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BASELINE DOCUMENT FOR TSPOOND 

I. Introduction and General Philosophy 

The purpose of the computer program TSPOOND (Statistical Program of 

On-Board Decisions) is to simulate the actual conditions of target visibility 

and activity, crew voting and consequent main optics selections for the 

MOL/DORIAN missions. The program will accomplish the simulation using a 

Monte Carlo technique that will take random samples of certain variables 

such as target visibility, activity, scope dwell times, crew recognition 

and false alarm. The distribution functions for the above random variables 

will be input so that studies may be performed to determine the effect 

that changes in the distributions have upon mission results. A primary 

output of TSPOOND will be a comparison of the manned vs. unmanned results. 

The primary input to the program will be the BIT (Binary Interface 

Tape) produced by TWONDER. In addition, various software control parameters 

and output commands will be processed via card input. 

The general structure of the program will be designed so that the user 

can specify both (1) the important input parameters that describe crew 

performance and voting strategies, and (2) the desired output quantities 

including those from a typical sample and/or statistical summaries for 

a large number of samples. 

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Notation and Definition of Variables 

Non-random- Variables Definition 

V 

.probability that a target is visible, 

target specific 

T

(i) Group decision time for group i (in 

machine time) 

At Recommended dwell time designated by 

mission planning 

(St

s 

Acquisition scope slew and settle time 

from previous target 

P

a 

Probability that a target is active, 

target specific 

P

R 

Probability that a crew member recognizes 

activity when it is present, scope specific 

P

F 

Probability of false alarm (i.e. probability 

thae7i crew member reports a target active 

when it is inactive), scope specific 

p Intra-group visibility correlation 

coefficient 

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The following set of random variables will be sampled from in the 

program: 

Random Variable 

V 

t 

a 

Definition 

Target visibility, Pr{V=1}= PV 

P

r 

{V=0} = 1 - P V 

Scope viewing time for a target St = St

s 

dt

r 

Target activity, Pr{a=1} = Pa 

P

r

{a=0} = 1 - Pa 

Recognition, P

r

(=1) = PR 

Pr(E=°) = 1 - PR 

False alarm, Pr(n=0) = PF 

P

r

(n=1) = 1 - PF 

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Monte Carlo Cycle Length 

The Monte Carlo cycle length for TSPOOND will be one rev, i.e., all 

the random samples will be processed for one rev before beginning on the 

next. One advantage of such a short cycle length is the possibility of 

looking at the results from one rev before making a decision to complete 

the entire run. It should be emphasized that this choice for cycle length 

does not preclude getting statistics for quantities over several revs. 

Random Number Generators 

Two types of random number generators will be used by TSPOOND to 

generate the (pseudo) random numbers needed to compute the random quantities 

involved in the Monte Carlo sampling. The first random number generator 

(RNG) generates random numbers between 0 and 1 with a uniform distribution. 

The second RNG generates random numbers from an input cumulative distribution function. This second RNG is needed to compute the random dwell 

times for the acquisition scopes. 

Many of the variables in the program are so-called "zero-one" random 

variables that assume the value 0 with some probability al and 1 with 

probability 1 - al. If Q is such a variable, its value for each sample is 

determined by generating a number r from the uniform RNG on [0, 1] and 

comparing it with al. If r < al, Q is set to O. If r > al, Q is set to 1. 

That Q has a probability al' of being equal to 0 follows from the relation 

Pr(r< al) = Pr(Q=°) = al 

since r is uniformly distributed on [0, 1]. 

Simulation of Acquisition Scope Dwell Times 

The amount of time that a crew member spends viewing a target is a 

variable that depends on whether or not the target is visible, and, if 

visible, on whether or not the crew member notices activity. TWONDER uses 

a recommended dwell time for each target which will be transmitted to 

TSPOOND on the BIT. This recommended dwell time will then be used as a basis 

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for determining a distribution function for the sample dwell time. Thus 

if (St

r 

represents the actual dwell time and 6t* represents the recommended 

dwell time, the distribution function for the random variable dtrwill be 

represented by some function f(6tr). X $.14) 

However, the amount of time required to recognize a cloud is less than 

that required to view a target and search for activity, so that TSPOOND 

will use a different distribution function for (St

r 

if the target is not 

visible. 

Since very little detailed information is available on the shape of f, 

TSPOOND will allow the form of the distribution functions to be specified 

by input so that they may be altered for parametric studies or to better \-\•04". 

simulate crew actions. 

Crew Voting Logic 

One purpose of the Monte Carlo program is to study the effect of 

different voting logics on the mission results. To that end, TSPOOND 

will maintain for each group a voting table V with four columns and up 

to eight rows. The four columns will correspond to the votes of active, 

inactive, reject, and override, respectively. TSPOOND will either fill-in 

the tables from weights on the BIT or will read them from input cards. 

The program will allow the user to specify what numbers he wishes to use 

for unobserved targets and whether or not any targets are classed as 

mandatories. A mandatory target (no more than one per group) will 

automatically be chosen for photography and no voting will be simulated 

for its group. One logical way of choosing the weights for the voting is 

, to let the active and inactive columns be the expected weights of the 4,1011' cs. 

targets given that they are in the reported state, the reject column be 

all zeros, and neglect the override column. This method would maximize the 

expected score. However, it should be emphasized that the program will be 

used to study different types of voting and will be designed accordingly. 

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Probability of Visibility 

One of the major factors influencing the decision of what target in a 

group to photograph is the weather conditions in the area of the group. The 

weather, or more strictly, the presence or absence of cloud cover, affects 

the vote that is made for each target by the crew members. The program 

requires for each target a probability of visibility, denoted PV, which 

reflects the percentage of times (on the average) that the crew member will 

see the target out of the total number of times that he views it. 

Since cloud cover is clearly a regional phenomenon, it is desirable 

to reflect this in the computation of PVfor each target. 

Two types of correlation may be considered, namely, inter-group 

"414. 

correlations and intra-group correlations. Inter-group correlations take c4w4).00 

re

4%4140. 

00.4t.11... 

into account that the weather of two groups close to each other is correlat-eb.Uebv4P 

ed. Intra-group correlations take into account that the weather of two e . 

targets in the same group (and hence geographically quite close) is 

correlated. 

A detailed account of the methods of introducing the correlations 

and the conditional probabilities that result are included in Appendices 

A and B. 

Conflict Analysis 

Although the determination of groups guarantees that the primary of 

group i + 1 doesn't conflict with any target in group i, and that the 

primary of group i doesn't conflict with any target in group i + 1, it is 

possible for an alternate in group i to conflict with an alternate in 

group i + 1. TWONDER will compute the conflict relationships for each 

group and pass the information to TSPOOND on the BIT. The simulation 

in TSPOOND will then remove from consideration any target in group i + 1 

that conflicts with the target choosen for photography in group i. 

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III. Inputs 

The primary input to TSPOOND is the BIT (Binary Interface Tape) 

generated by TWONDER. In addition, various software control parameters 

and the specification of output parameters will be input on cards. 

The BIT from TWONDER will contain the following information: 

1. The path selection for the main optics and two acquisition 

telescopes, including the recommended dwell time and on and 

off times for the scopes. The targets will be arranged 

by group. 

2. The active and inactive weights of the targets 

(1) (1) 

wA 9 wI 

3. The probabilities of activity, visibility. 

4. The target card parameters (priorities, visual intelligence 

only flags, etc.). 

5. The group decision time for each group. 

6. Identification of run and rev span. 

The second class of input data includes software control parameters 

and output specifications. The amount of data of this type is small and 

will be input on cards; the data will also be included in the output to 

clearly identify the run. 

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IV. Output 

The output from TSPOOND will consist of two general types of results 

for a specified rev span: (1) a typical sample giving detailed information 

about the actual status of the variables, voting, and decisions made, and 

(2) statistical summary data derived from all the samples including the 

average number of interdictions, average scores for manned and unmanned 

modes, and the average number of unobserved alternates. 

All of the output will be written on the output tape (SPOUT) in BCD 

and may be printed if desired. The rev spans and requested output parameters will be specified as input to TSPOOND, but will also be written 

at the beginning of the SPOUT to clearly identify its contents. 

In order to ascertain what a typical mission looks like, TSPOOND will 

output, on option, a detailed account of specified rev spans. Included 

in this output will be: 

1. The actual and reported states of every scheduled target 

2. The target choosen for photography in each group. 

3. The scope on and off times. 

4. The main optics on and off times for each target. 

5. The group decision time, group number, rev number, 

scope number. 

6. The mid-time stereo angle, slant range, obliquity and 

photo mode. 

7. The active and inactive priorities. 

8. The VIT and primary flags. 

In order to determine what the most common results and variations will 

be, TSPOOND will output statistical summary data for a large number of samples. 

Means, variances, cumulative distribution function plots, maximum and minimum 

values and percentile information maybe obtained for the important output 

quantities. The number of samples to be taken, a specification of what 

statistical information is desired, and the rev span of interest will be 

specified as input to TSPOOND and written on the SPOUT for reference. 

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Some of the quantities presently of interest are: 

1. The average number of primaries selected for main optics 

photography. 

2. The average number of alternates selected and why they were 

selected. 

3. The average manned and unmanned score. 

4. The average number of primaries not selected and why. 

5. The average number of adjacent alternates (if no conflict 

analysis is conducted) 

6. The average number of unobserved alternates. 

7. The number of mandatories specified. 

8. The average number of overrides. 

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APPENDIX A 

Intra-group Correlation Model 

1 if group is in a "clear" area 

New ramdom variable: G = 

0 if group in a "cloudy" area 

let PG = P

r

(G = 1) 

1 if T. is visible 

V. = 1 

1 0 if not 

let P = P (V. = 1) v. r 1 1 

The V. depend on G in the following way: 

Pr(Vi = 1 I G = 1) = r1 

Pr(Vi = 0 I G = 1) = 1 - r1 

Pr(Vi = 1 I G = 0) = r 

P

r 

 (V. = 0 I G = 0) = 1 - r2 

The non-conditional probabilities of visibility are: 

Pr i (V = 1) = (PG

)(P

r

(Vi = 1 I G = 1) + (1 - PG)(Pr 1 (V. = 1 I G = 0) 

= PGr

1 + (1 - PG) r2 = P rj (V. = 1) (4) 

Pr(Vi = o) = PG (1 - r1) + (1 - PG

)(1 - r2

) (5) 

(1) 

(2) 

(3) 

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A posteriori: 

Pr(G = 1 1 Vi = 1) = Pr(G= 1) and Vi = 1) 

Pr(vi = 1) 

P

G

r

1 (6) 

P

Gr1 + (1 - P

G

) r2 

Pr(G= 1 1 Vi = 0) = Pr(G= 1 and Vi = 0) P (1 - r2

) 

Pr(Vi = 0) P

G

(1 r

1 ) + (1 - P

G

)(1 - r2

) 

(7) 

Pr(G=01V.

1

=1)=Pr(G=Oariolv. ..1 (1 - P

G

) r

2 

Pr (vi = 1) P

G

r

1 + (1 - P

G

) r

2 

(8) 

Pr(G= 0 1 Vi = 0) = Pr = 0 and Vi = 0) (1 - P

G

)(1 - r2

) 

Pr(Vi = 0) P

G

(1 - r

1

) + (1 - P

G

)(1 - r

2

) 

(9) 

Joint: 

P

r 

 (V.=1andV.=1) = P

G

r

1

2 + (1 - P

G

) r2

2 (10) 

P

r

(V,=1andli.=0) = P

G

r

1

(1 - r1

) + (1 - PG

)(r2

(1 - r2

)) 

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"rid V. = 

P

r 

 (V. = 0 and V. = 0) = P

G

(1 - r1)

2 + (1 - P

G

)(1 - r2

)

2 (12) 

Conditional: 

Pr 1 3 

(V. = 1 1 V. = 1) = P

r 1 (V. = 1 and V. = 1) — 3 

Pr 3 (V. = 1) 

= P

G

r

1

2 + (1 - P

G

)r2

2 

PGr

l + (1 P

G

) r2 

Pr(Vi= 01 = 1) = P (V. = 0 and V. = 1 

r — 3 

P (V. = 1) r 

= P

G

r

1

(1 - r1) + (1 - PG)r (1 - r 2 2

) 

PGr

l + (1 - P

G

)r2 

P(11.=01V.=0) = P (V. = 0 and V. = 0 

r 1 r 1 - 3 

P rj (V. = 0) 

= P

G

(1 - r1)

2 + (1 - P

G

)(1 - r2

)

2 

P

G

(1 - r1

) + (1 - P

G

)(1 - r

2

) 

P 

r 

 (V. = 1 V

3 

 = 0) = P

G

r

1

(1 - r1

) + (1-PG)r2 (1 - r2

) 

P

G

(1 - r1

) + (1 - PG)(1 - r2

) 

The generation of the visibilities V. for each group is then accomplished 

by the general method described in Reference [1]. 

(13) 

(14) 

(15) 

(16) 

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APPENDIX B 

Inter-group Correlation Model 

PV 

(i + 1), will be determined by the time between their respective group 

decision times. A quantity tmaxwill be input and represents the time 

at which the inter-group correlation falls to O. A linear decay is 

assumed so that 

if t < t 

t

max 

p(t) = (1) 

0 if t t

- max 

where t represents the time interval between the two group decision times. 

One constraint must be applied to (1) that is inherent to the problem, 

P ::) P (i 1) (1 4. (11)-

(

:V

P) 

V V V 

P (t) min (i) P (1 - P V V V V 

Thus, the correct definition of p(t) is 

p(t) = 

min 1 - t max 

t P (i)(1 - 

PV(i 4- 1)

) 

P (i+ 1)

(1 - P (i)) 

( 

✓ V 

✓ V V V 

if t< t

' (i + 1) max (L - P (i))' P (i) - P (i + 1)) P 

0 if t

- 

 > t

max 

Once this correlation coefficient has been computed, the group visibilities 

will be computed by the general method described in Reference (2). 

The correlation coefficient between the probability of visibility 

for group i, PV (i), and the probability of visibility for group i + 1, 

t 

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REFERENCES ( 

1. P. L. McEntire, "A Method of Generating a Correlated Set of N 

Zero-One Random Variables with a Specified Mean," TRW Systems, 

Sunnyvale. 

2. P. L. McEntire, "A Method of Generating Two Zero-One Random 

Variables with Specified Means and Correlation Coefficient," 

TRW Systems, Sunnyvale. 

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SYSTEMS GROUP 

INTEROFFICE CORRESPONDENCE 

TO: T. A. Magness CC: DATE: 21 August 1968 

„„EcT, A Method of Generating Two Zero-One Random FROM: P. L. McEntire 

Variables with Specified Means and Correlation BLDG. MAIL STA. 

Coefficient Sunnyvale Office 

The purpose of this memorandum is to describe a method for 

generating two random variables, x1 and x2, so that: 

(A) P

r

{x1 = 1} = 1 [P

r

{x1 = 0} = 1 - p1

] 

P

r

(x2 = 1} = p2 [Pr{x

2 = 0} = 1 - p2 

(B) P(Xl, ) = P 

The values of p p

2' and p are numbers in [0, 1] subject only to the 

constraint 

min 

I /p1(1 - u2 ) /u (1 - u ) .2 -1 

I P2(1 - Pl) ' I P1(1 - 11 2) 

1 

(1) 

The constraint (1) says essentially that if pi and p2 are very far 

apart, then the correlation cannot be very high. 

The value of x1 will first be selected and then the value of x2 will 

be selected conditionally so that 

P

r

{x2 = 1 1 xi = 1} = P1 [P

r

{x2 = 0 1 x1 = 1} = 1 - P1

] 

Pr{x2 = 1 1 xi = 0} = P2 [Pr{x2 = 0 1 xi = 0} = 1 - P2] 

SEP 1 b 1968 

SYSTEMS 1100 REV. I-017 
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From condition (A) and the definition of expectation 

E(x2) = p2 = pi Pi + (1 - pi) P2 (2) • 

Also the variances of x1 and x2 are given by 

a 

2 

= p1 (1 - p 

xi 

a = p2

(1 - p2) 

x2 

and 

E(xi x2) = P1 P1 

Hence the correlation coefficient between x1 and x2 is 

p(x x2) 

COV(xl' x2) E(xl x2) - 

 E(xi) E(x2) 

a

X1 

a

x2 

a 0 

X

1 X2 

p-, P - P P ± 1 2 (3) 

J - P1) P2(1 - 112) 

And so from condition (B), 

P - (4) 1 1- 12 

/P1(1 - 

p1) P2(1 - P2) 

Thus the problem has reduced to the simultaneous solution of (2) and 

(4) for Pi and P2. Solving (4) for Pi yields 

(5) 
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Substituting P1 from (5) into (4), 

P

2 = 112 1 P p 

11 P1 ) P2 (1 - P2) 

- 1 

The solutions (5) and (6), however, must fall between 0 and 1. 

Subjecting (5) to this constraint yields 

(6) 

u

4 1 (1 - pi) 11 2 (1 - 

1 

+ p (7) 

2 < 1 

2_ 4 1

(1 - 111

) 112

(1 - 112) < 1 - 112 

1 

- < 

P

1 (1 - 11 2

) 

/111(1 111) 112 (1- 11 2) 

and finally 

which is one-half of the constraint mentioned above in (1). 

A similar computation from (6) results in 

- 

//11(1 - 112

) 

P < P2

(1 - W ) 1 

(8) 

1/1/2

(1 -) 1 

P < P (1 - u 2) 1 

(9) 

Thus (8) and (9) together yield (1). 

P. L. McEntire 

PLMcE:ne 
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L, 7 

SYSTEMS GROUP 

INTEROFFICE CORRESPONDENCE 

To T. A. Magness CC: DATE: 19 August 1968 

SUBJECT: A Method of Generating a Correlated Set of N FROM: P. L. McEntire 

Zero-One Random Variables with a Specified Mean BLDG. MAIL STA. VC 

Sunnyvale Office 

This memorandum describes a method of generating .N random variables 

x

1, x

N such that: 

(A) Pr{x. = 1} = x 

P. {x. = 0} = 1 - x for i=1, 2, N 

r 1 

(B) p(xi , x.) = p for i+j' 

The values of x and p are arbitrary but fixed. 

First an auxiliary zero-one random variable C is introduced with 

P

r

{C = 1) = P and P C 

r

{C = 0} = 1 - PC. Then the distribution ,functions for 

the xi are determined conditionally as functions of C so that:', 

P 

r 1 ix. = 1 I C = 1} = P1 (Prix. = 0 C = 1} = 

1 

Prix. = 0 I C = 0) = Pi (P 

r 1 ix. = 1 I C = 01 =1- P1) 
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The method consists of determining P

c 

and P1 so that Conditions 

(A) and (B) are satisfied. Condition (A) implies that 

E(xi) = Pc Pi + (1 - Pc)(1 - Pl) = Tc 

Now the covariance of xi and xj is given by 

COV(xi, xj) = E(xi xj) - E(xi) E(xj) 

where 

E(x k,) m P

C P1

2 (1 = PC ) (1 - P

1 )

2 

and 

E(xi) = E = PC P

1 + (1 - Pc

)(1 - P1) 

So carrying out the algebra, 

COV(x., x.j) = Pc(1 Pc)(2131 - 1)2 

Finally, the correlation coefficient between x. and x, isjgiven by 

p(x., xj) 

COV(x , x.j) 

a a 

X X. j 

(3) 

a2 = E(x.2) - [E(xi)]2 = E(x)[]. - E(xi)] x 1 i 

So (3) becomes 

p(x., x.) = 

P(1-PC)(2P1

-1)2 

E(x )[1-E(x)] (4) 

(1) 

(2) 
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If condition (B) is now applied to (4), 

P

C (1-PC )(2P1 -1)2 

p 

- 

E(x.)[1-E(x )] 

Rewriting (1), 

- = P x C P

1 + (1 - P )(1 - P1) = E(xi) 

Thus the problem reduces to the solution of two simultaneous equations 

(5) and (6), for Pc and P1. 

- - 

Substituting x for E(xi) in (5) yields 

(5) 

(6) 

Let K = p Tc(1 - 

P

C

(1-PC

)(2P1

-1)2 

T) 

Then (7) becomes 

K = (PC - PC 2)(2P1 - 1)2 (9) 

Solving (6) for Pi, 

x + P - 1 

P1 • 2PC - 1 (10) • 

Back to (9), 

[2Tc + 2P

c 

- 2 

K = (P - P 2

) C C 2P

c 

- 1 1 

• 2 
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Page 4 

) 2X - 1 2 

2PC - 1 K = (PC- PC 

2

) (12) 

Now let 

K' - K 

(2x - 1)

2 

(13) 

Equation (12) becomes 

K'(2P

c 

- 1)

2 = PC - P C2 

K'(4P

c

2 - 4P

c 

+ 1) = P

c 

- P

c 

Whence finally 

K' P 2 Pc + 0 (14) 4K' + 1 

Either root of (14) may be chosen for Pc, so, without loss of generalit 

let 

PC 

.0 

= 

1 2+ 

1 .1 _ 

2 

1 

— 2 

1 

2 

i/ 1 4K' 

4K' +1 

/ 1 

/ 4K' + 1 

Substituting Pc into (10) yields 

1 / P

1 = 1 —+ (x 2) /4K' + 1 2 

(15) 

(16) 
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The values of P and P1 

from (15) and (16) yield the N random 

variables x, satisfying conditions (A) and (B). The method is .useful 

for computational purposes because the generation of the xi can! be 

accomplished by a succession of samples from a simple Bernoulli random 

number generator. 

. Al 

P. L. McEntire 

PLMcE:ne