A composite scattering model, suitable for explaining the behavior of measured scattering cross sections of the ocean surface, is presented. Furthermore, utilizing this scattering model, the spectrums of the small gravity, gravity-capillary, waves will be predicted for MSA/MSC, 13.3 GHz Scatterometer data

Krishen, K.

English

NASA Technical Reports Server

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THE COMPOSITE SCATTERING MODEL FOR RADAR SEA RETURN
K. Krishen
14
Lockheed Electronics Company, Inc.
Earth Observations Department
Houston Aerospace Systems Division
Houston, Texas
In this paper a composite scattering model, suitable for
explaining the behavior of measured scattering cross sections
of the ocean surface, will be presented. Furthermore, uti-
lizing this scattering model, the spectrums of the small
gravity, gravity-capillary, waves will be predicted for
NASA/MSC, 13.3 GHz Scatterometer data.
i_,r w
N7,3- 15383
INTRODUCTION
From the viewpoint of radar scattering at high fre-
quencies, the ocean scene is best described by the composite
surface function E(x,y) given by
+ t: + _ + C + ...... , (i)
= Cs i 2 3
where _s is the swell, _i is the sea, _2 is the
gravity-capillary structure, and so on. The solution of '
i the electromagnetic scattering from the sea requires the
probability density function of the height and the correla-
tion function of each component of _ , as well as the
}i joint probability densities of the components of _ . The
interaction of the surface wind with each component of
must be formulated to study the wind dependence of the
scattering cross section. However, a mathematical model
of _(x,y) as a function of surface wind velocity is not
:i available.
• :,.'_
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https://ntrs.nasa.gov/search.jsp?R=19730006656 2020-03-17T07:40:51+00:00Z
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Most of the ocean studies in the past have been devoted
to developing models for ocean surface wave forecasts.
Toward this goal low frequency gravity wave spectrums for
fully developed seas for various windspeeds and fetches have
been studied (ref. 1). The dependence of significant wave
height, H1/3 , on surface windspeed has also been reported
(ref. 2). The measurement of high frequency gravity-capillary
waves has been reported in only a few instances, with the most
recent investigation reported by Dobson (ref. 3).
Due to the lack of complete mathematical description of
_(x,y), the composite model for sea surface scattering will
be studied using the NASA/MSC scattering cross section (o0)
data.
SCATTERING THEORIES AND COMPARISONS
Numerous approaches have been advanced to explain scat-
tering from rough surfaces. The three theories which have
received attention and show promise of efficient interpreta-
tion of experimental data are as follows:
• The Kirchhoff method
• The small perturbation theory for slightly rough
surfaces
The composite scattering theory
In the Kirchhoff method the field scattered by rough
surface is formulated according to Huygen's principle and
is given by the Stratton-Chu integral. The total field and
its normal derivatives are determined by tangent plane
approximation on th_ surface. These requirements generally
restrict this method to locally flat surfaces.
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The small perturbation method is valid for large values
of the angles of incidence and exhibits meaningful polariza-
tion dependence. It is useful in the low frequency limit
and can therefore be applied to a class of slightly rough
surfaces when very low frequencies are used. A comparison
of theoretical and experimental results over slightly rough
water surface has been given by Wright (ref. 4). As pointed
out in his paper, the measured average backscattering cross
sections are in good agreement with the calculated values.
The depolarized return from slightly rough ocean surface
has been obtained by Valenzuela (ref. 5) by using Rice's
small perturbation method. A slightly different approach
using the small perturbation theory has also been used for
slightly rough surfaces by Bass et al. (refs. 6 and 7).
It is interesting to examine closely what parameters
are needed in the formulation of the electromagnetic problem.
If only the large-scale structure is considered, the tangent
plane approximation can be used. With this theory _o can
be expressed as a function of tan 8o, the root-mean-square
slope of the surface, in the high frequency limit. Only a
few measurements of the sea slope distributions have been
reported. The most widely used of these are the optical
measurements of Schooley (ref. 8) and Cox and Munk (ref. 9).
Figure 1 shows the value of B o as a function of windspeed.
The curves C(1), C(2) have been calculated using the spec-
trums of Kitaigorodskii and Pierson and Moskowitz (ref. I0).
The lower value of C(1) and C(2), as compared to other curves
in figure I, is attributed to the fact that these curves are
not directiondependent (upwind, downwind, etc.) but involve
all facets of the sea surface.
14-3
_ i.-_ _, _ ._ r _;_.ii:_¸ • .:/_ i_'_i_.i_::, _, _ :_ _,_:_i_?:i_!_?: i :_;%11_ _ _, . , • , : _ _
For comparison, a value of tan 80 = 0.27 was taken
!ili! corresponding to a 20.5-knot upwind speed as given in
figure I. The value of _ was calculated for the
i VVGaussian height probability function. The calculated value
of the scattering cross section (using the Kirchhoff method)
given by I
= see O IRl1(0) 12 exp / -tan2 e (2)
(_vv) L tan 2 SO _ tan 2 S
for e = 55 ÷ j30.25 is plotted in figure 2. In equation (2),
i 0 is the incidence angle, and Rll(0) is the Fresnel reflec-
tion coefficient for normal incidence. Also shown in this
i figure are the NASA/MSC, 13.3-GHz, F4L8RI, 21-knot, forebeam
data. The evident disagreement is attributed to the fact
i_ that equation (2) is a limiting solution, and only one com-
_:_ ponent of the composite surface is considered.
It has often been suggested that, near the normal direc-
tion for backscattering cross sections, scattering of the
optics type (Kirchhoff method) predominates. In other direc-
:_ tions, however, the slight roughness on top of the large-
scale roughness constitutes the major source of scattering.
The scattering cross sections are calculated from the
scattered component of the field. In view of this, Wright
i_ (ref ii) and Guinard and Daley (ref 12) ignore the effect
:i! "
ii of large, structure to account for the scattering at higher
i
::_ backscattering angles. The procedure by Wright, Guinard
•!i_ and Daley parallels that of Rice (ref. 13), Barrick and
:_ Peake (ref. 14), and Valenzuela For Rice's method the
:!_
i!
backscattering cross sections for a slightly rough surface
using first order terms are given in reference 14.
(Oy6) = 4.n-ko4 cos4 el_y_12W(p,q) •
In equation (3)
k o wave number of the incident radar energy
C3)
,il 0 = the incident angle
W(p,q) = p,qthe roughnessareradianSpectralwave numbersdensity' of the surface, and
" s- 1
cos 0 + 4e - sin 8
i VV
_ (s - 1)[,(¢ - 1) sin 2 0 + e.]
[e cos 8 + _e- sin 2 O] 2
e = the complex dielectric constant of the surface.
The scat'tering cross sections can be computed from
equation (3) for the
correlation function.
the result is
OVV) =
S
where £
h 2
exponential and Gaussian surface height
For the Gaussian correlated surface,
4ko4h2_cos40l%vl 2 exp (-k02£2 sin 2 0) (4)
is the surface height correlation distance, and
is the surface-mean-square height.
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2]
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Several ocean wave spectrums have been proposed over
the last few years. However, no experimental spectrums of
the fine structure (the capillary waves) have been reported
for different wind velocities. In the equilibrium range the
isotropic spectrum is of the form
W(r) = kr -4
r
'2 2= + q
(s)
There is some uncertainty for the value of k , but
Phillips (ref. 15) gives the following estimates:
6xlO -3 for the equilibrium range spectrum for gravity
waves
l. SxlO -2 for the capillary wave spectrum
Evaluating at wave numbers satisfying the Bragg scattering
condition (ref. ii) equation (3) yields the limiting form
of the cross sections as
1.s_ × lO-31ar_sl 2 cot 4 8 (6)
In the case of the ocean, it is thought that the sea
waves, the small gravity waves, and the gravity-capillary
structure combined produce the scattering. The swells are
assumed absent. To the first order of approximation, the
composite shattering cross sections were calculated by adding
the average incoherent scattering cross sections from the
very rough surface (Kirchhoff method) to that of the slightly
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rough surface. The mathematical proof of this is given in
papers of Semenov (ref. 16) and Fuks (ref. 17). The com-
parison of composite scattering cross section and NASA/MSC,
13.3-GHz, F4L8R1, forebeam vv data is shown in figure 3.
The theoretical curve is the summation of (Ovv)L and (Ovv)s
as given in equations (2) and (4) respectively. The value
of the dielectric constant e is taken as $5+j30.25 and,
furthermore, tan 8 o = 0.27 . The result is encouraging.
Comparisons such as shown in figure 3 made it obvious that
the scattering by small gravity-capillary structure plays a
significant role at higher angles.
A comparison of equation (6) with 13.3GHz, NASA/MSC data
in the range of angles 20 ° ! @ & 50 ° showed that the angular
variation of the data _as approximately the same as that given
in the equation. It was therefore concluded that the direc-
tional spectrums of the small gravity and gravity-capillary
structure of the sea could be expressed as
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-k 3
W(r) = kr (7)
The values of k and k 3 are wind dependent. After substi-
tution of equation (7), and toward the goal of studying the
change of spectrum as a function of wind velocity, an
expression of the following form was used.
• k k3
gO (e) = klWl21avvl2(cos e)4(cosec 8) (8)
In equation (8), W 1 is the wind velocity reduced to a
19.S-meter anemometer height. After using equation (8)
it was found that the value of (cos @)4-k3 (in the range of
14-7 _
angles 20 ° & 8 & 50 °) could be taken as 1 for most data.
Consequently, the following simplified form of equation (8)
was also used:
k k
_0 (O) : kzW12 l vvl2(coto) 3 (93
By using algorithm 178 "direct search" from ACM communication
(ref. 18), a FORTRAN program was used to find the values of
kl, k2, and k 3 . The program then searches for a minimum
value. The value of dielectric constant was taken as
e = 55+j30.25.
Two typical sets of data will be analyzed here. One
set consists of NASA/MSC Mission 119, 13.3 GHz data for
vertical-transmit vertical-receive polarization combination.
The upwind forebeam data, for F9LIRI9 (flight 9, line 1 and
run 19), F2LIRI and F3LIRI, the corresponding surface wind
velocities are 6 knots, 22.5 knots and 33 knots respectively,
were processed, The values for the constants using equa-
tion (8) for this set were as follows:
k I = 0.026
= x 1 324k 2
k 3 = 5.47
In figure 4, the experimental and calculated data using
preceding value of constants is presented. Similar results
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•2
(
ii
7
ii
were obtained for other sets of Mission 119 data. The same
set of data using equation (9) gave the following values:
k I = 0.043
k 2 = 1.33
k 3 = 5.00
The second set of data was chosen from NASA/MSC Mis-
sion 156 data. The data is the aft beam 13.3 GHz (vertical-
transmit vertical-receive) data for upwind conditions. The
F2L8RI, FSL4R5 and F6L4R2 data corresponding to 33 knots,
15 knots and 3 knots of average wind speed respectively were
processed. The values for constants using equation (8)were:
k I = 0. 0207
k 2 = I.I
k 3 = 6.6
Figure 5 shows the comparison of the experimental and cal-
culated data using the preceding values of the constants.
It should be pointed out here that the flight 6 data was
for very calm conditions with extremely low surface wind
velocity.
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CONCLUSION
It can be concluded that a theoretical composite model
can explain the dependence of the scattering cross section
on the angle of incidence @ . The spectrum of the small
scale structure is found wind dependent. In general, as
the wind velocity increases, the magnitude of the spectrum
for high spectrum frequencies increases. The value of the
constant k 3 is also a function of the radar incident wave
length. Two sets of data (FTL3RI, forebeam Mission 119 data)
gathered under identical conditions for a surface wind of
15.5 knots using equation (9) gave the following values for
k 3
k 3 = 7.3 for 0.4 GHz , and
k 3 = 4.7 for 13.3 GHz data .
This dependence is expected since 0.4 GHz radar would be
relatively insensitive to small gravity-capillary waves.
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iREFERENCES
• .:!
5
(
1. Ocean Wave Spectra. Proceedings of a conference
sponsored by the U.S. Naval Oceanographic Office and
the Division of EarthSciences of the National Academy
of Sciences, (National Research Council in 1961),
Prentice Hall, N. J.) 1963.
2. Moskowitz) L.: "Estimates of Power Spectrums for Fully
Developed Seas for Wind Speeds of 20 to 40 Knots",
Journal of Geophysical Research, Vol. 69, No. 24, 1964.
3. Dobson, E. B.: "Measurement of the Fine-Scale Structure
of the Sea") Journal of Geophysical Research, Vol. 75)
No. 15, May 1970.
4. Wright, J. W.: "Backscattering From Capillary Waves
With Application to Sea Clutter'!, IEEE Transactions on
Antennas and Propagation, AP-14, No. 6, 1966.
5. Valenzuela, G.: "Depolarization of EM Waves by Slightly
Rough Surfaces", IEEE Transactions on Antennas and
Propagation, AP-I5, No. 4, 1967.
6. Bass, F. G.; and Bocharov, V. G.: "On the Theory of
Electromagnetic Wave Scattering From a Statistically
Uneven Surface", Radio Engineering and Electronics,
Vol. 3, No. 2, 1958.
7. Bass, F. G., et al., "Very High Frequency Radiowave
Scattering by a Disturbed Sea Surface", Parts I, II,
IEEE Transactions on Antennas and Propagation, AP-16,
No. 5, 1968.
8. Schooley, A. H.: "A Simple Optical Method For Measuring
the Statistical Distribution of Water Surface Slopes",
Journal Opt. Soc. Amer., 44, 1954.
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•!
J,!
9. Cox, C.; and Munk, W._ Measurement of the Roughness of
the Sea Surface From Photographs of Sun's Glitter",
Journal Opt. Soc., 44, 1954.
I0. Krishen, K.: "Saturation of °o With Increasing Wind
Velocity", Lockheed Electronics Company, HASD TR
No. 649D.21.007, Houston, Texas, 1969.
ii. Wright, J. W.: "A New Model for Sea Clutter", IEEE
Transactions on Antenna and Propagation, AP-16, No. 2,
1968.
12. Guinard, N. W.; and Daley, J. C.: "An Experimental
Study of a Sea Clutter Model", Proc. IEEE Vol. 58,
No. 4, 1970.
13. Rice, S. O.: "Refiection of Electromagnetic Waves From
Slightly Rough Surfaces", Comm. Pure Appl. Math., Vol. 4,
1951.
14. Barrick, D. E.; and Peake, W. H.: "Scattering From
Surfaces With Different Roughness Scales: Analysis
and Interpretation", Research Report, Battelle Memorial
Institute, Ohio, 1967.
15. Phillips, O. M.: The Dynamics of Upper Ocean, Cambridge
University Press, London, 1966.
16. Semenov, B.: "An Approximate Calculation of Scattering
of Electromagnetic Waves From a Slightly Rough Surface",
Radioteknika i Electronika (USSR), II (English Transla-
tion).
17. Fuks, I.: "Contribution to the Theory of Radio Wave
Scattering on the Perturbed Sea Surface", Izvestia
Vyshikh Uchebnikh Zavedeniy Radiofizika, 5 (English
Translation).
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18. llooke, R. and T. Jeeves: "Direct Search Solution of
Numerical and Statistical Problems", Journal A88oc.
Computing Machinery, Vol. 8, No. 2, 1961.
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15 --
i0 --
O
5
0
A(1)
/
/
/
/
/
. B (Z)
A(2)
c(1)
B(2)
A (!) UPWIND
A(2) CROSSWIND I COX AND MUNK
B (i) CROSSWIND I SCHOOLEY
B (2) DOWNWIND
C(1) KITAIGORODSKII I CALCULATED
C(2) PIERSON AND MOSKOWITZ
I I I I I I I
i0 20 30 -_
Windspeed, knots
40
Figure i. -- Values of 8 0 as a function of windspeed.
14-14
wi
2O
o
i0
0
-i0
-20 -
o Kirchhoff Theory,
_0 = tan-i (0.27)
-30 - [20.5 Knots]
A NASA/MSC, Svv
13.3 GHz, Upwind
-40 - (Forebeam), F4L8RI
21 Knots, Mission 119 \
Data
-5O I I I I I
0 10 20 30 40 50 60
Angle of incidence (8) in degrees
Figure 2. -- Comparison of NASA/MSC, 13.3-GHz data with
. the Kirchhoff theory.
I_-15
•ii
!
m
o
20
I0
0
-i0
-2O
-3O
-4O
o Composite Model Theory
-I (0.27) ,8 = tan
°vv' 0
k0h = 0.546 and k0Z = 3
(20.5 Knots)
A NASA/MSC, avv,
13.3 GHz, Upwind (Forebeam)'
F4L8RI, 21 Knots
Mission 119 Data
-SO I I I I I
0 i0 20 30 40 50 60
Angle of incidence (8) in degrees
Figure 3. -- Comparison of NASA/MSC, 13.3-GHz data with
model theory.
14-16
FFigure 4. -- comparison of calculated and experimental
scattering cross section for Mission 119.
14 -i'[
>''.',I
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EXPERIMENTAL
NASA/MSC MISSION 156 DATA
_ ---- CALCULATED
SCATTERING CROSS SECTION
F2L8RI
33 KNOTS
I I I I
o o 20.00 30.00 40 00 50.00 60 00
I u AFT BEAM INCIDENT ANGLE (DEG)
Figure 5. --Comparison of calculated and experimental
Mission 156 data.
18-18


The composite scattering model for radar sea return

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