AD-AI71 779 URBAN DISCRETE CLUTTER SOURCES(U) WEN HAMPSHIRE UNIV In1DURHAM K SIVAPRASAD ET AL. JUL 86 SCEE-PDP-83-35RADC-TR-86-78 F38682-Bi-C-8i9

UUCLASSIFIED F/S 17/9 NE1h7h7hiEhhhhhhhhmhhhEEhhhhhhhhhhhhE

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MICROCOPY RESOLUTION TEST CHART

NATIONAL BUREAU Of STANDARDS- 1963-A

i' %

RADC-TR-86-70Final Technical Report

July 196

URBAN DISCRETE CLUTTER SOURCES

University of New Hampshire

DTICELECTE

Kondagunta Sivaprasad SEP 10 1986Allen Drake

* Subramanyam D. Rajan

* API'ROiD FOR PUBLC RELEA" OIWRTION UNLIMITED

ROME AIR DEVELOPMENT CENTERAir Force Systems Command

Griffiss Air Force Base, NY 13441-5700

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17. COSATI CODES IS. SUBJECT TERMS (Continue on rgvww if necmiry and identfy by block number)FIELD GROUP SUB-GROUP" Scattering

17 09 SAR Imagery20 14 odic Surfaces

19. ABSTRACT (Contino on reies if necemiy an identfy by block number)

A possible model for the urban areas viewed by SAR imagery is to consider them as a gridded

periodic structure. Holford's integral equation method has been used to calculate the back-

scattered intensity and the results indicate a variation in intensity as a function of aspect

angle. This has been observed to be the case.

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Table of Contents

I. Introduction 1

II. Review of the Literature 3

1. Kirchoff Method 3

2. Rayleigh Method 4

3. Uretsky's Method 6

4. Method of DeSanto 10

5. Method due to Holford 14

5.1 Dirichlet boundary 14

5.2 Neumann boundary 16

6. Numerical Implementation ofHolford's Method 18

III. Results 23

IV. Conclusions and Recommendations 4:9

DTICd ELFCTE

S EP 10 1986 ).

:!.

List of Figures

Fig. 1 Plane Wave Scattering from a Periodic Surface 9

Fig. l(a-e) Reflection Coefficient vs. Scattering Anglefor Incident Angles 5, 10, 15, 20, 25(Kh = 5,A/X = 10) 24

Fig. 2(a-e) Reflection Coefficient vs. Scattering Anglefor Incident Angles 5, 10, 15, 20, 25(Kh a I0,A/X = 10) 29

Fig. 3(a-e) Reflection Coefficient vs. Scattering Anglefor Incident Angles 5, 10, 15, 20, 25(Kh =15,A/X = 10) 34

Fig. 4(a-e) Reflection Coefficient vs. Scattering Anglefor Incident Angles 5, 10, 15, 20, 25(Kh = l0,A/X = 15) 39

Fig. 5 Reflection Coefficient vs. BackscatteredAngle (Kh = 5,A/X - 10) 44

Fig. 6 Reflection Coefficient vs. BackscatteredAngle (Kh = I0,A/A = 10) 45

Fig. 7 Reflection Coefficient vs. Backscattered 46Angle (Kh = 15,A/A = 10)

Fig. 8 Reflection Coefficient vs. Backscattered 47Angle (Kh = 10,A/X = 15)

-II

* ' U ;

I. Introduction

The synthetic aperture radar on SeaSat Satellite has provided high resolu-

tion synoptic images of the earth's surface. However, the same geographical

regions viewed at different look angles sometimes provided different images.

This especially was the case of the images in the urban areas surrounding the

Los Angeles areas (1). In these images the areas of the bright returns did not

correspond while certain areas of low backscatter were common to both the

images. A study of the causes of this variation in the imagery is the subject

matter of the present report.

A possible model for the study of backscatter from urban areas due to the

synthetic aperture radar is to consider it as a periodic gridded structure with

the streets and the buildings forming the grid. The average dimensions of the

grid for our study must have dimensions large compared to the wavelength.

As part of this task a literature survey was conducted and the principal

relevant references were found to be due to DeSanto (2), Jordan & Lang (2),

Holford (4), Uretsky (5), and McCammon and McDaniel (7). Uretsky's method which

is based on the Helmholtz equation reduces the scattering from a periodic

surface to an infinite set of linear equations of the form [A] [X] = [Y].

DeSanto also arrived at the same result using a different approach. For solving

the infinite set of equations, the method of reduction was employed. In the

method of reduction a truncation of the infinite series is employed and the

correctness of the solution will depend on the order of matrix. However, for

equations of the first kind, there does not exist, at present, a criteria for

, truncation which will assure the convergence of the solution.

Holford's method, like that of Uretsky, is based on the Helmholtz integral

formula and results in a Fredholm integral equation of the second kind for the

scattered intensity. This reduces his integral equation to the form

2

(1+ [A] I [X) - EY]

and rigorous proof exists for the convergence of the solutions. We, therefore,

plan to use the approach due to Holford to study the scattering from a periodic

surface. The only numerical results available using Holford's approach is due

to McCammon and McDaniel and here they consider only relatively shallow struc-

tures, viz, for Kh , 1, where h is the height of the structure, K = 2w/A and A

is the periodicity.

Section II of the report contains a detailed review of the relevant liter-

ature. Section III has the results of the backscatter calculations for a

typical geometry suited for our observations using the exact method due to

Holford. Section IV has the conclusions and the recommendations for future work

on the problem.

1. SeaSat Views of North America, The Caribbean, and Western Europe with Imaging

Radar, JPL Publication, 60-67, California Institute of Technology, Pasadena, CA.

2. J.A. DeSanto (1981), Rad. Sci. 16 p. 1315.

3. A.K. Jordan and R.A. Lang (1979), NRL Report 8284.

4. R.L. Holford (1981) JASA 70 p. 1116.

5. J.L. Uretsky (1966), Ann. Phys. 33 p. 400.

6. D.F. McCammon and S.T. McDaniel (1985), JASA 78 p. 149.

9.,.

3

II. Review of the Literature

The general methods of treating periodic structures can be broadly classi-

fied into two general approaches. The first approach are the so-called approxi-

mate methods and the second are the more exact methods. Sections II-1 and 11-2

review briefly the approximate methods, while Sections 11-3 through 11-5 review

A the more exact methods. In Section 11-6, some of the details, that have to be

considered, for the numerical implementation of the calculations for the Holford's

method are discussed.

II-1 Kirchoff Method

The general Kirchoff Method is based on Helmholtz integral equation.

Consider a plane wave incident on a rough surface. The scattered field at any

observation point P is

E2 (P) - fjJ(E r 3) ds (1)

where the integral is over the surface and E is the field on the surface.

is the free space Green's function (exp(ikR')/R'),

eikR'

R' (2)

where R' is the distance from the origin to P and k = (2,/x). In the above

equation, E is an unknown. The Kirchoff approximation assumes

E- (I + R)E1 and (9E = ( - R)El(k-R) (3)

where is the normal to this surface at P and R is the reflection coefficient

of a smooth plane. In the case of an (,ne dimensional surface where L = (x),

and R = +1, Beckman (1963) arrives at the following for the scattering coef-

ficient, pA Beckmann and A. Spizzichino (1963). "The Scattering of Electromagnetic Waves

from Rough Surfaces", Pergammon Press.

'.24 - .

i4

A/2

0 (19 2 (E2 /O) e, coe +os A-J e iF4dx (4)0~~~~~ ~~ _e,2 E/E)=sc 1 A/2

e8 is the direction of the incident field, E2 is the field reflected in the

direction 82 and

F.:= [ (sine1 - sine 2)x - (cose1 + cose 2) (x)] (5)

This approkimation is valid when A>>x and the radius of curvature of the ir-

regularities is large compared to the wave length. Consequently, this method

will break down when there are sharp edges.

11-2 Rayleigh Method

In the Rayleigh Method, the field is expressed as a-sum of plane waves

whose directions correspond to the scattering angles and for a periodic surface,

the scattering direction will be given by the grating equation. So for a

periodic surface C(x)

S(x) - hcosKx (6)

The grating equation for the field is

sine 2m = sine0 + mK/k 0 = , +1, +2,... (7)

The total field in the region z>;(x), E, (eq. 8), is the sum of the incident

field, El , and the reflected field, E2 9

E 1 + E2

- e F + Ame A 2m'r (8)

, ma

"..

5

where A

2m k(ex sine2m + ezCOSe2m)

K1 =k(ex sine 0 - ez cose)

= (xe x + yey + zeZ )

and •x , ey and ez are the unit vectors in the x,y and z direction. For the case

of the Dirichlet boundary, EIz=(x) =0 . So, from Equations (7) and (8), we

hdve

e ik(x)cosee+imkx + ik (x)cose2m (9)K .M= -=

Expanding both sides of Equation (9) in terms of Bessel functions, Beckman et

al. (1963) arrives at the expression for the coefficient Am

J n(khcose0 ) = J_ (-i)m Am Jn+m(khcose2m) (10).' Ilm= -w=

From this set of equations, the coefficients Am are determined, which are used

in Equation 8 to obtain the field.

Uretsky (1966) notes that the major error in the solution given in Equation

8 is its implication that it describes the field everywhere above the surface.

This has also been pointed out by Lipman (1953) that the solution does not take

into account both up and down going waves excited in the "valley" of periodic

surface. We will now describe two methods of solving the problem, both based on

the integral equation formulation, starting from the Helmholtz equation, which

will take into account both the upgoing and downgoing waves in the periodic

structure.

B.A. Llpman (1953), J. of Op. Soc. Am., 43.

A(4

6

11-3 Uretsky's Method

This section describes the method proposed by Uretsky (1966) for solving

the boundary value problem. For simplicity we consider a two dimensional

problem and the scalar wave equation. The problem reduces to the solution

of the two dimensional equation

a"I2p(x'z) a2P(xz) 2S az2 (11)

1" subject to boundary conditions which can be either soft, i.e., P(x,z) = 0 on the-- " a P (x ,z )

boundary, or hard, i.e., 0, or an impedence boundary condition, i.e.anapTnP + AP = 0 on c(x), where A is a constant. Specifically, we will consider a

surface

Z a r(x) a hcosKx where K - (2w/A) (12)

The field is now the sum of the incident field Pinc(XZ) and the scattered field,

P, given by the Helmholtz integral.

P(x,z) - Pn(xz) +Pr an"

-H(')(kJF-FJ) anr- d (13)

where r = (x', (x-)) and the integral is taken along the boundary. In the

case of a soft boundary, (P(r') - 0)

k *.~ (x) H('1 rJ-r 1 , x (4P(r) -P inc(r) + H47k'' 0x(4

where

T an dxj z.(x)

aP - aP (15)k -. -Z = "(x)

7

As Equation 11 is unaffected by a shift of 21/A and the boundary condition

is also unaffected because it has a periodicity equal to 27/A, we can make the

following expansion for p(x).

" neikan x

i(x) = e n (16)n-w

where

sine = an = (aO + n K/k)

cosi cSn = Yn = (1' n2 )I/ 2; n<-I

= i(l+ov an2 ) 1/ 2 ; a nl (17)

, Substitution of the above in Equation (14) yields (Holford (1981))

P(r) = Pinc(r) + E n T eikamx I m,n(z) (18)". n - n= .o

whereI (Z 1 (k-r) e 2kZdr(9m'n() 21i Cm-n 'r 2 2

"n () 1 e-it (x)inKxdx (20)

0

Holford (1981) has shown that the integral Im,n(z) is given by

S(Z) Cm (ky )eikYmz z>(x) (21)Im,nZ 2y m --

Also the field P(x,z) can be written as_ 00 'k Reknx + ik k~z

P(x,z) = Pinc(x,Z) + I Ren' n n (22)n= -0

Inserting Equation 21 into Equation 18 and comparing with Equation 22, we

obtain

Rm 1 .Y n Cm-n(kYm) (23)

2m n=-o

h€ We now proceed to determine the ipn" In the Equation (18), we let the field

* 8

point, r approach the boundary.

P(r),) Pn(r)l ,x) k f,(x-) H~l)(kjr-r-i)dx- (24)

If we now apply the boundary condition that P(r) 0 we have

k

f J *(x-) H(') (kir-r-i)dx" = - Pn ) (25)

Substituting Equation (16) in Equation (25)

k r e ikan X') H l) (kp)dx' - -Pinc(X) (26)

where U- [(x-xj)2 + ((x) - ;(x'))2 1 /2

Applying the operator

v*. AIl k-i kamXdx (27)

0 ik x-iky z

to both sides of Equation (26), we obtain on substituting, Pinc(xz) = e

from Eq. 8.

n Vm -2C(ky) m =0, +1,2, ... (28)

where Vm,n A

1 etkCmXdx f H 1 (ku)e+lk nx dx" (29)

0

This set of linear equations (Equation 28) is used to solve for 4n which is

then substituted in (Equation 23) to obtain R

Note that the set of linear equations is of the form AX a Y. The procedure

recommended by Uretsky (1966) is to truncate the set at some value n-N and then

solve the linear equation by matrix inversion. However as Holford (1981) has

pointed out there is no guarantee that the solution for AX * Y will converge as

S7'*p*%***> * -. , V. "N.

z

aon

"-L-U2 U2 L

s .3 5

Fig. 1. Plane wave scattering from a surface s(x) of periodL and depth d. The closed contour (12345 1) is used for Green'stheorem. The incidence angle is 0,, and e,, is one of the discretescattering angles. Diagram lines are slightly offset for clarity.

pi

.1I

.d,. . %

.9.

10

imn- or will converge to a correct solution. This method is valid only for

equations of the second kind represented by (I + A)X * Y where I is an identity

operator. Holford's method of overcoming this will be discussed later in

Section HI-5.

11-4 Method of Desanto

Desanto (1981) has proposed another approach which is similar to that of

Uretsky. For a sinusoidal surface shown in Figure 1 and a plane wave incident

on the surface.

,*i(x,z) - e ikaox iky0Z (30)

where

" 80 - incident angle

a 0 - sineO,

YO a coseo

The reflected field is then

*r(x,z) " Z kanX kYnz (31)

r nnwhere

sinen a On = ao + n X/A - 0 + n K/k n 0, +1, +2,...Sn n o0cosen " n

The Green's function

G{-x,z) - (e(0koYmzko mX)) (32)

satisfies the wave equation in the region z>o. We now consider the contour

12345 in Figure 1 and apply the Green's theorem to the contour

f and 2C- -G±l) 4ann (33)

Si~ 5 . .. *~5 and** I

11

*(x,z) = *inc(x,z) + AneikoonX + tkynZ (34)

If we now let 23 and 51 shrink to zero, then the integral along the path 12

becomes+A/2

112 = I {+ikoymeiko(a°-m)x + ikoy; Aneik°(an-"m ) + ikoyoeik°(a°t@cm)x

-A/2

- ik0 jA.yneiko(anam)x} dx (35)n ,

For G+ along 1-2, Eq. 35 reduces to

ll2 = -4iriym(A/X)6mo (36)

and for G" along 1-2, Eq. 35 simplifies to

I12= (4-m ) Am (37)

Hence

yA )-amo for G+

I12 4 m- Am for G (38)

The integral along 23 and 51 are zero since we have allowed 2 to approach 3 and

5 to approach 1. The equation simplifies to

f - G-t -d + J an G " A) du 0 (39)1

345

or

I !'n -G n ) dU O -g- - G n ) dp (40)

35n 12

A )I am{ for +(4wiym No l-AmJ for'-

55 .S5.

12

We now need to evaluate the left hand side of the equation. Let us assume

Dirichlet boundary condition.

( dii - -( 4 riy A mo (41)mAo Am for -

Following the method of Uretsky, we now replace 2- O(xC(x))du by *(x)dx, where O(x)

is a periodic function with a period A.

2w e ikooXB#(x) = W- ex) (42)

where B(x) a 2 exp[-ik0 dy (1 + cos(2/A)x)] nan exp(in2wx/A). (42a)nou-

Substituting Eq. 42 to the right hand side of Eq. 41, we obtainf G±x,r.(x)) it du

an345

A/2 xek~x~dSG:x(x)) 2w ekoX(x)dx (43)

Now,.let c(x) - 1 + cos 2--) and let y - x,rAG±tx,C(x)) A du

345

O.* ;*~(Ixodym/2) Je ;(ik 0dycosy/2 - im3#)C(y)dy (44)

wherewhereCly) - Bl(;-y)

Using equation 41, we obtain for G+

r C(y)e(-imy - ikodmcosy/2) e m kod/ 2Ym) 6 (45)0, • mo

.0

"S, ,, . -.. "..... .,. ", .. , , .'. : .:. .. .., -. . [ -.'.'...

13

and for G-

F. C(Y) exp[-imy + i(kam cos y/2)] dy - -AM(4 7dy/X 0 exp( -ikdy2)(IT (46)

From Eqs. 44(a) and (42a)

2iy AC(y) =-(-- °)e(Oko dyo0 + cosy)/2) l + n inaneiny} (47)

Substituting for C(y) from Equation 47 in Equation (45), then yields the set of

linear equations for m#O

Jm(Tm) + I an Jmn(T ) = 0 (48)mm n#on - m

Similarly, substitution in Equation (46) gives

J.Y 1)M++A =- (0) Imexp(iT+) {J (T ) + Y a Jm(m ) (49)4M m m mm) an m-n rml

where-'+ kod

T m = (Y + Y ) 0

m a-in 2

We, therefore, solve the first set of linear equations to obtain an which is

then used in Equation 49 to obtain the Am.

Jordan and Lang (1979) solved the Equations 48 and 49 numerically. They

state that no numerical instability was experienced and the stability of the

method was checked by testing conservation of energy (i.e) JIRnI 2 Yn= 1 forn YO

real orders of n. Wirgin (1980) reports that the method of Desanto was also

used by Whitman and Schwering (1977) who reported no instability for Kh < I and

Kh < 2.36. However, in their experience this method does not work well for

Kh>,fw. Further Wirgin states that the solution will become unstable for Kh>i.

A. Wirgin, (1980), JASA 68, p. 692.

G. Whitman and S. Schwering, (1977), IEEE. PGAP 25, p. 869.

14

Though Jordan and Lang reported no instability in their numerical computation,

as the equations are of the form AX : Y, convergence to the correct solution

cannot be guaranteed..

11-5 Method Due to Holford

Holford (1981) has developed a method to overcome this problem. This

method is similar to the approach of Uretsky, where we start with the two

dimensional boundary value problem and seek a solution to the differential

equation satisfying the given boundary conditions on the surface and the radia-

tive condition at x2 + z2 - .

11-5-1: Surface Satisfying the Dirichlet Boundary Condition:

Applying the Helmholtz integral formula as in Section 11-3 we obtain for

the Dirichlet boundary condition,

P(r) P Pinc (r) + j (x-)Ho(1)(kF-F-j) dx' (50)

where*ix) - as (51)

k n x) Z.;(x) (1

--, Let the normal derivative operator ik'1- 2- + ;-(x) x] be applied to both

-. . sides and taking on the limit when z * (x), we obtain

*(x) OW + *(x-) C-2+ (x) a o H)(kF-F-)dx (52)

It has been shown by Meecham (1956) that the integrand on the right hand side

1has a singularity at F F - and this integrates to a term V(x) in the limit.

W.C. Meecham (1956), J. Rat. Mech. Analysis 5p. 323.

15

Thus, we have

1 ik 7*x- 1~(I-I (53)i= j(x) -4 Ix-

[(c(x') - (x)) - (x'-x)i'(x)]dx

where

P(x)=(+ik-1 ) - ;-+ c'(x) 2-]e ik 0 X - iky z

= 0 B'o + 0'(x)]e ika 0x - iky0 (x) (54)

We therefore obtain

O(x) + 2. J* (x') [( (x') - (x)) -c(x'-x)c'(x)]dx

= 2i(x) (55)

As explained in Section II-III, *(x) is periodic with the period A. So

F(x) = ZPne ikanx = * qineik(O+nK/k)x

nf-m no-=

= eikax JneinKx (56)n

Substituting for *(x) in the above expression

n ineikcnx + e ik nX'(H(')(kF-P)/Ir-r'l)

n ,-

[( (xl) - ;(x)) - (x'-x);'(x)]dx' = 2*(x) (57)

We now apply the operator

Af e ik ctx dxR (58)0

o~ ~~

="I

16

to both sides of the equation and obtain

',m+ Z ,nUmn =2m (59)

where Urnn A .1I e'ikamx dx IeikanX" HIF.-II(kIF'F))

0 --i

- (x)- (xx~x)x(60)andA

m 1 e- x ikamx dx

0

A1 ka x - I ky W~x (61)

A a( +0 (x))e 0 o dx

0

This set of linear equations (Eq. 59) is now solved to determine 'n and then ,ix).

This is then used to obtain P(). The specular reflection coefficients for

various orders of scattering is obtained from Equation (23) which is reproduced

below

as~'m " *n C m-n (ky M) (62)

11-5-2: Surface Satisfying the Neumann Boundary Condition-*

Applying the boundary condition 0 0, in Eq. 13, we obtainan

f( o~n- --2 [H( ) k x,-P ( r ) .- P i n c ( r ) + 4 T - 4 a( x - ' ( k j ' ) z " - ( x ') ( 6 3

"" (63)

where .(x') - P(xA,;(xi)). We now let r [x, (x)], then using the same

method as was used in deriving Eq. 52, we obtain

'I..

•.

17

00

W 4(x 1 OW)+-a () 1-00

[.(x) - 4(x')) - (x-x')'(x')] dx" (64)

k i HW ) C(x,x-)dx' (64a)O(x) = 2O(x) + - (x)

where

C(x,x') = [(4(x) - (x)) - (x-xlC,(xi)] (65)

U= [x-x') 2 + (x(x)-x(x'))2]11/2

where (x) = Pinc(x, 4(x)). We now write

*(x) = *nelkanx (66)nl=-a

Substituting in equation (64a), we obtain

ika ikanX') ik H1(1)(k,)

on - C (. flkc'r1xn 1k ax-= 20(x) (67)n -OD

AApplyig the perato f. e-ikm~dx to both sides, we obtain

0a A - [)Hjl)(k,)

n " fl e ik'mxdx {" (k ueikanx ) - 3 C(x,x') dx"n A nn

0

Cm " !nUm,n = Cm(69)

where Um~ is given by

Aa1 I e'Ik~mX ik H )k) ix

Urnmn = I dx J - e C(x,x) dx (70)

0 -

' ( ) - .

18

The set of linear equations obtained can now be solved to yield o n which then

can be substituted in the Equation 63 to yield P(r). The reflection coefficient

for this case is given by the following expression (Holford).

R C (K-y Xy (m-n)Ka M)(1m n n Cmn(KYm)(Ym + ky (71)

We note that in this case, the equation for on is in form (I + A)X = Y and

therefore, convergence is guaranteed.

" 11-6 Numerical Implementation of Holford Method

We will now describe the numerical implementation of the method due to

Holford, as we will be discussing the results in the following Section III.

Specifically we will consider the case where the surface is a sinusoidal surface

and satisfies Neumann conditions. We, therefore, have

;(x) - h cos Kx

The linear set of equations to be solved is

Om" On 4m,n = $m m O, + , . (72)

where Vm,n is given by

imin AV fn dx eim x Jk e i an~ X' I k~-(

(C(x) - (x) - (x - xl)(xi)) dx, (73)

Where,: ,,1/2

. E(x - x.)2 + ((x) - '(x')) ] (74)

Let T - (x-x)

A-

mn 1J e-tk m Xdx e i kanx' K('r,x')dx' (75)SVm'n "A" 0 -

0 .,

19

where

K(t,x-) iLkh H ((cosKx - cosKx') + KTsinKx') (76)2. P

and

p [T2 + h(cosKx - cosKx-) 1 /2

Then,A

Vmn= e feika nX, dx e- ika mx K(-r,x')dxVm, n A f

O -0

AW= 1 J e'ik(am-an 'dx f e-ikamX e+ikamx' K(t,x')dx-

A -

= e- (m-n) KX'dx f e-ikamTK(,r,x)dT (77)A

where k(a -an) = K(m-n)

and K(T,x') is given by Equation (75). To determine Vm,n , we consider Vm, n as

Vm,n Vm,n +(Vm,n - Vm,n) (78)

where Vm,n is obtained using K(T, x') in the Equation (74) K(T,x') is different frol

K(T,x') in that P is set equal to T. (i.e) the cosine terms in P are neglected.

This manipulation will simplify the final computation of Vm,n .

A C

e m,n ei(m-n)KX-dx' e-ikaMK(T,x')dT

0 -00

A 0I [e-iaKxx , e-ib4(T,x)dr

O -

0 CA

I~f e~' ibd e-iaKx' (T,xi)dx' (79)

:," ',w "',. , w- " ]-",,! "; ", , a T , . ,-,,' , ,-v,, .. , , ~ . ,., . ,,,,, ,, . ,, .'', " 'A'rFwF,,,- n T o ,

% j 'j - _ ...

20

ik H,1 (kj T )K(T,x) = - IpI E (x+ =) - c(x) - (80)

where a = (m-n) and b = -kam

Integrating with respect to x',

A k H (1)*" . l ibtdr i i ' (k)iax

Vm,n = j ed i E(x'+) - (x') - T'(x')] e-ialx ' dx"-- A (81)

Let A

a l J(x-)e 1iaKx-dx' (82)+-); {a A f x (2

0

Then from the periodicity of the terms in the integrand, we have

Hl( 1 )(kIjT)I' ITI [ eaKe -iaKr)] d-r (83)

This integral is in the form of Weber-Schasheitlin type (Watson 1944) and it

gives

m,n a i 2 (84)a (k2 .b2)1/2 + / - (b + aK) - /k(84)

Substituting for a and b, we have4. - (n-m)Kam

V,.n = I E kym + kyn] (85).rnV, n rn-n rn

In the case of a pure sinusoid, of the form hcosKx, the fourier coefficients are

h/2 for a * + 1.'.'::: .(n-n)lK m

:-" V +~ k(yn-Ym] Im-ni = 1*:' Vm,n I [ Y

0 Im-nI 0 1 (86)

G.N. Watson. A Treatise on Bessel Functions, Cambridge Univ. PressCanbridge, England, 1944.

44 - *. . .-." . . ** -~4

21

The next step is to evaluate (Vm,n - V m,n) = Vdm,n

A C

v d 1 e-ika mrx dx J e+ikannXf(x,x') dx' (87);m,n T A0.o

Hl1 )(kp) H,(1 (kP)

f(x,x) = ik [ ) (((x) - (x')) - (x-x-);'(x'))dx'

22

P = [(x-x') 2 + (;(x) - ;(x')) 2]11/2 (88)

= (x-x)

writing the integral over the infinite limits as a sum

A A

Vdmn =1-Je'ikamx dxA eikanxf(xxA + qA) dx-

o q=-.oo*A A

e n ( f(x,x' + qA) dx-) (89)

0 0 q=-

A A

e-ika0x-imkx dx eikaox' + iknx'( O f(x,x' + qn) dx-

o 0 q=-o

A A= e - i mkx dx f eiknx'f(x,x-)dx-

where 0 0

f(x,x') = e'ikao(x - x ' ) .f(x,x, + qA). (90)q

We can easily evaluate f(x,x-) and determine its fourier coefficients corres-

ponding to m and n. We will show that, as the point x-'x and q = 0, f(x,x')

is bounded. To show this, we take a sinusoidal surface represented by C(x) =

hcosKx(1 )'(kp) H(1k)

f(x,x-) = ( Hl_ _ -- 1) [(h cosKx - h cosKx') + h(x-x')KsinKx]p

(91

writing x = x' + T as before

22

f(xixi (P[h cosK (x- + T) - h cosKxl + hKrsinKx'] (92)fxx) (- p -

.'v. as x-x', T-0rO

h(cos(Kx- + KT) - cosKx' + KrsinKx'i

a h[cosKx'-cosK - sinKx'sinKr - cosKx° + KrsinKx'] (93)W 1,22

aT). Therefore, the right-! " For small values of r, sinKr - Kr and COSKr* "(1 - ') Thrfeteig -

hand side of Equation 93 now becomes C-cosKx-(hK2T2/2)]. By a similar argument,

we write for small values of T

pM T(I + h2K sin 2Kx-' 2 (94)

Also for small argument

Substituting all these in the expression for f(x,xo) we obtain

f(xxd) h3K4 cosKx" sin 2 Kx (96)as xbx iitk(l + h2K2sin 2Kx-)

q-11o

f(x,x') can therefore be evaluated without difficulty and a 2D FFT is performedto obtain the elements of the matrix A

m,n*

From Equation 71, we have

[I " 9 d]; 2 (97)

The expression for V has a singularity at ym 0 and therefore we multiplymn

each equation by the corresponding y m

- - ;Vd1; - 2y* (98)

where is a diagonal matrix containing ym along the diagonal.

The method of Holford can also be used to calculate the scattering strength

due to a beam of finite bandwidth by the following scheme.

23

The incident plane wave is split into its fourier components given by

B(a 0) = 2i Pinc (x,k)e-ikaondx (99)

The resultant scattered field is computed from

P(xZ)= I B(ao) Pscat(Xz;ao) do (100)

where Pscat (X,Zc 0 ) is the scatter field due to unit amplitude plane wave

incident upon the surface with a direction cosine 0 "

III. Results

A possible model for study of the urban areas with a SAR would be to model

the surface as a rectangular periodic surface with a period of 30m and a height

of abbut 15m. For an incident wave length of above .235m, the number of scattering

angles will be of the order of 300 and a corresponding 300 x 300 matrix for

obtaining the scattering intensities at these angles. Because of such a large

matrix inversion, a smaller period for the surface was picked. It is felt that

this scaled down model will also exhibit the principal characteristics of the

original model we are interested in. The computation involves estimating the

scatter intensities at discrete angles given by the grating equation using the

exact method due to Holford. This was done at different incident angles ranging

from 50 to 250, with the corresponding back scatter directions given by -50 to -250

respectively. Higher angles were not considered as bulk of the energy was

scattered in the forward direction at these angles. Also the look angles for SAR

is in the region 200 + 3.

The issues in the numerical implementation of the scheme are

(a) The infinite sum in Equation (2.89) can be approximated by a finite

number of terms and a length of IOA was sufficient to obtain con-

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24

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48

vergent results

(b) Nyquist criteria was used for sampling while performing the 2-D FFT.

(c) A rectangular window for the FFT did not perform well, while Hanning

window significantly improved the results.

(d) Conservation of energy was used as a test to check the validity of the

results and except for one case, the energy conservation was not

violated. In that case (50 angle), the reflection coefficient was

slightly greater than one at scattering angles close to grazing

and less than one at other angles.

Figures 1 to 4, the reflected energy is given as a function of the incident

angle (e) for various scattered angles. The back scattered angle is -e. As the

incident angle increases the back scattered intensity decreases significantly.

* Figure (5-7) shows the back scattered energy as a function of the incident angle

for Kh = 5,10,15 and A/X - 10 and Figure 8 for Kh = 10 and A/X = I5. The back

scattered energy decreases rapidly to insignificant levels beyond 250. However

• ?it seems to be having a minimum between 0 and 250. This may be a possible

explanation for the observed variation in the intensity of the two SAR images in

the LA area.

For the same Kh = 10, as A/X increases, (Figures 6 and 8) the reflected

energy decreases significantly. This indicates that the fundamental component

of the spatial structure contributes a significant amount of energy.

,%

49

IV. Conclusions and Recommendations

The numerical results indicate that there is a definite variation in the

back scattered intensity as a function of the aspect angle. However in order

for us to do a more realistic model, we would need the facilities of a large

computer like the CRAY, due to the computer intensive nature of the calculation.

In any future work, a request for such facilities has to be made, and only then• ..

can one compare the actual observations from the SeaSat images with the model

calculations.

Modelling of the back scatter as being from a series of regularly spaced

* -" flat rectangular radiators parallel to the earth surface was not pursued as it

was felt that significant errors would result from neglecting the height of the

-buildings. To include this effect, one would have to use some modified aspects

of ray tracing and GTD corrections to rays. This represents another alternate

approach that needs to be investigated and numerically implemented.

In order for us to get a more quantitative feel for the back scattered

radiation, it would be useful to perform experiments on scaled down models using

laser beams and optical frequencies. Measurements of optical backscatter can be

made by modelling the city as a grating or a crossed grating and simulating the

satellite radar with coherent optical radiation. The ratio of the optical

wavelength to the length of the gratings periodicity will be equal to the ratio

of the radar's wavelength and the dimensions of a city block.

MISSIONOf

Romne Air Development CenterRAVC ptanA and exewcue4~ te&eatch, devetopment, tezstand Aetected acquiL&Zion pxogitam6 in 4uppoxt o64 Command, Cont.'wZ, Commntication4 and Intettigence(C31) actvit4.. Techn.Leat and eny.Lneeting6Auppo~'t within axea& oj competence Zis pxovided toESV Pkogtai O6jcu (POh) and othe' ESV eteinent.6to peA%6o~m ejjectZve acqui.Lat.Lon o6 CJ! AytemA.The a~'eaA oj tecknicat competence inctudecommuni.cationA, command and conttot, batttemanagement, indo4mation p/Loeh4Zng, AWuxettance4enA6OA, intettigence data cottection and kandting,Aotid 6tate AcienceA, etect'tomagnetc,6, andp4opagation, and etct~onic, maintaLnabittyr,and cormpatbiLtt.

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