Extension of slope-facet model of radar backscatter from se

Rouse, J. W., Jr.

English

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Technical Report RSC-11
AN EXTENSION OF THE SLOPE-FACET MODEL
OF RADAR BACKSCATTER FROM THE SEA
by
John W. Rouse, Jr.
September, 1970
supported by
National Aeronautics and Space Administration
NASA Grant 6.
	
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Technical Report RSC-11
AN EXTENSION OF THE SLOPE-FACET MODEL OF
RADAR BACKSCATTER FROM THE SEA
by
John W. Rouse, Jr.
INTRODUCTION
The slope-facet model developed by Katzin
(1957) is frequently referenced in the literature be-
cause of the unique approach he employed, however it
is seleom employed in practice because of the several
restrictions inherent in his results. This paper pre-
cents an extension of Katzin's work which consists of
a new approach to handling slope variations. In so
doing an expression is developed for the normalized
radar cross section which (1) apparently applies for
all angles removed from the vertical, (2) expresses the
angular dependence as related to the wavelength depen-
dence, (3) is a function of both the mean sea slope and
the standard deviation of the slopes, and (4) provides
a mechanism for calculating the upwind-downwind ratio
as a function of slope statistics.
Technical Report RSC-11
AN EXTENSION OF THE SLOPE-FACET MODEL OF
RADAR BACKS CATTE R FROM THE SEA
by
John W. Rouse, Jr.
INTRODUCTION
The slope-facet model developed by Katzin
(1957) is frequently referenced in the literature be-
cause of the unique approach he employed, however it
is seldom employed in practice because of the several
restrictions inherent in his results. This paper pre-
sents an extension of Katzin's work which consists of
a new approach to handling slope variations. In so
doing an expression is developed for the normalized
radar cross section which (1) apparently applies for
all angles removed from the vertical, (2) expresses the
angular dependence as related to the wavelength depen-
dence, (3) is a function of both the mean sea slope and
the standard deviation of the slopes, and (4) provides
a mechanism for calculating the upwind-downwind ratio
as a function of slope statistics .
I
KATZ TN' S FACET-SLOPE MODEL
The scattering mechanism which Katzin develops
consists of modeling the sea surface using perfectly
conducting facets, that is, the surface of the sea is
considered to be the superposition of facets of various
sizes with orientations distributed about the mean sea
contour. The facets are assumed to move randomly.
The theory is developed for small depression
angles because of the comparative ease of dealing with
the scattering equations. However, Katzin advances that
the general approach is applicable to all depression
angles.
At the offset of this development, four basic
assumptions are made.
1. Backs catter due to diffraction is neglected.
2. The phases of the waves backscattered by the
facets vary randomly, so that the backscattered
rowers are additive.
3. The facet size and slope are statistically in-
dependent of each other.
4. The distribution of both size and slope embrace
a continuous range .
2
3The development consists of obtaining a radar
backscatter cross section per unit area, & 0 , for the
slope-facet model based on the above assumptions.
The backscatter cross section per unit is
defined as:
where:
	
N f = number of scatters per unit area
E(Fa) = mean value of the average cross section
of each scatterer.
The average scattering cross section is a
function of both the area of each facet and of its
orientation (slope), hence:
A ) o-	 (2)
where: p(s,A) = joint probability density function
Since it has been assumed that slope and area are inde-
pendent, substituting (2) into (1) yields:
(1)
.0
00 a =// WA ) Ip (a) 4^ (a, R ) dA ds 	 (3)
r4
where: N(A) = Nfp(A) = number density of facet areas.
The solution of (3) is obtained by (a) de-
termining the average scattering cross section, Q (s ,A) ,
of specific types of flat plates, (b) determining a
number density of facet areas consistant with the empir-
ical measurements, and (c) employing a probability density
function, p (s) , for the sea determined by some indepen-
dent means. The latter step in Katzin's work involves
incorporating the probability density functions obtained
by Cox and Munk (1954) and as such is incidental to the
purpose of this report.
Faced with the problem of determining the
scattering cross-section of various size facets, Katzin
introduced an ingenious idea. The techniques for calcu-
lating the scattering cross-section for flat areas which
are either large or small relative to the incident signed
wavelength are well. known. However, for areas on the
order of a wavelength, no easily manageable methods are
available. Therefore Katzin defined two distinct regions
in which facets were either all "large" or all "small".
In the region where facets are "small" the scattering
cross section and scattering coefficient, Q°, increases
as the facet size increases. In I;he region where facets
5are large, the number of possible facets decreases as
their area increases and hence the scatte-ring coefficient,
a' decreases. The transition between these two regions is
assumed to occur smoothly at some facet area A_ ' The pro-
cedure is to integrate (3) with respect to A independently
over the two reg:i.ons and then sum the results.
The average scattering  cross - se ct:ion from a
circular disc which is large relative to a wavelength is
v ,	 t ^6^ A 4 ^/"	 (4)I
wha re
4*7to A O ice
for a disc which is small relative to a wavelength
where
co s z -9y12 .at
The transition point is obtained when (4) and (5) are
equal that is
6EXTENSION OF SLOPE-FACET  MODE L
The following introduces a modification to
Katzin's method of handling the integration of (3) rela-
tive to slope variations. First Katzin's area integration
operation is reviewed and then the point of de j arture from
Katzin is explained.
The basic relationship of concern is
The area
00 	 421
 integration is handled independently
0.	 C3) r
	
o1A
-3 	 L 3 	 J
or
01 =	 PCs 104) dS	 (A- 1)
In order to evaluate I(A), some functional relationships
must be employed to represent N(A) and Q (s,A) . Following
Katzin
/4 ) = / V A _ <n t 4,/Z
and
o
. __
--A,
1
hence	
A
dig
Ao
As
a Iq
q^	 (A- 2)
T (,q ^' ^t 2 C^)	 (A- 3)
Consider Il(A)
^^) a /^Q
	^ < ®^ / F _ ^ / j^ r a t f ^/
AQ
Cr No a ^^J	A	 —	 (A- 4)
If (2 - n/2) > 0, i.e. n < 4, then I 1 (A) is finite,
and AO can be assumed to be zero under the assumption
that the contribution to I 1 (A) from very small facets
is negligible. Note however that ri cannot exceed 4
otherwise 1 1 (A) does not converge.
Using Kat.zin's value for A l , i.e. A l =
(f /f2) 2/5 x2
it M-) = A /* ^a
	
14
	 (A-5)
In a similar manner it is easily shown that
1/t)	 (A-6)Y p ^/^
If (-n/2 - 1/2) < 0, i.e. n > -1, then I2(A)
is finite, and A2 can be assumed to be infinity under
the assumption that the number density of facet slopes
that are large approaches zero, hence the contributions
to I 2 (A) for large areas is negligible. Note however
that n must not be less than -1 or I 2 (A) will not con-
verge. Consequently in Katzin method, n is	 restricted
to the range -1 < n < 4.
Substituting for A l as before
(j q^- 	 ^j 7'	 s
a
8
-Z^
n^ ^7!*
z	 ^1
Na Ft O)	 n	 (A- 8)
From Katzin,
=-
-/
<4rf 3.';L ) 4.44. z O scc 0	 (A- 9)
Y Vq + 0/.z) $ (A _ 10)
The problem now is to evaluate the slope
integral
©' 0	 <S	 .^"l^9 , q^s 	 (A-11)
9
10
Katzin handles this problem by employing the following
assumptions
SeC	 -- /	 (A-12)
+ Cos
	 (A-13 )
That is, he evaluated a° for small 6, (A-12), and
assumed that the slope dependence for small facets is
not a major factor (A-13) . This would seem reasonable
except for the fact that subsequently Katzin attempts
to evaluate the integral (A-11) for a particular p (s)
by employing slope limits from - oo to + oo. This is ap-
parently justified because p(s) is a filtering function
which suppresses the contributions to a' for slopes out-
side a select range.
In view of the fact that the method of dealing
with the slope integral is somewhat arbitrary, which is
understandable since little is known about the slope
variations of an actual surface, it seems reasonable to
deal with (A-11) in much the same way as was done to
obtain I(A),  (A-2). That is, assume that there exists
two distinct slope behavioral regions synonomous with
the two area regions. This may be expressed as
F
11
S,	 AZ
	
AAMA
	 (A-141
s.	 eqd
To handle (A-14) requires a slightly different
approach when expressing a (s,A) . It is arbitrarily
assumed that
e	 /
/	 ^	 4 
d'z - k2 S 	 A	 /	 e	 (A-16)
where r and t are positive constants. The relationships
are implied in hatzin's work, where t=0 and r=2. These
expressions, (A-15) and (A-16), show that the scattering
cross-section is somehow dependent upon the slope of
the facet, and it is assumed that the dependence can be
expressed as a coefficient of s just as the area de-
pendence was expressed as a coefficient of A; no assump-
tion is made as to the values of r or t except that they
are positive; raor are k  or k 2 known. Using (A-15) and
(A-16) the transition area, A. l , becomes
12
Z/5	 /
/S Y	 ^2
^^ o
(A--17)
Equation (A-6) becomes
r, rg ) r	 o2 Mo	 (A-18)
Equation (A-7) becomes
_p{.l
1'ft,I
-	 Y10 e s ^,	 ^	 3s-CS/(w-t/^-h (A-19)
and the sum reduces to
J
/O A4	 S	 '^	 /^^ s ^^ s CA -20)
Consequently
O' 0 :J,,P (S ) _rig) v(3 c /<'	 'A	 (A-21)
13
whe re
r (4 — m) + f ( m+/)	 (A-22)
5
In the same manner that N(A) is continuous
and independent of the two area regions on each side
of A 1 , p(s) is continuous regardless of which facet
region is being considered. Based on the measurements
of Cox and Munk (1954) it is reasonable to assume that
p(s) is Gaussian. According to Cox and Munk the mean
of the distribution is near zero for crosswind slopes
and slightly removed from zero for upwind or downwind
slopes. Regardless of the exact nature of p(s), it is
safe to assume that the integral (A-21) converges re-
gardless of the values of r and t, because p(s) must
approach zero for finite values of s. Since p(s) does
serve as a "filter" function in a bandpass sense, the
limits on s can be extended so that so = -^ and s 2 = +^
in (A- 14) .
However, it does not seem reasonable to give
equal weight to both positive and negative slopes. Clearly
for large facets the areas having a positive slope will
01
have a greater influence on a' than those with nega-
tive slopes. This will also be true for small facets,
but possibly to a lesser extent. However, it has been
noted that t will likely be small so that for all prac-
tical purposes the negative slopes can be excluded.
Consequently it will be assumed that p(s) is a Gaussian
function, and that the limits s  = 0 to s 2 = +- enclose
the region of primary concern for determining the scat-
tering coefficient. Assume
P (S) • 	 (/'s-,) t	 (A-23)
^TTO-s	 zQ,i
whe re:
	
Q 2 = variance of the slopes
Ti = me an
Using (A-23) , the integral (A-21) reduces to
O'er ^ K P tla+^) ^ (^tt/^ ^i` ^ ^^a^^XP^^^Yo+/
arras	 (A-24)
14
for the limits zero to infinity. D_ (a+1) ( - n/Q) is
the Parabolic Cylinder Function. The mean of the dis-
tribution, n, is approximately zero for the crosswind
direction, and is generally small for the upwind, down-
wind directions. The mean may be positive or negative
depending upon the wind direction. For a zero mean
and r = 2, t = 0, and n = 1, (A-24) reduces to
15
o	 •
P
/,i (r d. Z
(A-25)
From (A-20) and (A-24) the scattering co-
efficient is found to be
(A-26)
- (a -t o	 `
It will be assumed that k 1 and k 2 can be
related to f l , (A-9) , and f 2 , (A-10) , by an constant.
These constants combine with No and will be denoted as
N 1 . It is further assumed for the following calculations
that the angle 8 in (A-9) and (A-10) is the depression
angle. This assumption is not strictly true, but it is
r
I16
reasonably valid on the average and is consistant with
the approach used by Katzin.
RESULTS
In figure 1 the function (A-26) is plotted
for X-band using r = 2, t = 0, and n = 0, 1, 2. The
normalizing factor has been arbitrarily adjusted for
convenience such that the curves join at 85° depression
angle. Actually the curves are displaced vertically
approximately 10 db at 85* for each unity charge in n.
The values of r and t used for figure 1 were
selected to agree with Katzin. In figure 2 the curves
for n = 2 are shown for the four combinations of r = 1, 2
and t = 0, 1. In general the effect is a vertical shift,
and it is to be noted that the changes in r and t do not
appreciably effect the angular dependence of the nor-
malized radar cross section.
In figure 3 are shown data points reported
by Guinard and Daley (1970) obtained with an X-band,
vertical polarization radar over relatively high sea-
states near Iceland. It is evident that the curve for
n = 0 provides a reasonably good fit to the data,
17
especially for the mid-angles. The fact that (A-26)
led to a value of n =; 0 when fit to 'the angular depen.-
dence of actual sea clutter measurements is important
since two recently reported radar experiments have con-
firmed that high windspeed sea clutter is wavelength in-
dependent (Wright, 1968; Guinard and Daley, 1970) .
It is particularly interesting to note in
(A-26) that changes in the wavelength of the incident
signal, a, do not effect the angular dependence of a o ,
but variation in n, the wavelength dependence, do alter
the angular 1)ehaviour. That is, a change in the angular
dependence of a' strictly implies a change in the wave-
length dependence.
Schooley (1962) has shown that variations in
windspeed change the probability distribution of -the
surface slopes. The distribution width broadens as the
wind velocity increases. The term a s in (A-26) accounts
for this wind dependence and the predicted "saturation
effect" at high wind velocities at which point a, the
standard deviation of slopes, no longer increases with
windspeed, will lead to "saturation" of Q°.
The change in the value of the mean sea slope,
TI, due to wind direction and velocity are contained in the
y
18
parabolic cylinder function (A-26) . The upwind- downwind
ratio is approximately the ratio of two parabolic cylinder
functions having arguments which differ in sign. From the
Guinard and Daley (1970) X-band data, the average upwind-
downwind ratio for relatively high windspeeds was found
to be approximately 1.0 db over the range of depression
angles from 30 0 to 45 0 , using measurements from six dif-
ferent flights. This upwind-downwind ratio is achieved
from (A-26) for an n/a ratio of 0.1, which is in general
agreement with measurements reported by Schooley (1962) .
The surface slopes are determined relative to
the angle of incidence of the incoming signal, which ac-
counts for the fact that the upwind-downwind ratio is
angle dependent. in addition, the range of facet areas
which contributes to the slope  statistics is restricted
by the wavelength "size-filtering" effect (Rouse, 1968)
such that the upwind-downwind ratio is also wavelength
dependent.
.__*I
19
REFcRENCES
[1] M. Katzin, "On the mechanisms of sea clutter,"
Proc. IRE, vol. 4S, pp. 4S-S4, January 1957.
[2 ] C. Cox and W. Munk, "Statistics of the sea s arface
derived from sun's glitter," J. Opt. Soc. Amer.
vol. 44, pp. 838-850, November 1954.
[3] N. W. Guinard and J. C. Daley, "An experimental
study of a sea clutter model," Proc. IEEE, vol. S8,
pp. 543-SSO, April 1970.
[4] J. W. Wright, "A new model for sea clutter," IEEE
Trans. Antennas and Propagation, vol. AP-16, no. 2,
pp. 217-223, March 1968.
LS] A. H. Schooley, "Upwind-downwind ratio of radar
return calculated from facet size statistics of a
wind-disturbed water surface," Proc. IRE, vol. 50,
pp. 456-461, April 1962.
[6] J. W. Rouse, Jr., "Measurements of the frequency
dependence of backscatter from rough surfaces,"
presented at the 1970 Fall USNC/URS1 Meeting, Ohio
State University, Columbus, 14-17 September.
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An extension of the slope-facet model of radar backscatter from the sea

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