ad-a254 2539. mean and variance of scattered power for h-pol for: 16 a....

AD-A254 253

RL-TR-91-351tn-House ReportDecember 1991

ANALYTICAL CHARACTERIZATION OFBISTATIC SCATTERING FROM GAUSSIANDISTRIBUTED SURFACES

Lisa M. Sharpe DTCGI.,L EC - F-~UG 13 1992

APPROVED FOR PUBLIC RELEASE,, 01STRIRBT1O7N UNLIMITED

Rome LaboratoryAir Force Systems Command

Griffiss Air Force Base, NY 13441-5700

92-2251292• 8 1 0 0 35 ii11li1111111l1ml

This report has been reviewed by the Rome Laboratory Public Affairs Office(PA) and is releasable to the National Technical Information Service (NTIS). At NTISit will be releasable to the general public, including foreign nations.

RL-TR-91-351 has been reviewed and is approved for publication.

4

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Analytical Characterization of Bistatic Scattering From PR: 2305Gaussian Distributed Surfaces TA: J4

WTJ: 09

6. AUTHOR(S) PE: 61102F

Lisa M. Sharpe

7 PERFO)RMING ORGANIZATION IhAME(S) AN ) ADDAESS(f S) & PERFOfAMtNG ORGANIZATION

REPORI NUMBERRome Laboratory/ERCE RLTR-91-351Hanscom AFB. MA 01731-5000

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Approved for public release; distribution unlimited

13. ABSTRACT (Mavdmum 200 worda)

In a study by Papa and Woodworth, the mean and variance of the diffuse power scattered from aclutter cell were examined as a function of cell size for one-dimensionally rough surfaces. Theirstudy was performed for horizontally polarized incident and scattered fields. In this report theirwork is expanded to include vertical and arbitrary linear polarization. Scattering statistics andpolarization effects will be examined using a theoretical model based upon physical optics. Thepresent study shows that as the size of the clutter cell is decreased the statistical distribution of thescattered power deviates from Rayleigh. Also. it is shown that the tilt angle of the incident linearpolarization has little effect on the statistics of the scattered wave.

14 SUBJECT TERMS 15. NUMBER OF PAGESRough surface scattering Scattering statistics 54Bistatic scattering 16. PRICE COOD

1'7 SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT

OF REPORT OF THIS PAGE OF ABSTRACT

Unclassified Unclassified Unclassified SAR

NSN 7540-01-280-5500 Standard Form 296 (rev. 2-69)Prescrbed by ANSI Std Z39-11296-102

Contents

1. INTRODUCTION 1

2. THEORETICAL FORMULATION 4

3. VERTICAL POLARIZATION 8

4. ARBITRARY LINEAR POLARIZATION 33

5. CONCLUSIONS 35

REFERENCES 43

Dn QULMfY IJWI'ECTED a

Acoession For

NTIS GRAIIDTIC TAB flUnarnnounced []JU.;%itf icst ion._

ByDistr-ibuti-on/Availability Codes

Avail and/or

iDlet Special

tit

Illustrations

1. Rayleigh Distribution Functions 3

2. Scattering Geometry 4

3. Surface Geometry 6

4. a. Magnitudes of Horizontal Reflection Coefficients 10b. Phases of Horizontal Reflection Coefficients

5. a. Magnitudes of Vertical Reflection Coefficients 11b. Phases of Vertical Reflection Coefficients

6. Mean and Variance of Scattered Power for: 13

a. Ot-30, X =0.25m.=0.2.L=3T =30 +j2b. 0, = 300, X = 0.25 m. cT = 0.2. L = 6T1, = 30 +J2c. O8=300, X=0.25m. a=0.2, L= 12T. £=30+J2


a. Ot=300.X)=0.25m.a=0.5.L=3T,=30 +J2b. 01=30 0 .X=0.25m.o=0.5.L=6Tc=30 +J2c. 01 = 30 0. X = 0.25 m. a = 0.5. L = 12T. E = 30 +J2


a. Ot=75 X =0.25m. a=0.5,L=3T, e=30+j2b. 9--75°, X =0.25m.c=0.5.L=6Te=30 +J2c. i= 750. . =0.25m,aY=0.5,L= 12T.e=30.+J2

v

9. Mean and Variance of Scattered Power for H-pol for: 16

a. 0,=75 0.X=0.25m,o=0.5,L=3T,e=30 +j2b. 09=750,X=0.25m,c=0.5.L=6T.c=30+j2c. 0t=750,X=0.25m, c=0.5, L= 12T"•E=30+j2


a. 0 1=75°,X=0.1 m,a=0.2,L=3T,e=30+J2b. 0,=750 ,X=0.1m.cr=0.2,L=6TE=30 +j2c. O0=75 0,X=0.1 m.a=0.2,L= 12TE=3O+J2


a. t=750, X=0.1 m. o=0.5, L=3T. e=80+J9b. 0i=750.X=0.1 m,o=0.5.L=6T.c=80+J9c. O=75 0,X=O.Im,o=0.5,L=12Te=80+j9

12. Comparison of Variance for Different Cell Sizes for: 20

01 = 300, X= 0.25 m, o = 0.2, e = 30 +J2


%9= 300, X= 0.25 m, o = 0.5, e = 30 +J2


% =750 .X=0.25m, o = 0.5, = 30 +J2


0,=750,X=0.1 m,=0.2.e=30+j2


a. 01=300.X= 0. m.r=0.05.L=3T,£=30+J2b. ,= 30 X.-=0.1m. =0.2,L=3T,e=30 +J2c. ,= 300, .= 0.1 m,o=0.5,L=3TE=30+J2


a. 0t=300.X=0.25m,a=0.2,L=3T,E=30+J2b. 0t=300,X=O.l m,to=0.2.L=3T,r=30-+ J2


a. 0,=300, = 0.25m.o=0.5,L=3T,e=30+J2b. 0,=300, X= 0.1 m,a=0.5. L=3TF=30+J2


a. ,=-750 ,X=0.25m.o=0.5.L=3T,e=30 +J2b. t=,750.=0.I m.o=0.5.L=3T.,=30-+J2

20. Mean and Variance of Scattered Power for H-pol for: 29

a. 0,=75 0,X=0.25m,o=0.5,1L=3T, =30+J2b. 0,=750,X=.0.1 mo=0.5.L=3T.=30 +j2

vi

21. Comparison of Normalized Variance for Different Surface 31Roughness for 01 = 750, X = 0.1 m. L = 3T, F = 30 + J2

22. Comparison of Normalized Variance for Different Surface 32Roughness for 01 = 300. X = 0.25 m. L = 3T. E = 30 + J2

23. Mean and Variance of Scattered Power for: 36,37

a. 01 =.300, X =0.1 in.o=0.2. L=3T, E=30+J2, =0 0

b. 0t=300.X=0.lm.c-=0.2,L=3T,E=30+J2,NV=30 0

c. 0i=300,.=0.1 rn.c=0.2,L=3T,E=30+J2, V=450

d. 0I=30',.=0.1in,a=0.2,L=3TE=30+J2, v=600

e. 0,=30O..,=0.1 1n. =0.2. L=3T,.r=30+J2, 41 =900

24. Mean and Varianct- of Scal tered Power for: 38,39

a. 01=75 0,•.=0.25m,ca=0.5.L=3T.E=30+J2,'p=0 0

b. 01=75 0.9.=0.25m, =0.5.L=3T.e=30+J2,iV=30°c. 01 -=750,X=0.25m, a=0.5. L=3T.1=30+J2.W =450

d. 01 = 750. X = 0.25 m. o = 0.5. L = 3T, E = 30 +J2, W = 600

e. 0t- =750. X = 0.25 m, a = 0.5, L = 3T,.E = 30 +J2. = -90 0

25. Mean and Variance of Scattered Power for: 40.41

a. 0,=300,X=0.1 mo=0.2. L=Tr, E=2+jl.6. W=0 0

b. 0=300, X =0. m,o=0.2, L=3T.e=2+j1.6, =300

c. 01= 300, =0.1 m.,=0.2. L=3T.E=2+J1.6,1=500

d. 01=300,?.=0. 1 mo=0.2. L=3T, e=2+J1.6, W =700

e. 01 =300,X=0.1 m.o=0.2, L=3T, e=2+Jl.6, =-900

vii

Tables

1. Surface Dielectric Constants 8

2. Variance as a Function of Wavelength 25

3. Comparison of Measured a' with Calculated o0 33

4. Complex Polarization Factor 34

Ix

Analytical Characterization of Bistatic ScatteringFrom Gaussian Distributed Surfaces

1. INTRODUCTION

Accurate statistical modeling of the clutter encountered by a radar system at a particular

site is critical to the proper design of the system. Clutter is any unwanted radar echo, such as

reflections f-om the terrain. When the clutter Is as large as the expected target return, the radar

assumes the return is due to a target being present, and a false alarm occurs. Clutter Is,

therefore, a significant factor in determining the optimum value of the minimum detectable

signal. (smallest target) for the required false alarm rate. 1 The clutter returns are not equal but

are distributed In magnitude. If there are more high energy returns due to clutter than a model

predicts, the false alarm rate will be unacceptably high. Conversely, fewer high energy returns

than expected means that the probability of detection could have been improved by lowering

the detection threshold.

Also, theoretical modeling of scattering from rough surfaces that cause clutter is very

important because accurate measurements of clutter are very difficult and expensive to

perform, and are sometimes unreliable. The mean and variance of clutter can be determined

experimentally, but this requires many measurements of similar, independent clutter cells to

be made. These measured values would then only be valid for the specific geometry and type of

Received for Publication 19 November 1991

Skolnik. M.I. (1980) Introduction to Radar Systems. McGraw-Hill Book Company, NewYork.

1

clutter being measured. Any changes in polarization, frequency, radar deployment, geometry,

seasonal variation, or even moisture content of the ground will result in a completely different

set of statistics. Even measurements made under identical conditions can vary by up to 10 dB. 2

For these reasons, theoretical models of clutter statistics are important. The power scatteredfrom a rough surface is usually assumed to be Rayleigh distributed. The Rayleigh distribution

function is:

x 2 e_V/22

f (x) = -e- U (x). (1)

Figure 1 is a plot of this distribution function for several values of a. The assumption ofRayleigh distributed clutter is valid for an Infinite scattering surface of many similar sized

scatterers. If the scatterers differ significantly in size or if the number of scatterers is

decreased by limiting the cell size, the probability density function of the scattered power willno longer have a Rayleigh distribution.

An application of this is in short pulse radars where the clutter patch is limited in the

range dimension. For such radars, the requirement of a large surface with many scattering

centers is violated. In this report, the statistics of the power scattered from a rough surface

with a finite range dimension will be determined using a physical optics model.

The effect of polarization on rough surface scattering will also be examined. Because the

reflection coefficient of a surface is polarization dependent, the statistics of the scattering will

also be a function of the Incident polarization. In a study by Papa and Woodworth, 3 the meanand variance of the scattered power were investigated for horizontal incident polarization.Arbitrary linear polarization states will be studied to determine the effect of polarization on

the scattering statistics.

2 Long, M.W. (1975) Radar ReJlectivity of Land and Sea, D.C. Heath and Company,

Lexington, MA.

3 Papa, R.J.. and Woodworth, M.B. (1991) The Mean and Variance of D(ffuse ScatteredPower as a Function of Clutter Resolution Cell Size. RADC-TR-91-09.

2

C\J C1) LO

1111110

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2. THEORETICAL FORMULATION

The normalized cross section of a rough surface, a" is

o (47rR2) 2]A . . [<(E 8 /E1 ) (Es/E 1 ) > - < (Es/E 1 ) >2] (2)

where Ei is the incident field, E, is the scattered field, Ro is the distance from the surface to the

field point, A is the area of the illuminated cell, * indicates the complex conjugate, and the

angle brackets indicate an ensemble average over the surface variables. Figure 2 depicts the

scattering geometry. For a one dimensionally rough surface only in-plane scattering occurs

= 00). and there is no shadowing.4

•Zo

7 i• ,Y)

Figure 2. Scattering Geometry

4 Beckmann, P.. and Spizzichino. A. (1963) The Scattering of Electromagnetic Waves fromRough Surfaces, The MacMillan Co., New York.

4

The surface geometry is shown In Figure 3. The points x, and x2 are two random points

separated by the correlation distance. T. with heights ý, and ý2. The correlation distance

defines the density of the irregularities. The local slopes of the rough surface at these points

are p I and p2 . The (list ributlon of the heights is described by a Gaussian dlisiributlon function

because this is the most typical distribution of a rough surface. This distribution function Is

given by:

w() - 1xp (3)oV2i 2 a'

where Y is the standard deviation of the surface heights and C2 is the variance of ý. The

variance can be chosen to represent the desired degree of roughness and the correlation

distance chosen for the desired density of the irregularities.

For a one dimensionally rough surface used in this study. the expression for

normalized mean cross section, aO. is:

a= ( .9 (<E/Ell >-<!EIEI> 2). (4)

The scattered field is a cylindrical wave calculated by evaluating the Helmholtz Integral

equation:

"En dL (5)

where: W~ 41 (k) 9//Ok.R') exp (I k R')

k, = scatteied wave vector

R'= Ro -ro

R, = vector from origin to field point

r = vector from origin to source on the surface

E = (I +R) E is the total field on the surface

R = Fresnel reflection coefficient

n= normal derivative on surface, S

L = range dimension of the scattering cell

and the time dependence, exp (-Jcot). is supressed.

5

X0

€4

N N• N

6

The most general expression for o"(I involves a six-fold integral over the surface variables.x 1 , X2 .ý t2, p1 .and l2 . However, Papa and Woodworth show that because the surface heightsare regarded as a stationary random process, the integral is a function solely of the separationbetween the two surface points, 'r = x 1 - x 2 , which simplifies the six-fold integral to a five-foldintegral. The five-fold ititegral can be further simplified to a four-fold integral because the

expression for '" is a function of the height differences, ý = ý - ý2. Then, by assuming that thesurface slopes are small, V2 cT/ T < 1, the integrals over ptl and 92 can be performed analyticallyand the expression for ro can then be written as a double integral given by:

C = dxi dx2[x2- Xl *Cos(V T) (6)12Lk j.IL/2 1 (6/2

where: I = x 1 - x 2

X2 = exp [4_2 (I - C 1 2)] is the characteristic function for bivariate height distributionX, = exp (_Y2/2) is the characteristic function for univariate height distributionC12 = exp (-(x1 - -2}2/T2

12= C2 v 2 = Rayleigh parameter squaredZ

c = rms surface height

v= (2 nt/ X) Isin 10t) - sin(%))

Vz= (2 Yt/ ;Q (cos (0,) + cos(0j)2R 1 + cos(0÷.)F = - ' o(0 1

Cos 0 + Cos 0

Note that it is the reflection coefficient in the factor F that causes CFO to be a function of the

polarization of the incident wave and the dielectric properties of the surface.

The variance of the scattered power, aC° is given by (see Papa and Woodworth):

L/2

0 2F f f f f dx dx2 d x 3 d x 4 cos v (xl - x 2 + x3 - x 4 )] *[X4 - X1 2 34 ] 1(7)

L/2

where: X12 = exp -XY2 (1 - C12)1

X34 = exp i-Y2 (1 - C3)i

Cq = exp I-xi - xj)/T 21

7

For a Rayleigh distribution, the variance is equal to the mean squared. This definition,Coo = (00 )2 is the definition used to calculate the Rayleigh variance in this study.4

3. VERTICAL POLARIZATION

The analysis done by Papa, et al is limited to the case of horizontal incident polarization. To

extend the analysis to include vertical and arbitrary linear incident polarization, new

reflection coefficients must be calculated.

For a perfect conductor the reflection coefficient is RH = -1 for horizontal polarization,

(electric vector perpendicular to the plane of incidence) and Rv = + 1 for vertical polarization,

(electric vector in the plane of incidence). Because the reflection coefficient is squared in

calculating the scattered power, the results for horizontal polarization and vertical

polarization are identical for a perfect conductor and will not be repeated here.

Many surfaces, however, have a complex dielectric constant, and will therefore have

different reflection coefficients for vertical and horizontal polarization. A complex dielectric

constant is expressed as c = r'+ J",. where E' is the capacivity of the medium and e" is the

dielectric loss factor. For a good dielectric the term c' will remain fairly constant over all

radio frequencies and e" will be small. The relative dielectric constant is defined as Er = E/Eo

where e, is the capacivity of vacuum. Table 1 shows the relative complex dielectric constants

for the surfaces which are used in this study as well as the dielectric constant for glass, which

is considered to be a good dielectric.

Table I. Surface Dielectric Constants

Surface C_ IEll

Moist Ground 30 2

Sea Witer 80 9

Dry Sandy Loam 2 1.6

Glass 5.25 0.0115

8

The Fresnel reflection coefficlent for horizontal polarization is given by:5

cos o-Ve inJ

where 0 is the local angle of inicidence. The magnitude and phase of R11 are shown In

Figure 4. Results for this case given by Papa. et al will be used for comparison with

results obtained for vertical polarization.

For vertical polarization the reflection coefficient is:

C Cos05-'N\E - sin 0Rr ()

E cosO0+-\ c -sin 0r" r

Figure 5 shows the magnitude and phase of Rv for the dielectric constants used.

In comparing the horizontal and vertical reflection coefficients, important differences can

be seen. The magnitude of the vertical reflection coefficient has a minimum value for a

particular angle of incidence. This angle is called the Brewster angle and is defined as

0 B = tan- 1 fir. Also. the magnitude of the vertical coefficient is always less than that of the

horizontal, except at normal Incidence where they are equal. The differences in the reflection

coefficients will have a direct effect on the scattering from a surface.

5 Ruck. G.T.. Barrick, D.E.. Stuart. W.D.. and KrIchbaum, C.K. (1970) Radar Cross SectionHandbook. Vol. 2. Plenum Press, New York.

9

.- + +\

+00

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0

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o i '

(6epC) aSI4

00

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-o -oo r.

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00

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LL

00

cm Z

... .. ... . C

3an.LN!D0

Many of the other results, though, are valid for both horizontal and vertical polarization.

Some of the other aspects that are considered are the statistical dependence of the scattering

on the angle of incidence 01, the wavelength X, the length of the clutter cell L, and the standard

deviation in the surface heights a. The surface roughness is defined as a/T and F2a/T Is the

rms surface slope. T is the correlation length and Is set to one meter for this study.

There are two factors that cause the variance of the scattered power to deviate from

Rayleigh, the number of scatterers and the roughness of the surface. The number of scatterers

is a function of cell size, incident angle, and the slopes of the rough surface. The roughness of

the surface depends on the frequency of the incident wave as well as the surface parameters.

How these parameters affect the statistics of the scattered field will be examined.

One condition necessary for Rayleigh scattering is that the illuminated area, or clutter cell.

be very large with respect to the correlation distance, so that it will contain a large number of

independent, nearly equal, homogeneously distributed scatterers. Decreasing the size of the

clutter cell will decrease the number of scatterers and should then cause the distribution of a0 0

to deviate from Rayleigh. When there are only a few independent scatterers in a particular cell.

their relative positions or the presence or absence of one of the scatterers causes a significant

change in the reflected signal from one sample of the surface to another, resulting In a large

variance.

In Figures 6 through 10. several cases are shown and the effect of changing the cell size is

examined. The scattered power Is normalized to both the incident field and the illuminated

area, and is plotted as a function of the scattering angle. Each plot shows three curves. The

curve consisting of long dashes represents the normalized mean of the scattered power, which

is equivalent to the mean cross section. The normalized variance of the scattered power of a

Rayleigh distributed scattering process is the curve of small dashes. The third curve with the

crosses is the calculated normalized variance of the scattered power for the parameters shown.

Figure 6 is for 0j = 300,. X= 0.25 m. a = 0.2 m, Er =30 + J2 with L increasing from L = 3T

(Figure 6a) to L = 6T (Figure 6b) and finally L = 12T (Figure 6c). These parameters correspond to

a relatively smooth patch of moist ground. Figure 7 is similar to Figure 6 but represents a

rougher surface. Figures 8 and 9 are for a rough surface with a 750 angle of incidence, for

vertical polarization and horizontal polarization, respectively. The effect of the different

reflection coefficients Is noticeable when comparing these two figures, particularly at the

location of the Brewster angle, 0 B =850. In Figure 10, 01 =750. X = 0.1 m ai dA = 0.2 m

corresponding to a relatively smooth surface. Figure 11 represents a fal ly rough water surface,

Er = 80 +J 9 .

12

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In all cases, it is seen that for the smaller cell size, L = 3T, the scattering is more non-

Rayleigh than lor the larger clutter cells. L = 6T and L = 12T, in the scattering regions where

differences in the variance can be seen. The variance of the scattered power is consistently

higher for the smaller clutter cells than it is for a surface that gives rise to Rayleigh scattering.

Figures 12 through 15 show a comparison of the variance for the different cell sizes

normalized to the Rayleigh value so that a value of one means the calculated value of the

variance is equal to the Rayleigh value. The higher variance causes difficulties in radar design

because for a constant mean value of clutter power, a larger variance corresponds to more

frequent large clutter returns. This Is a significant result that must be considered in the design

ola short pulse radar system.

A noticeable trend .-een in all of these plots is that the deviation from Rayleigh scattering

becomes more severe as 0, approaches the backscatter region, particularly when the angle of

incidence is 750. Since scattering facets must have steep slopes to cause backscatter, especially

for 0, =750, and the distribution of the slopes is the derivative of the height distribution, and

therefore also Gaussian, 6 there will be relatively few facets oriented properly to cause

backscatter. Once again, the effect of having a small number of scatterers is to increase the

variance of the scattered power, causing the scattering to become more non-Rayleigh.

The second factor that affects the distribution of the scattered power is the surface

roughness. A wave that is incident on a rough surface will scatter in many directions and it is

the relative heights and slopes of the scattering surfaces that will determine the angular

distribution.

For a very smooth surface, the Incident wave will be almost entirely reflected in the

specular direction defined by Snell's Law. As the surface roughness increases, the scattering

will become more diffusely distributed over the entire angular space. 4 Figure 16a shows the

scattering for a very smooth surface, a/T = 0.05. It is quite clear that the majority of the

scattered power is in a small region around the specular direction. In Figure 16b, a = 0.2 m

corresponding to a relatively smooth surface, and the peak of the scattering in the specular

direction is no longer well defined. Figure 16c is for a rougher surface with a = 0.5 m. In this

case the scattering is quite uniform over the entire region with no discernable peak in the

specular region.

6 Papoulis. A. (1965) Probability. Random Variables, and Stochastic Processes, McGraw-HillBook Company, New York.

19

C)LOE + E E'

I~l O

z U)

C)V (0

0 LOZ

< CV )

wN -0 WJ

U')

T- cc

0 0

0(Ga)~~~~~L CO N 0 )l C) CJ - 0

T- 1-

u 000/ 0OO.D

20

oEM E

o E +

00

0)I

C) U) -

z a

o 0

Nc W

0J 0 01< COO

00 if

N 00

0)~L (0)0C) ~

U 0

000/ 0

21

Ii• II II II

0

0)0

LO-

� 00 (0

0 to

o.- :J .V

CV) -Jc

CD 0

oo0 ooJO2iU 00

~CI/ u0

..... .... .22 .. ..

E'lEE+" EEEiII II II

"U U

0h

0)

> 0O

N 0~i

002

0 ~ ~ ~ ~ ~ ~ I <OC • 40 OC to

z0C)00( 4f) -0

0

z ay

0 w w t N 0 w0 C)cC\111oo-0-oo.

23

((D

0 0

-LJ

z CSI

10? 0 0)< 0(~ 0 C) C - tI

o 0 o i

-r~~~I 11 h v ~o

ID0-1p 00. PH 0.0(00

,(0i

cr)

00

< 0

ar TE Er

C)~C LA 0rC A 00 LA 0 L I

r ~ '- C\J

(OP ?o <UP c,)(S)ypu?

24

The relative roughness of a surface is a lunction of the wavelength of the Incident wave so

that the frequency of the incident wave will also affect the statistcal distribution of the

scattered power. Decreasing the wavelength increases the Rayleigh roughness parameter 1, sothat the relative height-, of the scatterers is Increased. Figures 17 through 20 show

comparisons for cases where only the wavelength is changed.

In Figure 17. the statistics are for 0, = 300. o = 0.2. and L = 3T. Figure 17a is for the longer

wavelength. X = 0.25 m while in 17b. . = 0.1 m. Figure 18 is similar to Figure 17 but for a

rougher surface, and in Figure 19 the angle of incidence is increased to 750. Figure 20 is thesame as Figure 19 but f'or horizontal polarization. In all cases, the variance of the scatteredpower is seen to be closer to Rityleigh when the incident field has a wavelength of 0.25 m. For

the shorter wavelength, or higher frequency, the scattering becomes less lutyleigh. Table 2

shows the variance of the scattered field for several wavelengths normalized to the Rayleighvalue for 0, = 300. 0 = 0.2. L = 3T. 0s = 00. and 0r = 30 + J2. Over the range of wavelengths tested.

the scattering becomes less Rayleigh as the wavelength decreases, or the roughness parameter

of the surface increases.

Table 2. Variance as a Function of Wavelength.

(. m) 0D dB) 00o (dB) 00o/a 00_______R (d__ __ __ _ _ C R

.25 2.891 3.772 1.22

.15 2.939 4.526 1.44

.10 2.999 5.038 1.60

.05 2.987 5.614 1.83

25

C5 0

o C\J C 0 0 0\ o-C0 0\ .- <

-. 1-

+ -- n

(0 +

w If

-~ 0

(0 U

0~~~: if) 0 CD t ) I f

266

0 0 05 C)o 0 0 0 0 0

LO (D C) 6 Lfl 0

(Y) 0 - C'Y) 6 cr-

0 IS E E Cs

+ .---

0 +

cc 0

1 0(DO

C-1Kr LU) I,) C, 1)'-j .

lj z

LU L

Co 0 0C -

0f Uf)

27

X00

C'J w

o C) 0 0C9 D 0 0) 0 C)

rl- 0) CW c; 6

00- E +E IE

~wi

C) I

Ir I( I

N IIco o

C~)*0e

z ii

0 4 (n 0 L I)

(aP) 00. Pu 00 (9P) 00 pue OD

28

Cý LLI0 0 -

0 0 0 0 )0>

LO LC? 9l < ~r- C) -06 0

E _ _14

-6 o ~ ~ . + -- nIo +

wIf

.- )

C:)__ __ 0 L

o~ 1o

0ýIn) 4) -c 0 0 )

6 .0 0

I -0 0 0 C) II 010 In 0

0~~ <n 035 CC0In0 In 0

(8cc: c u o(p arp~ 0 ).

Il 29I (

Figures 21 and 22 are plots of the normalized variance as a function of rms surface heights,These values are calculated by dividing the standard deviation by the mean value. Because theRayleigh variance is defined as cOo = (ao)2, a value of one means the variance is equal to the

Rayleigh variance, while a value greater than one means the variance is larger than theRayleigh value. These plots show that as 0. approaches the backscatter region the variance islarger for the smoother surface. This is because the smoother surface will have so few facets

with slopes large enough to cause backscatter that the small number of facets becomes the

dominating factor in determining the scattering distribution for this region. In comparingFigures 21 and 22 it can also be seen that the variance is larger for 01 = 750. This would be

expected because the slopes required to cause backscatter would have to be even steeper for thiscase. In the near specular and forward scatter regions, the scattering from the smoothersurface is closer to Rayleigh than the rough surface scattering because there are many more

facets oriented favorably to cause scattering in these regions.Table 3 compares the calculated mean in the backscatter direction with measured values

given by Ulaby 7 and Long. 2 The measured backscatter from soil given by Ulaby is compared tothe calculated value for a dielectric constant of 6r = 30 + J2. The measured backscatter fromArizona desert given by Long Is compared to the backscatter calculated with Er = 2 + J1.6.Seawater measurements from Long are compared with calculations for cr = 80 + J9.

For the higher sea state calculation, corresponding to 16-ft waves, the shadowing function

was included. For pure backscatter. 0, = 0,, the shadowing function is given as:8

S = (I + C2)-'

where

" --4-a 1/2 (-T2 cot 2 02 erfc(T cot 0.(2 2 tan") - erWc2 a (10)

For the parameters used in this calculation, one obtains S = 0.928. This means that even for

high sea states at a 700 incident angle, shadowing is not a significant factor.

7 Ulaby. F.T., Dolbson. M.C. (1989) Radar Scattering Statisttcsfor Terrain. Artech House.Dedham, Massachusetts.

8 Sancer. M.I. (1969) Shadow Corrected Electromagnetic Scattering From a Randomly RoughSurface. IEEE Trans. Antennas Propagat.. AP-17:577-585.

30

C ~E EE0 C'\! r)

_jI I IL.-+

0)

0)

(0 I4

o a)LC) +)

0 >

-o 0~

CY)0

c'Jo

Id0 D 0 O

31

C'*j E Eo E *-*j C~LO

LO~ ci ci11 11 Co

0)

0)

o 7L-

ICD

0 -

w wo

z :4

0 z

.0 u 2

0- 00) OD *-C (0 LO 1 C10 N~ i

32

Table 3. Comparison of Measured YO With Calculated o".

Cr 0111C. a (m) T (m) L (m) a calc. oa meas.VY Vy

30 +J2 300 .1 1 3 -15.96 dB -13.9 dB

2+ J1.6 300 .1 1 3 -23.61 dB -25.75 dB

80 +J9 300 .61 6 12 -19.94 dB -20.0 dB

80 +J9 70" 1.225 6 18 -21.82 dB -31.0 dB

For the given terrain surfaces, as well as the lower sea state, the agreement betweencalculations and measurements Is extremely good. The calculation for the higher sea state,however, does not agree with the measured value, but this is not unexpected. High sea states do

not have a Gaussian height distribution or a Gaussian correlation function, so the model used

in this study cannot accurately describe scattering from very rough seas.

4. ARBITRARY LINEAR POLARIZATION

When the incident field is linearly polarized at some arbitrary angle, ip, from thehorizontal, the calculation of the reflection coefficient becomes somewhat more complicated.

The incident linearly polarized wave can be separated into a horizontally polarized componentand a vertically polarized component. The horizontal and vertical reflection coefficients and

the magnitude of the overall reflection coefficient, IRALI can then be calculated.

'RA ~COS 4f ~~)o t+ (R R7snf,(1

The scattered field will no longer be linearly polarized but will have some ellipticity due to

the unequal phase change of the horizontal and vertical components. The polarization ofthe scattered field can be found by examining the complex polarization factor, p. defined as:

E 'E 1Ep -1 _V cos ++j E sin * (12)EH HI H

where: 6 - phase difference between vertical and horizontal components of the scattered

field

33

ERe(p) = i cos0

EH

Im (p) = YE sin 0.

EH

The magnitude and phase of the components of the scattered field are determined directly fromR, and RH. Table 4 shows the real and imaginary components of p for 6, = 300, X = 0.1 m.o = 0.2 m. L = 3T. vi = 600, and er = 30 + J2. The very small magnitude of Im (p) shows that the

polarization of the scattered field is very close to linear. Case 2 is for Er = : - 1- 6 with y = 500.The scattered field in this case is slightly more elliptical due to the higher relative magnitudeof the dielectric loss factor, s". but it is still highly linearly polarized. These two casesrepresent the highest degree of ellipticity for the values of p that were examined.

Table 4. Complex Polarization Factor

03 Case 1 Case 2

Re (p) Im (p) Re (p) Im (p)

-45 -1.7212 -0.0003 -1.1678 -0.0084

-30 -1.7321 0 -1.1918 0

-15 -1.7212 -0.0003 -1.1678 -0.0084

0 -1.6888 -0.0015 -1.0970 -0.0326

10 -1.6550 -0.0026 -1.0255 -0.0564

20 -1.6116 -0.0039 -0.9360 -0.0849

30 -1.5582 -0.0054 -0.8304 -0.1168

40 -1.6116 -0.0039 -0.9360 -0.0849

50 -1.6550 -0.0026 -1.0255 -0.0564

60 -1.6888 -0.0015 -1.0970 -0.0326

75 -1.7212 -0.0003 -1.1678 -0.0084

90 -1.7321 0 -1.1918 0

Im (p) = 0 Linear polarization p = Vertical polarizationhmn (p) > 0 Right elliptical polarization p = t Right circular polarizationIm (p) < 0 Left elliptical polarization p = -1 Left circular polarizationp = 0 Horizontal polarization

Case 1: Ot = 300. X = 0.1 m, a = 0.2 m. L = 3T =600 , r=30 +J2Case2: 0l=300.X=0.1 m, =0.2m. L=3T, =500,Er =2+j1.6

34

Figures 23 through 25 show the mean and variance of the scattered power as the tilt angle %V

of the linear polarization increases. While the mean does decrease as the polarization goesfrom horizontal, Wy = 00, through several values of Wi and finally to vertical polarization, %y y

900, the tilt angle of the polarization does not affect the Rayleighness of the statistics.

5. CONCLUSIONS

The main objective of this study was to determine whether cell size has any effect on the

mean normalized cross section and variance of the scattered power from a rough surface.

Several other variables were also studied including wavelength, dielectric constant, angle of

incidence, surface roughness. and polarization.

Throughout the figures shown, several general trends can be seen. It is quite clear that the

cell size does affect the scattering statistics. For the largest cell size, L = 12T, the calculated

value of a00 is almost an exact match to the Rayleigh value of a00 over the entire range of

scattering angles. As the size of the cell decreases, the variance of the scattered power becomes

larger than the Rayleigh variance, especially in the backscatter region, due to fewer facets

scattering in this direction. This means that the problem of non-Rayleigh scattering for finite

clutter cells is a more significant problem for monostatic radar systems than for large angle

bistatic radars.

The wavelength also affects the statistics of the scattered power for a surface containing

many scatterers. As the wavelength increases, the Rayleigh parameter, Z, decreases and the

surface becomes less rough, so that the scattering is closer to Rayleigh.

Other parameters stich as dielectric constant and tilt angle of the incident polarization

affect the mean value of the scattered field, but do not affect the Raylelghness of the scattered

field.

35

IL. O0)0

uLJC-J

0 00i 00 <e cc

<1U

zcc 9 ewU 0 1

0V C 0 0 60C 0 0 0 0ý 04Cq

C) (1) cV) 0 0 S- 62 6 ai4) 0c

0 F I~ co

C) t) 0) U') 0 It) 09,ii

(ap) 0.0Pue OD .0 N N c4

0 ID 0-

0 ~0 0 0

0' C]) C1

wj W i

C)V

V. 0

0 It 0 It) 0 I) 0 O Cf) 0 O 0 LO 09

(8P) 00. pue OD mu (Ep) OOD Pue OD L)

36

C> 0 0o C 0 C)

ý o E E E 0 0

~~~-C c' 0 cJ

e',~~' 6CO _ o c~00

00

C-i cli

.00ocv cv v

cr) (Dcr) M'D s 6wc

LU 7-C) 0 Q >

z 1A z 0 0

< < 2 2

cc cc

0 Cl) 6 6.00) 4(0

'n CD 0n a ~ 0ý LO > U)) 03 LO C

(13P) 00 puB OJ '0 (OP) 00D PUB opa

37

CDD

01

cr) -a>

o~-. CD r- 6 . ?- o00

/? 00 0 c

C)1 C) / 7i LOi i

/7V))

CD

0-v oo o o

6r CVD a)

0 I I ' 0C)z z~0I) 0 L A A ~ -

(8< < u 0C a)~pe0

X3

0

0 LO) 0= 0 0Lei N LO) a c; ur) CNý 0 0i C-2 66

Ii clii

0011. 11 1

C).

CD II Iq0 0

7 ~00

00) 0 L U f4

w w-J - >

o~ ~ ~ C)C

(j) Cr 35-0

0 0) LA

LO 0 Ln OT 0O a to)LO O

(EIP) 000 pue OD" (sp) 00 pue, 00C

39

I 0

'00

Co 0 0c

< a 0 0 0) 6 C 0

o) C/ 6f N 0- + +\ +6V C l) c9 C1 6 NI c

a- E E E E W~

-6 P

(UP) C ueOD.

11 I t II I

7 AK (0 Dc/ (AD C9 1 1 f

cC' CF) M

00D pe a)

400

0 0 0 0 C0 w C)o N 9 o 0 0 <

ce) 0 6 '- 0I ce) cr-

o 0 -

*~~~ T+ ~ -

II c11w 1

0 0

0 C)

4 4

(0 C **--

**1

a) 0 0 a0 Le 0 l) 0 LO 0.

10p)00 PU O)1 (8p) 1 00 Put, opC~

41

References

1. Skolnlk, M.I. (1980) Introduction to Radar Systems, McGraw-Hill Book Company, NewYork.

2. Long. M.W. (1975) Radar Reflectivity of Land and Sea, D.C. Heath and Company,Lexington. MA.

3. Papa, R.J., and Woodworth, M.B. (1991) The Mean and Variance of Diffuse ScatteredPower as a Function of Clutter Resolution Cell Size, RADC-TR-91-09.

4. Beckmann. P., and Spizzlchino, A. (1963) The Scattering of Electromagnetic Waves fromRough Surfaces, The MacMillan Co., New York.

5. Ruck, G.T., Barrick, D.E., Stuart. W.D., and Krichbaum, C.K. (1970) Radar Cross SectionHandbook, Vol. 2. Plenum Press, New York.

6. Papoulis, A. (1965) Probability, Random Variables, and Stochastic Processes, McGraw-HillBook Company. New York.

7. Ulaby, F.T.. Dobson, M.C. (1989) Radar Scattering Statistics for Terrain. Artech House,Dedham, Massachusetts.

8. Sancer. M.I. (1969) Shadow Corrected Electromagnetic Scattering From a Randomly RoughSurface, IEEE Trans. Antennas Propagat., AP-17:577-585.

43

ad-a254 2539. mean and variance of scattered power for h-pol for: 16 a....

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