12-8
SURFACE RAY CONTRIBUTION TO BISTATIC RADAR CROSS SECTION
OF A GENERAL PARABOLOID OF REVOLUTION
RM J h a , SA Bokhari, V Sudhakar and PR Mahapatra
D e p a r t m e n t of Aerospace Engineering
Indian Institute of S c i e n c e
Bangalore, 560 012 India
ABSTRACT
A comprehensive method h a s been proposed f o r t h e d e t e r m i n a t i o n of t h e RCS
of general paraboloid of revolution utilizing t h e Geodesic C o n s t a n t Method (CCM)
developed by t h e authors. The analytical expressions h a v e been obtained for a l l
t h e s u r f a c e ray p a r a m e t e r s required in t h e b i s t a t i c RCS c o m p u t a t i o n i n onep a r a m e t e r form. The inclusion of t h e diffraction d u e t o higher-order geodesics
in t h e analysis increases t h e a c c u r a c y of t h e t h e o r e t i c a l RCS predictions.
Introduction
The radar cross section (RCS) is a n i m p o r t a n t p a r a m e t e r for consideration
in t h e design of a e r o s p a c e bodies. The monostatic a n d b i s t a t i c RCS essentially
depend on t h e shapes of t h e a e r o s p a c e vehicles which a r e o f t e n q u i t e complex.
However, t h e s e s t r u c t u r e s a r e essentially hybrids of simple canonical shapes
such a s t h e quadrics like t h e cone, cylinder, s p h e r e a n d ellipsoid etc.
The theoretical study of t h e b i s t a t i c RCS in t h e high frequency domain h a s
been a n a c t i v e a r e a of r e s e a r c h d u e t o i t s potential applications. While t h e
simpler quadrics like t h e r i g h t circular cylinder [ I ] and c o n e [2] h a v e been
extensively studied, no t h e o r e t i c a l study h a s b e e n c a r r i e d o u t f o r t h e RCS of a
general paraboloid of revolution (GPOR). A GPOR is a n i m p o r t a n t s u r f a c e f r o m
aerospace s t r u c t u r e s point of view a n d o f t e n c o n s t i t u t e s t h e leading sections
s u c h a s r a d o m e s in a i r c r a f t , n o s e c o n e i n s a t e l l i t e l a u n c h v e h i c l e s a n d
missiles.
In this paper, we present a method t o t r e a t t h e problem of R C S of a general
paraboloid of revolution based on t h e Geodesic C o n s t a n t Method (GCM) in
conjunction with t h e UTD.
Bistatic Radar Cross Section of GPOR
A general paraboloid of revolution (GPOR) is described (Fig. 1) by a set of
p a r a m e t r i c equation [31
x = au cos
in which
"a"
b
y = au s i n
b
z = -U2
(1)
is a shaping p a r a m e t e r which c o n t r o l s t h e sharpness of t h e GPOR.
In c o n t r a s t t o t h e cone, a GPOR does n o t exhibit t h e t i p singularity, SO
t h a t a tip-diffraction t r e a t m e n t is not required for t h e RCS solution. However,
just like a finite cone, a GPOR f r u s t u m also has a circular rim f r o m which edgeCH2864-2/89/WoC1.0219
$1.00 @lSSS IEEE
219
diffraction would t a k e place. Marhefka e t a1 [ 2 ] h a v e developed a method f o r
predicting t h e edge-diffracted field c o m p o n e n t s which may be readily applied t o
this c a s e as well. Consequently, f r o m t h e high-frequency s c a t t e r i n g point of
view, in this paper we c o n c e n t r a t e on t h e surface-diffracted component of t h e EM
field.
The UTD developed by P a t h a k et a1 [ 4 ] predicts t h e s u r f a c e d i f f r a c t e d field
o n a n a r b i t r a r y c o n v e x s u r f a c e , i n t e r m s of i t s s u r f a c e r a y g e o m e t r i c
parameters. Since, a GPOR f r u s t u m r e t a i n s i t s overall convexity, t h e UTD c a n be
readily applied t o c a l c u l a t e t h e s u r f a c e diffraction from" a GPOR, o n c e a l l t h e
s u r f a c e ray g e o m e t r i c a r e known. The r a y g e o m e t r i c p a r a m e t e r s for a GPOR a r e
presented h e r e in a n e x p l i c i t form.
In t h e high frequency limit, t h e b i s t a t i c RCS s c e n a r i o c a n b e described
(Fig. I ) as t h e propagation of t h e EM field originating f r o m t h e source,
launching on t h e s u r f a c e of t h e GPOR, and finally arriving a t t h e observation
point a f t e r being shed in space. This a r r i v a l could e i t h e r b e by t h e phenomenon
of reflection or by t h e g e n e r a t i o n of s u r f a c e waves which c r e e p along t h e
s u r f a c e of t h e GPOR which shed in f r e e space. The e x t e n d e d Fermat's principle
requires t h a t t h e s e p a t h s be'extremal. Hence, while t h e r a y propagation in f r e e
s p a c e is along s t r a i g h t lines, t h e s u r f a c e propagation on t h e GPOR is along t h e
geodesics.
I t may be observed t h a t t h e feasible s u r f a c e r a y p a t h s could b e both in
It
anti-clockwise (right geodesics) a n d clockwise (left geodesics) directions.
has been shown by t h e a u t h o r s t h a t in c e r t a i n situations t h e contribution of t h e
higher-order geodesics (i.e., geodesics with multiple e n c i r c l e m e n t s round t h e
GPOR) could b e c o m p a r a b l e t o t h o s e by t h e primary geodesics [51. H e n c e a
c o m p l e t e analysis m u s t i n c o r p o r a t e t h e higher-order geodesics a s well which
could b e d e l e t e d only if t h e i r contribution t o t h e t o t a l s u r f a c e ray field h a s
been found t o b e negligible.
The analysis presented h e r e is for t h e m-th order right geodesic denoted by
t h e subscript "rm". Using t h e geodesic coordinate system, t h e equation of t h e
m-th order right geodesic is given as
db
- _
du
+h,((r~~+a~)'/~
au(a2u2-h%1'2
which h a s been i n t e g r a t e d a s [31
T h e First Geodesic C o n s t a n t h
is obtained in t e r m s of t h e launching and
shedding points (e.g. 5 , and P l f % y a s i m p l e univariate search. I t is possible
t o e x p r e s s all t h e s u r f a c e r a y g e o m e t r i c p a r a m e t e r s required for t h e EM field
computation by t h e UTD analysis, in t e r m s of t h e First Geodesic C o n s t a n t hr
alone, leading t o t h e n a m e of this m e t h o d as t h e Geodesic C o n s t a n t Method (GCMY
W e prespnt s o m e of t h e key p a r a m e t e r s without derivation. T h e s u r f a c e normal
v e c t o r N i s expressed as
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Similarly, the unit tangent vector along the right m-th order geodesic i s given
as
t = t x i + t
Y
~
t
j + t Z k
2u ( a2u2-h2m,)
=
au~a2+4u2)1/2
The radius of curvaturep along the geodesic on the CPOR i s derived as
g
G=
-a’(a2+4u2)3/2
2(a4+4h&)
On the other hand, the torsion of the m-th order geodesic Trm i s found to be
Equations(6) and (7) a r e r e q u i r e d i n obtaining several i n t e r a c t i o n dependent ray geometric parameters of which the blendings functions are
examples. The arc length and the generalized Fock parameter for m-th order
right geodesic have already been derived i n Ref. C31. Finally, the specular
point for the reflected ray and the individual launching and shedding points for
the surface rays are determined by the algorithm outlined by Bach i n Ref. ‘f61.
Once these ray geometric parameters are available, the RCS of the general
paraboloid of revolution can be readily determined.
In conclusion, a theoretically comprehensive method for the bistatic radar
cross section calculation of a general paraboloid of revolution (GPOR) frustum
has been presented i n this paper along with the- closed-form solutions for the
surface geometric parameters needed i n these calculations.
REFERENCES
111
-
C.T. Ruck, L E . Barrick, W.D. Stuart and C.K. Krichbaum, Radar Crass
Seaion HandbodG VoL 1. New York, Plenum Press, 1970.
221
[2] R.J. Marhefka and W. Ebihara, "UTD bistatic scattering from a cone
frustum", IEEE AP-S International Symposium and URSI Radio Science Meeting
1987, vol. 2, pp. 1094-1097, Blacksburg, VA, USA, June 15-19, 1987.
[31 R.M. J h a , V. Sudhakar and N. Balakrishnan, "Ray analysis of mutual
coupling between antennas on a general paraboloid of revolution (GPOR)",
Electronics L e t t e r s (GB), vol. 23, pp. 583-584, May 1987.
[41 P.H. Pathak and N. Wang, "Ray analysis of mutual coupling between antennas
on a convex surface", IEEE Trans. Antennas & Propagat. (USA), vol. AP-29,
no. 6, pp. 911-922, Nov. 1981.
[51 K.M. Jha, S.A. Bokhari, V. Sudhakar and N. Balakrishnan,
"Closed-form
expressions for ray geometries on a cone", IEE Conference Proceedings,
Fifth International Conference on Antennas and Propagation,York, U.K.,
ICAP 87, vol. 1, pp. 557-560, 30 March - 2nd April, 1987.
[61 H. B a c h , " E n g i n e e r i n g A p p l i c a t i o n s of T h e G e o m e t r i c a l T h e o r y of
Diffraction", Chap. 6 of Modern Topics in Electromagnetics and Ant-.
R. Mittra, Ed., Herts: P e t e r Peregrinus Ltd., 1977.
SpecuLar
Ref iectiorI\
2ndorder
Right
Geodesic
>('
--E\
1 st-order
Geodesic
Fig. I Bistatic radar cross section (RCS) of a general paraboloid of revolution
(GPOR) frustum i n t h e high frequency domain. Only t h e right (anticlockwise) 1st and second order geodesics have been shown. The complete
solution includes t h e l e f t (clockwise) geodesics of all feasible orders.
The b i s t a t i c RCS of GPOR may also contain t h e specular reflection
contributions.
222