM

CO CO CO CO CO CO CO•^ <r <r co co co coo o o o o o o

r-~ r-» r-. in uo m 1/1

\0 vO \o vD vO \0 vO MD CO CO CO CO 01 on—(

•—1 —

4

O O i-l i-l o o o o o o o o oco co co

o oooo oooo ooo oo oo ooo oooo oo ooo oo

1/1

CM

UOON o co

oCO

CMo oo

CMoCM

CM

o

u

X

COO o

CO -3"

CD

co

On

co

ococoO

mM3O o

COo

OnO

COCM

O

r-» r-. r-»

CM CM CMco co r-- ooO O i-l o

1/1 1/1 1/1

ci ci oi o oCO OUO r—

I

i/1 i/1 oCO CO CM

CO CO r-« r-o o oi oi

o oo o.si CM

CM OI OnOOOCM CM CM

in ulCM CN

CO

V3

O >0 vOCO CO CO

vo ^o i—i r^-T < -3- CM

in in i/i

OOO r-- r-~

\D \0 O i-H i—

*

*n r-~ \D \D ON vO \0 r—l f—

1

co co -3" <" <T in m r-» r-~

O O r-~ -3- <r CO COCM CM

Un

CD

CO

CO OI 00r-H COrH UO i-l CM rH >-(

OOO OOOO OOO OO OO OOO OCT.OO OO OOO OO

o co o no in m en •* <j h co <r m oo co <r m oo oi o •—i coo oimco <r r—COCOCO COCOCMCM COCOCM .-HrH H M CM CM CN CSICMCOCO iHr-l r-H r-H

U

X>•w

(0c_>

mo mm

Figure 13 - CHANGED DATA POINTS

70

Method 4 (data transformation by Cox)

This method is analogous to one suggested by D. R.

Ccx [Cox, p. 33]. The derivation for the variable yi

shewn on these pages can be applied correspondingly

in the following manner:

Y. = In~i

S + a(3.10)

where the random number Si

is the number of

suppressions (successes, given ni trials)

.

Ihe constant a, which represents an unfciasing

adjustment, is derived in appendix E. It is taken to

te:

a = i (1 - P(S/8,r)) (3.11)

This is suggested by an auxiliary analysis similar

to that of Cox.

Hence the transformation results in the formula:

Xi= £n

S±+ Y (1 - P

i(S/0,r))

(3.12)

This formula enables one to include data points that

involve zero observed frequencies cf suppression;

the embarassment of taking the logarithm of zero is

no longer present.

71

Method 5 (unweighted iterative regression)

As an extention of the method discussed last, this

method replaces the value of P(S/6,r) in the formula

S, •+ \ (1 - P. (s/e,r))

Y\ = (— -

} (3.13)~i \ n.

After each iteration by an estimate of the

probability of suppression P (S/9,r) , the number of

iterations was determined by the appearance of

convergence of the parameters K and e.

The two following methods are weigatad regressions which are

necessitated ty the earlier described ae tercscedast icity.

Method 6 (weighted iterative regression Var(Zrr)~r)

This method is suggested by the fact that the

observed variability in residuals increases with r.

[Pindyck, p. 1C0] Since the error variance is not

known, it is reasonable to assume the existence of a

simple relationship between the error variances

Var (Zrr i ) and the values of one of tne explanatory

variables in the regression model. In this

analysis, the distance from tne foxhole to the

explosion was chosen as the important explanatory

variable. By using

72

Var[Err ] ~ r (3.14)

the resulting regression formula is:

(Var[Erri l)

1/2(Var[Err

±])

Err.

1/2+ B

2(VarfErrJ)

1/2

(Var[Erri ])1/2

(3.15)

This equation can be reduced to the following

regression function:

y'. = 3, + B-v!i 1 2 i

(3.16)

where

YI

=

in

S. + j(l - P.(S/d,r))

(3.17)

•i-5

Afterwards, the iteration procedure is equivalent to

method 5. This method tends to make the variance of

the residuals around the fitted line of more nearly

constant variance; estimates of the model parameters

should be thereby improved.

Method 7 (weighted iterative regression)

73

Using the error variance derived in appendix C,

which has the form;

VarUn f,]

1 - Pi(S/9,r)

(3.18)

where

f, = — (3.19)1 n

i

the regression formula will be modified as follows:

= 3,

(VarUn f±])

1/2 1(VarUn f

±])

1/2 2(Var[£n f )]

1/2

(3.20)

where y is again:

y. = In

S. + y (1 - P.(S/9,r))(3.12)

as derived in method U . The resulting equation has

the form: /u±

Sl 7, r777, s\l/2

In

S. + ± (1 - P.(S/8,r))i 2 i

n.

1 - Pi(S/9,r)

iK- P^S/e.r).

1/2

1 - PjL

(S/9,r)

n. • P.(S/8,r).

+ 3.

'l - P.(S/9,r)

>n. • P.(S/8,r)x i

(3.21)

T72-+ **!

The iterative procedure in this method is performed

by using the estimate of the conditional probability

of suppression as input for each succeeding

iteration. The number of iterations was determined

as in method 5 and 6 by the apparent convergence of

7<*

the estimators of the parameters K and e .

In order tc perform the multiple linear regression, we

switch over to matrix notation, where we can write the

normal equations in the following form:

T TZ -Z-B = Z «Y (3.22)

The matrix Z is the design matrix which consists of vector

0, the square of the distances and V, the product of the

cos8 and the square of the distances. The variables y and S

represent a (sample size l x l) and a (1 x 2) vector respectively,

for the experiments analyzed, the entries of the design

matrix were obtained by solving the normal equations for 8.

He find:

T -IT= (Z-Z) -Z-Y (3.23)

Remark: Capital letters used for matrix notation.

Ihe parameters S and e can be evaluated by using tae

equations

:

c = B2-K

(3.24)

As a modification of the suppression function (2.1)

in chapter II. B. 1 the function (2.4) in the same chapter

75

was considered.

HP(S/0,r) - L«exp --r (1 - e cos 0)>]

(2.4)

Ihe inclusion of the parameter L allows for the possibility

that suppression may not occur for shots that fall in very

close proximity to the foxhole.

Method 8 (siiiilar to method 4 but using model above)

In this method the regression procedure and the

method 4 were applied to:

Y, = S.u. + 6 v. + S + Err,i 1 l 2 i 3 i

(3.25)

where

y . = In.l

S1+ j (1 - P

i(S/0,r))

u. = r.l i

v = r. cos (3.26)

Sl '

" " K

2 K

63

= e

The ccrmal equations have tne same form as before:

(3.23)T -1 T

8 = (Z-Z) -Z-Y

76

but 1, the design matrix consists this time of the

vectors U,V, and a column vector consisting only ofA

ones. The resulting 8 vector consists now of 3

elements, from which it is possible to evaluate the

different parameters K, s , and L.

The calculation of the parameters for all methods was

performed fcr each type of ammunition separately by applying

the computer language APL. The programs which where used

for this regression are listed in appendix I. The appendixes

D through G show the analysis of the residuals after the

regression cf the original data of appendix A.

Ihe analysis for each ammunition type was performed

according to the same pattern, and includes the following

steps.

For each earlier developed method (1 tarough 7) :

Determination of BETA(1) and BETA (2)

Determination of K

Determination of e

Plot of the residuals as a function cf y as defined

in each method earlier. The APL function which was

used has the name SCAT and belongs to the library

package OA 2 3660 (available at 8. R. Church

Computer Center) .

Plot of the residuals of the regression as a single

array with the function BOXPLOT of the same library

package. The plot characterizes the guartiles, the

interquartile distance, the median, data points

inside and outside the 1 and 1.5 interquartile

distance and outliers [McNeil, p. 13 and 71,72].

Numerical values of the residuals.

77

Histograms for the residuals of the regression which

show the relative frequency and statistical

features.

For method 8, which is an analysis based on different

assumptions for the probability cf suppression P(S/9,r), the

same analysis was performed, with the exception of the

histograms fcr residuals.

The methods 1-7 describe the attempt, to master the

problem of zero-probabilities in the original data set as

well as tha problem of a possible heteroscedastisity

.

For the later methods, in which irore appropriate

statistical tools were used, the systematic structure of tne

residuals seems to disappear up to a certain point. The

residuals concentrate themselves more and more symmetrically

around their mean and median (compare particular the

boxplots and the histograms in the appendixes D, E, F, and

G) .

The relative large remaining range of the residuals

is determined by single outliers. The analysis showed

differences for different kinds of ammunition. Among the

three ammunitions considered, the analysis cf 81 mm showed

tne smallest spread for the residuals, for almost all

regression methods.

The appendix H shows for the iterative regression

methods 5,6, and 7 the development of K and e for the

different kinds of ammunition. In each method , a

convergence of the values K and e with increasing

iteration can be observed. The starting value for K is in

method 5 smaller and in method 6 and 7 larger than its

corresponding value after convergence is obtained. This is

true for all types of ammunition. An equivalent observation

can be made for the e -values.

A graph for the iterating e for method 5, 155 mm

could not te plotted because of the very small change cf the

£ -values along each iteration.

78

The third page of appendix [1 displays the numerical

values for K and e produced in each method for the three

different types of ammunition.

The analysis of the parameters K and £ showed the

following result;

for the iterative methods (5,6,7) the K and

e-rvalues approached with increasing iteration a limit value

(see page one and two of appendix H) ; this occured after

about the sixth iteration. The final values for these

aethcds are also presented in the tableau of appendix H

(third page) . The plots for K and e show different shapes

for different methods and partly also for different types of

ammunitions . The approach to the final value may occur from

a relative snail or a relative high value. Among the three

ammunition types considered, the analysis of 81 mm showed

tne smallest spread for the residuals, for almost all

regression methods. The scale parameter K influences the

probability of suppression P(S/9,r) as stated under formula

(2.1) in chapter II.B.1. in the following manner:

An increasing value of K leads to an increase of the

probablility of suppression E(S/9,r) .It is reasonable to

assume that tne suppression effect increases with the

increase of the calicer. This behavior was confirmed by the

data analysis; except for method one and three where an

inversion could be ooserved between the K-values for S 1 mm,

105 mm, and 155 mm and for 81 mm and 105 mm respectively.

This distortion results from the fact that in irethod

one all data points with probabilities of suppression equal

to zero are disregarded and in method three the cluster

procedure emphasizes the average values produced by the

clustering. He prefer the latter methods.

Contrary to these observations on the K- values, a

general trend for the £ -values can not be related to

different methods or different kinds of ammunition. The

most reliable value of K and e seems to be found by method

79

7. This method considers the different variabilities for

different sets of distance and angle.

In the course of this paper, the value of K and £

derived in method 7 are taken as input for developing the

rates for the model of section II. B.

Ihe third parameter L developed by method 8 is valid

only up to a probability of suppression evaluation equal to

1.0. Since for 31 mm and 155 mm L is larger than one, the

function for the model should be decomposed. This

alternative approach to model suppression is beyond the

scope of this paper and will not be considered further at

this point, although it is certainly a topic worthy of

further research.

As a possible further step, confidence intervals for

K and e could be developed.

2- Suppression Time

The suppression time data, which were collected in

seconds, represent the time for which an individual remains

suppressed as a reaction of a singla round. During this

time, the individual was unable to carry out his mission; he

is in hiding in an effort to survive. According to the

setup of the experiment, it was possible for the suppressee

to react to detonations, which he was able to observe and to

hear or to hear only (visual/auditory and auditory

perceptors) . This is consistent with tha model design in

chapter II.

This analysis is an attempt to compare the

suppressicn interval gained from the experiment Kith a

certain distribution whose parameters are estimates of the

data. The GAMMA distribution was selected, because the range

cf the random variable X, representing the time, is limited

below by zero and by + °° above.

The data of suppression interval for the calibers 81

mm, 105 mm, and 155 mm are plotted in histograms in appendix

J.

80

The estimates A and r for the Gamma distributionare derived from the expectation and the variance of the

data.

X -gjx]

Var[x]

in

n j i i

r = X • E(x] =

-.2

i=l

n 2

( I Vi=l

n _2

nJ,

(x.-x)i=l

1

(3.27)

fcith these parameters, the Gamma density can ce computed in

the considered range, by the formula:

r(r)r-1 -Xx

x e < x < «

f(x/X,r) =

otherwise

cor comparison, the related density was computed by

„ . . frequency in intervalf (x) = -

,

;

N • interval

(3.28)

(3.29)

within the same range, where the frequency in a particularinterval and the interval itself is taken from the

Histograms of appendix J, and the constant N is the sample

size of the measured suppression intervals.

The numerical values for the estimates X and r andthe values for the sample size M, the interval Ax, and therange are displayed in the following table.

Caliber X r N

INTERVALAx RANGE

81 mm 0.686 2.174 445 0.5 to 16.15

105 mm 0.468 1.974 348 0.4 to 13.7

155 mm 0.587 2.591 101 0.5 to 16.75

81

Ihe following Fig 14 compares the Gamma density (3.28)

with the height of the related estimated probability cf the

data (3.29). The sketches show that the Gamma density for

each ammunition type respectively underestimates the related

freguency of the data because of its long tails. In fact,

any distribution which meets the requirements above could be

taken for a comparison, although the Gamma distribution

seems to te a particularly good choice for the 81 mm caliber

ammunition.

82

RANGE OF X: 20

RAN OF. OF Y: 0.3 5

frequency in intervalH* interval

--OO-- — o — o o — o— oo — — — o o — o — o — o o — o — —

105 mm: A = 0.468, r = 1.974, N = 348, interval = 0.4

(r-1) -Xxx • e

frequency in intervalM« interval

freguoncy in interval

W« interval

• |-4-oo-o— oo— o— o-o ± — ±

Figure 14 - COMPARISON GAMMA DISTRIBUTION AND DATA

83

In order tc intensify the comparison, as a next step

the skewness and the kurtosis of the data and of the Gamma

distribution with the estimates above will be contrasted.

These characteristics express the symmetry of the

distribution about its point of central tendency and the

relative concentration of cases at the center and along the

tails of the distribution.

For the derivation of the skewness and the kurtosis

for a Gamma distribution, we need the first four moments

about the origin:

vi

= E[x] = Xr „ r 2, f

2 -Xx (Xx)r-1

p = E[x ] = / x eZ T(r)

By changing the variable of integration

z = Xx

dz = X dx

CO 2

r,r 2, ! Z _Z r ~ l J^

L A~p2

- E[x ] =J^e z ^X-^dz

we find

X dx

(3.30)

(3.31)

(3.32)

r (r + 21 rt+2-i -z l

dzz er(r + 2)

dZJ

x r(r)

but the above integral is equal to one, hence

U2

- i (r + 1). (3.33)

In a similar fashion, the third and the fourth moment can de

derived.

y, = — (r + 2)(r + l)r

X

U, = ^- (r + 3)(r + 2)(r + l)r4

x4

Converting these moments to moments about the mean

using binomial expansion:

U3

= u3

- 3u2

E[x] + 2(E(x])3

u, = y, - 4y. E[x] + 6y„(E[x])2 -3(E(x])

4 A 3 *

(3.34)

by

(3.35)

84

we can compute the skewness and the kurtosis

M/,

'3 3 l

4 A(3.36)

For a Gamma distribution we get the following formulas'.

r. .„

(r+2)(r+l)r (r+l)r rJ

3_lx3 "

x2

*+/V (rA

2)3/2

°3=

/r"

and by similar operation:

3r + 6

Using the above estimates for X and r we get:

(3.37)

a„ =

^a4= 3r + 6

(3.38)

The formulas for the skewness and kurtosis derived from the

data itself are:

N1 v ( -v3m 2.

a3

=

(x - x)

i=l

N -x 2 N 3/2

(£ I (x - x)2)

i=l

1N

-s4 (3.39)

a4

= i=lN

Ii=l

-v2 N 2

(f I (x - x)2

)

The resulting numerical values for the skewness and kurtosis

derived fcr the Gamma distribution and from the data are

displayed in the following table:

85

81 mm 105 mm 155 mm

SkewnessGAMMA

Q3

1.356 1.423 1.242

SkewnessDATA

S3

2.156 1.032 2.103

KurtosisGAMMA

a4

5.759 6.039 5.315

KurtosisDATA

S4

8.004 1.309 6.750

In discussing the numerical features we can see that the

skewness of the Gamma distribution and of the data itself

shows an asymmetric right-skewness (positive values).

Although the values differ considerably . So is e. g. the

skewness of the data for 155 mm almost twice as big as the

skewness based on the Gamma distribution.

The same can be said for the kurtosis. There is

significant difference between their numerical values.

This analysis supports the fact that the Gamma

distribution can only be a rough fit to the data given, and

as already earlier expressed, a fit of another distribution

probably would have been as successful as this one.

C. VEBIPICflllON OF THE MODEL

In order to supply general features as input factors for

the decision process for military leadership, the purpose of

this paragraph is to compute numerical values for the

probabilities and rates developed in chapter II.B.1. and 2.

For the computation of numerical values, it is necessary

to make the following reasonable assumptions:

It is assumed that an artillery unit consist of six

weapons all either 105 mm or 155 mm, or that three

81 mi launchers are combined to a mission unit.

Hence the fire rate ^f

for such units will be

concluded to be [ Wiener, p. 189, 21 1 ,2 13 ]:

81 nir unit with 50, 55, and 60 rcunds/min

105 in unit with 28, 32, and 36 rounds/min

86

155 mm unit with 18,21, and 24 rounds/min.

The standard deviation a for the density for

artillery hits will be considered to be [ FH 6-161-1,

p. 53]:

25,50,and 35 m for 81 on unitr

20,25,and 30 m for 105 ram unit

30, 35, and 40 ra for 155 mm unit.

The numbers are taken from the field manual 6-161-1

and out of working papers of CDEC. Ihey are rounded

fcr convenience.

The parameter H of the probability of kill is chcsen

completely arbitrarily with 100 for 81 mm, 150 for

105 mm, and 200 for 155 mm and has nothing to do

with experimentally observed values fcr the weapons

here in question.

The parameter estimates K and e are taken from

appendix H (tableau) with the values:

81 nun 105 mm 155 mm

Ak =

Ae =

1360.498

0.871

1991.151

0.769

2349.845

0.359

Ine followinc figure displays the conditional probability of

suppression with the atove estimators for the parameters.

P(S/9,r) = expL x

r (1 - e cos 9)]

for 8=0 i.e. along the main direction of sight.

P(S/e = 0°,r) = exp - - r (1 - e)

(3.40)

(3.41)

87

Range of r i

P :

•100 2001

81 mm

K - 1360.498

Ep3 - 0.871

: p

105 rnm

K = 1991.151

Eps = 0.769

• P

155 mm

K - 2349.845

Eps - 0.859

Figure 15 - FUNCTION P(S/8 = 0°,r) =• cxp[- £ r2(l - c)]

88

For the above selected values of the standard deviation a,

the fire ratef

' and the parameters K, e , and H the

values of the probabilities

KP(s) = (3.41)

2a 1 + - e

2a'

PK(S)= P(S) -

K

- G

(3.42)

POO =

a /H + 1

(3.43)

and rates

X (wiLh no killing) = \ P(S)

X (with killing) = X r P„(S)s f K

(3.44)

X = X P(K)

K

are displayed on the succeeding two figures cumber 16 and

17. Numerical values for the rate of rise ^ are alsou

computed and displayed in figure 17. The collected data

(suppression intervals) were not distinguished among

individuals as in the course of tha foregoing analysis,

aecause of this fact the numerical values fcr X are basedu

en the expectation of duration time with regard to

population.

E[t] =XfP(S)

X P(S)te fpCOdt (3.45)

It might be worthwhile to evaluate the rate of rise

by applying the Gamma distributions for f (t) which were

discussed in chapter III.E.2, and contrast them to the

values of X gained by the data itself.u 3 1

89

Hence solving equation (3.45) by having set

fp(0 = ?^ (^)

r_1x (3.46)

we find:

E[t] =XfP(S)

- XfP(S)t -Xt

/ e It^t (Xt)r-i

X dt - I

r(r)(3.47)

if X> X P(S) and by changing the variaole cf integration:

(X - XfP(S))t = z (X - X

fP(S))dt = dz (3.48)

we get:

EM-z r-1

e z

XfP(S)[X - X

fP(S)T

_ r e z

r J r(rdz -

XfP(S)

= 1

E[t] =

XfP(S)[X - x

fp(s)]

r Afp(s)

(3.49)

Ey using the earlier developed estimators, we receive the

mathematical expression for the rate cf rise based on a

Gamma distribution for the suppression interval data.

r r

X =u

LXfP(S)[X - x

fp(s)]

r ^fP(S)

-1

(3.50)

3emarJc: Ihis derivation is only true for A > X P(S) , which means

that for applying this formula, the fire rate X- may not

exceed the value:

X, < (3.51)f - PCs)

Otherwise the integral (3.47) explodes towards infinite

and the resulting rate of rise would oe

X =u

(3.52)

90

which is consistent within the set of assumptions.

Instead of using the Gamma distribution the data

themselves were applied in a second step to calculate an

estimate foi E[t] and ^u

based on the number of

observations n.

E[t] =XfP(S)

\n

1=1

n XfP(S)t

±\

- 1 , (3.53)

Ihe approximate value for the rate of rise is then

X = -U

E[t](3.54)

If killing as an additional event is considered, the rate of

rise is computed with the same formulas (3.49) and (3.53)

except that this time P..(S) is used instead of P(S) . TheK

computation of the values for the formulas (3.50 ) and { 3.54)

was performed by APL and FORTRAN respectively. The programs

are displayed on appendix K.

91

PROBABILITIES P(S), P (S) , AND P(K)

81 mm

P ^"\. 25 m 30 m 35 m

P(S) 0.573 0.495 0.4 30

Pk(S) 0.503 0.446 0.394

P(K) 0.137 0.100 0.075

105 mm

P ^\^ 20 m 25 m 30 m

p(s) 0.731 0.643 0.564

Pk(S) 0.582 0.542 0.492

P(K) 0.272 0.193 0.142

155 m

P N. 30 m 35 m 40 m

PCS) 0.573 0.497 0.432

Pk(S) 0.480 0.428 0.378

P(K) 0.181 0.140 0.111

Figure 16 - NUMERICAL VALUES FORP(S), P (S) , AND P(K)

92

3 r—

N

r-N r~\ r-S r—

l

r*^ no r-H ol en ST in NO r~.u N^ Nrf* »^ v_^ ^-^ ^^

ON rH Oi en oi h pi CO sr lO ^J H uri in <r <r o in <f CN Ocm in r~ r-i en in en pi sT <r en cm in ^r m -I Ol rH m vt cnrH r-H rH

O O OrH r-H rH O O O rH rH »H rH rH r-l rH <-< i-i rH rH rH

O O O O O O o o o o o o O O O O O Oon <r ro CO Oi H Csj oi O in rH fs r-. «5 c-i rH vO rH »T r-H 1^

e <r i-s cn cm sr i

—

-.T - 1 in cn cm o <T m CM en rH a NT MH

mmrH

D rH »H rH r-t f-i t-i o o o rH rH rH rH .H ^-i rH rH rH r-l r-H »H

o o o o o o o o c o o o o o o O O O O O OCM O ON -T CO CM <r en p., OI VO rH CO -T r-H rs o -1 <r on m

p r- O oi sr vO on in no in CM O On OI e*l r-^ •H O CO Cn rH oo rH Oi Oi rH r-H rH O O O r-l rH O rH rH r-H r-l rH O rH rH rHto

o o o o o o o o o O O O o o o o o o o o o

mhI" in in in -cr in m m m1w

en en -J

O O Oen en v-r en en o cn en vj- O r~, -7 cn en <f cn en ^j-

O o o o o o o o o O O O o o o o o oOi co 00 io o in in m in •O O CTi -T m co on en co Nf DNvf

£ITi ON m CM mo on o r^ CO o <r cn co no -r co r- m O co rscm oi cn CM CM CM o O O o o o o o o o o o rH O O

enO o o O O O o O O o o o o o o o o o o o oin rH m on r- in CO r-j u"\ co co en rH oi in ^T r~ en Ntsn

R Cv <f CO <r co cm CO O rH ^T CN rH r- in m o- in <r on rs no

1

mo

D m cm m m cm c-j en O H rH o o o o o o O O O O O OO O O O O O o o a o o o o o o o o o o o o

--t NO Os CO r- oo on in <j m O CM CM c-j en in On OJ O no ctn ~j-

F= cn oo oi iO o <r <^j <r o en rH O no -.-j- c-j in <r en co no ino m n vt

O O Ocm en en r-H r-H r—

'

O 3 O O O O o o o o o oo o o O O O O O O O O O o o o o o o

no cn o o m o ^c m o o cn o NO ON O >0 cn C> no co o

tUl o o o

-j- m o vj iTi \3 ^r in no <i m nc <i in no j in io

O O o o o c O O O o o o o o o o o o

MO O Ps CM -.T OJ CT* lO h rj n no co cn rH CM 00 CO CO OJG m on r~> cm id en >^D vO r^ c f^ in O ce r^ i— l'i n co no in

U~l

enen m vj-

o o oen en en o o o O O O rH O O O O O o o Oo o o o o o O O O o o o O O O o o o

rH in in O O vO n cm o in r- en -T no O m on on <T in CMF; rH -n a* r^ -h ^j- M ^O no <r m CO NO 1/1 -3- OI -H no ~j- en

D o ~:r ^r <r en vt <r O O rH o o o o o o O O O o o oirHCO

mo o o o o o o o o o o o o o o O O O o o o

in i— p-t n el n ro vO r^ O rN CJ e I <r O ^7 en r^ n n ne p* cm r-% rH lO O h o; r-i m r- j rH lO vr n ~l rH O ~J- Cn| rH

cm<r m mo o o

--T nt in 1—1 r-< r—

*

o o o o o o O O O O O Oo o o o o o o o o o o o o o o o o o

en cm o cm m c-j en CM m oi cn oi cn Oi

1m

CO cn co av co a^ oo on CO ON CO ON CO O-i

.

O O .H O O rH O O rH O O H O O rH O O rH O O rH

co CH ,-n O ^-^ r*NUl CO H 3 c") J 3 co 3Ul c Ul C-Q < -< C l.«' <r< C lr<CJ -H '•> r- c "^ *H -1- ---

r ^ .5

•H ^^ul 1j .H u oo r< rH •""* rc -< r3 CO0) O. rH a. c. f-l el nh ,h CJ U rH cn ij cu Q. -H O. -H i-l U": ^ -H Ul fj -r-l CCi I'J 'ri

CJ 3 r* 3 -H •H H O -<C H (J rH H TJ j! •H --) rHerf VO Ul Ul" H -^ ^C as crt -h i rrt DC rH

ir! •"> <r< -H , ^< C u C -H c u r: -nu-i :) 'M .* '*-4 'h o rj 'u o ^: m o n U-l o ^o o o o o o o o

.1-1 jr. •o .r. TJ £ t> .t: -o .n0) u 01 1< CJ CO CJ vj V rj u CJ "J I

J

nj -ii uiJ -.H IJ ',H iJ IJ 1/1 -H 1 <i W «iH i-i 10 -J i-l I0 -H<1 J "J U

nlm n 3 1

- in J 173 i-l \l 1 1 ,t

Bi <—

'

nri v> a: .a -~-,-r. .,j ~ U. SZ -^ Oi J3 ~-r

Figure 17 - RATES FOB THS MCCZL

93

Row (4) and (6) and row (5) and (7) of figure 17 display

comparable values for the rate of rise. As it was stated

earlier, the Gamma distribution underestimates the related

frequency of the data and hence the comparable values for

the rate of rise Au in row (4) are larger than the values in

row (6). The same is true for row (5) and (7). They differ

in general by 10 - M%. If we are willing to live with this

fact, the rates of rise evaluated by the fitted Gamma

distribution are a good approximation for the values

computed by the data.

In order to verify the four Lanchester equations (2.76),

(2.77), (2.96), and (2.99), presented in chapter II. C, it

is necessary to specify particular rates given in the figure

before. In case of a certain known composition of fire

units, specific rates could be developed as inputs fcr the

model equations.

By this, the model equations receive their specific shape

and scale, their general behavior remains the same, as can

be seen in figure 9 and 10.

94

IV. DISCUSSION

Eased on the research effort of this paper it was found

that the phenomenon of suppression as defined earlier', is a

multidimensional problem. It is influenced by

psychological, physiological, and environmental variables.

Ihere exist many possibilities to model dependencies in

general form, however to make quantitative statements about

suppression the modellers have to restrict their efforts to

those variables which are observable and measurable. Since

the main objective of this paper has been to establish

models which are able to express relationships

quantitatively, the main thrust has bean to formulate

suppression as functions of weapon systems and their

dispersions.

For the evaluations of the models, a set of simplifying

assumptions was necessary in order to guarantee a

mathematical transparency. Ihe dependencies developed in

this thesis postulate some satisfying results in modelling

suppression. Ihe models reflect sufficient accuracy of:

The physionomy of the human being and its resulting

behavior with regard to suppression. Suppression is

mainly caused by visual and auditory perception.

Ihe influence of the weapon systems i.e. their size

and their firing capability are determining factors

for the amount of suppression.

Cf course the detailed results, i.e. the estimation of model

parameters, are based on selected weapon systems and

scenarios. Many possibilities exist for further work in

this area particularly under the aspect of including

extended human factors components in the construction of

95

suppression models.

The authors feel the paper may provide a contribution to

future design of vargames and simulations as well as weapon

systems. It also supports a more careful analysis cf the

combat situation.

96

APPENDIX A

ORIGINAL DATA 81 mm

r*ioor»io<Nir»CN.^r>a>:rr~i0r*or*inm<nr*mJ'-Tr*<Ne«j'Oooo ^o o o

J1/1

o

u(->

o

r~ cn cNr*CNCNp*CNCNr~CNCNr»CNCNr-<NCNr*<NCNr-cMCNr^<MCNr- cm cnp- cn cnc-» cn <n

a>

cr>

a.

in ID aifl?(CJtDinr»lD^Wn(NHi/nfllNu)rtr1(NOOj) r4

r-fOO^-tOOOOOOOOOOOOOOOOOOOOOOOOOOOO oo o o

<u

+)

E

c•H

LT> O O O O O Or^ cn o U3 o1 n 'O

'CNcni/JCNr-tnCNr*ir>JCOCjloClJ'flC 010u30300C3CCr>riCNC7>0 CNCn O Ol

ccc

0)

U

«

c•H

Q

ooooooooo mp*^in*HJLn»H^CN»-<p*CNu3r^cNkOr^nHHnNHnCJHinnnuirjninnw o «-* o o co oo a) toa j n >a j nio ^ n

T-«CNtnjt/>ujr»cDO'0^tCN(nJ'»/>»or» cacflOHrtnjuxOhoocftOrtNnjwm)HHCiNfinNcmncinnnnnnon

97

APPENDIX A

ORIGINAL DATA 105 mm

U) U) CS (N jt J (N (N trt U1 fcO <N CN CN CN J (N IN lO J- t/> ut <N <N J N (N (N (N i/) i/l i/) irt

a

o

«oU

' Ln»i3j-ju3ir»oj-aiuijj-j'jojcnoOnc*<r»ooo*coj-oo*oor*«-*i-» ,-< CN CN -3" .* r-* CO rt rH r-i

^ id n J iq co 1/1 uin rt n r^ go en «-« c** t*~ in en en co ul^ mnr-j-rnenr- o-t *h ^h ,-< cnuticn cn r^ en thcn^h '

,h co cn

tn OOOOOOOO-rHtHOOOOOOOOOOOOOOOOOOOOOOO

b

E

ooooooooL/)i/icr)r^r^jjvr>r**r^cootDuiu->iDOi/ii/5coa3r'r^cici""TJT i nr)OOC0a3OOr->f0^*Hr-(OOC0iOvD»^TH3--3'mr0CNC0C0OOCnC,»-"

c

tt!

0) noooooooootn«-!«--!r-r-cNcDcDu3p*CNC\iiotoe^r^r~iou3tH*HOco

a>

«-( c«*r)jTinvop^aDa>o «-« <n r> ^ ui «5 r* co o o «-« cn n j W) ^o r* co cr» o .h

98

u

o

u

X>

S

5)

-a

©

APPENDIX A

ORIGINAL DATA 155am

f-i f> r-» CS h rl W H C< «-« «-*

(N CNC*<N(N<NCNCNCN<N<NCN(NCNCN(NC<<MCNCN*

O OOr-tOOOOOOOOOOOOOOOO

ooooooor, <N(0-j-cN(Nm(jtcNuia>j3-r» r-i0r-^or'^-.or-' r-f0ocN00<NO^«T-4

«-« (N r( CN h (N r-tCN .H CN H OJ rl H

o oooooco<Nr*03<Nr*nnj « n J n rt

«-* c* o ^vntor*-co<7»o^«<Nn3"i/»or*cocT»o

99

APPENDIX A

SUPPRESSION IWTiKVALS IN SiiCONDS

mm

O* io«-»H*^'H —••-• •*«*-'*~«^«#--*-«<v'Nry'>jrv(>jrs;fN,r\jrNjr)r^)0>(pr^r'>rorriNT >r sr-ru\mu*-o -dr^ v>

+0<J^-*T -O fc^—-cC sOf*>O i u*»^—r-<1Ovj'Nt><'iu^r^*r'ir'OC~»—«fMO r""»<r*rl-*<>u'i tO'*Trri-— f'tr- *TvT ^u^o •c<r

S J> <r-o ^^-cxijor^crinf^fL^cON.* i>or^a^-^o-j*-Oa^ir»rifrTrrtinr •^U^US^J- r^^r,*f-L J- ^ pj i/s cn) C7 a

'

• «H • •:

Oi^rJO(>-^—^o-ocfcm—*'- cd^-md^o—•stacrt/,.«-^-com —>ccrF^*?-jr~-<rJ-"<~<T'£C>*»'ir^C}KnoiuM^-C r--cr:ooajC~o — r^r^m-fl-ir-ot^ coco — f-jiTim^rcc OO Nv' ci>(j -,— i.Trtujoo

OQo "-••-< •-'^'-'-*'Y—'^^>-'«-^J<M<V<M<^, <>-'C.'(vrMr\jr|r<'irntr, ror^r^rOo-i-^r^rsTa-iU'i-o-iJ-OO^

i^f-Tctof-'<~u l-nir>in-©f— ccajcncc-* —imrosrm-or-coco » —* 1 ' u~i -ooc t-J r--' -J" a." ;?> —(in^xiajo'

I

rOlfVHCtO 1^-ON0iiM^--Ncaf^0'00^-*r^*C C C~U~ —^^C L

^rycoo prjp~\t/Mj-\^ r-cn^a^c^ioo- rncnv^i/><:r-c0CEJO.-«<>lrf-\NL.0C-~C}'^-~3-C~ — ^•-''^"CCC-

v^ <J^ U~ -*- *C •(JV-J

a:^-.^-vr*00'^o-M^'~t^OOs rM"*'<N', "

oq>o

<rrtcoo-

.'C-JCJCNCM jfMp:r\i^jn m r^T'n r^ roro^r ->r ^r -ir.^-O-: <;r- • >

0O00 -i —i—J—'w—< ,-*— ^^,— (-jf-jM i^jrjf ifNjr-jt^^r^r^^Tfp^r^t-nrr,^^^-* .i- •j-ir.lnir' ^

co"—io-od-oi/1- "ocr o odj'"3—< ( >|lt,l-n^i -Oc-'XCCCOIOO-t-JrO

2 t^tir* —'***» o T moo-r-

coccri-

OOOOM-<» ~j«sJ< —c-i^-4.-ji\j<- jr I i-g.-jr J rxjropjf^ of.-nrr -ifn^in fy~.rnw-r •v,* m n u> < ot

<Ci,\<DiT>,-.

QOoop^^^-^^^^^— '-» ryrt'Nryt[j''N ryrNry r1 rlf,1(rl'fI f,ini,1rfl ^^T^^'k1*^^

100

APPENDIX A

105 m SUPPttSSSION INTERVALS IN SECONDS

^^r r-1r-(^»-ir^<,'<rsi(Nrv*(vrgrwr^mr»ir^vj' sXvj-^-u^u^u'Au-*o>OsO*Of^oOaQa3

i —ifNl mr—c-Jrninr*'iincom—. cm vt-

Of\j^in>oa:r-<—icj'j-j in -o co o r\j .£> cco »- m'-o — rMro>i/.-CNj>rr— mrMmartf*

^(^(r<^<^-i>j{>jojtM(Vi(NiC\/c>jojr":'ri', irr> j-o" ^*rmminin>0'0<J-or-coaJ'Cr''i

^0»r»>oa;r^rMt^tnrnuNOair— Or^c7, --<^ci*<?invj-in,-JC*jrninc*1ini— m«— ojnTcOr-4Vtn<JLU —< •^{\vTvru'«0iOrM>uci;r?--i/\f*-yN rvii*i'0(-ic\jvj-r— r*>r\jm<T}o»

Oi-if-'»-<'-i'-<f^jcvjr>,'iNjruc\/<Ni<Nj"imr'>(n«j' ~r J"vr .j-inininvo-o.o-or— ajcacorn

-'Of^OtCHyNroifir^LA Or^NOO'r'^rn>r'ir ri.n.—»fs;ctnc- mr-- rvj~ rvj*i*

0«-ir<'-<H—' fv'fvJ<>J<vJrurjcvj<,vJrr>rrirO(o^- >r vj-vr -rinmm>co.ot<3r— or. cc cc m

)-d~<cMO»HiCXJ^*<"vjmmfnvr oroinOC*c»"»in vrr^^m vj- in in .ccLinr^tor—r-—* rnvr

OHrc-ir ^rviCMCNCvrsjcvr\jcvrnrnmm<^vrvr*,rNj-ininin-o*0'C*<)i— (— ajco<*i

r-^f^0^insOrtiCr, rr\o"Or'"<ino'N/ir^ir. <r r^(^ Ju"\ *j-a-mvCC>ir-r— ror*-r-*-~«rviin

©-arnininp-o —«_*<-.fcrin>or-c-"<'-o r*- (r^u>r- os -wc\jinor\iro*G» r> o3r'"iN{jcvj

0^^^^i^rjrlii^r^f\jcvjr^4c>jror^rrirnrno'vr^r^rurv ir,.tr»\0>0^'ijr*rTajaOr*

m<\jnO-- SJintf

0~<

o- -.—

r>o.rsjmu

0«-(— -

vmu

inorht-or-

^rjrufMrNjfMrvir-JiMr'-imnirnm-r <r

TOntON fOMtaOOvtlAifOO o**i~nMrin-0-OOC>-rr- p»otr

—'CNjryr.j'Nj Njr\irsir\.pirirnov ri~JW

— rOrurNry^^rNojfNr^rnr^^n^ *r

'-x> -sT cr in a:o <J* r

r -r j>mmio-o ^NH CO=U 3

x>-j-^incop<>c>l-'->cr«f4 —*m^

T -j-u^>nin>n-<eO-0''-r+;e'D 2

a? pi *n f** <** CT*—

-j- v/ ut> in *n *n * o-or4 cO <V'U

- J—<c

155 nro

r- J-c IV—+— srO— in r*-

• '••: «•••••^w Cm rv, rnro ^ mm »o *}

I

fc> 'n/ •— h-OoO u~» oinoo^'ic^Ci/ r- o

. ,. . .

n *T in in *o o

I

I

O.- CUi-^Cn

"»% %T Ifl in -O On

ONr-o^-tntj-f\jr--Ovrin on in cr p- —'Cjimny

n^g-'nOr^

r\rJr;tN-~covttDrnas^-cafiiPo

"-i— rvrNifOvr NJlnoc^N

ocbc0l'^lnr•Jr-JOccomconu^~mci-Chhivoj "n >r vr «n vo ui

co <c m, t .i In c -O c x

u

A.

f^> 1)i>jo-

n^j- v.rmiN0r-

j>tt> <)c _>ai

"\rr\.J.n.c.r»-

rs/t*- o^o ?**- 3^c r>

'OI/lCC D

^«.*-lCNjr\J

o«Po-c

<r -s

InwuiNOr--

\<*"W miner-

101

APPENDIX B

DERIVATION OF CONSTANT A

The constant a can be determined corresp ondingly to the

following derivation:

Modifying equation (3. 10) in Cox, The Analysis Of Einary

Data, p. 33, we can define a transform as:

Zn

S. + a'

(1)

Starting with the original model

f(6,,r.)P. = P

i(S/9,r) = e

l l

and using the log- transformation

y . = Zn p

.

i i

wa subtract

/ 3i+a

\yi

- yi

= in \~nT~y "" n P

i

and choose the constant a such that

z[y±

- y±] = o

Since s-Einoaial (n p , "iP-^ 1 ~ pi^

which can be approximated by

S = n.P(S/9,r) + /n. Ui ^ *

where U is a random variable with

E[U] =

E[IT] - P1(l-P

i)

Inserting S in equation (3)

(2)

(3)

102

Ely, - yt

] - E in

S^ + a

- *n P,

= E In

n.p . + /n7 U + a1 i i

~ Vi ~''

= E An 1 + JL_ +a

/—

'

n.p.

ir i

Approximating this by Taylor series

= EU

/n'

+

i Pi

Vi )

-\^r?i

ni%

31 /!-• _ n.pn.p. 11/

- 11 '

It is sufficient to consider the firs- two terms of the

Taylor expansion since U and a are small in comparison to n.

= E JL_ +a U Ua

/—

'

n.p. 2n . p

.

3/22 „ 2 2'n.p. ii li n . p . 2n

.p

.

11 ii ii-1

This equation has to be zero according to (3)

Hence

terms cf crder E[U] =

terms of crd = r —n.i i 2p

i J

a - -j (1-PjL

) =

a «f <1- Pl )

terms of higher order are neglected.

103

APPENDIX C

DEBIVATION OF VARIANCE

The weighting factor Var (In f) for method 7 in chapter

III B is computed as fellows:

Expressing the conditional probability of suppression

E (S/9,r) as a guotient of the nuinDer of successes Siand the

number of trials ntleads to:

i n.1

where the expectation cf f is:

Eff.J = P.(S/8,r)

in f . = In \—

Vn.

Var[£n f .] = Var In

l -1

Since the random number S is binomial with mean

E(S) = nP(S/Q,r)

and variance

Var(S) = nP(S/0,r)-(l - P(S/0,r))

it can be stated as a function of a random variable U with

mean

E(U) =

and variance

Var(U) = 1

104

i.e.

} = n.-Pi(S/0,r) + /i^P^S/G , r) • [1 - P.(S/G,r)]-U

Inserting this in the equation above:

Var[£n f.] = Var

Var[£n f . ] = Varx

Var [Jin f . ] = Var

In

n.P (S/0,r) + /n.P. (S/0,r) [1 - P . (S/0 ,r) ]. U '

In P. (s/6 ,r)- I 1 +'[1 - P.(S/0,r)]

n.P. (S/0, r)x x

[1 - P.(S/8,r)]

J,. P.(S/9,r) + JLn(l +n P ( S/9>r)

- •i;

i l

Since P(S/9,r) is constant we know that

VarUn P(S/0,r)) =

which leads to:

Var[2n f.

] = Var la 1 +'1 - P

i(S/0,r)

n.P. (S/0, r)i i

knowing that for small x 1 s

£n(l + x) ~ x for x <<

we can state

Var [Jin f '] ss Var

1 - P.(S/0,r)

n. P.(S/0,r)l l

105

APPENDIX D

ANALYSIS OF RESIDUALS 81 mm

Method tMethod 2

ironsitrt

"o.oooia.ao.otnr.

• Jit . JM.'Otin

O.J JI»H«1'«

Mtcr nr ii "j

*4*c« or ii *j

J.UHIHIlf I

• 1

• •• 1

• 1

• ....1

• • • 1

• 1

« • 1

• *1

• 1

1

• 1

|

tOIFLOt At Hil.»

11

1 1

0.1??*llh*ll *0 70312»H«T»0.O774M14139 *0 927513151 J

• »tsrp(M,'.; . f-OM ;mm "o 521(."027S6

o.i iJiilMii 9 1 707;351fi»

0.77913:8131 7B1S211 3*40.71145 7-685 1 2118717310.06 Of. 3« 05738 9 14842303130.1 1 BMOlGa "o 95»»9210Sa9i

0.2M». "•05-78 *0 353GG51 294o,iott>*eri093 "o 6h71538**«>

, * tt 1 7 S » 5 C 5 <• "l 7781109*70.17187671 17 "l 111 845011

*0 .1 983751308 8731177153*0 .06CC027337B JflJ588848$*0.J1213*I B71

IIOIIS

"a, ooo-i ji ut.i i,oflO)?sti»HilI

Illl ,>JJ1<«trt

O.JHOMKIIscat I

Ittct or xi "i.l •

iaicc or it "i.» i.l

lOITLOt It ill

1 I

11

1 1 »17?*»06 -o 61915*417)073HMI171 79 P 1 1 G 1 1 4 * 1 7

iJ8314G*37*0 41<4S0137«

ll*7??*t0& 570713*871

3 <• e t : b 7 } 1

1

1 0*1109*1 7

{61 J-3 *7 3"«l*2"'373«947

0428"' 0169 ',4 35 3 191

087371061 07 T C7t?CS«*3«211 521 *135 : 09151 *2I *67

1 till 01 714 3 7411*19441• r,i"5M 1

913861 351

|33'.bG3S9G 118 WGJ131

"o 03*1 15975*8 8*71187533

"o 3232001871

Method 3Method /»

*O.OOOtoM?S01»t. o'.OOOTin.JJ.Jljfjv

«.<IO0i;iH«"H 0.00O1! 1 IJJ1J13

jtTq.ttitit

O.U JJ JJMitSCAT A

mncr. or tt "J.l o .

tijcr. or n "i.J i.l

„l

• • 1

3 ••• 1

• • *• |

• * • 2

• •• 1

• 1

• t

» 1

1

1

BOIPLOT 4ClU-!.•

0.1813-14144 C

• Kr.1tnU*t.3 0.1*1l»*72»7.101174 737*

0.13 33'- I50448.377717110*8.B71307IH

, U ' » ' 1 <j 1 77

8.1 A? !•»« )«Ofc

B.3> »311117J0.1*1.37*191f , 1 ) ». 3 • » *» » 3 7

1 .1 MM' Mi*8.11 • '11030*'i,4«*r.i it Lit

"o.i4*'"i*4<»ii" • , 8, 7 * *. <• I *i R

1

*8.M 11 »«01l"8,J'.t1M^*0l

1.1

8.5*** *'.4«7l•.1* I l*l 3

Bj m M*j*>1489* B

. 4 3(.1*<- l*i«

, f I !••»• 74410,1318'.', *3i*5

fl . 7114 JS^Ifl*

I»1fclM»l141t )»tl»*l

0.««:34777378,t/1ft?H1»l1 .OM 8' 4171*4.111 H4171**

8. 89/74777178.I477W77 )l«. •» /71/7M79. 1 701711 1*7#.7708311 »»*

SCAT AHAir.r. or fi "J oniter or /t "i.J J

• 1

1

1

I "1

•1 1

II * ** . . 1

)

lozrinr if i 1)

1

*i lot1

....1

1

• .811 T»l*»4* 10* 1171 713i .MlmT)M J 07l371«<4«7

ki t • 10M 01 9 »-l. fit! 7HIB i

:

1713**

7

11- )H1 31 )*0 •

/

• 01141 * ?|*»w 1C14OM •7771111 9 4 T»» 11 113 1

10* *.M 7 1 • • (.'» .•*.1(.07»

*l ai < 711*11 1 4 l 1 4 4 7 » » 1

"o I'S 1 • 11/19, 114 •••1^48> >v. '. lot, 8 711 «lt'l7>"0 •»)lVki.? 41 1 1 1178 71

"a .7 'I*.j||»n '.1 •1\»I18

'0 4j1«%tll«189 • 1 » j 7"»* J0»9|'0 .971 J1 ' 8«4» 1 aq; '»ll 1* 11*

1 6 J- 8i 7417 3 37 I*. ^* 7 1'9 >1» • OHIO'

3 li« 21(8 >CI•f 81 101 ( 8*41 ' T 11 '.711*1•• 10481 1*81 9 , • .71178

106

APPENDIX D

ANALYSIS OF THE RESIDUALS 81 mm

Method 5 'at Iteration

rrrp rPFTIT

"0. 00063 I 7*. 11 -.1 1 0003173737M6

«1S77.«3»47

SPSo.tnsuisiiT

SCAT APIKCS OF II i

PA.ir.F or /I t .1 1

• 1

1

•

1

1

3 >l• •7 t

73 . .. . 1

1 • • 1

SOlPlOt it; 2]

"l1

1.

After 8 Iterations

itfh r

~0, 00 0633518)171 0,. 0005171736613r.

H78.7I703J

0,llt2«SStSscar 4

/?4.7f;r or ji "s o

fl^,7f;.r or ri "i .s a

BOXPLOT 4Cl2)

1

I-

o . I >. i i 5 ? 5 a o i

1 .190:5351

2

O.S'i 917*756$. B <• ! I i'M'iII

"0."?lHl 70-»19

.03-*i<51 *563

. 1 1 7 16 f. ( 9 7

*1 .00l*7JC4*S*0 .5*00580001

"O.B735<*C59l7~O.6a71CG560l*0.S6-Jl«.2S«»l~o.9 5ai6i6531 ,«9iti--aa236

*. 5 1 t 2 5 « I

~o.95aoai653"0, 6879965601

0.1 00921 E77*O.0:?75**>.f.3T

. H 9 J 7 ).,.,', )

0.1 0-73:1677. ? 1 1 1 )C<« )1

.V.HC81 09O.0J1i:3'17ll3.051 (.5599J5H3 . i 3 i j -. 6 n0. 713 7 7. 51550.51 }<,*>!;) G

. 1 C *, 2 7 J 5 6*0.0&1 * • ? ) J«55*0. 00311 lf*.?5783"0.:75555C"401*9,5973G79>tt30. 78 S?

. 5 8 * 1 c l, 7 5 5 8

Method 6 1st Iteration After 3 Iterations

ITF.R PBF.TIT

"0.000586O7I.353

r1703 .79708?

zrs0.8006073771

SCAT A

PARGR 0? Xi "S

RAtlGF. OF 7 1 "l .J

0.0OOH7'»S91329»

37 I

•7323

TTF.R PBFTIT

'0.0005871105689 00O«755aG602ir. 1011635701 75 1 5 S3 It X*'

1700.7075S *Q<* 5 3 7 6 8 7 5 1 093UM Ofl7

EPS 4501 712773 t.uTunao82S0.808333725 3 1 011635701 3 7 5 1 :i 1 7 3 nH

SCAT A 200f62l028 3 50/e?fi 1.OO3RAtiGF Or I: 5 5<t2799*33 C CG05;S:i771RAI1GE OF 1: 1.5 3 325>*01 80950 3 0773r.7i32B8

03&"B158735 1 0131 59£OS• 1 1031 i 61852 S'-S 87x6828

1 ?79l3X»t17 2 5$39597<il|• •»*.(;<*•. 17717 1 0:3701717

3 *| 85 fj2u703'.a 3 n.10131278• •2 1

"o"a

101 7997208031 761 891 1

7

J729 )1t".19

7959343M331 53991261301 735508• .. •

|1

52 • • 61 277T3331 "n 7 95 13*1<033185507038 i 15 309926*

*0 G3>.6?1 U566 3 iB0939?78

P01PL0T A( ;3] BOXPIOT A( ;2)

Method 7 I st Iteration After 8 Iterations

nrrrr*0. 000611*108971 . 000601 t»3".0*lS

K1H79.71161I

cp:O.I&?>76291»

PAnr.r. nr Xt i

I

• I

I

I

I

] -I

•7 I

• 737JI

I

I

I

rrrit pnr.riT

~0. 00071 rj07<*'^QOl 0.0006HOHS6707*K

1360 . H1S66Itps

0.1711007331

SCAT A

KAHC. IP I, ~

RAicr. or i\

fl•(55055366? 01503«.*O787

t 150A7G03 : » i f. f. 5 C 6 1 9

-50G51763 I 95 «t59357OiCl 31553 0f5nn'*'»t7B7• O'l !'I77B11 1 JlOtt'jf 9J805 170 153718 11108751 a

3 : 3 I ( r, » 5 5 1 00-13 7? 7071070OM7O7 /«G3 0J71H.H7I.I. 8•.'im',710031

3 OM 113735383 1 <* 8 1 1 L 7 7 7 71«*011«857

71 Ul 3 (.71 4 }i'.'.«ou'<ig

"o f. r, I 1 h 3 h 6 5 <• 7".7?I 5^119*0 7017871 5 91 3 ea070',ir,*u

* 1

.J. 1

775UT371 07 0* ?79t* 11131 571531131 »i*0O71 «••• 18

"3 JOl 7171519 5*»l OjI Ji»,fc

J , 77 *t 09 3 7 1 It 7 « N > S 1 1. o '.

«

.601 1 i ll.fc5« Q 751.'.57'-595

poiptnr At i7)

rntn.nt a( i3)

I I

107

APPENDIX D

ANATSIS OF THE RESIDUALS 81 mm

i a • • « o « t a t e • i «i « t • t i • t it i • t t • z t

Hottvxl I Sample Size 29* *•

.

•II

m •«

• • , «*• • . *•• • . ••

• •• —— • • • . »• '

-1.73 0.97

9 1 9 1 I

Method 2

Sample Size 29 ~;~

p . •••

«•••••

•*" ••• *

•••

{*•:::

IB!

••* •••

t i

-1.44 1.04

108

APPENDIX D

ANAU3IS OF THE RESIDUALS Q) ma

.... • • . - - — ----- .....Kothod 3

.-1• ««

• •• • M

__!«*«—!•••!••*

.«*•••*!•••!•••!•••!•••

••• • ••

• i • *- *

• :i:{ Sample Size 3°

— • . •••

»•• • % • • *

• ••• ••

...i• •* I

•|

-1.39

:::h

1.14

.-. ,- 9- |" o""4— l"" l""9' "6 0" • ".2 ?._-.-l2.~i.

iMcthoU U"- Sample 3iae 36

— t:

| ••* ••• ••• *•,

.1 *«f *"• * f • *

•

!*•• •** ••• •••

Ec ji • •• •• •••

»• - ••*1 ••

.!••• •*• • • •

!•••*•••••••I I

-1.08 2.73

109

APPENDIX

ANALYSIS OF THE RESIDUALS 81 mm

,..;.-...-".—-....«—•— •---•- ...............— .— •—«—)•»« e'» o • >> ^

.19

J

I

Kothod 5:::

. . »IIV...

• •• • •• •• •

»•• • •

• • • f -. »*-.• • •*• •• • • • •*

(*••- *!•••

••_—!•*• • • • • •* .« t ..

• ••• •«

•»*•*•• ••* "" •• .

[•" • •• • •• •• •'" !••*

*-•— »••• •»• • •• * • .

I •••••« ••••*•••.»

• * »• •• •

• ••• • ••

• • #» •*•

••• #*•••• •••

• • • ••«••• •••»*•••••*•*>*•«•••*•"•

• •• *

•«•*•• «*•

Sample Size 36

::::

::::

i—• i

-1.012.78

'Zizzzxzi\^?.:^:r^-r.z'L..i..s.:.i^.i:~i:-^.^:-lli.~'-..j.^.j

At

.IS

z

Kathod 6." — Sample Size 36

#«• •• •••!••• •"• •*I••

•

••

•

• ••

I*««

•• ••*«•• ••« •••• •• ••• • •*

*Afci*** .—- *""•• -•*•!•«* ••• ••*| ••* ••• •••,••!••« ••• <•.••!•• * •• • ••».••

••• ••• •*••**,

| •« m ••• •••,••I •«• ••• • • « •>«.*•!••• ••• ••

•

•••,*•,U|M • ••• ••• •••,•

| *• • ••• ••• ••»•«|**w •- • •" • •••»••

I ••• ••• ••• •••,•«

-1.19' '

2.55

110

APPENDIX D

ANALYSIS OF THE RESIDUALS 81 mm

1

1

«...a.4«> -^

C- «4

1

|

T> :

1

1i^

O 1 <n

-C * T* '

*>: !

0)'

i

B3[

i

i

i

i

•1

•- •

j

Ii <N

i |

j

i

|

1

i

T1 r»

1 *^i'

|

•*•

11 INt

j

I|

1

1 1 1

1

i!

*

1!

i |-*

i

I |

i

1i

I!

i

!

in

1i

!

j

I j

i !

1

*

I IA| . i

i OU1 1

— •

i

•j

j!

|i

t !

i ! 1 j ci

i !1 1 in

.

i

*— •

1

i

!1

*»»•-*» V » » i —; I

«.*«•*»* * # '

i

I

1

:

! 1

1 , t •*

; i

I i rti i *— •

;

j i|

|'

!

-

! i i

ii

-3

O;>a-a»»*-j->»

: j; —

»«» l>o „- > ff > » « »!| > » < »

*i

*

! !

I ;<v

'

i

i1 3>

i

i • #*•-#»< » # oa it » * -a * * « ** < e a. -3 • «. » *

|

i

i

;

i T i

1\

:

|;

| j1 -

ii ; m*— •;ii i , :

<*»*-** ,!J »»iO3 • l. B * * * p C

:!

• »#saro-*i.* *

;i>}tan/x * j ^>^iA^v>iijj^)

|

i

i

i

1 i

!

i > • ' * > e »»>i*(»-v»»I*>l • J ,:

i •

i

1

.;.J..;..L.

i

i

i

1

1

i

i +

• i .' a »-.,-*, > * w » i* i

1

i

i

'! r

:

"

'

', i, '.

,

'.

.

'•>..<!•• , , m • • * * *• • - - - i

:i

•••••"""v:i

i

! ,

o 1 O |O

•| I • 1

*> •

- 0.

111

APPENDIX E

ANALYSIS OF THE RESIDUALS 105 mm

Method I

Method 2

ato itsaSTA

"0. 00331 1753957ft 0.0007 3197 79173«

3137. 1173)7SPS

0.77777SS75SSCAT A

RAaot or it '1.1 a

macs or it "1 i

atoatsBtTA

"0.000.11013373.I

3717.30.573IPS

».IJ-)Sli)IlSCAT A

RA1ZZ Of It "•)

RAliOl Or 1\ *2

0. 000371717ft. 79

•I

•7-7I

1

2 •

• •|

• •

•

1

1

1

BOXPCOT 4C:2)0.

1 *

1 1

991957*59'*5602293*78 3

*16**17S1**51 161506

r, 3332327032 3 0239336327272S379oaaa 3 04692915095005137*078 3 3639285 719C1275200998 3 '•66 1990376

3 68*S2397*3 3 13*M*121S3 626737339** 3 02*76«:*206*3 22o2S73lSl 3 136 31 89396

333193792* 1063 1493963 2*3*33173,5 3 3 3293*03533 3136 53**81 3 39857710983 3700568963 3 3335771098

roiriot Al-.x\

1

1

0.

r--I

1 - -.

3627351*3*1 1 70271571731*357512 5*260 1 37360536123 1613 005521929**

3 03062321791 5370253S113 »10-.6:0832 3922170332

5158126 3*9 026923919*31566952* c 52*8380*130 3 1 5 J*6095* 521*12*988

3 01631376363 039391935253 0968 1 976 363 J 61*53*5772

9*25032303 513706130*233996613<* 3756930127230936613"* 3 0*t30173231%

Method 3 Method k

acoatsBETA

"3.00O.591993O77 9 . 0003797"*.7a6»

7177. 751059IPS

0.9359307351SCAT A

rahgs or X: "*

RAacK or 1: "31

i

3 7 •;

i

•i

I

• j.i

i

i

i

i

aoirioT iCi7]

1•*"

1871017*61-3 618 1680213

RESIDUALS 3 323*1 1 li**> 3 3676* 3 15*78

0«*57*8',37O3 3 531381)1113 1215351*1* snamsoi**

*0*(i * 5 1386 3 on3U2*>*351 JOOo 1 I*} 1 68 1*5*0*3153i'.61035 1 fc89*5*0%)1001 1 87075253 * 11* 1939t»

3 33713731801 *1 1* 19119»»

J7T170 J891 575711 5 7C39T,01C*0»i3 • 757H6741

*0 ;0*3*G'M34 3 175 719576970*1*611 76 3 •13881 7051

3 5).y 71* '*6l 3 *5t8l705101 3*

1

779047 3 »188l 7053

3 •W177389Q5 3 »^**170339 3337111*305

atcatssf.ta

"0.0301*927156355K

7079. .05B71CPS

0. 930051.703SCAT A

?Aacz or I: "5RAaot or 1: "« 3

7

•7

3. 000.017957051

••73.-2.22*

BOIPIOT JC.2]

I I

J 05733773*79 3 5)110551*3 6035138139 3 **7<il723i 1

3 57*7171 383 a 0133777V m5338(168 173 0)6 10 78 5 8)0815275561 3 3361 7077 4

3 576025)007 *3*1691*t *

a 35*7713355 a 1 /575803" 5

3567753355 0133757) 35

i 170781706 132581 15* 6

9 31*7290131 10258115b 4

l)f.05«2829 1 119)115 1

1 7 1760333 0C1 273730 <4

1 0»37'. 0283 3 3057217)0*41 • I* 79 T6 SI 9*17*371 1 1 6 5 2 ", 3 7 ? 1 059859431a J»?27?37?» 3 33*.34 333

101 13637 7*

112

APPENDIX E

ANALYSIS OF THE RESIDUALS 105 mm

Method 5 1 st Iteration

m» rSITIT

'0.0001963393137 0.0001113747003I

2011. '1170ftIPS

o.8?as?Q7ia.SCAT A

hahos or Xi "5 o

mace of ri "« a

2

•2

SOIPIOT 4C;21

I

I

After 8 Iterations

:rra rtxrtT

"a 0QJ*1UJ?)J1) 0.000111191097.(

2011. 130781tfS

0.826117787SCAT A

BAA'.t OF XI *1

HAAGK Or 1\ "l 2

30XPLOT JClJ]

3 6010 1 Dl*l

J

9 33«8&7]07«MM.'jnn *ft7)<i17J«)55HS1 7-65 «}$tJl5>J0S*5'-*..' i940Cl1 j * M ) ,

. 1 -. (a

? WJM'liJ 1* ) 3", 9«735'-e2J7?^81J3«» 1 k l (. 6 7 i t ; »

133CJ0J lit a 1 7 1 ,i>7iCJ«1 6 - a i a 9 1 :

1

07 318 jjttsa718191330?

1, , i . > i -J J

382130311 2 1 1 17 J 1 3 J

t 05 3:i« ia«* 1 J77 "0,7'. 7

1 os i; )i ia<* J 0?5si777j:i1 <n*10CU<t39 ajs«i7?r)2a1 o9oi: i sa* J 877 iCt<i "'l*

S7>.0713373 ~3 3 4-: OS 1.7711701«*2»» 3 3a77576o88

06 3ti<.. ,«0W?

Method 6

iTia r3STIT

~0. 0001873 37107K

2073.217SUKP3

0.730036177SCAT A

meet or ri ~s,7<.VCS Of 7: "l

1st Iteration

O.0O03762S95S91

30IP10I xC;21

After 8 Iterations

ITS* ?atTiT

"3.0001830392895 0.0003773 759877K

2070.13976SPS

0. 7873479875SCAT A

RANGZ or I: *S8ARG5 Or /! "3 2

aoiPLOT ^C:2]

I

o oJ

s: -c ;:-*- 396 2953314 9

Hicnnor. S09 39 J37-.ll

717S0QM56 27335 1S76-03913)75511 ::Jb33.37)l

3 S -.756610 11 J noi) 1!?'.?J 1 ]*>iac<*3<t1 9 2 ". h 9 2 r. <; 2 7 8•3 1 39l06<*2<*9 7571S11?1?1 77*. ',30167 C9C16993101

7370530559 1746<ta 3197312S792GG 1 2 B fi 't 8 3 ; '. 7

99S15-2297 1 S7-J1G .7^.73 3915-42297 3 2 3 7 3 ; 2 3 9 2 7

1 - 1 J068678 7379123n77I 1 132<iO*S 3 6?'* 5277733

5331.6 57T.1 *2 9<./„>0711295739891 *3. :58>.W5127

0. 1 3 72*212-.7

Method 7 1st Iteration After 8 Iterations

UTIT*0. 00051 11101181 0.0001363981118

r.

1953 .360326.r Pv

0.771IH66206SCAT A

mr.cz or i, "5 o

/M7Gf 0/" fi '3 3

fOIPtrtT A( 12)

2

/rrr ,»

«.:r/r 0.21 267706010.00O502222O2 »3 3

.30", 3 db i 1*0 .5068*735 7

X 1 .7680051 761991 .151271 . 927690001 3

W9J 0.11961378330.7633 6 83214 0,19 no 395 6"*

5*"iP 4 . 1 37307555Mjreje *ir /. *s 0.1 021O755SMVC* Or* 7t *J 7 1 . • ?•> 778221

• • 0.3fl"13667Bt»• *

• 2

"0. 75153 6 7136"0

. avi I 91 78662 • »2 • . 971 1 91 7866

1,516 4668 HI

*3. V,«03;i>iU2 • «

.6005 100639

. 1 80k6365<i6

10IPLOT it i2]J. 7 .

oo

1 ,1

•

1"'•1

"l

1 . 1 78SGP1091 .9:96738060.351 75-7018

_ . 1 5 1 n * t 75~9.05 /?08»5086"*0

. 77',?fl«. 7*010. 7*5583 117*0.1 ) » « *1 J 5 6 "9

O.lMflVJlOt*?0.1M 8H11987I .81 21 81 6710. 3tkT51 1 ) 1 fc

. Jt.»85 11 1 1 6*0. 5*.17917517"7. '.iv.,.("0.1 75301 1 901

f

113

APPtNDIX E

ANALYSIS OF TILE RESIDUALS 105 mm

» a • «• • i • a ' a~v " « »~ ri i a • i i 9 i • * | i •.• •.„!...J...!...;..'.5.r.«.:-!...!:..!...5—J.-.'...«...!. ..2. ..! ...1

u

Hothod 1

— • - •-

* *•• ••

* • *

• •*

• -•• ••• ••• . •

*.•*.•• •*

• ••• ••• ••

•*)• • • •

••• •••*•• ••« "•

MU*2

-0.99*

Sample Size 26

__-.••-

0.83

, „ _« 9 a a.„a__»,„2._2,,_J 2 I I 1~ - -—-—

-

—P*

Hothod 2J

:

- Sample Size 26 *:"

5- i -\- o-"l T"i""'0 "'« "a" "j>~ i.

oii

—

* .•• •

:n

-»-— •

»

* a*** . •• • ••

. M • ••

•• *,••4 « * .• _• •• • . •

• *• «,*>

••• .•• •* • ,•

)**» "-" • — * *""

-1.70

««• • .« •••• •« *> .• ••*•• •.» •••

0.94

114

APPENDIX £

ANALYSIS OF THE RESIDUALS 10? mm

M f,_:,..:^."..."i...i- —'- »••'• —•-•!•• — •*"

MaUwl 3 Sample Size 33 ::•

ii

• •••

41...

... ... ..

—('"-

i...

s .,!.-.!. .-5.."?'. j a • j • a a a t

* • • •••• • .... m

*"*' •

•••'•" .

• •• I

HI M :

• »•»>* .

m in i :: :;:••• «•• — — *

••• •• t — •— t.<

..' «« •**

-

::: ::: ::: ~ :!:;

_:." :" ;~ ^:!

1.1. .CI* I- ..---••.*• :;.__..:.":.-...—.

—

.'.'.'•

•1.69 0.97

**™t* a ' a s"' o" a "0 •»•* o * .,*, j »_j »;";" ,r.. .», /,»7~?:::*:::»~l-.r? /~!CiCr?Z?.--?.--".—~!

Method &

J3

41

at- .

.

-3.68

Sample Size 33

• .••

.••• •

a** • *•

•

••* • . •••

**•• ••• •••••• •****• ««• ••• •«

>»•-*•••••«

1.42

115

APPOJDIX S

ANALYSIS OF TRS RESIDUALS 105 am

i o * » e • " a a a • 6 * » 3 oi i i » «vi i"

i""«. e * ' o— a- 'i i "i«»«••»••

Mothod 5 •*• Sample Size 33 • , ...• A •••* , •••

. •

«

. p • •••

• *• -

• •••••

is:—i— i -i-

-3.05 1.65

l e • a o aosOQI2>«2tlia-»4oi iitoeii'istf

.lit

Method 6

L

r

Sample Size 33»• . » •• •».. • • • •

.*• . • • •• .

.«•••• • •• ••

-i.80'' * — i

1.77

116

APPENDIX E

ANALYSIS OF THE RESIDUALS 10$ mm— —— Cs.* —

j~j

l

i

j

. --— 1—

.

Method

7i

1

i

i

i

i

1

1

i

1

i

!

j:

i i

I

;i:i::i:i

1 1

v « • • • ft

-

II1 '

* •

» »

1

i

|

!

i

1

,

*

:

•

i»

1 !

1

'

i

ii

. t » « » - * *i

; !

1

! j j j a.?. > a k>u > a. ii r r > t i. i ? i i- * a * >. : > a. r > i > > » i > 5 i 2 J I s J i ?:>. v. -

? i

I

* 4 * • i> < *a-»?fi *«*.**•«•»*** >

!

^;

:

*

1

I

!

1

J * t : « . - c • » 3 -i --- > 1ft < * ft <L »«-»*«**«*<w-»ft

'

• * • »«-»«-C«-fr 1 s ft fc- 3 *

1

1

! i.

I

j i

vi > » r * :-v~*#*-* ** »«« i »* » *-* *

f

j 1

i i

1 :

1

: i

|

•*:*:!

| i

• 3»>*>i»3i

!j

1

; ,j

i:

j

> I

1 *-'

1

;

J

1

i

;:

!

!

i T'

!

t

i i r

1

1

i :1

, !

i

1

,i

i

i1

i

i

1i

1 i 1 1

**1

1

.;..'.. J.ii

1 -o «r T -tJ

O T O

—

117

APPENDIX F

ANALYSIS OF THE RESIDUALS 155 mm

Method t

Method 2

'o.ooojnj-aao:* o.oo-waistoTi?r.

.»8b20S1S91

• • •• 1

• • 1

'1

1

BOXPLOT At (J]"1 0.1

11

-

0.1057796123i REStflUALS *3.131*073B63

3.1783501751"o ,3ao5<i***ll"3

. 70*81 01 855. "a. 1101081006

"3.-1 *05*t 886,60101 *7039

"0. 0531 316*1792

"0.1306173*29.1 8510*369

0.32*0777187"l .0*530377"l.l358**lS19

.90O56"1632"fl ,61 6758*5*3

nrr.ffrs

PtTA"0,000795 3 7 St 311 0.00 03*68153*7*

K1386. 06*071

0,1357323*11sr * r a

stAtnr r>F Xt "j o

1ABGK or X: *7 1

BOXPLOT Al.71

J0.1650.0070. 337

"0.233"0.-13"0.053"0.2670. 363

. 0070.007

. 6680.376

*1 .53"*

"0.073. 330

"0.587

1 *?*7*21 87-886971 3167197*56813627389303815

01 6*96561*3579*9*21 3505K?03*97l721172755737531G7*0l*3».3tl76067U261755197*

Method 3Method k

HETA"0.0003372*75178 0.0007Ba57 9*376

K3965.181201

EPS0.865C902939

SCAT A

ttaanf. "p li *J o

"A/Mr IP T: *2 1.5

fiOXPl"? A{ :2l

1.

a PC5TSUA13U.lf,51071*3.0010<-*31'«M T

'O. 2 170 V.I.'. 11

*0.J5 5 6M*f<«.5"0.15 5NM*<J&5'0 .0177*11 / 777'

. 3M I 3 * 7 '• 3 •

0. 11 V3 »05l 1 J

0.11 137«511

2

0.1151705117

.070197700*3

. 071 21.1001 31

.t^732«10755

.0*91^3231

.5*137 1255,5*0371 J55. 01 50O0»*6i*.n7 M1601 *

,*31*350O96

REGPSSBFTA

"0.00017*231256 2 0.0003 0993558*7

r.

2673.1**519FPS

0.12819267*SCAT A

pi.nr.z cf it "s o

RA/tGS OF Ti "3 2

BOXPLOT Alii)

\"|« o •

11 7

0.25t67675561 .71 *761 6116

"t .5870567350.31 50*87101

*0 .0711403 311»0.1'.3lt131050.0*56038036"O.01171761J931 .762*425*0.1 3503770*10.7*516)330*1 . 37730167 6

0.5197'. 0571*"0. 5877*1 1*77"0 9*833860010. 1 60537 773 7

*7.7693«IM 7

"0. 9*173460091 .35*1876*

"0,05787161535

118

APPENDIX F

ANALYSIS OF THE RESIDUALS 155 mm

Method 5 1st Iteration

tSTIT"0

, 000 1 fll -I 6S 01 0.000116»994t<ll<

1*71. 110)711

0.1197111191SCAT A

kahcz or t\ 'i o

Miter, or r> "i i

I

• I

I

.. i

i-i-lI

BOtPlOT J fill

After 8 Iterations

/rrf rrrr:?

~o . ooo i* ; ionsl .00011 t}$lt90lr

1411 '.'UMr.r-s

0.811671992SCAT A

flj.ijr or t\ "1

.»</!.;» or 1, "1 1

tOlFLOT <t ill

0.3S973630.7771671

*1 .ir.l 91770.11 flt ?f

1

*0.0?t79.-*.

q. 9«i(l]'. 110.011

'CO.': 7111*1 . 71')0 n-n0.1 tOtCQlGo.?m:i\fl .-40H971CC.61 1C9711

"0.17C06B71*0.11 21796-0.1 7H»aa7s

*1.7S1615i.t*0.9'.21796«1 .1CUX2067

'O.0777J16*

117111

1 91 A

caar.aai

Method 6 1st Iteration

ITER PPFTIT

"0, 000 1681 ',49i->a Q. 0002050*99609X

J779.11127S£T.S

0.76*670155

5C4f 4

M.tf;r of iiRAItCC OF T:

After S Iterations

TTF.lt PBETIT

*0.Q002G3 95 62133 0.0002066715375Jt

370*. 333316EPS

0.76557*0619

SC4 7 i

.Mc/cr or r, ~

fl4TG,T OP 7:

0.2*613.20?

"l .6270.310

"0 .n»i

0.0!0~0 .071

1 .30'

0.*50"0.731*1 .1263.010

"2,313"1.1263 , 9 Q

"0.*3l

72277875 205532*3?77203 0313

2

13037 CGI!

7333**571 ;71321 81 81 1072 5 7 9 7

001 1 72*»Q )1 *9S6 ».

* '» 7 3

1 77B87690295J83927971.039913*961*3S2751621976330559

BOXPLOT A{ i2)

BOXPLOT A( ;2]

Hehtod 7 1 st Iteration

ITER PBF7 IT

"0.0003 3*8*855 5 8 0. 00033341 966*4X

2532.616 5 82EPS

0. ••.51677629jcir 4

M.TCff OP 7i *3 2

I

02*-»I

I

(

I

I

I

ROXPIOT A\ |7l

After 3 Iterations

ITER P

BET IT"0.000*255590866 0.000:6591*833

21*9. 1*55 3iEPS

9.8538*33369

SCAT A

RABGF OF It ~iPA/tnt OF 7l "l 2

antrwT At ,2 J

1

• I'

.220*88229*

"l , 77301033 7

0. J977f. J7W3"C. 36517073 f, 7 6

0.7771*', 178*0.01-.560U 121 7

"O.07(,i ,33'i«52

1 .5a972-fc'i2

0.1233 I >-*17?O.201H716I.31t . 23»3-« 1710.tl'8H.3?7»

"i3.5'«3-37'.77ir*1 .0787'.fl»r,9

. 1 3 1 6 1 3 I 7 1

1

"2 . 7 7 2 t 7 7 3 8

"l .07 < 7 « I •• -f, ?

|,..'-.|"iOM

0.07703081 271

119

APPENDIX P

ANAUSIS OF TOE RESIDUALS 155 ma

iiz.iei«t.ot°e*B0*'i 7.?"

*. MJ,

Sample Size I6... -•• ••

• «•

* •*•

••

-•—• ••• c

• * a

? •?

—

•••i

1 I

-1.04

;: tiii- ::;|

0.90

—y- »- ^— a—o-- 8- '

3— 9' " I

""9""I <t~ 1

—1 3 1 "J"

Hothod 2Sample Size 16

4-!

•M •••

• ••t ••«*•• •••••• *•«••• •••«•• •••••••*.

••• ••• •••.•*• « * ••• •#•.• *

«•• ••• •••.••* * * ••« ••*,••

• • • * •

••• «•• ••*.••• ••• ••••••• ••• •*••••• •»• ««*.

...t

-1.89

*..._.« •«•I I

0.93

120

APPENDIX F

ANALYSIS OF THE RESIDUALS 155 nun

... 10 »~ 1 6 «' 5"" >

—Kottwd 3

Sample Size 20.

.

• * * * * •

— •: - •.14

mm

• — - *•••* -

—

jjat

:!— _ . .

-,„

!fi ••• •• ••* — - «••*.»• .•• »••.«• .*• ••••••• „• «*...• „•• • ••«.

.

••«_• *• •J

-M•••

:: —*•• •

-it •.* — — ;• ••

•*

- ••• ».* **• ... - -- -- —

•

.:: ::

•i

:'.—..

.. ... •••<*-• IZl z

:: :- "*••• ••« •

;; ;;; ";••• ill *••• _„ . ... ••** - - — ——

—

.« .a ... • •"

"*••• _ „ — .

;:.;'.:.—„::::.; vi—:—•."*".r_-rr;^r-"i

-1.54 1.06

©' 3 3 10 9 1 9 3 " t ' 9 3 V 1 43 i e" ~o 1 ~c "J ""i »" T*

.19

Method 4Sample Size 20

.»« •••....— »••»„....

Is::::

( * - • »

-2.77

i tit trt ••• ••*»

1.78

121

APPENDIX ?

ANALYSIS OF THE RESIDUALS T 55 ma

-oo i e • t- ' • '

'

.„;...?...^.?:..»...^.g^..^^~j" MuthoU 5

Sample Size 20

ji •••Ml

i* •»

• |...

• ••

1 •**

** • •

*« •*• • •• "

*• •.• • •• . ...».- - —** • , • •» * a

•••a***" • i i . ii —— —— — — —— "

• • •. • • »• •

*• •„•*.«,••*,»*••".•••»•••••**«.*••• a

•J « • , » ••• ,

.. .;. ... ; ... •— ••• n

•:.:]:. :::..:./.::.—_.--;

:::»;---rv^rr :

-2.75 1.79

•i-o- r-»;-a;,< ,"«,' "», ' o o, «

,* 2 2_2_JZJIiL.2;JXL3.--2".-JL.Zl.r.^7.?.72~...TZ12a

—i— a— ^

"

1i

-all

Mothod 6Sample Size 20

ji

»}?•-

i...

J•*•

•••

• ••••

•••

• ••«••

. t • • ..

• ••

• •**••• •*

• •• "**- •**••• •• ••~* ••«-* •• •••• •• •»•••• ..... ••

.

• •• ••«•*• •••... ' * •*•••• •**• ._^^^__„ • • "

t • *

-2.981.49

122

APPENDIX F

ANALYSIS OF THE RESIDUALS 155 mra

!•*'

e -r » « « * <

« t - • e *,

I !

3 '

Hi .

ii i 7l i !

'

I: ! i

I

i I

i

!

i

1

1

« • -> * » * « * p + • » * e • «-**-*»-4-e»e-<

| I

I

II

I

!' !

!

}> I

I

I t

I

I

i T

i i i

123

APPENDIX G

ANALYSIS OF RESIDUALS (METHODS)

DATA 81 MM

av.u*3 0O0C >» ?- U J * 7 a.aVliOlMTjKH 0.1307*61111

K1666.7163(3

£.

I .III'lliU

o.7»:<.tio73<*5

."<r <

»<»<;/• or r : "i.s j

I

• I

I

I

• •!33 I

32 3 3

731

7>.?7u«t *3 Jt •,< J?l 1 9*

1 177 71 ?>.*a 7 o p ) j 1 7 a ^ o j i

1 1 n a ., * T ? •. : >> r. i 1 n a A it ] 1

7 • 3 if ) 7 1 », 1

1

0iM*>37*J«t«>.7 J0lM» Jt 3 1 /:C713i ?7OtOTt-l ti 7"OG 3 ii'i«»l I 7 Tin01 37 ?? 71 88$ t 17 l<Ji.«7<lk

I 0S7* 0OQ*0 1 ; . i ' •> '^ : o 7 7

1

S?«« jr.7*i«.07 c IIN.'OHMJ7 7 ?;si mo J..7 1 (.?7f.".C

71.11 8-)«*776 »j )1<*0'J?G9*> M S3J1 017* <i">7<OT711 J

tJliU'iO*.!.! "o 7l7S$Ot OSSIt J 3 Mil C* a CM3?i9:mi

'. S07J". 717* 'o 71- 7 J lot. *}6 7738 .'I- - s "r. 71&H7J717Ii*33$3 ,)3 a* *p. * f ;il 3030861 1331 1778 S?r,3f>?s<M

fOXPl.OT Al |3)

DATA 105KM».rCS£S

"©.AGO"!;;! 35*586 0.000*011333073 '0.336775350JT

7163 .867717t-.ar:\<f 3]c

0.77W.57163

0.166676935*

FAACt OF /! "5

^mr or / "3 3

aotflOT K ;2 ]

I I

1 C$73 37730 S)310SSM6C3Sn8l31 »b?M72167S7U71?I385 -

t 3 *, 7 7 7 $ l| 7 C

5 3 .1 P r, r. 3 j 7 3 03MC73S7B3Of. S22SSG3 3}dl7Q7??6S7t0?S3032 "0

>• :m $oi <«a»

3S«7?533SS 3 1 JS T'.OO'WS3S67r-i335S i: «073 7353*5

1 €503*1 :06 a ;-3SflU5Cb70* ??*01M i T7$<i; : $66iofcos^nra i 1 1 111 1 SI 5

1 07i?*.orsa 5 OoS 72*1 730*61 071 3007 r. a JTS?:'< 730*61 M »797tSl *0 ". k ; 7 <* 1 7

1 1

1

6S7S372 *j M78S1M*3 SI J?7 JQ707 "o 3?Vlt»3tO3 10338CO771

wryl l 5 5////

aftilPS

'o.ooonnis^ii o,ooo33ositci)i o.oas.?6ajs«o*.jr

711 > .lltl'iSo«/>rr*f3]r.

1 .06TC0127Jrrs

.-:<* r i

"J.»flflirr.nr At ,? I

•

0.177S. 1 S 7 7

•l.iMI. ?»n

j.ofli.. 'iC*fl

"0.070'*».0M i

1 ,77| J

,0(. » >

t.llM1 .)

.S

' It"

. '. I 1 »

*«.i*.»

1

0.1737"j.'ll J

"O.*07)1 ,17(1*

*0.4«1

$

Srt^TlQ1710Ibill

»11 I

at H •I

' ', .• H $ C

11 *30J( *.%! ••»*

(.7m1JI11S$«1 -*$0*

71 nn; - o / 1 j

7«.-Cl J

I CSOli7U 7rCI* ft 1 37M (, *71IOC4I

124

APPENDIX H

ANALYSIS OF PARAMETERS K and e

Parameter K

DATA 81//// DATA 105MM DATA 15 5///-/

Method 5

SCAT KYA'l'S OF Xl 8

a-::; or t: 1577.9 1573.3

SCAT KYAnr.r or Xt a t

i.lr.r. OF J\ J0K.7S JOK.SS

• • •

SCAT r.v

PA'lCr. OF X: 8/..VCK Of r: 2S21.5 2623.5

Method 6

?A1p.a:i

l

I

1

SCAT KY:r or X-. a

;r or r-. 1700.5 i7o<t

SCAT xvraxss or X: 3

XA/IC-r. Or ti 2070 2073. 5SCAT XV

RA:icr. or X: 3

PMIGF. Or /: 3700 3730

Method 7

SCAT KY.V.'Tf *r X: 8

'^VT 1- Of r : 1JC0 Into

«X//C£ Of X: 8

/M.7CS Or It 1955 1995

5C^7 XY?Ar:cn or Xt o a

ta/tg/: or it 2300 2550

« « •

125

APPENDIX H

ANALYSIS OF PARAMETERS K AND e

Parameter £

DATA 81 MM

Method 5

SCAT EPSY

USGT. OF X: 3

IAH0E OF I: 0.8162U 0.8163

I

!

I

i

i

i

i

i

i

i

i • • • •

i

i

i_....—.—.— ——

Method 6

SCAT SPS9USSS OF X: 8

Xt39F CF 1: 0.6086 0.80e8SI

I

I

I

I

I

I

I

I

I

I

Method 7

SCAT P.PCY

Hi*.* if x: o a

•A!-:* nr T: 0.85H 0.87?

• • • • •

I .

I

I

I

I

I

I

I

I

I .

I

I .

DATA 105MM

sr.it trsrannex or it o i

tAncs nr i> {.inn urn,

SCAT EPS

7

rauge or X: 8

RAI1CE OP I: 0.78 0.782S

SCAT FPSYttAxc.r or I: 8

r.Axcr or :-. o.76i 0.77s

DATA 1SSMM

could not beplotted,change of£-values tosmall.

SCAT P.PSY

RAnnr of x-. o a

RA:/CE Or I: 0.76U6 0.76S6

scat rrsvMilan or X-. o a

fiAiicr. or Xi o.m o.»6• • • •

126

APPENDIX H

ANALYSIS OF PARAMETERS K AND e

*o m m CO CT. m OS MCO r-i m (N CN lO m cn

U 00 CD CO CO CO p* CO CO

P. o O o o O o o o

in cn ** H *» U3 <*» m Ul cnH p* u> CO *» fN o t CO CO

en O f-4 «H ^ rn CO 00 oX 10 U3 m C4 m *r Cft r* .H

rs CO vo p* n o ^ r*•h r> cn vO \0 p^ m in*n <n CN <N <N m (N *N

p* «* m o 00 «N on CO<N cn (N n (N CO *a vo

(J P* CO CO CO CO r* p* CO

t= O o o o O o o os

no r* «* r-t in o CT\ »H r* Mr-* h o in CO o *n in \o CN

H CN F* ^ <T» ^H H CO P^

X r* r*> p* <J\ *t O M *n Orn H p* <N •-t P*> cn tar-* CN H o o O a* rHn cn fN <N CN CN ^H (N

rH CO PI ^ U3 CO M CNl/l .-t <n i-t .-H o P» a\

(J m r- vo CO CO CO 00 r*

M

o o o o o o o o

00 CO .n rH 03 p* r* CO m <no <n CO <n CO o on cn 10rn r* *r <3\ CN p* v r» M

* *r <-t «* i-t CD o o VO M\a .-i p^ cn r* o 143 >0r* CN *r m m p*» <n m** CN CN <H ^H ^^ rH H

K i-J

-iC~" -^C~"•H —1

a* a ^a. ^ ^^c c B c e k

i c -.o o o u *w ^* -j x, ^-* <-+ hH •H -4 •H O "H > -H -—

•

** -H a« cn Ul 01 ul «-• CU ~i O. 1) H 1) -H 71 >• 1

Ul rj 01 Ul Ul Ul i jj 1 > i-4)CN c > ^h IrM C ui J3 •-*

0) -H rj (N rj cn o C »H TJ r-1 -H •H Oj —IJH U U u u n — J-l 4J t 4J + ui e -HC7t<-H H •j> rn mTJ a^ ^i H u H a id (Ti Ui iHtC4 B(J) C H a o OJ o D U rHlfN B u ^|CN C u WH Ul -i • D

TJ W -H M -G Ut -G i-i H i) o •^j ^i H H *M -*

O **-» -3 *j 4J >. + + jj '"'^— - —** u ~^— .——tr Bt. a Ul

X "O O «l 73 O 73 O T3 4J TJ c H C C •H C c 1 •H ^S <oH

4J <U J2 GJ ti 'J c <U -H -^ a O , ~H O ml arf —* co u c o 4J 4J U ^H ^> w r-t '>» •O -H . CN 73 -I » ****N«^_ 4J Ul ^£." J- O i-i ^: Jh .c u JT -H *- " rj Ul ^^-——"*

Of ul IO cc CD U> |o ^> x: *J

<?-H H pp i> ry» oj t?.Q X G U1 w c *J Ul (HIM ^> -0 -H hH d 1 G•H (.» 1 H * J •H 4_> ^rjo «M H 1) -v* -c a) lui II ^ a* lc/) a •H « o*

V <U rj ol 1) Ul 9I.QU a Ul tr> ui rj» Ui!V -a ^,

3 .h M 3 a IS 3 3 O II ^ Ui u H CT» D O •w rj» it o 3 o x |

coo c —

*

(.' "H3 a.a >i

c D at a) n 1J v M C M3 au >*3TJ N J o 3 O 3 U >i 3 u N V) 3 ^ >i CO

m

x: ^ CN rn * in id r* CO4J01

i.

«-4 T~ V" T- T" ^~ t* T- ^i-O

q

• • • * • • •CM CM CM CU CM CM CM CM

127

APPENDIX I

API REGRESSION PROGRAMS

NON-ITERATION METHODS

V BEORESCl] Z«- 36 2 pi

T2] Zlill+U[3] Z[;2]*-7[4] zt-<-§z

[5-] BETA-{ (HCZT+.xZ) ) + .*Z?)+.*Y[6] K*Cl)*SETAZll[7] £PS«-#xfl£T/tn2 3

[ 8 ] YHA T+ ( n.?T.4 [ 1 3 x £/ ) + ( Bff2M [23*7)[9] RES+Y-YHAT[10] /I* 36 2 pini] d C : 1 3+J[12] At ;2]*-RES

ITERATION METHODS

V ITER P[I] ST<7-«-l

[2] 7«-(©( (/?+(0.5*(l-P) ) )* ,v) )v5I<7

[3] U+UOiSIG[4] y+-70vSIG[5] Z-*- 36 2 pi

[6] Z[;l]«-0[7] ZT;2>K[8] ZT*-§Z[9] BFTIT^i (SKZT+.xZ)) + .xZ2,

) + .xy[io] k«-(~i hbhtitTi 3

[II] EP5<-K*BET'ItT2 D

[12] YRAT+(BETIT[1 ]*UO) + ( BETITl2~\*VO)[13 3 RES+-YC-YHAT[143 /I* 36 2 pi[15 3 4[;13-YC[16 3 4[;2>-ffff5

REMARK: GIG CHANGES IN ACCORANCE TO THEDIFFERENT ITERATIVE METHODS.

128

APPENDIX J

r

s ~

til O"1 f^<.»

u h-

u. M

^r

HISTOGRAMS OF SUPPRESSION INTERVALS 81 ram

II

i

II

• « •

• « *f • «

a * * -- B -

t> < t. a c rB 6 v fr ' *

•4-

-o

* it »

Ir fi * " t- >

• c c e c• * « IT »^«l I. f

l» 6• ft *

» a i

» = I

* I

* *

I

• r»»l • * » o

« i H' ' i i r

*

« r >

« a o e- » ft -J ft i C- *» ft ft t ft » »

' i >" i :' > t > >

I

> 5

»Tt J -'t !*>« J* ;;< . f i . u • j r »

ft— <v»•#•? i .

i 4 * i - « r : , « - •

129

APPENDIX J

HISTOGRAMS OF SUPPRESSION INTERVALS 105 mm

i

i

OlI

I

o,

I

O I

I

I

+O I

I

I

Z i

i'\I ^

I

1

I

i

I i-c

B- * * * C *• C "J Cfl * c « c-

C *> 1 <.*««» £• * 4 « # « *

t ' -. C < t? *

« B ** -0- ** * *

tf a r ; e ft c £ »«*•»*ft « * «

C « { >i 1 M S"

# a a e r •

*.t « • r- t r (• -a

* < t: * »

a r « * »* K K * & „ « «

—•Oj

>>. > > > ? X •* * .

i.' t ^ . i - < a -a

c o«w •- r « a » a-

« e c c » \

9 fl <: i -." '

ft 4C J t * J

: » C » o cr j a u « *f*

1 W I

" < a - . a <| a - • v .'->-* j •

-h t

I

*

O I

t

V * > I

H0-«-r»^«.'.j-i«y

n •* * ' i.b<**'Jt.44r. W* i»h 1 * „ >/ c j .. • (.

t J • I -o t « « I •*»•(* a

O |

**«!>»«*

^IT -a

o

130

APPENDIX J

HISTOGRAMS OF SUPPRESSION INTERVALS 155 mm

i :

i

t

I oI •

)— sr'.

i o,

c * c » »

? » O * B •

6 < * *>

s. v* c * »a

—

1

'

J9

3 °'

-1

I

* » « *t n» if *«(•*«

» C- t* -

t 9 <J < P :

i » * # e-o-r a **»«<*-»-'*a(*B--'t» "- -^ e r

****» \

• » * fl t

» « f> f # V rt *

5 C r t 5 ,- c - B

1 fi!

"*!

!r-1

4 I »»« 4* 1 I) I l> ft*

. >. 7^2. x >>».>>>->>. is?: -t >. ii : ii> * > > a^> i:- i >. > >. ? i j > > > >: * * > ? t t

. i. (! * 4 £ 1 * n*«**-i**c(i'

iOi*<i.-*«*«^»a»»i

uj eg |

jl'tlr-ti'B * <- r i « < u o r

o— Psl

"3"o

131

APPENDIX K

PROGRAMS FOR COMPUTING RATES OF RISE

Program for formula (3.54)

FORTRAN IV G LEVEL 21 MAIN

0001 DIMENSION TIM(500) r SLA(20) , RETAU(20)0002 N=3480003 READ (5,10) (SLA(I) ,1=1,18)0004 10 FORMAT (F10.5)

0005 READ(5,11) (TIM(I) ,I=1,N)

0006 11 FORMAT (F10. 5)

0007 DO 1 1=1,180008 SUM=0 .

0009 DO 2 J=1,N0010 C=EXP( (SLA(I) *TIM(J) )

)

0011 SUM=SUM+C0012 2 CONTINUE0013 A=SUM/FLOAT(N)0014 B=A-1.00015 ETAU=B/SLA ( I

)

0016 RETAU(I)=1.0/ETAU0017 1 CONTINUE0018 WRITE(6,12) (RETAU(I) ,1=1.18)

0019 12 FORMAT(1X,3F10.5)0020 STOP0021 END

Program for formula (3.50)

V LAMV[ 1 ] LA+ (LAMS* ( {LAM-LAMS ># ) H (LAM*R )

[2] L4£AKU((1tL4 )-(ULAMS))

132

LIST OF FIGURES

1. Structure Of Fire Suppression Process 12

1 22. Function P(S/6

Q,r) = exp[- - r (1 - e cos 0)] 22

-. T , . 2 -K £n 0.1-i3. Isolines r - •= — 24

1 - e cos 8

1 24 . Function P(K/r) = exp(- — r ) 31

25. Isolines r = -H-2,n P(K/r) ,. 33

6. Summary of the derived formulas 38

7. Schematic Description Of The Model 41

8. Schematic Representation Of The Model 49

9. Functions XJt) And X(

(t) .• . . . 54

10. Functions XJt) And X:(t) 59

11. Schematic Range For 81,105,155 mm 62

12. Deleted Catapcints 68

13. Changad Datapcints 70

14. Comparision Gamma Distribution And Data 83

1 215. Function P(S/6 = 0°,r) = exp[- - r (1 - e)] 8 8

16. Numerical Values For P(S),PK(S), And P (K) 92

17. Rates For The Model 93

133

BIBLIOGRAPHY

1. Abram, H., S. , Psychological Aspects Cf Stress,

Charles C. Thcmas, Springfield, 1970.

2. Ad Hcc Group On Fire Suppression, Report Of The Army

Scientific Advisory Panel, Department Cf Tne

Army, Washington, 1975.

3. Agrell, J., Stress:. Military

Implication §2 Psychological Aspects, Proceedings

Of An International Symposium Arranged By The Swedish

Delegation For Applied Medical Defense Research,

Stockholm, 1965.

4. Appley, M. H. , Trumbull, R. , Psychological Stress,

Meredith Publishing Company, New York, 1967.

5. Barr, B., Techniques For Including Suppressive Effects

Into Lanchester Type Com bat Models, M. S.

Thesis, Naval Postgraduate School, Monterey, 197a.

6. Combat Developments Experimentation Command,

Suppression Experiment , Final Report, Fort Crd,

Califorria, 1976.

7. Cox,D. E . , The Analysis Of Binary Data, Methuen

& Co. Ltd, London EC 4,

8. De Groot, M. H., Pro bability And Statistics,

Addison Wesley Publishing Com-pany, Reading,

Massachusetts, 1975.

9. Gaver, D. P., Lectures On Evaluating, Human Factor

Data^ Unpublished Notes by Mueller, M. P.,

134

10.

11

12.

13.

14.

15,

16.

17.

Monterey, 1978.

Gaver E. P., Thompson G. L. , Programming And

Probability Models In Operations Research^

Brooks/Cole Publishing Company, Monterey, 1973.

Graham < C. H.,, Vision And Visual Perception,

John Wiley Son, Inc., New York, 1965.

Grinker, R. R., Spiegel, J. P., Men Under Stress,

McGraw-Hill Eock Co. Inc., New York, 1963.

Hald, A. r Statistical Theory With Engendering

JUS-Elic^iiS-S 5., p. , John Wiley Sons, Inc., N.Y.,

1965.

Headguarters, Department Of The Army, BM 6-40-5,

Washington D. C, 1976.

Headguarters, Department Of The Army, |M 6-161-1,

Washiigton D. C, 1976.

Katkin, E. S. , Relationsh ip Between Manifest Anxiety

And Two Indices Cf Autonomic Response 1c Stress,

Journal Of Personality And Social Psychology, 1965.

Keegan, J., The Face Of The Battle,

House Inc., New York, 1976.

Random

18. Leibrecht, B. C, Lloyd, A. J., Field Study Of

Stress ; Psychophysiological Measures During P£2J§£*

SUPEJf, CDEC, 1977.

19. Leibrecht, B. C, Lloyd, A. J., 0«Mara,P.A., Field

Study Of Stress^ Psychological Measures During Project

SUP P EX, 1977.

20. Ljungberg, Stressj_ Military Implications-Medical

Aspects, Proceedings Of An International

Symposium Arranged By The Swedish Delegation For

Applied Medical Defense Research, Stockholm, 1965.

135

21.

!2.

23.

24

25.

26.

27.

23

29.

30.

31.

McNeil, D. R., Interactive Data Analysis,

Wiley And Soqs, Inc., New York, 1977.

John

Nie, N. H., Hull, C. H., Jenkins, J. G., Steinbrenner

,

K. and Eent, D., Statistical Package Per The Social

Sciences, McGraw Hill Bock Company, New York,

1975.

Pindyck, R. S., Rubinfeld, D. L., Economet ric Models

And Economic Forecasts, McGraw-Hill Eook

Company, New York, 1976.

Beddoch, R. , Manchester Combat Models Kith Suppressive

HtS And^Or Unit Disintegration, Master's

Thesis, Naval Postgraduate School, Monterey, 1973.

Richards, F. R., Lectures On Data Analysis ,

Unpublished Notes By Pietsch K. H., Mcnterey, 1977.

Sprent, P., Models In Regression,

Noble, Inc., Londcn, 1969.

Barnes And

Swann, E. G., Background And Approach For Studj Of

Combat Suppression, Naval Weapons Center, China

lake, California, 1972.

Taylor, J. G., Attrition Modelling,

Postgraduate Schocl, Monterey, 1978.

Naval

Taylor, J. G. and Brown, G. G., Numerical

Determination Of the Parity Condition Parameter For

l§H£J}J=ster Ty__ge Eguations Of Modern jjacefare,

Naval Postgraduate School, Monterey, 1977.

Taylor, J. G. and Brown, G. G., P£tiJ53l Commitment Of

Forces In Some Lane h ester TyjDe Combat Mod els,

Naval Postgraduate, School, Monterey, 1976.

Taylor, J. G. and Brown, G. G. , A Table Of Manchester

Clifford Schaefli Functions, Naval Postgraduate

136

32.

33,

34

35

36

37

38,

School, Monterey, 1977.

Taylor, J. G. and Brown, G. G., A Mathematical lh3.Q£l

For Variable Coefficient Lanchester l^je Equations Of

Modern Warfare,

Monterey, 1974.

Naval Postgraduate School,

US Army CDEC, Suppression Experimentation Data Analysis

Report, Fort Ord, California, 1976

US Army CDEC, Suppression Experiment, Final Report,

Fort Ord, California, 1977

Weltmann, G., Chr istianson, R. A., Egstrom, G. H. ,

iiijisi] l3£i°I"s# 19 65.

Wiener , F., Die Armeen Per NATO S taat en ,

Verlag Carl Ueberreuter, Wien, 1968.

Yale, W. W. , Mills, D. L., An Exploratory Stud_y_ Of

Hum an Reactions To Fragmentation Weapons

,

Stanford Reasurch Institute, 1975.

Young, P. T. , Motivation And Emotion,

& Sons, Inc., New York, 1961.

J. Wiley

137

INITIAL DISTRIBUTION LIST

No. Copies

1. Defence Documentation Center 2

Cameron Station

Alexandria, Virginia 22314

2. Heeresamt BW 2

Tra janstrasse 128

Koeln, fcest-Germany

3. DOKZENT BW 2

Friedrich Efcer-Allee 34

531 Eonr, West-Germany

4. Department Chairman, Code 55 2

Department of Operations Research

Naval Postgraduate School

Monterev,. Calif crria 93940

5. Prof. G.K. Poock, Code 55Pk 2

Department of Operations Research

Naval Pcstgraduate School

Monterev, California 93940

6. Prof. D.P. Gaver, Code 55Gv 2

Department of Operations Research

Naval Pcstgraduate School

Monterej, California 93940

Library, Code 0142 2

Naval Ecstgraduate School

Monterej, California 93940

Prof. Cr. R. K. Huber 1

133

Hochschule DW Muenchen

Fachhereich Information

Werner-Heisenberg-Weg 39

8014 Neubiberg

9. Ass. Prcf. J.G. Taylor Code 55tw

Departm€nt of Operations Research

Naval Postgraduate School

Monterey, California 93940

10. Dr. M. Eryson

Technical Advisor, US Army

CDEC

Fort Ord, California 93941

11. Dr. D. Mac Donald

Executive Vice-President, BDM

CDEC

Fort Ord, California 93941

12. LTC. B. S. Miller, Code 55Mu

Department of Operations Research

Naval Postgraduate School

Monterey/ California 93940

13. Commander US Army Combat Developments

Experimentation Command

ATEC-PL-M

Fort Ord, California 93941

14. Capt Michael Peter Mueller, FGA (student)

23595 Ecu Place

Carmel, California 93923

15. Major Karl-Heinz Pietsch, FGA (student)

1153 Wildcat Canyon

Pebble Eeach, California 93953

139

Th4

M88c.

Thesl

s

M8838c.l

i Ton^l

Muel ler: 78021

30WAY7921 APR £8

Human factors in

field experimentationdesign and analysis of

an analytical suppres-

sion model

.

2 52803 2 6 11

Thesis 1 73021M8838 Muellerc.l Human factors in

field experimentationdesign and analysis ofan analytical suppres-sion model

.

lhesM8838 ,

Human factors in field experiments ion d

3 2768 001 92521 7

DUDLEY KNOX LIBRARY