A simulation of radar imaging of ocean waves with synthetic aperture techniques is presented. The modelling is simplistic from the oceanographic and electromagnetic viewpoint in order to minimize the computational problems, yet reveal some of the physical problems associated with the imaging of moving ocean waves. The model assumes: (1) The radar illuminates a one-dimensional, one harmonic ocean wave. (2) The scattering is assumed to be governed by geometrical optics. (3) The radar is assumed to be down-looking, with Doppler processing (range processing is suppressed due to the one-dimensional nature of the problem). (4) The beamwidth of the antenna (or integration time) is assumed to be sufficiently narrow to restrict the specular points of the peaks and troughs of the wave. The results show that conventional processing of the image gives familiar results if the ocean waves are stationary. When the ocean wave dispersion relationship is satisfied, the image is smeared due to the motion of the specular points over the integration time. In effect, the image of the ocean is transferred to the near field of the synthetic aperture

Swift, C. T.

English

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NASA TECHNICAL NASA TM X- 72629
MEMORANDUM COPY NO.
A SIMUIATION OF SYNTHETIC APERTURE
RADAR IMAGING OF OCEAN WAVES
Calvin T. Swift
October 29, 1974
(NASA-TM-X-72629 ) A SIMULATION OF
SYNTBETIC APERTURE RADAR IMAGING OF OCEAN N75-12172
WAVES (NASA) 14 p HC $3.25 CSCL 171
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NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
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https://ntrs.nasa.gov/search.jsp?R=19750004100 2020-03-23T03:15:50+00:00Z
1. Report No. 2. Government Accession No. 3. Recipient's Catalog No.
NASA TM X-72629
4. Title and Subtitle 5. Report Date
A Simulation of Synthetic Aperture Radar Imaging of October 29, 1974
Ocean Waves 6. Performing Organization Code
7. Author(s) 8. Performing Organization Report No.
Calvin T. Swift
10. Work Unit No.
9. Performing Organization Name and Address
NASA Langley Research Center 11. Contract or Grant No.
Hampton, VA 23665
13, Type of Report and Period Covered
12. Sponsoring Agency Name and Address
NASA Technical Memorandum
National Aeronautics & Space Administration 14. Sponsoring Agency Code
Washington, DC 20546
15. Supplementary Notes
Paper presented at the 1974 USNC/URSI Meeting, Boulder, Colorado,
October 14-17, 1974
16. Abstract
A simulation of radar imaging of ocean waves with synthetic aperture techinques
is presented. The modelling is simplistic from the oceanographic and electro-
magnetic viewpoint in order to minimize the computational problems, yet reveal
some of the physical problems associated with the imaging of moving ocean
waves. The model assumes: (1) The radar illuminates a one-dimensional, one
harmonic ocean wave that propagates as sin(2 X - t) where L = ocean wave-
length and g - acceleration due to gravity, La = L. (2) The scattering
is assumed to be governed by geometrical optics. (3) The radar is assumed to
be down-looking, with doppler processing (range processing is suppressed due to
the one-dimensional nature of the problem). (4) The beamwidth of the antenna
(or integration time) is assumed to be sufficiently narrow to restrict the
specular points of the peaks and troughs of the wave.
The results show that conventional processing of the image gives familiar
results if the ocean waves are stationary i.e., if a) 0. the specular points
are resolved within the resolution element E = AH/2vT where X = electromagnetic
wavelength, H = aircraft altitude, v = aircraft velocity and T = integration
time for the synthetic aperture. When the ocean wave dispersion relationship
S= V777T is satisfied, the image is "smeared" due to the motion of the speculer
points over the integration time. In effect, the image of the ocean is trans-
ferred to the near field of the synthetic aperture.
17. Key Words (Suggested by Author(s)) (STAR category underlined) 18. Distribution Statement
Communications Electromagnetic Rough
Surface Scattering, Synthetic Aperture Unclassified-Unlimited
Radar, Radio Oceanography
19. Security Cassif. (of this report) 20. Security Classif. (of this page) 21. No. of Pages 22. Price'
Unclassified Unclassified 1 2 $3.25
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*Available from
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INTRODUCTION
A side-looking imaging radar can obtain good azimuth resolution
either by utilizing a long narrow beam antenna, or by developing a
"synthetic aperture" which requires a time integration of the phase of
the scattered signal. The problem with using real apertures from space
is that resolution of a few tens of meters requires an antenna that is
a few hundred meters long. The problem with the synthetic aperture is
that the ocean waves are moving as the synthetic aperture is developed.
The latter problem is the subject of this paper.
The statistical theory of rough surface scattering has been used
with success to define the average microwave scattering cross-section of
the ocean. Such a statistical theory; however, is not appropriate for
interpreting radar imagery since the scene is deterministic. Therefore,
in order to analyze the synthetic aperture radar, an ocean-like surface
must be generated, and the calculations of the magnitude and phase of the
scattered energy must be undertaken on that particular surface.
In order to derive a closed form solution of an ocean wave image, it
is assumed that the ocean wave propagates only in one direction and that
the wave is monochromatic. The antenna is assumed to illuminate the
surface in the near nadir direction, and geometric optics is assumed to
govern the scattering process. • The image is reconstructed by doppler
ordering of the phases of the scattered signal. Parametric calculations
are presented as a function of integration time, platform altitude to
velocity ratio, ocean wavelength and electromagnetic wavelength. It is,
2shown that unless the system parameters are judiciously selected serious
distortion of the image can occur.
Discussion of Figures
Figure 1 shows the two-dimensional geometry for the imaging of
specular points by means of doppler ordering the back-scattered electric
field as the antenna beam moves over the specular points. In this figure,
R is the radius vector to a point along the X-axis, H and v are
the altitude and velocity of the moving platform and Xi  is the position
of the ith stationary specular point. The time reference is chosen so
that at t = 0 h is the perpendicular distance to any given ith specular
point.
The phase relationship for the return from the ith specular point
is given by the equations in the figure, where the latter is an approxi-
mation for large h. Note that the phase contains a t2 term due to the
variable doppler shift of the specular point as it passes through the
antenna beam.
Figure 2 shows the simplified system model used in this paper for the
analysis of the scattered return. The received signal S1 is mixed with
a reference chirp to re-order the doppler components. The signal then
goes through a low pass filter to produce S 3, whose phase depends
linearly on time. This signal is then integrated over time T and
squared to produce the intensity, S4 . The resultant signal S4  is a
series of sinc2  functions of width X = h/2vT, where Xo  is the
resolution of the image. For analytical simplicity, a zero IF stage
which passes both positive and negative frequencies has been postulated.
Figure 3 shows a very simplistic model of a moving ocean wave that
is being imaged. The wave is a single harmonic gravity wave of wave
number K = 2jr/L (L = wave length) and radian frequency C =N/g ,
where g is the acceleration due to gravity. In this geometry, the
antenna beam eB is assumed to be small enough so that only those
specular points at normal incidence are illuminated. Therefore, reflec-
tions occur at the peaks and troughs of the ocean wave; i.e., a double
harmonic is imaged. Also the amplitude of the return from each specular
point is the same: only an uninteresting phase shift of 900 occurs
between the peaks and troughs.
Because of the motion of the wave, the specular points will move
in time in proportion to the phase velocity vph = /g7 , as given by
the equation in the figure. Since X - X. (t) is squared additional terms
proportional to t2  will appear in the exponential. The time integration
of signal S3  will therefore be Fresnel integrals instead of simple sinc 2
functions.
The details of the derivation explicitly give
X r )2 [ 121 i is s 1
whr 4vepT
where
S2
C( + j S() =f e u du
020
h
X = -
o 2vT
As the antenna scans off nadir, the specular points will move
towards the even zero crossings of the X-axis, so that the illumination
approaches a spacing equal to L, the ocean wavelength. Indeed a simple
calculation shows that the specular points are identically spaced L
apart when the e, the viewing angle, is
S= = tan (- )
c L
There are no specular points for incidence angles greater than that given
above. Therefore, the specular points occur at distances I apart at
nadir; and approach a spacing of L at the critical viewing angle 0c ,
beyond which, they entirely disappear from view. There is no unique
relationship between the amplitude and length of an ocean wave; however,
the rms slope (A7T) generally increases with increasing wind speed
and decreasing depth of the bottom. Under these conditions, some specular
returns will occur with reasonable probability for viewing angles up to
20 or 30 degrees. At greater angles of incidence, Bragg scattering will
dominate.
Figure 4 shows the radar image of a "frozen" ocean wave (vph = 0).
The value of 40 sec. for the h/v ratio is approximately the value for
a high flying transonic jet aircraft. Intensity peaks occur every 20
meters, or one-half an ocean wavelength. This figure shows that the
resolution of the image improves as the integration time increases.
Figures 5-8 show how the image of a moving specular point is distorted
as the integration time increases. The calculations assume that the ocean
wave propagates with a phase velocity vph =/g L/23. For T = 0.5 sec.
(figure 5), the intensity distribution closely resembles the sinc2 function.
As T increases to 1.0 sec. (figure 6), the width of the central peak is
only slightly greater than Xo , the width of the sinc2 function. However,
large side lobes are beginning to appear. When the integration time
increases to 1.2 sec. (figure 7), the pair of side lobes are almost as
intense as the central lobe; and severe image distortion occurs. Finally,
for T = 1.4 seconds (figure 8), destructive interference causes a minimum
to occur at the specular point position.
Figure 9 shows the distortion of the specular point image as a function
of the electromagnetic wavelength A or h/v for L = 40 m and T = 1.0 sec.
With h/v = 40 sec., the image degrades as the electomagnetic wavelength
becomes shorter than 0.5 m. A re-scaling of parameters shows that with
A = .25 m., the distortion is reduced as h/v increases. However, an
increasing h/v will result in a larger value of Xo; i.e., poorer resolu-
tion.
In figure 10, T = 1.0 sec., X = 0.25 m and h/v = 40 sec., and L is
allowed to vary. As the ocean wave increases, the phase velocity also
increases; and the image is distorted. On the other hand, as the ocean
wavelength becomes smaller, the specular points are closer to each other
and the image of a given specular point smears into the image of its
6neighbor. This figure therefore implies that there is both a minimum
and a maximum ocean wave that can be imaged.
Figures 11 and 12 outline methods by which the imaging of an ocean
wave can be optimized. Figure 11 indicates how a distorted image may be
reconstructed by hardware implementation. If the Chirp frequency is
tuned by an amount equal to 1 - 2Vph/V, then the mixing and filtering of
signals SI and S2  results in a signal S , which contains no quadratic
time varying phase terms. The integrated signal then becomes a sinc2
function, and the specular point is resolved to within xo. The diffi-
culty is that one must know what the ocean wavelength is in order to
measure it! Anotherdifficulty is that this method will restore only one
wavelength of what is generally a spectral ensemble of wavelengths.
The other method (figure 12) is to make an appropriate selection
of the electromagnetic wavelength to image the range of ocean wavelengths
of interest. This can be done by utilizing the two conditions given in
the figure. The first criterion xo < L min/4 resolves the minimum
ocean wavelength as reviewed from near nadir. The second criterion
x > ph max T follows from inspection of the Fresenel integral expressiono - ph max
given earlier. Basically, the latter inequality states that a given specular
point must remain within the resolution cell during the integration period.
Otherwise image distortion will result. These criteria are conservative
because the specular points will tend to be spaced a full wavelength apart
as the viewing angle increases; and the calculations show that latter
criterion can be "slightly" violated before serious image distortion occurs.
If the inequalities are replaced by equalities, and if T is
eliminated, one obtains the result in the box. If h/v is fixed, this
equation states that the greatest range of ocean wavelengths can be
illuminated by choosing the shortest practical electromagnetic wavelength.
CONCLUDING REMARKS.
Geometric optics has been used as a scattering model in order to
interpret the synthetic aperture image of a moving ocean wave. The results
basically show that the image of a monochromatic ocean wave will be distorted
if the product of the integration time and the phase velocity of the wave
exceeds the resolution of the synthetic aperture.
When the monochromatic wave is viewed in the nadir direction, specular
points (images) occur at increments of one-half an ocean wavelength. As
the viewing angle increases, the images occur at one-wavelength intervals
and then disappear with increasing viewing angle.
Numerical techniques can be used to generate a surface which spectrally
resembles an ocean wave, and calculations performed within 20 to 300 of
nadir should be physically meaningful. At steeper viewing angles, the
model must include the Bragg Scattering Mechanism.
IMAGING RADAR GEOMETRY
BI I
2-1
-X--.
v k V2 2
E' Ix. t)-exp -j [2k x x +vt) 2  h2  exp -j o h
Figure 1.
IMAGING RADAR SYSTEM
TX ~--c DUPLEX ol
CHIRP
DEMOD
sl = exp -j [k0 (x-xi) vt/ h +k0 v2t21 h-wt
s2= exp j [k0v2t2/h -wt
S3 = exp -[2k 0 (x-xi) vtl h
S4 Pi j Xhh r(x-xi) ge 2Fi o 2vT
Figure 2.
IMAGING OF OCEAN WAVES
eB
SPECULAR
H(x, t) A sin (kx- V' t)
-- xi(t)----
(2i +.)b i
x-xi (t) = x- + D t2k V k
S4= A COMBINATION OF FRESNEL INTEGRALS
Figure 3,.
IMAGE OF A STATIONARY WAVE
. . T a
- -0.5 sec A 0.25 m
8 1.0 sec h/v = 40 sec
.. 2.0 sec L= 40 m
INTENSITY '
S / /
0 5 10 15 20 25 30 35 40 45 50
SPECULAR POINT
Figure 4.
IMAGE OF MOVING WAVE IMAGE OF MOVING WAVE
T= 0.5 sec T= 1 sec
1.0- 1.0-
X = 0.25 m
.8- hlv = 40 sec .8- = 0.25 m
L= 40 m h/v= 40 sec
L= 40 m
.6 .6-
INTENSITY X INTENSITY o 0
.4- .4-
.2 .2-
-10 -5 0 5 10 -10 -5 0 5 10
DISTANCE, meters DISTANCE. meters
Figure 5. Figure 6.
IMAGE OF MOVING WAVE IMAGE OF MOVING WAVE
T= 1.2 sec T 1.4 secT= 1.4 sec1.0 1.0
X = 0.25 m h = 4'
h/v= 40 sec =4L= 40L =40 m8.8
.6 .6
INTENSITY I NTENSITY X
.4- 
.4
.2 .2
.2
0 I I I I
10 -5 0 5 10 10 -5 0 5 10
DISTANCE, meters -10 5 0 5 10DISTANCE, meters
Figure 7. Figure 8.
IMAGE VERSUS X OR h/v
1.0-
.8- \ -h/v= 40 , = 0.25 m
-- -I_ 0.125m 20sec
-_0.25 m 40 sec
.6 - --- 0.50 m 80 sec
INTENSITY 11 \
.4 /I
T = 1 sec
..2 L = 40 m
0
-10 -5 0 5 . 10
DISTANCE, meters
Figure 9.
IMAGE VERSUS OCEAN WAVELENGTH
1.0
. L= 10 m
.8
T = I sec
.6 X = 0.25 m
Sh/v = 40 sec
INTENSITY
. YL = 80 m
.2
L = 40 m
-10 
-5 0 5 10
DISTANCE. meters
Figure 10.
SYSTEM OPTIMIZATION
A. HARDWARE IMPLEMENTATION
LPF 3
2= exp I kv2  
- - wctj
CHIRP v L
DEMOD Vph 27r
Figure 11.
SYSTEM OPTIMIZATION
B. SELECTION OF PARAMETERS
L i
CRITERIA: 1. X < min
0- 4
2. X > v phmaxT
max
WITH X 2v-T Vph a
max
REPLACE > BY = , ELEIMINATE T TO OBTAIN
=R1 m in
OR, x( h min
Figure 12.


A simulation of synthetic aperture radar imaging of ocean waves

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