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U1 N C~ I - A 1; ST T T i, rInoSECURITY CLASSIFICATION OF THIS PAGE (Wien Date Sntered)

REPORT DOCUMENTATION PAGE BFRE COMPRLETINORML OP0R T NUM BER2.OVA

1 AA-ATR-9:)$4

6' A Closed Form Expression for Line-of- S~~ __ _ _ __ _ _ __ _ _ __ _

d. PERFORMING ORG. REPORT NUMBER

7. A THOR - G . CONTRACT OR GRANT NUMBER(&)

J.JSheldonS. Smi

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK

US Army Materiel Systems Analysis Activity ARE WR UNIT NUMBERS

I14. CONTRORING OFFIICE NAME AN ADRESS dlfr2 fAmC~toln Ofc)I.SCRTE tIW~f7ot

U NC LA S SI F IE D

IO. ECLASSI FICATION/ DOWNGRADINGSCEDL

16. 0DIST RIBU TION ST AT EMEN T (of this Report)

Approved for Public Release, distribution. unlimited.

17. DISTRIOUTION STATEMENT (of the abstract entered In Btock 20. If different from Report)

IS. SUPPLEMENTARY NOTES

19.- -KEY WORDS (Continue on reverse side it necesarmy and Identify by block number)

Line-of-Sight Probability

observer altitude H and observer target range R.. The re ult for a somewhat rougtterrain classification is (0.689606)

~~ L~ -.~,LOS ' 1 exp -9.96)~/and the result for a smooth table-top terrain classification is

S,., 6L0 L 1 exp -(770.748)H(O 6092 )/R.

D O JN 1473 ze"flON or I ovSItis 008MET U-NC CLAS S I F ISECU1111ITY CLASMPICATOW OF TH18 AGE Do#te tWd)

TABLE OF CONTENTS

Page

ACKNOWLEDGEMENT ........................................... PagePURPOSE ....................................................... 5

1. BACKGROUND ................................................ 7

2. ASSUMPTIONS AND HYPOTHESIS ................................ 7

3. DEVELOPMENT OF PLOS EXPRESSION ............................ 7

4. RESULTS ................................................... 9

5. RECOMMENDATIONS ........................................... 10

6. COMMENT ................................................... 104 REFERENCES .................................................... 11

APPENDIX ...................................................... iS

A.l. PROCESS TO OBTAIN PLOS EXPRESSION ..................... 15

A.2. TABLE OF ACTUAL TERRAINS STUDIED ...................... 16

FIGURES ....................................................... 17

DISTRIBUTION LIST ............................................. 3S

A>Vor>O for;

NTIS V cl!on IDDC B ffr Se-tLionUNANrNOUNCTD[JUSTIFICATIOI

.... °.oo........................................ .

. BYDSRONINAMP: 7 t'noC IA

ACKNOWLEDGEMENT

The authors acknowledge R. Kaste (TOAO), Stephen Monroe (TOAO), G. Reed(TOAO), and W. Olson (Formerly TOAO) all of whom were instrumental in thedevelopment and implementation of concepts discussed in this report.

4

A Closed Form Expression for Line-of-Sight Probability

PURPOSE

This report describes a closed form expression for the probabilitv ofline-of-sight (P ) for a given terrain type as a function of observer altitudeand observer-to-iget range R.

Figures illustrate the goodness-of-fit of "PLos vs R" curves to actualterrain and terrain classification type data.

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A Closed Form Expression for Line-of-Sight Probability

1. BACKGROUND

For several years, those involved with modeling of line-of-sight orobability(P o) have been investigating analytic approximations for specific ard classi-

fica tion type terrains. The authors propose the expressions in this text as ameaningful first step toward the accomplishment of establishing such analyticexpressions.

2. ASSUMPTIONS AND HYPOTHESES

It is assumed that P is 1.00 when an observer looks down where lookinadown defines A zero eleva on angle, 0 = 0 degrees. The expression selectedshould yield "P vs. R (observer/target range)" curves that exhibit a slightLOSrounding at the near range. That is, P OS is approximately 1.00 for elevationangles less than 60 degrees (Ref. 1). An initial hypothesis is formed by takingthe exponential of a cotangent (ctn) function on the elevation angle

PLOS = 1 - exp (-a ctn e) (1)

where elevation angle 0 is interpreted as the arctangent of the ratio of range

and observer attitude.

0 = Arctan (R/H) (2)

The condition that 0 < PLOS 1 1 requires that "a" be positive.

3. DEVELOPMENT OF PLOS EXPRESSION

An "a" family of "PLos versus 0" curves is shown in Figure 1. The "a"

parameter characterizes a rate at which line-of-sight is lost. This rateroughness factor reflects the combined degradation effects of terrain curvature,target height/profile, and vegetation.

Comhinina eauations I and 2 yield.

1- exp (-a H/R) (3)

Figures 2 and 3 illustrate equation 3 as an 'H" family of "PLOS vs. R" curves.

Figure 2 applies for a rate roughness value, a = 10 while Figure 3 applies fora rate roughness value, a = 20. Maintaining a constant product of a and I' willmaintain the same PLOS value. For example, if a is doubled and II f-halqvd,

PLOS remains unchanged for a given range.

7

Figure 4 illustrates the behavior of a factor b which when multipliedby elevation angle 0 increases the PLOS asymptote as R ornroaches infinity.The expression plotted in Figure 4 is

PLOS = 1 - exp (-a ctn (b arctan (R/H))). (4)

Since b affects the asymptotic value of the "P0o vs. R" curve, it is referredto as the asymptote roughness factor. The neeb-for considering non-unitvvalues (O<b<1) for this asymptote factor was hypothesized as necessary sincereal PLOS data appear to show curves that approach values greater than zeroat far ranges (Ref. 3).

Equation 4 was fitted to three specific maps of the 50d classification forvarious observer altitudes (Ref. 2).* Figures 5 through 10 illustrate thatequation 4 fits P data well for each specific map. However, these samefigures demonstra 5 the variation in rate roughness factor A with observeraltitude H. This variation of a versus H challenges the hypothesis thatP LOS is fundamentally a function of elevation angle 0. That is, PLS remainsconstant for a given ratio of R and H. Therefore, with this allowance of afalse hypothesis on 0, a is fitted wTth H. In addition, Figures 5 through 10illustrate that parameter b is constant it 1.00.**

To characterize the terrain roughness for the 5Dd terrain classification,three terrain samples in the 5Dd classification were analyzed.***

*This classification is detdi led in Table 1 as extracted from Ref. 2. In adescriptive sense, 5Dd is rough terrain.**For 19 of 21 terrain fits (one fit for each of seven observer altitudes H foreach of three terrains), the b value was 1.00. On the other two cases, the bvalue was 0.99.***Presalection process of what terrains were to be analyzed was based on teavailability of tapes with digitized terrain characteristics and the knowledcethat the terrains selected were different from each other by geographiclocation and general appearance. The Appendix gives a listing of terrain mansanalyzed.

The a values were averaged over these three terrains at each observer altitudeH. Then these a values were least squares fitted as a function of observeraltitude H. Equation 5 is the result of this fitting process.

a = (97.59267) H( 0.310394) (5)

The resulting PLOS expression for the 5Dd terrain classification is*

PLOS ' 1- exp (-(97.59267) H(O'689606 )/R). ()

Figures 11 through 17 illustrate the fit of equation 6 to the "P0 vs.

curves of the three 5Dd terrains frcm which parameters for eouation v werederived.

4. RESULTS

The results from Figure 1 are:

a. Larger values of a imply that the probability of line-of-sight isgreater.

b. The rate roughness factor a causes marked changes in curves ofPLOS vs. elevation angle when given values between 0.01 and 10.

Results from Figure 5 through 10

a. For specific terrain and observer height, equation 3 does fit date oflpLOS vs. R" well.

b. Equation 3 is not sufficient to depict the PLOS characteristics as botha function of range and observer height. To obtain such an expression invlvesfitting the dependence of the rate roughness factor a on observer altitude H.(See Appendix)

a = AHB

*-The-Appendix provides a description of the programs involved to conduct a fit

of equation 4 to digitized terrain data. Also provided is the subseouent analysito arrive at this expression.

9

Results from Figures 11 through 17 are:

a. Variations in "PLOS vs. R" curves among terrains in the sameclassification appear large.

b. Equation 6 does yield a "PLOS vs. R" curve within the variation ofthe three terrains of classification 5Dd from which the two parameters inequation 6 were obtained.

5. RECOMMENDATIONS

a. A similar analysis/fit should be conducted on all of the Natickclassifications to form a data base for immediate use by expected value and/orMonte Carlo simulation/wargame nidels.

b. The Natick classification scheme could be improved by grouping terrainsinto categories determined by a specified variance about given "PLOS vs. R"curves.

6. COMMENT

A similar analysis was conducted on three terrains of the Natick ClassificationlAa - a smooth, table-top terrain. Results/fits are as good as, if not betterthan, those shown in this report on the 5Dd classification. The correspondingequation 6 for this lAa classification is

PLOS = 1 - exp (-770.748 H(O'6092)/R) (Fa)

:10

REFERENCES

1. Burge, Carol J., "A Revision of Surface-to-Surface Masking Studies,"June 1975, NWC-TP-5773.

2. Anstey, COL R. (Q11 USAR), "The Natick Landform Classification System,"US Army Materiel Systems Analysis Activity, TRIO0.

3. Olson, Warren K., "A Terrain Analysis of Four Tactical Situations,"TM158, Dec 1972, AD90934.

4. Defense Mapping Agency, "Catalogue of Maps, Charts, and Related Products"Map Series M745.

Spes

I

g1

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11L

TABLE I. CLASS INTERVALS OF CLASSIFICATION DESCRIPTORS

Maximum Hill Height (Local Relief)

Descriptor Class Interval

1 0 -10 Meters 0 - 33 Feet2 10 -30 33 - 9q3 30 -50 99 - 1654 50 -100 165 - 3305 100 -300 330 - 9906 OVER 300 OVER 990

Modal Hill Height (Local Relief)


A 0 -10 Meters 0 - 33 FeetB 10 -20 33 - 66C 20 -35 66 - 115D 35 -50 115 - 165E 50 -75 165 - 248F 75 -100 248 - 330G 100 -125 330 - 413H i25 -150 413 - 495I 150 -175 495 - 578J 175 -200 578 - 660K OVER 200 OVER 660

Number of Pcsitive Features Per Mile


a 0 -0.8/Kilometer 0 - 0.5/Mileb 0.8 -1.6 0.5 - 1.C,c 1.6 -2.4 1.0 - 1.5d 2.4 -3.2 1.5 - 2 0e 3.2 -4.0 2.0 - 2.5f OVER 4.0 OVER 2.5

Next page is blank.

K13

APPENDIX

A.I. PROCESS TO OBTAIN PLOS EXPRESSION

Obtaining a closed form expression for PLOS involves three stens:

Step 1: Generating PLOS curves from digitized terrain da*a and fittinqequation 4 to these curves,

Step 2: Obtaining a correlation function between rate roughness factora and observer altitude H, and

Step 3: Calculating/plotting of the theoretical/fitted curve.

s kStep 1 consists of the program "real curve fit." This program is acombination of two programs. The primary program "LOS Rings," written byWarren Olsen, generates the l vs. R" curves from digitized terrain data(Ref. 3).* The secondary program "Minmax" modeled by J. Sheldon and written/executed by Sean Smith, fits equation 4 to the digitally processed "PLp vs. P'curve by using the Kolmogorov-Smirnov, maximum difference test for toodness-of-Fit.

The program "Real Curve Fit" is written in FORTRAN IV compatible onlywith the BRLESC II computer at Aberdeen Proving Ground, Maryland. Table IIillustrates the informatior' necessary to run a single case - a case consists ofa "pLOS vs. R" fit for each of seven observer altitudes.

Table II

Computational Information on BRLESC II Program - Real Curve rit

Range KH6 8

Run Time(Mi) 45 711

Output(Lines) 600 8735 70

*The PLAC result from digitized terrain is an average result. The averaaino

occurs Ler observer location on the terrain over a 360 degree scan. Thisprocess accounts for the somewhat smooth curve representing th" actual

Los vs. R" result.

15.is

Step 2, finding an equation which relates rate roughness factora to observer altitude H, uses a standard/library program "RegressionAnalysis with Plotting.W This process begins by averaging all the rateroughness factors a over the three terrains for each of the observerheights H. These average values of a are least squares fitted to Hin the form*

a = AHB. (7)

Step 3 consists of the plotting of equation 6 which is the resultof combining equations 3 and 5. The general form for equation 6 is

PLOS = 1- exp (-AH(B+I)/R ) (8)

A.2. TABLE OF ACTUAL TERRAINS STUDIED

Table III lists the terrains studied in this report. These terrainsare shown in Figure A.1.

TABLE III

Actual Terrains Studied

Natick Terrain DMA MAPClassification Numbers Actual Terrain Code**/City

5Dd L6134 - L6132 G-93/BayreuthL5120 - L5118 G-4/Stadt AllendorfL4920 - L4918 G-44/Bad Wildungen

II

1Aa L7130 - L7128 G-70/NordlingenL7332 - L7330 G-76/NeubergL7334 - L7332 G-77/Ingolstadt

*The k orrelation function between A and HLwas tested in two different forms(a=AHD and a=AB ). The best results were obtained by using the formerexpression, Equation 7.**Terrain codes are standard DMA (Defense Mapping Agency) codes (Ref. 4).

16

Z5 (0

00

.-

C14

17 a

BPI

4ce.

N 0L8 b.

ID

18

00

t

4U

Le

00

19L2

rr

°4IC

.00E -0

0-

s 0044SI -V

U, £ 8

0Z 20

1.0

0.9

0.6

1 . 0.7

~0

0.1

OA --

00 1 2 3 4 5

RANGE (KM)

OBSERVER HEIGHT : 3.00 MTARGET HEIGHT -.00AREA G93CLASSIFICATION 5D d

"I RATE SHAPE FACTOR a 2 43.11ASYMPTOTIC SHAPE FACTOR B: 1.00MINIMUM MAXIMUM DIFFERENCE 0.03

Figure 5 Line-of-Sight Probability vs. Range.

(Artificial Curve Fit)

21

1.0-

0.9

0.8

0.7

0.6

0.5

0 .3 -

0.1

0 10 1 2 3 4 5

RANGE (KM)

OBSERVER HEIGHT a 1.67 MTARGET HEIGHT : .00AREA G93CLASSIFICATI1M 5DdRATE SHAPE FACTOR a u 61.96ASYMPTOTIC SHAPE FACTOR B 1.00MINIMUM MAXIMUM DIFFERENCE 2 0.02

Figure 6 Line-of-Sight Probability vs. Range.(Artificiai Curve Fit)

22

1.0-

0.9

0.8

30.7

0.5 \

S20.3

a2

0.100 I 1

1 2 3 4 5RANGE (KM)

OBSERVER HEIGHT ' 10.OOMTARGET HEIGHT: .00AREA G44CLASSIFICATION 5D dRATE SHAPE FACTOR a x 40.76ASYMPTOTIC SHAPE FACTOR B x 1.00MINIMUM MAXIMUM DIFFERENCE : 0.05

Figure 7 Une-o-ight Probability vs. Range.(Artificial Curve Fit)

23

1.0

0.8

>- 0.7

0.5

S0.4

'~0.3

1 ' Q2

0.1

0, I0 1 2 3 4 5

RANGE (KM)

OBSERVER HEIGHT * 10.0OOMTARGET HEIGHT : .00AREA G48CLASSIFICATION SOdRATE SHAPE FACTOR a m 2767ASYMPTOTIC SHAPE FACTOR B = 1.00MINIMUM MAXIMUM DIFFERENCE a 0.03

Figure s Une-of-Sight Probability vs. Range.(Artificial Curve Fit)

24

1.0

0.9

0.76

cc0.6

0.5C. 0.4

a0.3

01

0 12 3 4 5

RANGE (KM)

OBSERVER HEIGHTS 500.OOMTARGET HEIGHTz 0AREA G93

N CLASSIFICATION SDdRATE SHAPE FACTOR as 15.36ASYMPTOTIC SHAPE FACTOR B-: 0.99MINIMUM MAXIMUM DIFFERENCE a 0.03

Figure 9 Uine-of-Sight Probability vs. Range.(Artificial Curve Fit)

25

1.o

0.9

0.8

0.7

O0A

2

0.1

o 1 2 3 4 5RANGE (KM)

OBSERVER HEIGHT a 20.00 MTARGET HEIGHT a .00AREA G44CLASSIFICATION 5DdRATE SHAPE FACTOR a x 29.70ASYMPTOTC SHAPE FACTOR B x 1.00MINIMUM MAXIMUM DIFFERENCE 20.06



26

1.0

0.9

I0.8

0.7 .... ACTUALTHEORETICAL

0.6

0.4

Uj0.23'.1"-- ._.

0.12 - .._o

01 2 3 4 5

RANGE (KM)

OBSERVER HEIGHT " 1.67M

TARGET HEIGHT = .00ROUGHNESS FACTOR a z 83.24ROUGHNESS FACTOR B :1.00CLASSIFICATION 5Dd

VFigure 11 Line-of-Sight Probability vs. Range.


27

1.0-

0.9

0.7 ....- ACTUAL

~Q6 THEORETICAL

0.5

0.4 A0.2 "

0.1 -I'

0100 1 2 3 4 5

RANGE (KM)

- OBSERVER HEIGHTa 3.00MTARGET HEIGHT x .00ROUGHNESS FACTOR a , 69.39ROUGHNESS FACTOR B 31.00CLASSIFICATION 5D4



28

-

1.0

0.9

0.8.... ACTUALITHEORETICAL

0.7

0.6

0.2

0.1

1 2 3 4 5

RANGE(KM)

OBSERVER HEIGHT = 10.00 MTARGET HEIGHT x .00ROUGHNESS FACTOR a a 4775ROUGHNESS FACTOR B--1.00CLASSIFICATION 5Dd

Figure 13 Line-cf-Sight Probability vs. Range.


29

1.0

0.9

i° 'I0.7... ACTUAL

THEORETICALO.6

0.5

aQ3

0.2

oL I00 1 2 3 4 5

RANGE (KM)

OBSERVER HEIGHT a 20.00 MTARGET HEIGHT x .00ROUGHNESS FACTOR a • 38.51ROUGHNESS FACTOR B :1.00CLASSIFICATION 5Dd

Figure 14 Line-of-Sight Probability vs. Range.(Artificial Curve Fit)

30

1.0

0.9

0.9 - - -- ACTUAL__ -THEORETICAL

C 0.7 _ _ _

a6a

0.5

0.5

0.2 -,

0.1-

00 I 2 3 4 5

RANGE (KM)

OBSERVER HEIGHT " ,50.00 MTARGET HEIGHT : .(0ROUGHNESS FACTOR a z 28.98ROUGHNESS FACTOR B :1.00CLASSIFICATION 5D d

Figure Line-of-Sight Probability vs. Range.(Artificial Curve Fit)

31

i

1.0

0.9 ACTUAL

THEORETICALS0.8-

0.7-

0.6 -\ \

0.3-*0.4

0.2

0.1-

00 1 2 3 5

RANGE (KM)

[ OBSERVER HEIGHT z 100.00 MTARGET HEIGHT -.00ROUGHNESS FACTOR a 23.37ROUGHNESS FACTOR B 1.00CLASSIFICATION 50 d


32

0.9-

0.7

0.6

0.4

ACTUALa 3- THEORETICAL

0.2

0.1I I , I !

00 1 2 3 4 5

RANGE (KM)

OBSERVER HEIGHT 500.00 M

TARGET HEIGHT - .00ROUGHNESS FACTOR a s 14.18ROUGHNESS FACTOR B :1.00CLASSIFICATION 5D d


33

WEST GERMANY

SERIES MAP M745

i

E3IG-44 ROUGH TERRAINMEJG-48[

ROUGH TERRAIN

LGG-93

G-76 FLATG-77 1 TERRAIN

V Figure A-i. Location of Terrain Areas.

34

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