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Experimental study of the microwave radar backscatter from transient deep-water breaking waves

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Experimental Study of the Microwave Radar Backscatter
From Transient Deep-Water Breaking Waves
by
Eric Brian Dano
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
(Electrical Engineering and Atmospheric, Oceanic, and Space Sciences)
in The University of Michigan
1997
Doctoral Committee
Research Scientist David R. Lyzenga, Co-Chair
Professor John F. Vesecky, Co-Chair
Professor Anthony W. England
Assoc. Professor Marc Perlin
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UMI Number: 9732064
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To my wife Vivian
and son Brian, for
all o f their support
ii
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ACKNOWLEDGMENTS
I would like to express my deep gratitude to the numerous people who nurtured,
encouraged and educated me, on my journey to a doctorate degree. I would like to thank
Professor Anthony England for both creating and encouraging me to apply to the
combined Electrical Engineering and Atmospheric, Oceanic and Space Sciences degree
program at the University of Michigan. His unyielding dedication to the students in the
combined degree program was truly inspirational. I would also like to thank Professor
Marc Perlin, who was extremely generous with my research time in his gravity wave
facility. His insight on laboratory investigations of breaking waves and conversations on
hydrodynamics were extremely thought provoking and helpful. I would like to thank Dr.
Vesecky for bringing me into the University Research Initiative (URI) program at the
University of Michigan. Without his financial support my research would not have been
possible. Most importantly, I would like to thank Professor David Lyzenga for providing
me with constant support during my research. His willingness to share his breadth of
knowledge on both the theoretical and experimental aspects of oceanic remote sensing
was instrumental in my research. More importantly, his conduct as an advisor, scientist
and a person of uncompromising integrity will forever serve as a source of inspiration for
me.
I would like to thank Eric Ericson for many insightful discussions on
electromagnetic scattering. Eric’s insight during the development of my scattering codes
was greatly appreciated. I would also like to thank David Kletzli of the Environmental
Research Institute of Michigan (ERIM) for teaching me how to build research quality
radars. Dave’s dedication to the design of the URI radars, and generous use of ERIM’s
iii
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radar calibration facilities, was extremely appreciated. I would also like to thank Irfan
Azeem for sharing with me his vast knowledge of computer software.
I would like to express my gratitude to Guy Meadows, Lorrelle Meadows, Hans
VanSumerin and Messon Gbah for their assistance during the tow-tank experiment, along
with Robert Onstott who generously provided the X-Band FMCW radar. I also thank
William Kirkpatrick, Joe Krasny, Stuart Cohen and Gary Phillips, of the Marine
Hydrodynamics Laboratory, for their assistance in the design of the radar sled, radar
calibrator and radar mount used for my research.
Finally, my research would not have been possible without the unyielding love
and support from my wife Vivian. Her dedication to work and family made the long
hours of research and studies possible. For this reason, I would like to dedicate this thesis
to her and my son Brian, who also gave up his family time so that my research could
advance rapidly. I would also like to thank my parents Philip and Marilyn Dano for
encouraging higher education from early on, along with all my friends and relatives who
have supported me during my studies.
This dissertation was funded by the Office o f Naval Research through the
University Research Initiative for Ocean Surface Processes and Remote Sensing, Grant
Number N00014-92-J-1650. I am grateful to technical officers, Drs. Dennis Trizna and
Frank Herr, who provided overall guidance and strong support for this research task.
iv
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TABLE OF CONTENTS
DEDICATION.................................................................................................................... ii
ACKNOWLEDGMENTS................................................................................................ iii
LIST OF APPENDICES................................................................................................. vii
LIST OF FIGURES........................................................................................................ viii
LIST OF ABBREVIATIONS........................................................................................xiii
LIST OF SYMBOLS....................................................................................................... xv
CHAPTER
I.
INTRODUCTION........................................................................................... 1
1.0 Overview................................................................................... 1
1.1 Field Investigations................................................................... 4
1.2 Laboratory Investigations......................................................... 7
1.2.1 Hydrodynamic Investigations.......................................... 8
1.2.2 Electromagnetic Investigations...................................... 10
1.2.3 Current Investigations.................................................... 13
H.
RADAR SCATTERING FROM THE OCEAN......................................... 15
2.0 Overview................................................................................. 15
2.1 Physical Optics........................................................................ 16
2.1.1 Geometric Optics...........................................................24
2.1.2 Specular and Resonant Scattering.................................. 25
2.2 The Small Perturbation Method.............................................. 26
2.3 Two-Scale Models..................................................................28
2.4 Exact Methods........................................................................29
2.4.1 The Iterative Method...................................................... 29
in .
THE TRANSIENT BREAKING WAVE EXPERIM ENT...................... 36
3.0 Experimental Design............................................................... 36
3.1 Experimental Equipment........................................................40
3.1.1 Radars............................................................................40
V
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3.1.1.1 The K-Band CW Radar......................................... 40
3.1.1.2 The X-Band FMCW Radar................................... 41
3.1.2 Wave Characterization...................................................45
3.2 Experimental Results (Moderate Anglesof Incidence)...........46
3.2.1 Temporal D ata...............................................................46
3.2.2 Spectral Data..................................................................56
3.2.3 Wave Probe D ata...........................................................58
3.3 Experimental Results (Grazing Angles of Incidence)..............60
3.4 Numerical Modeling...............................................................62
3.4.1 Pre-Break (Moderate Angles of Incidence)...................62
3.4.2 Pre-Break (Grazing Angles of Incidence)......................69
3.4.3 Post-Break...................................................................... 70
3.5 Conclusions............................................................................. 72
IV. THE HIGH-RESOLUTION TRANSIENT BREAKING
WAVE EXPERIMENT................................................................................. 74
4.0 Experimental Design and Equipment..................................... 74
4.0.1 Wave Generation............................................................76
4.0.2 Radar Measurements......................................................80
4.0.3 Wave Characterization................................................... 84
4.0.4 Specular Facet Detection............................................... 85
4.1 Experimental Results..............................................................86
4.1.1 Temporal D ata...............................................................86
4.1.2 Spectral Data..................................................................95
4.2 Numerical Modeling............................................................. 103
4.2.1 Pre-break...................................................................... 103
4.2.2 Post-break.................................................................... 112
4.3 Conclusions........................................................................... 117
V.
SUMMARY AND CONCLUSIONS.......................................................... 119
5.0 Summary............................................................................... 119
5.1 Conclusions........................................................................... 121
5.2 Future Work.......................................................................... 125
APPENDICES.................................................................................................................127
BIBLIOGRAPHY...........................................................................................................154
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LIST OF APPENDICES
Appendix
A.
CW AND FMCW RADAR THEORY....................................................... 128
A.0 CW Radar Theory................................................................. 128
A. I FMCW Radar Theory........................................................... 129
B.
RADAR CALIBRATION........................................................................... 134
B.O General................................................................................. 134
B.l Calibration Setup.................................................................. 134
B .2 Internal Calibrations............................................................. 135
B.2.1 DC Bias....................................................................... 136
B.2.2 Phase Error.................................................................. 137
B.2.3 Amplitude Error.......................................................... 139
B .3 External Calibrations............................................................ 141
B.3.1 Normalized Antenna Gain Pattern..............................141
C.
RADAR CROSS SECTION DETERMINATION....................................143
C.O General................................................................................. 143
C. 1 The RCS Integral.................................................................. 145
D.
LINEAR WAVE THEORY........................................................................ 148
D.O General................................................................................. 148
D. 1 Equations of Motion............................................................. 148
D. 1.1 Conservation of M ass................................................. 149
D.1.2 Conservation of Momentum....................................... 149
D.2 Boundary Conditions........................................................... 150
D.2.1 Kinematic Boundary Conditions................................151
D.2.2 Dynamic Boundary Conditions...................................152
D.2.3 Linearized Boundary Conditions................................152
D.3 Linear Wave Phase Speeds.................................................. 153
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LIST OF FIGURES
Figure
1.1 A breaking wave time series o f Doppler spectra for V-pol (top) and Hpol (bottom) at 45 degrees angle of incidence. Each plot represents 15
seconds of data with the time increasing from the bottom to the top of
each plot [from Jessup et al., 1991a]...............................................................6
1.2 The velocity field in the pre-breaking wave crest of a plunging
breaking wave [from Perlin et al., 1996].......................................................10
1.3 The numerically predicted RCS versus measured RCS for nonspecular scattering events at various angles of incidence [from Kwoh
and Lake, 1984a]............................................................................................11
2.1 Ocean wave frequency spectra for a fully developed sea at various
wind speeds [from Wetzel, 1990].................................................................. 17
2.2 Illustration of scattering mechanisms which dominate the radar
backscatter from the ocean at various angles of incidence [from
Valenzuela, 1978].......................................................................................... 17
2.3 Scattering geometry for the physical optics and iterative method
derivations...................................................................................................... 18
3.1 Diagram of the laboratory experiment. The side view demonstrates the
configuration for the upwave 40 and 85 degree angle of incidence
measurements. The top view demonstrates the configuration for the
downwave 40 degree angle of incidence measurements, along with the
radar’s locations during the other azimuth measurements............................ 39
3.2 Normalized antenna gain patterns for the (a) conical horn antenna of
the K-band radar and (b) pyramidal horn antennas of the X-band
FMCW radar..................................................................................................42
3.3 Schematic (a) and photograph (b) of the K-band CW radar (on the
right) in the downwave configuration. The FMCW radar (on the left)
was used to observe secondary reflections while the radar absorber was
being placed................................................................................................... 43
3.4 Schematic (a) and photograph (b) of the X-band FMCW radar in the
V-pol upwave grazing configuration.............................................................44
3.5 Cartoon of the low energy breaking wave’s (LEBW) spatial and
temporal evolution, with respect to the radar’s 1-way 3dB pattern on
the water, at 40 degrees angle of incidence...................................................47
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3.6
Cartoon of the high energy breaking wave’s (HEBW) spatial and
temporal evolution, with respect to the radar’s 1-way 3dB pattern on
the water, at 40 degrees angle of incidence...................................................48
3.7
Representative plots of the K-band backscattered power for the LEBW
at 40 degrees angle o f incidence and azimuth angles of (a) 0 degrees,
(b) 45 degrees, (c) 90 degrees, (d) 135 degrees and (e) 180 degrees, Hpol and (f) 0 degrees, (g) 45 degrees, (h) 90 degrees, (i) 135 degrees
and (j) 180 degrees, V-pol........................................................................49-50
3.8
Representative plots of the K-band backscattered power for the HEBW
at 40 degrees angle of incidence and azimuth angles of (a) 0 degrees,
(b) 45 degrees, (c) 90 degrees, (d) 135 degrees and (e) 180 degrees, Hpol and (f) 0 degrees, (g) 45 degrees, (h) 90 degrees, (i) 135 degrees
and (j) 180 degrees, V-pol........................................................................ 51-52
3.9
Plot of the peak radar backscatter as a function of azimuth angle for
the (a) LEBW and (b) HEBW measurements at 40 degrees angle of
incidence. Values were averaged over 3 samples (6 ms)............................. 55
3.10 Plot of the Doppler spectra as a function of time, for the (a) LEBW
upwave, (b) LEBW downwave, (c) HEBW upwave and (d) HEBW
downwave K-band measurements at 40 degrees angle of incidence.
The spectrum were each based upon 64 points (125 ms) of radar data........ 57
3.11 Plot of wave height versus time for the (a) pre-break (break-55 cm)
and (b) post-break (break+37 cm) waveprobes, along with the
corresponding low frequency wave height spectra....................................... 59
3.12 Plot of the backscattered power as a function of range and time for the
(a) HEBW V-pol upwave, (b) HEBW H-pol upwave measurement and
(c) HEBW V-pol downwave measurement, at 85 degrees angle of
incidence. Each image consists of 150 consecutive 512 points (16 ms)
FFTs of radar data..........................................................................................63
3.13 Plot of the X-band RCS as a function of time for the HEBW (a) H-pol
upwave, (b) V-pol upwave and (c) V-pol downwave measurements,
averaged over several runs, at 85 degrees angle of incidence..................... 64
3.14 Laser sheet enhanced video image corresponding to the (a) peak
upwave return (time = 1.25 seconds) and (b) period of significant
upwave backscatter (time = 1.29 seconds) for the LEBW at 40 degrees
angle of incidence.......................................................................................... 67
3.15 Theoretical bistatic physical optics scattering solution corresponding
to (a) the surface profile in Fig. 14a and (b) the surface profile in Fig.
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14b. V-pol results are represented by a solid line and H-pol by a
dotted line. The experimental results, corresponding to Fig. 14a, are
also indicated for V-pol (x) and for H-pol (o)............................................... 67
3.16 Plot of the (a) surface profile, (b) slope profile, (c) PO surface current
phase slope and (d) PO surface current magnitude corresponding to
Fig. 14a the peak radar return at 40 degrees angle of incidence, in the
upwave direction............................................................................................68
3.17 Image o f the surface profile used to model scattering from (a) the
incipient ‘toe’ feature, (b) the prominent ‘toe’ feature and (c) the
diminished ‘toe’ feature.
Corresponding polarization averaged
physical optics backscatter RCS for the mean surface profiles in Figs
3.17a-c, at 85 degrees angle of incidence in the upwave direction, are
shown in (d)................................................................................................... 71
4.1
Diagram of the high-resolution transient breaking wave experimental
setup, with the radar in the upwave configuration.........................................75
4.2
The spilling breaking wave image sequence at 25 ms intervals. The
spatial reference is indicated in black, with distances in centimeters to
an arbitrary reference point. The imager was repositioned at times 0.6
and 0.775 sec..................................................................................................77
4.3
The spilling-plunging breaking wave image sequence at 25 ms
intervals. The spatial reference is indicated in black, with distances in
centimeters to an arbitrary reference point.
The imager was
repositioned at times 0.5, 0.65 and 0.8 sec.................................................... 78
4.4 The plunging breaking wave image sequence at 25 and 100 ms
intervals. The spatial reference is indicated in black, with distances in
centimeters to an arbitrary reference point. The imager was
repositioned at times 0.6, 0.7,0.85 and 1.125 sec......................................... 79
4.5
Schematic (a) and photograph (b) of the X/K-Band CW radar, radar
sled and laser sheet optics (far right side of photo)....................................... 82
4.6 Normalized antenna gain patterns for the X/K-band radar’s (a) K-band
pyramidal horn antennas and (b) X-band pyramidal hom antennas............. 83
4.7 Plot of peak RCS as a function of time at (a) K-band and (b) X-band
for the spilling breaking wave at 30, 45 and 60 degree angles of
incidence........................................................................................................ 87
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4.8
Plot of peak RCS as a function of time at (a) K-band and (b) X-band
for the spilling-plunging breaking wave at 30,45 and 60 degree angles
of incidence................................................................................................... 88
4.9 Plot of peak RCS as a function of time at (a) K-band and (b) X-band
for the plunging breaking waves at 30, 45 and 60 degree angles of
incidence........................................................................................................89
4.10 Images from the specular facet detector showing facet detection at (a)
the peak radar returns, (b) the interference dominated returns and (c)
the large post-peak radar returns................................................................... 93
4.11 Plot of the radar return after wave face slope exceeds the radar angle of
incidence for the spilling-plunging breaking wave at 45 degrees angle
on incidence...................................................................................................94
4.12 Series plots of the K-band Doppler spectra for the upwave (left side)
and downwave (right side) measurements, at 30 (top), 45 (middle) and
60 degree (bottom) angles of incidence, for the spilling breaking
wave...............................................................................................................97
4.13 Series plots of the X-band Doppler spectra for the upwave (left side)
and downwave (right side) measurements, at 30 (top), 45 (middle) and
60 degree (bottom) angles of incidence, for the spilling breaking
wave.............................................................................................................. 98
4.14 Series plots of the K-band Doppler spectra for the upwave (left side)
and downwave (right side) measurements, at 30 (top), 45 (middle) and
60 degree (bottom) angles of incidence, for the spilling-plunging
breaking wave................................................................................................99
4.15 Series plots of the X-band Doppler spectra for the upwave (left side)
and downwave (right side) measurements, at 30 (top), 45 (middle) and
60 degree (bottom) angles of incidence, for the spilling-plunging
breaking wave..............................................................................................100
4.16 Series plots of the K-band Doppler spectra for the upwave (left side)
and downwave (right side) measurements, at 30 (top), 45 (middle) and
60 degree (bottom) angles o f incidence, for the plunging breaking
wave.............................................................................................................101
4.17 Series plots of the X-band Doppler spectra for the upwave (left side)
and downwave (right side) measurements, at 30 (top), 45 (middle) and
60 degree (bottom) angles o f incidence, for the plunging breaking
wave.............................................................................................................102
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4.18 Plot of the numerically calculated bistatic and measured RCS at 30
degrees angle of incidence, at (a) K-band and (b) X-band, for the
spilling-plunging breaking wave..................................................................106
4.19 Plot of the numerically calculated bistatic and measured RCS at 30
degrees angle of incidence, at (a) K-band and (b) X-band, for the
spilling-plunging breaking wave..................................................................107
4.20 Plot o f the numerically calculated bistatic and measured RCS at 30
degrees angle of incidence, at (a) K-band and (b) X-band, for the
spilling-plunging breaking wave..................................................................108
4.21 Comparison of the numerically calculated theoretical RCS and
experimentally measured RCS over all wave profiles and angles of
incidence for (a) K-band and (b) X-band..................................................... 110
4.22 Plot o f the (a) surface profile, (b) surface slope, (c) physical optics
surface current phase slope and (d) physical optics surface current
magnitude corresponding to the profile determined to cause the peak
backscatter at 45 degrees angle of incidence, for the spilling-plunging
breaking wave.............................................................................................. I l l
4.23 Plot of the measured RCS (solid line), roughness corrected PO
solution (dotted line), SPM with Walker spectrum (top dashed line)
and SPM with Phillips spectrum (bottom dashed line) for (a) K-band
and (b) X-band at 30 degrees angle of incidence.........................................115
4.24 Plot o f the measured RCS (solid line), PO solution (dotted line), SPM
with Walker spectrum (top dashed line) and SPM with Phillips
spectrum (bottom dashed line) for (a) K-band and (b) X-band at 45
degrees angle of incidence........................................................................... 116
A. 1 FMCW radar RF output from the VCO...................................................... 130
B.1 Radar calibration setup............................................................................... 135
B.2 Calibration correction sequence for data gathered against a constant
RCS spherical calibration target. Raw data (a), DC corrected data (b)
DC and phase corrected data (c) and DC, phase and amplitude
corrected data (d) are shown........................................................................ 140
C. 1 Pseudo-spherical coordinates used for RCS calculations........................... 147
xii
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LIST OF ABBREVIATIONS
Abbreviation
A/D
analog to digital
AOI
angle of incidence
CW
continuous wave
DC
direct current
EA
echo area
EW
echo width
eft
fast Fourier transform
FMCW
frequency modulated continuous wave
GO
geometric optics
HEBW
high energy breaking wave
HF
high frequency
I
In-phase channel
IF
intermediate frequency
KA
KirchhofF approximation
LEBW
low energy breaking wave
LED
light emitting diode
MFEE
magnetic field integral equation
OMT
orthomode transducer
PEC
perfect electric conductor
PIV
particle image velocimetry
PO
physical optics
PRI
pulse repetition interval
PTV
particle tracking velocimetry
Q
quadrature channel
RA
Rayleigh approximation
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RCS
normalized radar cross section
RF
radio frequency
RMS
root mean square
SAA
small amplitude approximation
SPA
stationary phase approximation
SPM
small perturbation method
TE
transverse electric
TPA
tangent plane approximation
TSM
two-scale model
TTL
transistor-transistor logic
VCO
voltage controlled oscillator
xiv
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LIST OF SYMBOLS
Symbol
A,'
DC corrected maximum amplitude of in-phase channel voltage
A,
DC corrected maximum amplitude of quadrature channel voltage
As
illuminated area on surface
c
speed of light
Q
in-phase channel scaling factor
cq
quadrature channel scaling factor
Cp
wave phase speed
d
maximum aperture dimension
E
electric field
Ein
incident electric field
E0
magnitude of the incident electric field
Erx
received electric field
Etx
transmitted electric field
Es
scattered electric field
fd
Doppler frequency
G(f,f)
free space Greens function
G0
maximum gain
G sys
radar system gain
Sn
normalized antenna gain pattern
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Shh
H-pol small perturbation reflection coefficient
Sw
V-pol small perturbation reflection coefficient
h
height above water
H
magnetic field
Hin
incident magnetic field
H0
magnitude of the incident magnetic field
Hs
scattered magnetic field
J
electric surface current density
Jin
physical optics electric surface current density
JN
Nth iteration of the electric surface current density
5 JN
change in electric surface current density between iterations
h
electric surface current density
K
cutoff wavenumber
k
vector wavenumber
k
unit vector wavenumber
^in
vector wavenumber of the incident radiation
K
magnitude of vector wavenumber
k 0ut
vector wavenumber of the scattered radiation
K
x-component of unit vector wavenumber
ky
y-component of unit vector wavenumber
kr
radar wavenumber
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k?
z-component o f unit vector wavenumber
K*
x-component of ocean wavenumber
Ky
y-component o f ocean wavenumber
L(x’,y’)
two-dimensional weighting function
1
surface correlation length
n
outward unit surface normal
nx
x-component o f unit surface normal
ny
y-component o f unit surface normal
riz
z-component o f unit surface normal
p
polarization vector
P
pressure force
Prx
radar received power
PK
radar transmitted power
r
reference to observation point vector
r'
reference to surface point vector
rcaI
calibration range
rtgt
target range
Rpp
far field range
Ro
mean distance to surface
s
surface rms height
S
scattering surface
S’
pierced scattering surface
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S(k)
wavenumbr spectra
t
time
td
time from scattering to reception
Vj
in-phase channel voltage
Vq
quadrature channel voltage
WFX
antenna weighting function in x-direction
WFy
antenna weighting function in y-direction
P<t>
H-plane 3 dB beamwidth
P0
E-plane 3 dB beamwidth
e
surface complex permitivity
p
surface complex permeability
X
wavelength
A©
RF bandpass from VCO
to
radial frequency
tod
Doppler shift
ps
electric charge density
r](x’,y’)
surface elevation in Cartesian coordinates
Vrix
surface slope in x direction
Vr|y
surface slope in y direction
Vr]z
surface slope in z direction
0(x, y)
two-dimensional covariance function
0
angle of incidence
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4»
4»ap
<j>s
azimuth angle
actual phase between in-phase and quadrature channels
velocity potential function
H^(KX, Ky) surface elevation spectrum
t
surface tension coefficient
k
surface curvature
ct
radar cross section
ct°
average radar cross section per unit area
X
scattering size parameter
T
reflection coefficient
4
roughness parameter
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CHAPTER I
INTRODUCTION
1.0 Overview
Oceanic breaking waves play an important role in the hydrodynamics and energy
balance of the Earth’s oceans. Breaking waves transfer energy, momentum and heat from
the atmosphere to the ocean, and create currents and turbulence in the upper ocean
[Philips, 1966]. While it is well understood that breaking waves may be generated by
wave instabilities, wave-current interactions or nonlinear interactions in a random wave
field [Melville and Rapp, 1985], the actual physical processes and scattering mechanisms
involved in wave breaking are not well understood.
In fact, the high degree of
nonlinearity and topographical complexities make analytical solutions of breaking wave
dynamics virtually impossible.
Present wave propagation models incorporate the energy dissipation due to wave
breaking by correlating the amount of wave breaking to one of the model parameters such
as wind speed or friction velocity.
While these models have greatly enhanced our
understanding of wave growth and spectral evolution in developing seas, many
discrepancies remain between model predictions and experimental observations. The
ability to remotely detect the distribution and amount of wave breaking with radar, would
lead to more accurate models of breaking wave energy dissipation, and ultimately, higher
fidelity ocean wave models.
Radar experiments in both the field and laboratory have shown that breaking
waves produce increased radar backscatter, and under high winds this contribution may
1
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dominate the overall radar return from the ocean. The strong backscatter from the
breaking waves can bias remotely sensed parameters, such as scatterometer derived wind
vectors and synthetic aperture radar (SAR) derived ocean wave spectra. The scatter from
breaking waves at low grazing angles can hinder real aperture radar (RAR) surface target
detection in high seas. Therefore, improved accuracy o f remotely sensed products will
require a more thorough understanding of the radar backscatter due to breaking waves on
various scales will be required.
Modem radar backscatter models rely on resonant scattering as the primary
scattering mechanism in the open ocean. Most scattering models approximate the oceans
surface as “patches” of short wavelength resonant scatterers tilted by longer wavelength
ocean waves. Models o f this type, known as two-scale models [Wright, 1968; Fung and
Chan, 1969; Plant, 1986], have proved quite successful in regions where little or no wave
breaking exists. In regions of wave breaking, the two-scale models tend to underpredict
the magnitude of the radar return, as well as the horizontal to vertical polarization
backscatter ratio.
Breaking waves have long been suggested as one of the possible
explanations for the additional radar backscatter and larger polarization ratio [Jessup et al.
1991a,b]. However, the inability to simultaneously characterize the ocean surface and
obtain radar backscatter measurements has limited detailed analysis of the scattering
mechanisms occurring in oceanic breaking waves.
Laboratory wave facilities can generate highly repeatable breaking waves that
may be both spatially and temporally characterized while radar backscatter data is being
obtained. This thesis describes two comprehensive laboratory experiments investigating
the radar backscatter from mechanically generated, transient deep-water breaking waves
of various energy states. The major objectives of this thesis are to
1) determine the scattering mechanisms dominating the radar return from
transient breaking waves,
2
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2) investigate the frequency, angle of incidence and azimuth angle dependence of
the radar backscatter from breaking waves of various energy states,
3) characterize the temporal and spectral radar returns from breaking waves of
various energy states and
4) determine the scattering mechanisms for the observed radar backscatter from
deep-water breaking waves.
Although breaking waves have been observed for years, they still elude a formal
definition. For the purpose of this thesis, “breaking” will be defined as the onset of small
scale roughness on the forward wave face. The breaking waves examined in this thesis
were generated by wave superposition, similar to that occurring on the open ocean.
However, there are some significant differences between the laboratory generated and
open ocean breaking waves that should be noted. The mechanically generated laboratory
breaking waves considered in this thesis did not have wind generated capillary waves on
their surfaces. This was done intentionally so that the scattering mechanisms directly
attributable to the breaking waves could be isolated. Also, the degree of focusing of
wave energy achieved in these experiments is unlikely to occur on the open ocean.
Nevertheless, there are similarities between the radar backscatter from mechanically
generated breaking waves and that from breaking waves on the open ocean.
This thesis is organized into five chapters. The remainder of Chapter I focuses on
relevant field and laboratory work on breaking waves.
electromagnetic scattering theory is presented.
In Chapter
n,
a review of
Chapter HI describes the transient
breaking wave experiment, equipment, procedures and results. In Chapter IV the highresolution transient breaking wave experiment, equipment, procedures and results are
described.
Chapter V contains the summary and conclusions for both experiments.
Several appendices of relevant topics are also included. Appendix A describes CW and
FMCW radar theory, Appendix B describes radar calibration, Appendix C describes radar
cross section (RCS) determination and Appendix D describes linear water wave theory.
3
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1.1 Field Investigations
Open ocean experiments have the advantage of involving “real” breaking waves,
but present difficulties with regard to sensor stability and wave characterization. Some
researchers have avoided the sensor stabilization problems by locating on fixed at-sea
platforms instead of research vessels [see e.g. Plant and Keller, 1990; Jessup, Keller and
Melville, 1990]. The problem of wave characterization, however, can not be solved so
easily. Wave staffs, buoy based sensors, and more recently, laser-optical slope scanning
devices do not have the ability to routinely intercept the highly localized breaking wave
events. Even if the sensors could be positioned in regions of regular breaking, they most
likely could not stand the forces of energetic breaking waves. Because of this, field
experiments can not characterize the ocean surface within the radar footprint with the
precision and accuracy necessary for input into scattering models.
Some of the first observations o f the radar backscatter from breaking waves were
at low grazing angles of incidence. Long (1974) using a C-band pencil beam radar and a
telescope, which observed the radar footprint, found that as the sea state increased the
peak radar returns became polarization independent. He found that the HH and W peak
radar returns (or “sea-spikes”) corresponded to visually observed whitecapping in the
radar footprint about fifty percent of the time. Long also found that the duration of the
“sea-spike” was correlated with the size and duration of the whitecapping event.
Lewis and Olin (1980) examined shallow water breaking waves at low grazing
angles of incidence, with a dual polarization X-band pulsed radar and a video camera
which observed the radar footprint. They found that the “sea-spikes” corresponded to the
observation of visual whitecapping in the radar footprint. The maximum radar returns
were found in the upwave direction, and produced very narrow Doppler spectra. V-pol
returns were found to follow the waves in range as they propagated towards the beach,
4
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and to persist longer than the H-pol returns. A multipath scattering mechanism was
identified and used to model the experimental results. As with Long (1974) some peak
radar returns were observed in the absence of whitecapping. These returns were found to
be three orders of magnitude lower in power than the “sea-spikes” observed in the
presence of whitecapping.
Recently, Sletten et al. (1996), using a 10 GHz centered ultrawideband radar
[Sletten and Trizna, 1994], took radar backscatter measurements from a trihedral comer
reflector which was located above the ocean surface at a height approximately equal to
the height of a breaking wave crest. They showed that the unity or greater HH to W
polarization ratios which are observed at grazing angles may be explained by multipath
scattering effects.
They found that the V-pol radiation was highly attenuated upon
reflection from the water surface due to Brewster angle effects, and was therefore
primarily limited to direct path scattering. The H-pol did not have this limitation and was
shown to have significant direct and indirect scattering components, which tend to
interfere destructively at small grazing angles.
The significant backscatter due to breaking waves has also been observed at
moderate angles of incidence. Jessup, Keller and Melville (1990) and Jessup, Melville
and Keller (1991a,b) examined breaking waves at 45 degrees angle of incidence, using a
tower mounted dual polarization Ku-band CW radar and a video camera that observed the
radar footprint. They found that the peak returns were polarization independent and that
most were associated with breaking waves in the radar footprint. The peak radar returns
were found to contribute between 10-15% of the mean RCS for V-pol and 15-25% for Hpol. The maximum Doppler shifts were found to be coincident with the times of peak
radar backscatter. The Doppler spectra reached their maximum widths slightly after the
peak radar return, as the waves broke in the radar footprint (Fig. 1.1). They inferred that
5
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0836513480
9999999999999999^
-500
0
Frequency / Hz
500
Figure 1.1 A breaking wave time series of Doppler spectra for V-pol (top) and H-pol (bottom) at 45
degrees angle of incidence. Each plot represents 15 seconds of data with the time increasing from the
bottom to the top of each plot [from Jessup et al., 1991a].
the sea spikes were most likely caused by specular scattering from the wave crest just
prior to wave breaking. These results are consistent with Keller et al. (1986) who also
analyzed the evolution of the Doppler spectra associated with breaking waves. Jessup et
al. (1991a,b) also found that the amount o f wave breaking was proportional to the cube of
the wind friction velocity, which is in agreement with theoretical predictions of Phillips
(1988).
Lee et al. (1995) examined the radar backscatter from the ocean at both moderate
and grazing angles of incidence using a ship mounted dual-polarization, eight-frequency,
X-band CW radar. From analysis of the Doppler spectra of the radar returns, they found
6
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that the scatterers could be categorized into Bragg “slow” and non-Bragg “fast” scatterers
which were spectrally separated for analysis. The non-Bragg “fast” scatterers were found
at all angles of incidence in the upwave look direction and were attributed to “bound”
scatterers that were created in the process o f wave breaking. Polarization ratios of the
non-Bragg scatterers were independent of the angle of incidence and could not be
modeled using the small perturbation method (SPM). The “free” Bragg scatterers were
found to contribute to the radar backscatter at all angles of incidences and azimuth angles.
At moderate angles of incidence, the Bragg returns were found to be described in
amplitude and polarization ratio by the SPM. At low grazing angles, the HH and W
returns deviated from the predictions of the SPM. These deviations were attributed to
multipath scattering or preferential diffraction of the V-pol energy.
1.2 Laboratory Investigations
Field experiments have provided a great deal of insight into the radar backscatter
from breaking waves on the ocean. However, the inability to simultaneously characterize
the ocean surface and obtain radar backscatter measurements has limited detailed analysis
of the scattering mechanisms in oceanic breaking waves. Laboratory investigations have
the advantage of being able to generate and characterize highly repeatable breaking
waves, while simultaneously obtaining radar backscatter data.
Breaking wave characterization can be accomplished using both intrusive and
nonintrusive techniques. Intrusive methods include the use of capacitance wave probes
which take point measurements of the wave amplitude at set spatial locations. Wave
probe data may be used to determine the wave group energy, or transformed to yield one­
dimensional frequency spectra.
Nonintrusive techniques include particle image
velocimetry (PIV), particle tracking velocimetry (PTV) and flow visualization with high
speed imagers and laser sheets. PIV and PTV techniques [see e.g. Perlin et al., 1996]
7
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analyze the fluid flow by seeding the fluid with small particles and illuminating the water
surface by a laser sheet. Photographic techniques are used to capture the movement of
the suspended particles and determine the instantaneous velocity field of the particles
inside a developing breaking wave.
Recently, two-dimensional wave slope
measurements have been obtained using a high-resolution surface slope imager (HRSSI)
[Meadows et al., 1995].
High speed imagers may be used in conjunction with a laser sheet illuminated
water surface, to obtain high-resolution one-dimensional images of the breaking wave
evolution. The extracted surface profiles serve as the inputs for numerical scattering
calculations, and provide the primary means of isolating the scattering mechanisms in
transient breaking waves. The ability to differentiate between scattering mechanisms is
extremely important in light of the numerous mechanisms which have been proposed.
Current theories include wedge like diffraction [Lyzenga et al., 1983; Wetzel, 1989],
specular reflection [Jessup, Melville and Keller, 1991a,b], scattering from the wave
plume or ‘toe’ [Wetzel, 1986; Sletten and Wu, 1996], and scattering from spontaneously
generated Bragg waves [Alpers et al., 1981], just to name a few.
More likely, the
scattering comes from “families” o f scatterers which evolve both spatially and temporally
as proposed by Phillips (1988).
1.2.1 Hydrodynamic Investigations
Several hydrodynamic wave characterization studies have aided in understanding
the evolution of breaking waves.
Bonmarin (1989) used a movie camera to film
mechanically generated breaking waves illuminated by a strobed sheet of light. Both
spilling breaking waves (characterized by minimal post-break small scale roughness on
the wave face) and plunging breaking waves (characterized by a prominent jet of water
which protruded from the wave crest, impacted forward o f the wave face, and generated a
8
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large splash-up with significant post-break small scale roughness) were examined. He
found that the rates of asymmetry growth of the waves were consistent with theoretical
predictions of Longuet-Higgins and Cokelet (1976, 1978), and could be used to predict
the type of breaking wave observed. While wave asymmetry has not been proven to be a
reliable means of characterizing breaking waves [Schultz et al., 1994], the Bonmarin
studies illustrated the rapid temporal and spatial changes o f breaking waves of various
energy states.
Similar techniques were used by Rapp and Melville (1990) to perform a detailed
hydrodynamic study of mechanically generated breaking waves. They found that the loss
of wave group energy and momentum flux due to wave breaking was approximately 10%
for spilling breaking waves and 25% for spilling-plunging breaking waves.
Ninety
percent of this energy loss occurred within the first four wave periods after breaking.
Wave breaking was also found to generate surface currents with mean velocities equal to
approximately 2-3% of the characteristic wave phase speed. It should be noted that the
spilling-plunging breaking wave (characterized by a prominent wave crest which
collapsed into the wave face causing a large splash-up and significant post-break small
scale roughness) is referred to as a plunging breaking wave in their paper.
Duncan (1994) used a movable high-speed camera and laser sheet to obtain highresolution photographs of mechanically generated spilling breaking waves. He found that
a bulge or ‘toe’ forms on the crest, near the forward face of the wave. As the wave face
steepened, parasitic capillary waves formed beneath the ‘toe’ of the wave.
These
parasitic capillary waves eventually became turbulent and broke down to form the
random post-break small-scale roughness commonly referred to as whitecapping.
Most recently, Perlin et al. (1996) used PIV and PTV techniques to examine the
particle velocity fields in mechanically generated plunging breaking waves. Analysis of
the derived velocity fields showed that a focusing of particles creates the jet in a plunging
breaking wave, and that the maximum particle velocity in their most energetic wave was
9
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about 30% greater than the phase speed of the wave using linear wave theory. A pre­
break velocity vector plot for the plunging breaking wave analyzed in Chapter IV is
included as Fig. 1.2.
1.2.2 Electromagnetic Investigations
Kwoh and Lake (1984a), using a dual polarization X-band CW radar at angles of
incidence from 40 to 70 degrees, examined mechanically generated spilling breaking
waves. They identified several scattering mechanisms which were divided into specular
and non-specular components. The observed specular scattering occurred prior to wave
breaking and was approximately 3 dB greater than the non-specular scattering. The nonspecular scatterers were observed after breaking, and were determined to be the summed
100
ISO
200
250
300
350
400
x(iran)
Figure 1.2 The velocity field in the pre-breaking wave crest of a plunging breaking wave
[from Perlin et al., 1996].
10
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results of wedge-like diffraction from the mean surface profile (described by the
geometric theory of diffraction) and near-resonant scatter from steep parasitic capillary
waves (described by the SPM). The deterministic wave profiles, corresponding to the
non-specular events, were analyzed using the moments method and were shown to have
good agreement with experimental results (Fig 1.3).
Kwoh and Lake (1984b), using the same experimental setup as in Kwoh and Lake
(1984a), investigated wind generated spilling breaking waves. The radar returns from
wind generated waves were shown to initially exhibit a single spectral peak which was
described by Bragg theory. As the wind was increased, the Doppler spectra exhibited
Bragg and non-Bragg spectral peaks, similar to the findings of Lee et al. (1995). The
non-Bragg peak was attributed to the scattering from breaking waves. Based on these
R -
R_
-zs
Figure 1.3 The numerically predicted RCS versus measured RCS for non-specular scattering events
at various angles of incidence [from Kwoh and Lake, 1984a|.
11
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findings, they concluded that the backscatter from breaking waves was primarily due to
specular scattering, Bragg scattering and diffraction effects.
Loewen and Melville (1991) using a circularly polarized X-band CW radar at 65
degrees angle of incidence, examined piston generated transient breaking waves. They
found that the backscattered microwave power was correlated to the maximum wave
slope and energy dissipation of the breaking wave. This correlation was the strongest for
wave slopes commonly observed in ocean waves (slope <17 degrees). The peak radar
returns were observed prior to any visible whitecapping and were attributed to the
steepened wave geometry prior to wave breaking. The maximum Doppler shift was
shown to occur at breaking and corresponded to the radial component, in the direction of
the radar, of the phase speed of the center component of the wave packet.
Walker et al. (1996), using an X-band CW radar manually rotated between
polarizations at 45 degrees angle of incidence, examined the radar backscatter from
stationary breaking waves, using techniques similar to those of Banner and Fooks (1985).
They found that the radar returns from the roughness on the crest of the breaking wave
yielded normalized radar cross section (RCS) of approximately -3 dB and polarization
ratios near unity. The wave height spectrum of the same region, was shown to “saturate”
for wave numbers greater than 100 rad m*1, at a value of approximately ak'3'5, where a =
0.003mI/2.
Laboratory measurements have also been conducted at grazing angles of
incidence.
Sletten and Wu (1996) using a time domain reflectometer based
ultrawideband radar, examined the radar backscatter from mechanically generated
breaking waves at grazing angles of incidence. They found HH to W polarization ratios
fluctuated as the radar frequency (and thus the relative interference path length) was
swept. Polarization ratios were sometimes greater than unity, which was attributed to
multipath scattering from the ‘toe’ of the wave, and Brewster angle effects which
primarily limited the V-pol radiation to direct scatter.
The H-pol returns were not
12
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affected by the Brewster angle effect and showed large fluctuations as the direct and
indirect scattering paths interfered.
1.2.3 Current Investigations
The laboratory studies of breaking waves have further identified the
characteristics of transient breaking waves of various energy states. However, like field
experiments, laboratory investigations have failed to isolate the actual scattering
mechanisms dominating the radar backscatter from breaking waves of various energy
states.
This is primarily due to most investigations limiting their analysis to point
measurements of a single wave type, at a single angle of incidence, with a single radar
frequency. This “undersampling” prevents a definitive isolation of the actual scattering
mechanisms in breaking waves. Many studies have also been hampered by the lack of
simultaneous high-resolution imaging and radar probing of the breaking waves.
This thesis describes the results of multispectral radar backscatter measurements
on transient breaking waves of various energy states.
The transient breaking wave
experiment (Chapter LH) investigated the K-band radar backscatter from spilling and
spilling-plunging breaking waves at 40 degrees angle of incidence, and at several
different azimuth angles.
X-band grazing measurements were also conducted at 85
degrees angle of incidence in the upwave and downwave directions.
Wave
characterization was accomplished via waveprobes and video/laser sheet techniques.
The high-resolution transient breaking wave experiment (Chapter IV) investigated
the X- and K-band radar backscatter from spilling, spilling-plunging and plunging
breaking waves at 30, 45 and 60 degree angles of incidence, in both the upwave and
downwave directions.
Backscatter measurements were taken at numerous spatial
locations to isolate the scattering mechanisms and determine their relative importance
throughout the entire evolution of the breaking wave. To ensure high fidelity surface
13
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data, radar measurements were coincident with high speed flow visualization techniques.
Extracted surface profiles were analyzed using numerical electromagnetic techniques to
isolate the dominant scattering mechanisms.
14
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CHAPTER II
RADAR SCATTERING FROM THE OCEAN
2.0 Overview
Waves on the ocean are initially created by the transfer of momentum from the
wind as it blows over an unperturbed sea surface. The first waves to appear on the ocean
have wavelengths on the order o f a centimeter or less, and are known as capillary waves.
Capillary waves are dominated by the forces of surface tension. As the wind continues to
interact with the ocean surface over longer distances, or fetches, energy is transferred
through nonlinear interactions from the short wavelength capillary waves to longer
wavelength waves. These long wavelength waves are known as gravity waves, and are
on the order of meters to hundreds o f meters in length. Gravity waves, as their name
implies, are dominated by the force o f gravity. The wind will continue to input energy
into the ocean and create progressively longer wavelength waves until the energy input is
balanced by dissipative forces, such as wave breaking and nonlinear wave-wave
interactions. At this point, the seas are said to be fully developed. Examples of ocean
wave frequency spectra for fully developed seas, at various wind speeds, are shown in
Fig. 2.1.
The distribution of waves on the ocean is further complicated by features such as
long wavelength swell, which may be created from distant storms, and surfactants, which
may dampen out entire portions of the wave spectrum [see e.g. Huhnerfuss et al., 1994;
Espedal et al., 1997]. These features, along with the inherent multiple scales of waves on
the ocean, highlight the complex character of the ocean surface, and the difficulty in
15
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analytically modeling the radar backscatter from the ocean. Because of this difficulty,
ocean scattering models have been dominated by the use of two simplifying
approximations.
The first approximation, used in the physical optics (PO) formulation, assumes the
ocean surface consists of waves which have large radii of curvature compared to the radar
wavelength. Various PO based models have been used to accurately predict the radar
backscatter at near nadir angles of incidence [see e.g. Daley, 1973; Barrick, 1974]. The
second approximation, used in resonant scattering models such as the small perturbation
method (SPM), assumes that the root-mean-square (rms) wave heights and slopes are
small when compared to the incident radar wavelength. The SPM has been used to
explain the magnitude and polarization ratio of the radar backscatter from several field
and laboratory experiments [see e.g. Kwoh and Lake, 1984a; Lee et al., 1995].
Models also exist that apply both assumptions by assuming the ocean surface is
comprised of short waves (as defined by an arbitrary cutoff wavenumber) which scatter
the incident energy through resonant mechanisms, and long waves, which act to tilt and
modulate the short waves and provide quasi-specular scatter at small angles of incidence
(0 < « 25 deg). Models of this type are called two-scale, or composite models and have
proven quite successful at predicting the radar backscatter from the ocean at moderate
angles of incidence and low sea states [see e.g. Plant, 1990; Plant and Keller, 1990]. The
various scattering mechanisms, and corresponding angles of incidence where they
dominate the radar backscatter from the ocean, are summarized in Fig. 2.2.
2.1 Physical Optics
This section will derive the theoretical PO (or Kirchhoff approximation (KA)),
solution for the radar cross section (RCS) per unit area, for the scattering geometry shown
in Fig 2.3. The foundation o f the scattering model to be derived, is the Stratton-Chu
16
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28 kn
T5kn
-
-3
5 kn
f~ ‘
ASYMPTOTE
0.01
1
0.1
10
MHz)
Figure 2.1 Ocean wave frequency spectra for a fully developed sea at various wind speeds
[from Wetzel, 1990J.
£
10;
3
o
Sptcalar -Point Modal
\
\ p» t.0 0 0 , a * - 0 0 4 0 1
» 9 « GHz
Vortical Poloriration (u p -v ia d )
• - All Sm Conditions
V K\ \
*
Slightly ftovgti
y'tw .sxio-sk-*
\ .\
" r '
' vi
K
c *30
•
Spacvior 0)
voosi-Spoculor
QlffvtO ScOlMrMg
----- ----B » o g g -S c a tte rin g , W ave-W ave Modol,
Composite - S at tfi« o , Wovo - F a c t ' Modol
S c a tte r in g
Spacatar
“ Point Modol
20
30
A ngle o f
40
50
Incidence
60
t
I o*\t
: Shadowing
> Oil traction
Trooping
70
so
(degrees)
Figure 2.2 Illustration of scattering mechanisms which dominate the radar backscatter from the
ocean at various angles of incidence [from Valenzuela, 1978].
17
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integral equations. The Stratton-Chu integral equations may be written as [Stratton,
1941],
E(r) = EfcCr) + | {-jcop(n x H )G (r,r’) - (fix E) x V G (r,r')- (no E)VG(r,r')} dS
S
( 2 . 1)
H(r) = ! !„ !(? )+ / {-jcoe(n x E)G (r,r') - (n x H) x VG (r,r') - ( no H)VG(r,r')} dS
s
where
E
H
n
8
CO
r
r’
=
=
=
=
=
=
=
=
Surface electric field
Surface magnetic field
Outward unit surface normal
Surface complex permittivity
Surface complex permeability
Radial frequency
Reference to observation point vector
Reference to surface point vector
and G(r,r') is the free space Greens function, for an e1051time dependence, given by
G (f,r’) =
J k lf-f’l
(2.2)
47c|f-r'|
-x
x ,y ,2
— <—
r- r
+x
Figure 23 Scattering geometry for the physical optics and iterative method derivations.
18
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In words, Eq. 2.1 states that the fields within a given volume are completely
determined by the incident fields ( E in, H in) and the fields (or charge and current
densities) on the surface surrounding the volume. While Eq. 2.1 is extremely powerful
and allows the for the solution of radiated fields without the tedious conversion to
potentials, it does have limited usefulness in its present form.
The Stratton-Chu
Equations represent a pair of coupled integral equations which are difficult to solve
analytically for all but the most basic geometries.
To decouple the equations, an
assumption is made that the surface is a perfect electric conductor (pec). Due to the high
salinity in the ocean this is a good approximation at low frequencies (HF), but does not
hold as well at microwave frequencies. However, theoretical results derived using the
pec assumption have compared well with experimental results even at microwave
frequencies.
The classical electromagnetic boundary conditions for a perfect electric conductor
are,
nx E = 0
,
nxH
(2.3)
no E = —
e
,
no H
=
0.
where Js is the electric surface current density and ps is the electric charge density. As
can be seen, the surface can only support normal electric and tangential magnetic fields.
Applying the pec boundary conditions to the Stratton-Chu equations yields,
E (f)=
E ^ r H f {- jcop(nx H )G (r,r')- (noE)V G (f,r')} dS
H(r) = H in(r) -
S
J
_
(nx H )x VG(f,r') dS.
S
19
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(2A)
The magnetic field integral equation (MFIE) is now decoupled from the electric field
equation and will be used for further analysis.
The fields on the surface may be obtained by letting the observation vector ( r ) be
evaluated on the pierced surface S', where S' represents the entire surface S except the
singular point r = r’. From Holliday et al. (1986), the surface fields may be written as,
H(r) = 2H in( r ) - 2 J (fix H )x V 'G frr’) dS'.
S'
(2.5)
Eq. 2.5 is simplified by neglecting the integral term, resulting in the tangent plane
approximation (TPA).
The tangent plane approximation assumes that the surface
magnetic field or current density, at any point on the surface, is the same that would exist
if there were an infinite tangent plane passing through that point.
Because of this
assumption, the PO formulation will only yield exact solutions for flat surfaces in which
the integral in Eq. 2.5 is identically zero. However, the PO approximation is quite good
when, kl > 6 and I2 > 2.76 sX, where 1 is the surface correlation length (in cm) and
represents a measure of the surface horizontal roughness, and s is the surface rms height
(in cm) and represents a measure of the surface vertical roughness. [Ulaby et al., 1982].
Using the TPA and inserting the resulting field into the MFIE in Eq. 2.4 yields the
PO scattered field ( Hs) integral equation,
Hs(r)=
- J (flx 2H in(f))x V’G (r,f) dS.
s
(2.6)
To obtain the theoretical backscatter, a form of the incident field must be assumed. I will
assumed a weighted plane wave of the form,
_
H in(f)= p .H 0 -L (x',y')-e
J-^in 0 ?
.
(2.7)
20
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where p is the polarization vector, H0 is the magnitude of the incident field, L(x’,y’) is
the two-dimensional weighting function over the surface coordinates and
is the
vector wavenumber of the incident radiation.
In the far field, the term | r - r' | can be approximated as | r | for magnitude terms
and jr'| - ( r o f ') / |r | for phase terms. Applying the far field approximations to the
gradient in Eq. 2.6 yields,
Hs(r)=
- j - i — — J ( n x H in) x k 0Ut •e‘ j’k°“ or’dS
2rc|r| £
(2.8)
where kout is the vector wavenumber of the scattered field. Using the vector property
A x B x C = (A ° C)B - (Bo C)A on Eq. 2.8 yields,
Hs(f)=
- ;L^ ^ J [ ( a « k „ „ , ) H ill- ( k <>ut»H in)n ]e-j'E~ - f'dS.
s
For backscatter,
equal zero since kout,
(2.9)
k ^ will equal - k out. This will make the second term in Eq. 2.9
as well as k ^ , will be perpendicular to H ^ . Denoting the
backscatter wavenumber as k (where k = k Qk), and inserting Eq. 2.7 into 2.9 yields,
Hs(f) =- - Hf p ;.J, °- 1 J(no k) • L(x',y') •e- 2J kof’d S .
27t i r l
(2.10)
g
The surface unit normal may be expanded as,
n=
nxx + nyy + nzz.
(2.11a)
where
~
nx =
V
t
1x
=»
VVT1x+VT1y +1
(2-llb )
21
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- V ti
ny = - ~i~ • ' — - - ,
VVTi x + v ny + i
(2.11c)
nz = - - j~ ,
,
VVt1x +VTly +1
(2.1 Id)
and Vrj is the surface slope in a given direction. The unit vector backscatter wavenumber
can be defined in a similar manner as,
k = kxx + k yy + kzz.
(2.12a)
where
lr x_
X~ X> U - y ~ y ' anA U _ ^ ( ^ y)-T l(x \y O
I r - r 'l ’ ^y ~ I r - r 'l ’
I r - r 'l
’
(2-12b,c,d)
Applying Eqs. 2.1 la-d and 2.12a-d to Eq 2.10 and expanding dS in Cartesian coordinates
yields,
H s ( f )
=
~
J ~
2 n ^ |
K
k
z
"
V
n * k , t "
V T l y k y ]
• U
x ’ . y 1)
- e
2 j [ k < x '+ k y
x ' -y' > 1 ^ . d y .
(2.13).
Eq. 2.13 may be integrated by parts to yield,
ZTt-lrl
an )
Kz c
The expectation value of the squared magnitude of the backscattered field (<| Hs |2>) will
ultimately be needed to calculate the radar cross section (RCS). Assuming the wave
heights to be a Gaussian random variable [Kinsman, 1984; Phillips, 1966], and using the
property for a Gaussian random variable [Cramer, 1946],
22
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(2.15)
yields
where
x = x'-x" , y = y'-y" , z = r|(x ',y ')-ri(x ",y ")
and 0(x,y) is the two-dimensional covariance function of the surface and \
(2.17)
is the
illuminated area on the surface. The antenna weighting function has been dropped under
the assumption that the illuminated area is large compared to the region where
<t>(0,0) - <t>(x, y) is not small.
Since Gaussian statistics were assumed, the covariance
function may be written as
4>(x,y)=J ¥ ( K xKy)cos(Kxx + Kyy)dKxdKy
(2.18)
where Kx and Ky are the ocean wave numbers in Cartesian coordinates and ^(K*, Ky) is
the surface elevation spectrum.
The average RCS per unit area may be written for a large illuminated area as,
4ttR2 <
|H s(r)|2 >
(2.19)
where < > represents the ensemble average and R<, is the mean distance to the surface.
Inserting Eq. 2.17 into Eq. 2.19 yields the theoretical physical optics RCS equation,
23
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Z
S
As can be seen, the PO formulation has no polarization dependence. This explains why
its applicability has been mostly limited to near nadir angles of incidence, where
polarization independent specular scattering dominates.
2.1.1 Geometric Optics
The geometric optics (GO) formulation is merely the high frequency limit (Iq -»
oo) of the PO formulation. In the GO formulation, the surface is represented by small flat
facets. The radar return is assumed to only occur for those facets that are normal to the
radar. This is known as the stationary phase approximation (SPA). Since this model
considers the backscatter as a bundle of rays, no polarization or frequency dependence is
predicted.
The GO formulation is derived by noting that as the frequency gets extremely
large, the exponential term containing the covariance functions in Eq. 2.20 is nonzero
only over small values of x and y . In this region, the covariance function may be
expanded by Taylor series and truncated to,
From Eq. 2.18 this may be reduced to,
By applying Eq. 2.22 to Eq. 2.20, and using the properties of Fourier transforms and the
SPA, it can be shown that Eq. 2.20 reduces to,
24
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ctq =7tsec4 0-p(Vrix ,VTiy)
(2.23)
where p(Vr]x, Vqy) is the joint slope probability density function (pdf), Vr]x and Vr|y are
the specular slopes and 0 is the angle of incidence. Eq. 2.23 has been written in various
forms.
Some authors multiply the solution in 2.23 by Fresnel reflection coefficient
evaluated at nadir [see e.g. Barrick, 1968].
However, in this derivation the pec
assumption was made, making the Fresnel reflection coefficient unity. If the slope joint
pdf is assumed to be isotropic and Gaussian, with zero mean, Eq. 2.23 may be written as
[Valenzula, 1978]
A
-tan2 9
sec 0
CTo =
2
s
2
’e
S
(2 -24)
The GO approach is derived from the PO approach, and is therefore subject to the
same roughness criterion for applicability. However, the stationary phase approximation
places an addition validity constraint that the vertical roughness parameter (ks) be greater
than 2.2 [Ulaby et al., 1982].
2.1.2 Specular and Resonant Scattering
Specular and resonant (or Bragg) scattering can also be derived from the PO
solution by taking the low frequency limit (lq -> 0). In the low frequency limit, the rms
surface height is considered to be much less than the radar wavelength.
This is
commonly called the small amplitude approximation (SAA), and allows us to rewrite the
exponential term containing the covariance functions in Eq. 2.20 as
l-4k*[a>(0,0)-<D(Xfy)].
25
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(2.25)
By applying Eq. 2.25 to Eq. 2.20, and using the properties of Fourier transforms, it can be
shown that Eq. 2.20 reduces (for incident wavenumber in the x-z plane) to,
ct°
=87tk4 ['F (2k x ,0) + 4 /( - 2 k x ,0)] + 87tk4[l-4 k ^ < T i2 > ]-5(k x ,0)
(2.26)
where kx = lq sinO, and lq is the radar wavenumber. The second term in Eq. 2.26
represents the quasi-specular scattering component. As can be seen by the delta function,
the quasi-specular term only acts when the radar looks straight down.
The first term in Eq. 2.26 is due to resonant (or Bragg) scatter. Resonant scatter
occurs when a water wave’s wavenumber component in a given direction is twice that of
the radar’s wavenumber component in the same direction.
When Bragg waves of
sufficient amplitude exist, the water surface acts like a diffraction grating and yields a
surprisingly large amount of backscatter. The resonant term in Eq. 2.26 is similar to the
result derived from the SPM (Eq. 2.28). Holiday (1987) showed that if a higher order
slope terms are carried in the PO derivation, then the results are identical to those from
the SPM.
Theoretically, for the SAA to be valid, both ks and kl must be extremely small
when compared to the radar wavelength, and ks must be less than kl [Ulaby et al., 1982],
However, experimental observations have shown resonant based scattering models to
match experimental observations well beyond these limits and to even predict the first
and second order Bragg lines in Doppler spectra [Wetzel, 1990].
2.2 The Small Perturbation Method
The Bragg scattering relations can be derived by an alternative technique called
the small perturbation method (SPM). The SPM is based on a mathematical formulation
developed by Rice (1951). Basically, the surface fields are expanded into a series of
terms with the first order terms corresponding to the mean surface and the higher order
26
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terms corresponding to the roughness. The small scale roughness is replaced by effective
surface currents on the mean surface and a scattering solution is derived.
The SPM relies on the Rayleigh approximation (RA), which assumes that only
up-going scattered waves from the surface are included in the final scattering solution.
Valenzuela (1978) found by analyzing the radar backscatter from corrugated surfaces,
that the RA is valid for slopes less than 0.448. This holds for gravity waves, but not for
capillary waves which are the primary scatterers at microwave frequencies above
approximately 15 GHz. A roughness-based validity criterion was developed by Brown
(1978). The SPM was found to be valid when the roughness parameter £ , given by,
(2.27)
is much less than unity, where < r|(x \ y’)2> is the surface height variance.
Using Rice’s formulation, Peake (1959) developed the first working small
perturbation model. The first order RCS was found to be given by,
a ° ( 9 ) = 167dc4 -Igy-te)2 - T ^ s i n ^ O ) ,
where
(2.28b)
[cos(0) + (e - sin (0))v ]
rm
and
^
(2.28a)
cos2 ^
'^g ~ ^ ^ + sh]2 ^ ~ sifl2 ^
[s cos(0) + (e - sin2 (9))12 ]2
(2.28c)
4/(2krsin(0),O) is the two-dimensional wavenumber spectrum evaluated at the Bragg
wavenumber, and e is the complex dielectric constant given by the Debye relation for
water [Ulaby et al., 1982].
Assuming a pec surface, the SPM reflection coefficients reduce to ,
ghh(0) * cos2(0)
m d’
g
w
( 9
) S5[ l
+ sin2(0)].
(2.29)
27
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As can be seen, for all angles except nadir, Bragg theory predicts that the V-pol radar
return should always be greater than the H-pol return. This is contrary to the observations
of several field experiments conducted at both grazing [see e.g. Lewis and Olin, 1980]
and moderate angles of incidence [see e.g. Jessup et al., 1990]. These observations have
cast doubt on the universality of Bragg scattering based models, and have highlighted the
need for the inclusion of such features as breaking waves.
2.3 Two-Scale Models
To account for some of the discrepancies discussed with the Bragg scattering
theory, a composite or two-scale model (TSM) was developed Wright, (1968). The TSM
makes the assumption that the ocean is made up of patches of Bragg scatterers that obey
the small amplitude approximation and are modulated by the motions of longer waves on
the surface. The TSM is not a true theoretical model and therefore, will only be discussed
briefly. For a thorough derivation of the TSM one is directed to Plant (1990).
The TSM works by using the sea slope distribution to yield a sea slope averaged
RCS about the mean grazing angle. This is illustrated in the form of a 1-D model
[Wetzel, 1990],
00
J
g ° ( 0 o) =
-
a ° (9 0 + a)p (a)d a
(2.30)
00
where 0Ois the mean grazing angle, a is the local surface slope and p(a) is the sea slope
distribution.
Although the TSM is not derived from rigorous theory, it agrees with
observed data and has become somewhat of the defacto standard of ocean scattering
models. However, like the SPM, the TSM tends to overpredict the observed V/H-pol
ratio in heavy seas, and fails to take into account features such as breaking waves.
28
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2.4 Exact Methods
Attempts have been made to develop scattering models that avoid the inherent
inaccuracies of the models previously described (PO, SPM, TSM).
Some methods,
known as unified theories, involve rigorously deriving the scattering equations from
advanced mathematical formulations which reduce to the PO and SPM solutions under
certain limiting conditions [see e.g. Rodriguez, 1989, 1991]. Other methods expand out
the cross and dot product terms in the Stratton-Chu integral equations and make various
simplifying assumptions to yield scattering solutions. These techniques are collectively
known as the integral equation method (IEM) [see e.g. Mitzner, 1967; Fung and Chan
1969],
The method used to make numerical scattering calculations in this thesis was
developed by Holliday et al. (1986). In this method, the MFIE is converted into a surface
current integral equation and then evaluated iteratively on the surface, until convergence
is achieved. Solutions from the iterative method have proven quite accurate in explaining
the radar backscatter at low grazing angles of incidence [Holliday et al., 1993] and for
wedge like surfaces [Holliday et al., 1995]. Iterative solutions also have the advantage of
linking higher order scattering effects to the higher order iterations. Thus, the number of
iterations required to reach convergence may be used to identify the type of scattering
mechanisms present. For this reason, the iterative method was chosen to analyze the
deterministic wave profiles obtained during the experimentation in this thesis.
2.4.1 Iterative Method
29
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The MFIE (Eq. 2.5) may be written in terms of the surface currents by taking the
crossproduct of both sides with the surface unit normal. This yields the surface current
integral equation,
J (r)=
JfnCO- 2nx J { 1 (f)x V 'G (r,r')}dS\
S'
(2.31)
where
!* (? )=
2hx HinCf).
(2.32)
Rewriting the surface current integral equation in iterative form, yields,
JN(r)= J in ( f ) - 2nx j {JN_ ! ( f ) x V‘'G(f,f)}dS'
(2.33)
S’
where N represents the Nth iteration of the surface current integral equation. For the
zeroth iteration,
J 0 is approximated by J in. The zeroth order solution is, infact, the PO
solution described in section 2.1. As previously stated, the PO solutionwill only yield
exact solutions for flat surfaces. Higher iterations of the surface current integral equation
are required to correct the surface currents for higher order effects such as diffraction,
shadowing, finitecurvature
and multiple scattering [Liszka and McCoy, 1982]. The
iterative process is repeated until J N - J N_, is less than a predetermined convergence
criterion.
In practice, it is easier to find the difference in surface currents after each
successive iteration (5 J N). Thus, by letting 8 J 0 = J jn , the change in surface currents
between iterations may be written (for N > 0) as,
8JN(r) = - 2 n x
J
{SJn . j ^
x
V 'G (r,r')}d S '.
S'
The final surface current density will then be,
30
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(2.34)
f
N
JN(r) = Z
(2.35)
SJnOO-
n= 0
Once the final surface current is found, the scattered field may be obtained by inserting
J Ninto
the MFIE (Eq. 2.4) yielding,
Hs(f)=
(2.36)
- j J N * VG(r,r’) dS.
s
The scattered field can be related to the average radar cross section per unit area by Eq.
2.19 which is rewritten here for convenience as,
4t:r 2 < |H s(r)|2 >
lim
c t° =
R0
—>
(2.37)
°o
As|Hin|2
The denominator of Eq. 2.37 can be found by integrating the incident field over the
surface,
(2.38)
A s |H to|2 = / IHjnCOI2 dS.
s
The incident field may be calculated from the antenna dimensions and aperture
illumination (TE10 or TEn), as described by Balanis (1987), or from a Gaussian weighted
plane wave of the form,
- 2 .7 7 2 ( - |- )
Hin =
p - H 0 -e
***
-e
:
(2.39)
31
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where primed coordinates represent points on the wave surface and WFXand WFy are the
along and crosstank weighting functions respectively. The polarization vector p is given
by,
H-pol:
V-pol:
p = cos(<|>) cos(0)x + sin(<j>) cos(0)y - sin(0)z
p = sin(<j>)x - cos(4»)y.
(2.40)
The iterative technique requires N squared calculations, where N is the number of
grid points over the whole surface, for each higher order (beyond PO) iteration. As can
be imagined, the iterative technique becomes quite computationally burdened when
analyzing large two-dimensional surfaces at the large (5-10 per wavelength) grid point
densities required for accurate scattering solutions. This problem may be avoided in
cases where the surface is two-dimensional and is being illuminated by a weighted plane
wave in an upwave or downwave direction. For these surfaces the scattering formulation
may reduced to a one-dimensional problem, as shown by Holliday et al. (1986).
The conversion of the scattering problem from a two-dimensional to a one­
dimensional formulation is based upon the assumption that for a two-dimensional surface
infinite in the y’ direction, all of the points for a given x', z' will possess the same current
density. In other words, the surface currents have no y' dependence. Because of this, the
grid in y' direction may be collapsed by treating all the points in the y' direction (for a
given x', z' point) as a single line of charge which radiates cylindrically, vice individual
point sources which radiate spherically.
The mathematical derivation follows from Holliday et al. (1986), and is initiated
by separating Eq. 2.33 into x' and y' components,
00
00
e j k 0 |r - r '|
dy'
SJ>l(r) = ~ 2 I n(x)x J N_ !(x')x
-0 0
4 7 t|r-r'|3
-0 0
dx'
—(2-41).
nz(x')
32
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Note how only the expanded Green’s function has a y1 dependence. The y1 integral in Eq.
2.41 may be converted to cylindrical functions by applying the Hankel function property,
00 j 'V x 2 + t 2
J
<2 -4 2 )
-00 VXZ + t 2
where
(x) is a Hankel function of the first kind and zeroth order. The Hankel
function for large argument and order n is given by,
ji
H n >(x) =
jt
-eJ<X~n2
(2.43)
Carrying out the inner integration leads to the one-dimensional surface current iteration
equation,
00
SJnC?) = ^ k 0-n(x)x J
/_
J n - i ( xO x
_ \
L
H{1)(k0| r - r ,| ) - ^ *— .(2.44)
-0 0
For the upwave or downwave illumination, the cross products in Eq. 2.44 may be
carried out to yield separate equations for each polarization. The H-pol illumination will
only create y directed electric surface currents given by,
J
°f f
(x -x ')
6JN(f) = - k 0 J ( n x(x)- + n z(x )•
(z —z'))
m .
j -j n - i ( x') - h i (k|f ~ f ‘
—00
.
dx'
z
(2.45).
The V-pol illumination will create electric current densities in both the x and z directions.
The x directed surface current density is given by,
8Jn (?)= 2 k° 1 J N~i(x<) •n z(x) •f e .
—
~ Vt1x(x>) •
) •H(!1}(k |r- r 1) •
00
2
(2.46).
33
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The z directed surface current density is merely the product of the x directed current
density (Eq. 2.46) and the x directed slope (V qJ.
Once the final surface current densities are obtained, they are inserted into a one­
dimensional, far-field, weighted plane wave version o f the MFIE, as given by Holliday et
al. (1986),
H no -
i4 y n k 0|r| e{k"lsl'r)
P
P
*
k . e ^ ' .e 2'772(WFY} jdx^77- (2-47)
nz(x')
j
J Jn
1
-0 0
The backscatter is described in terms of a one-dimensional echo width (EW) given by,
< |Hs(f)|2 >
E W i_d =27t[r|— —-------- y—
(2.48)
|H fa(r )|2
To validate the iterative scattering codes, one-dimensional echo widths, calculated from
Eq. 2.48, were determined for various wedges and then compared to the exact
Sommerfield wedge scattering solutions. This validation procedure is described in detail
in Ericson (1997) and results from wave-like surfaces are given in Lyzenga and Ericson
(1997).
To compare numerical results with experimental observations, requires deriving
the average RCS per unit area from the one-dimensional echo width equation. This is
accomplished by first converting to a two-dimensional echo area (EA) using the relation,
EA2_ d = EW!_d —
K
r
-2.772(——)
wf '
dy'
(2.49)
-00
The two-dimensional echo area is then converted to the average RCS per unit area by
dividing by the illuminated area,
34
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In Chapters IE and IV, these techniques are used to numerically calculate the
theoretical RCS from deterministic wave profiles extracted during the experimental
portion of this thesis. The theoretical RCS are compared to the measured RCS obtained
from the same deterministic surface profiles. This technique provides a means to isolate,
identify and model the radar backscatter from transient breaking waves.
35
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CHAPTER III
THE TRANSIENT BREAKING WAVE EXPERIMENT
3.0 Experimental Design
The transient breaking wave experiment was the first of two comprehensive
laboratory experiments designed to isolate and identify the scattering mechanisms in
intermediate to deep water breaking waves of various energy states. The experiment was
conducted in the 110 m x 7.6 m x 4 m deep model basin at the University of Michigan’s
Marine Hydrodynamics Laboratory. The width of the tank permitted the study of the
azimuthal and grazing dependence of the radar backscatter without contamination from
secondary reflections. The breaking waves were generated using a wedge shaped wave
maker with a 30 degree face. The wavemaker was programmed to produce a sinusoidal
output whose frequency was linearly decreased over a small bandpass of frequencies.
The frequency components were chosen, based upon linear wave theory (See Appendix
D), to create waves that would add constructively at a given location in the wave tank, as
described by Davis and Zamick (1964). The wavemaker was controlled by a “master”
PC which provided the command-signal to the wavemaker, and sent TTL (transistortransistor logic) trigger signals to the “slave” PC based radar data acquisition system and
the video camera.
For this experiment, a “low energy” spilling breaking wave, characterized by
minimal whitecapping, and a “high energy” spilling-plunging breaking wave,
characterized by considerable whitecapping and a large splash-up on the wave face, were
chosen for analysis. The spilling breaking wave was generated using frequencies from
36
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1.75 Hz to 1.08 Hz, and the spilling-plunging breaking wave from 1.2 Hz to 0.75 Hz.
Waves produced at these frequencies ranged from deep to intermediate depth gravity
waves. Using linear wave theory and the deep water approximation, the maximum phase
speeds were found to equal 1.44 m/s and 2.08 m/s. respectively, (See Appendix D). An
important point to note is that the laboratory breaking waves used in this experiment
differed from ocean waves in that they did not have wind generated capillary waves on
their surfaces. This was done intentionally so that the scattering mechanisms directly
attributable to breaking waves could be isolated.
For ease of notation, the spilling
breaking wave will be referred to as the low energy breaking wave (LEBW), and
similarly, the spilling-plunging breaking wave as the high energy breaking wave
(HEBW).
For the moderate incidence angle measurements, the K-band CW radar was
aligned so that wave breaking (as defined by the appearance of small scale roughness on
the wave face) occurred near the radar’s boresight on the water surface. To prevent
contamination from secondary reflections, a rotatable 16 sq. ft. wall of radar absorbent
material was positioned behind the radar footprint on the water surface. The K-band
radar had a 1-way 3 dB footprint on the water surface of 37.9 cm by 25.8 cm for HH
polarization and 34.1 cm by 28.7 cm for W polarization (along tank by crosstank). Each
polarization produced a continuous time series of complex radar backscatter data that was
sampled at rate of 512 Hz per channel by a PC based data acquisition system utilizing a
Metrabyte DAS-8, 12 bit A/D board. Data acquisition was triggered via a TTL high-low
signal from the master PC. Four runs were conducted for each wave, at each azimuth
angle.
For grazing measurements, the X-band FMCW radar was aligned so that wave
breaking occurred near the radar’s boresight on the water surface. Pulse repetition
intervals (PRIs) of 10.0 ms (6.6 m unambiguous range) were used for H-pol, and 14.3 ms
(4.2 m unambiguous range) for V-pol. Radar absorbent material was used to prevent
37
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contamination from secondary reflections. The 1-way 3 dB crosstank beamwidths on the
water surface, at radar boresight, were 81.4 cm for H-pol and 63.6 cm for V-pol. The
radar was swept over a frequency range of 2.2 GHz, giving it a theoretical range
resolution of 6.8 cm. The actual range resolution was found from calibration to be 26.2
cm. The radar was only capable of single polarization operation and had to be manually
rotated between HH and W configurations. Antenna heights were not equal in these
configurations. Therefore, the slant range and PRIs were chosen to maintain the 85
degree angle of incidence and ensure the breaking wave was within the unambiguous
range of the radar. The X-band FMCW radar produced a continuous real time series of
radar backscatter data that was sampled at a rate of 32 KHz by a PC based data
acquisition system utilizing a National Instruments MIO-16X, 16 bit A/D board. Data
acquisition was triggered via a TTL high-low signal from the master PC. Four runs were
conducted for each wave, at the downwave (0 degree) and upwave (180 degree) azimuths.
The video recordings of the breaking waves were referenced in time by placing a
red light emitting diode (LED) just inside the field of view of the video camera. The
LED was driven by the same TTL signal used to trigger the two radar data acquisition
PCs, and would extinguish when the TTL trigger signal went from a high to low logic
state. With this reference, the video data could be compared directly to the radar returns
within the 30 Hz accuracy o f the video camera. Video data was initially stored on
commercial tape and then transferred to a PC using commercial software. Images were
320 horizontal pixels by 240 vertical pixels. Points on the wave surface extracted from
the images were accurate within 1 pixel or about 1.7 mm horizontal and 2.3 mm vertical.
Runs were commenced by determining the exact point and time past commandsignal for each wave to break. For the LEBW, the radar, video camera and laser sheet
were positioned so that their fields of view coincided and were centered at the point of
breaking (Fig. 3.1). The HEBW measurements offset the laser sheet from the point of
38
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i1
1
1
1
!
1
Radar
40 deg AOI
r- ■
«-------►
--
^*l>K
>,
V*
i
i
Laser
ig ta M K l
4-------►
Radar
Absorbent
Material
Radar
85 deg AOI
Video
Camera
Side View
It
Video
Camera
Top View
180 deg.
135 deg.
U
Radar Absorber
Wavemaker •
Figure 3.1. Diagram of the laboratory experiment. The side view demonstrates the configuration for
the upwave 40 and 85 degree angle of incidence measurements. The top view demonstrates the
configuration for the downwave 40 degree angle of incidence measurements, along with the radar’s
locations during the other azimuth measurements.
39
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breaking by distances of 35 cm and 95 cm. After a wave was generated, the master PC
would drive the TTL signal to its low state (triggering data acquisition), one second prior
to the determined wave breaking time. Four seconds of K-band CW radar data and ten
seconds of X-band FMCW radar data were collected for each respective measurement.
The water level was maintained at a constant level throughout the experiment and cleaned
with several surface skimmers between each series of runs. Runs were initiated twenty
minutes apart to ensure a still water condition at the beginning of each run.
3.1 Experimental Equipment
3.1.1 Radars
A comprehensive description of CW and FMCW radar theory is included as
Appendix A.
All experimental radars were internally and externally calibrated as
described in Appendix B. Internal calibrations determined gain errors, phase errors and
DC biasing inherent to the radars components and circuitry.
External calibrations
determined the returned power from targets of known radar cross section and the
antenna’s normalized radiation pattern.
These calibration parameters were used to
determined target RCS within +/- 1 dB [Ulaby et al., 1982]. The 2-way normalized
antenna gain patterns, determined from calibration, are shown for the K-band (Fig. 3.2a)
and X-band (Fig. 3.2b) radars.
3.1.1.1 The K-band CW Radar
A K-band CW radar (Fig. 3.3a,b), was used for the moderate angle of incidence
(40 deg) measurements. The K-band radar was a transceiver based system which utilized
an orthomode transducer (OMT) for single antenna, dual polarization operation.
To
obtain simultaneous polarizations, the radar alternately transmitted HH and W
40
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
polarizations, using a 5 KHz switching rate and a 50 percent duty cycle. Thus, the HH
and W transceivers were excited by square waves that were 180 degrees out of phase.
The switching rate was chosen to be an order of magnitude faster than any anticipated
Doppler frequency in the water waves under analysis.
The output from the HH and W transceiver units were the basebanded In-Phase
(I) and Quadrature (Q) signals. The received I and Q signals had to have their square
wave components removed prior to signal conditioning. This was achieved through the
use of active filters and sample and hold circuits. The sample and hold circuits were
driven by the same timing circuits as the transceiver’s Gunn diode oscillators and
removed the intermediate frequency (IF) signal’s square wave components. After signal
reconstruction, the signals were passed through DC removal circuits, a selectable gain
amplifier and a lowpass filter.
The K-band scatterometer included an internal calibration source [Fliss and
Mensa, 1988]. During calibration, the transceiver output was directed to a high speed
microwave switch which was alternated between an open and loaded condition
(simulating a moving target). The received energy took the form of a sqaure wave, and
therefore, bypassed the DC removal circuits.
This technique provided a means of
obtaining both the calibration power and the Doppler frequency resolution of the radar.
3.1.1.2 The X-band FMCW Radar
An X-band FMCW radar (Fig. 3.4a,b), was used for the grazing angle of
incidence (85 deg) measurements.
The radar utilized a voltage controlled oscillator
(VCO) and discrete RF components for transmission, reception and mixing of the radar
signal. The voltage input to the VCO was linearly swept with a triangle wave, created by
internal signal generation circuitry. This generated a swept radio frequency (RF) output
(9.22 - 11.42 GHz) which was directed to the transmitting antenna and radiated.
41
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1C
—
l a n d CW Radar M o nn atiad Antarma Patlam
t
00
0.0
0.7
*a
<»
1e
0.9
0 .1
o3 0
20
o
to
to
20
30
20
90
A ngla fdaflj
(a)
1
X—B a n d FMCW R a d a r M o rw a th a d A ntan rta Pattarrt
0.0
0.0
0.7
0 .0
*a
0
Q-
1|
04
0 .3
O .t
0■30
20
to
t0
(b)
Figure 3.2 Normalized antenna gain patterns for the (a) conical horn antenna of the K-band radar
and (b) the pyramidal horn antennas of the X-band FMCW radar.
42
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Signal
Conditioning
HH
Xcvr
5 KHz
Calibrator
Open
OMT
Matched
Load
W
I _
Q -
Signal
Conditioning
Xcvr
(a)
e
£
(b)
Figure 33 Schematic (a) and photograph (b) of the K-band CW radar (on the right) in the
downwave configuration. The FMCW radar (on the left) was used to observe secondary reflections
while the radar absorber was being placed.
43
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Cal/TX Sw.
VCO
TX
Cal.
Line
RFout
Signal
Conditioning
RFin
RX
(a)
(b)
Figure 3.4 Schematic (a) and photograph (b) of the X-band FMCW radar in the V-pol upwave
grazing configuration.
44
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The scattered energy was collected by a separate receive antenna and directed into
the mixer. The received signal was mixed with the RF signal coupled off o f the transmit
line. This produced an IF based upon the frequency difference between transmission and
reception, which was a function of target range. The IF output was high pass filtered, low
pass filtered and amplified prior to sampling.
The X-band FMCW radar had a switch selectable internal calibration mode.
When calibration was selected, the VCO’s RF output was directed into a fixed length of
semi-rigid coaxial cable, and then into the mixers. The resulting intermediate frequency
and power were recorded throughout the experiment to ensure accurate RCS
measurements.
3.1.2 Wave Characterization
To ensure the repeatability o f each run, both intrusive (capacitance wave probes)
and nonintrusive (laser sheet and video) techniques were used to monitor the
mechanically generated waves. Wave probes determine the instantaneous wave height at
a point by comparing the outputs o f an internally mounted local oscillator and variable
oscillator. The variable oscillator is connected to a wave probe element (acting as a
variable capacitor) and changes its output frequency based upon the instantaneous water
height on the probe. The outputs o f the two oscillators are then mixed and converted to a
voltage, linearly proportional to the wave height. The wave probes were sampled at 100
Hz, and were accurate to 0.2 mm.
The temporal data from the waves probes were used for several purposes. Time
series plots of the waveprobes nearest the wavemaker, were used to monitor the
consistency of the generated wave groups and ensure the repeatability of all runs. The
wave probes nearest wave breaking were used to analyze the spectral content of the wave
as it evolved though the breaking sequence.
45
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The laser sheet was produced by a Spectra-Physics 164, 5W, Argon-Ion laser.
The laser light was passed through two cylindrical lenses (Newport f34, CKX025), which
fanned the beam in the along tank direction, and illuminated approximately 55 cm of the
wave surface. The illuminated area was filmed at a 45 degree angle of incidence with a
commercial camcorder at a 30 Hz rate.
Video data were calibrated by imaging a
calibration target etched with 5 cm squares at the intersection of the laser sheet and the
water surface.
This provided a means of focusing the video system as well as
determining horizontal and vertical scaling factors.
3.2 Experimental Results (Moderate Angles of Incidence)
3.2.1 Temporal Data
The spatial and temporal evolution of the breaking waves, found from the video
data, are shown in cartoon form for the LEBW (Fig. 3.5) and the HEBW (Fig. 3.6), with
the small scale roughness indicated as thick lines. As can be seen, the LEBW generates
only minimal roughness on the wave face and is similar to the waves analyzed by Duncan
(1994). The HEBW generated a large splash-up region as described in Bonmarin (1989).
It should be noted that the HEBW entered the laser sheet at time =1.0 seconds. The time
= 0.8 and 0.9 seconds profiles are approximated based on visual observations and
waveprobe data.
Representative plots of the backscattered power versus time are shown for the
LEBW (Figs. 3.7a-e H-pol, Figs. 3.7f-j V-pol) and HEBW (Figs. 3.8a-e H-pol, Figs. 3.8fj V-pol), at all measured azimuth angles. As can be seen, the 180 degree (upwave)
measurements produced the largest radar backscatter for both breaking waves. These
returns were detected as the wave faced steepened through specular or near-specular
slopes, relative to the radar, prior to wave breaking. For the LEBW, the peak RCS
46
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Low Energy Breaking Wave Sequence
t= 1.5 sec
t = 1.45 sec
t= 1.4 sec
t= 1.35 sec
t = 1.3 sec
t= 1.25 sec
t = 1.2 sec
-30
-20
|«=
-10
0
Radar 3dB Footprint
10
20
=>|
30
Distance From Radar Boresight (cm)
Figure 3.5 Cartoon of the low energy breaking wave’s (LEBW) spatial and temporal evolution, with
respect to the radar’s 1-way 3dB pattern on the water, at 40 degrees angle of incidence.
47
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
High Energy Breaking Wave Sequence
t = 1.5 sec
t = 1.4 sec
t= 1.3 sec
t= 1.2 see
t= 1.1 sec
t= 1.0 sec
t = 0.9 sec
t = 0.8 sec
-20
-10
0
10
20
|<= Radar 3dB Footprint =>|
30
40
50
60
70
80
90
100
110
Distance From Radar Boresight (cm)
Figure 3.6 Cartoon of the high energy breaking wave’s (HEBW) spatial and temporal evolution,
with respect to the radar’s 1-way 3dB pattern on the water, at 40 degrees angle of incidence.
48
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
le w Energy OraaUtQ W ava.tO dag. H-pol
E apsed T ina (M e|
0
E b p tad T toa (m c |
02
04
0#
00
t
12
EbpM d T ina (Me)
14
>0
II
Figure 3.7 Representative plots of the H-pol backscattered power for the LEBW at 40 degrees angle
of incidence, and at azimuth angles of (a) 0 degrees, (b) 45 degrees, (c) 90 degrees, (d) 135 degrees
and (e) 180 degrees.
49
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- i
.
...
a3
at
(g)
(f)
r T— 1 I
'. ! '
......i ......L.....A
■ ..
r •
•
• •.
: : : : : :
...... :...... :......:___:.... •__ •..... „
;
i - - i ..... | ......
:
!
’
!
..... 1......
;
'
i
l xlli
i
:
y iL
n
Figure 3.7 Representative plots of the V-pol backscattered power for the LEBW at 40 degrees angle
of incidence, and at azimuth angles of (f) 0 degrees, (g) 45 degrees, (h) 90 degrees, (i) 135 degrees and
(j) 180 degrees.
50
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0
02
04
04
0J
I
12
14
1.4
II
2
0
EhpMd Tln« (m c |
04
04
04
04
I
12
14
14
ti
2
EkpMO Tki« (m c |
(d)
(e)
Figure 3.8 Representative plots of the H-pol backscattered power for the HEBW at 40 degrees angle
of incidence, and at azimuth angles of (a) 0 degrees, (b) 45 degrees, (c) 90 degrees, (d) 135 degrees
and (e) 180 degrees.
51
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Wigft Cn«ror Braahtae W a*a. Odag. V-pol
ilij-
i •
Mfclt in«gr Irookttg Wava. 4l<
j
Mgh Enargy I n M i g Wava. t JCtfag. V-pel
Mgh Enargy Bratfiktg Wavo. tBOdag. V-pol
-2 f
04
oi
13
-40
(0
02
04
O')
Figure 3.8 Representative plots of the V-pol backscattered power for the HEBW at 40 degrees angle
of incidence, and at azimuth angles of (f) 0 degrees, (g) 45 degrees, (h) 90 degrees, (i) 135 degrees and
(j) 180 degrees.
52
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
observed were 5.4 dB for H-pol and 5.5 dB for V-pol (averaged over 6 ms). Video data
showed that the peak return corresponded to the wave face reaching a nearly specular
steepness of 37 degrees (time = 1.25 sec). This implies that the backscatter was nearspecular scattering due to the reduced local angle of incidence with the steepened wave
face and confirms the findings of Jessup et al. (1991a,b).
The peak return in the HEBW (time = 0.8 sec), was attributed to specular
scattering from the forward face of the wave which was visually observed to obtain
specular steepnesses with respect to the radar. The peak RCS observed were 4.6 dB for
H-pol and 5.8 dB for V-pol (averaged over 6 ms).
Spatially, the peak return
corresponded to when the wave was positioned at the far edge of the antenna pattern. A
higher RCS could be expected if the wave had been boresighted at this point. At the time
of peak backscatter there were no significant returns observed in directions other than the
upwave direction, for either breaking wave.
The peak upwave return for both the low and high energy breaking waves were
followed by approximately 0.2 seconds of significant radar returns, which corresponded
to the time it took the breaking waves to transit the remainder of the radar footprint.
Initially (time = 1.25 to 1.31 sec), the LEBW’s crest height and facial steepness remained
relatively constant, with localized small scale roughness on the forward wave face. The
average RCS for this period was 4.2 dB for both polarizations, and implies that the nearspecular scattering, which caused the peak return, is still dominant in this time interval.
Beyond time =1.31 sec, the wave crest collapsed causing a reduction in both wave
steepness and upwave radar backscatter. At this point, the radar return was observed at
all azimuth angles for the LEBW.
The post-break radar returns exhibited a very spikey appearance (Figs. 3.7a-i,
Figs. 3.8a-i), therefore, to get a representative measure of the backscatter, the RCS were
averaged over the time the wave transited the radar footprint (0.3 sec). This yielded
values of -8.6 dB H-pol and -7.3 dB V-pol for the 0 degree azimuth, -16.2 dB H-pol and
53
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-16.2 dB V-pol for the 45 degree azimuth, -19.3 dB H-pol and -22.2 dB V-pol for the 90
degree azimuth and -14.9 dB H-pol and -12.8 dB V-pol for the 135 degree azimuth. The
velocity of the LEBW was such that the wave fully evolved within the radar footprint.
Thus, the radar data can fully characterize the LEBW.
The HEBW exhibited strong radar returns from time 0.8 to 1.1 seconds. This
corresponded temporally to the growth of the roughness covered splash-up beneath the
‘toe’ of the wave. The average RCS for this period was 0.8 dB for H-pol and 1.3 dB for
V-pol. Although slightly higher in magnitude, these values are interestingly similar to
those observed by Walker et al. (1996) for the roughness covered crests of stationary
breaking waves. The significant drop in RCS during this period may be explained by the
increased incoherent scattering from the small scale roughness and splash-up.
This
conjecture is supported by the significant peak backscatter (2.9 dB H-pol, 3.8 dB V-pol),
observed during this period at an azimuth angle of 135 degrees.
After the wave crest collapsed, the radar return was observed in the 0, 45 and 90
degree azimuth measurements. The RCS, averaged over the time the wave transited the
radar footprint (0.2 sec), yielded values of -5.4 dB H-pol and -5.0 dB V-pol for the 0
degree azimuth, -14.7 dB H-pol and -15.6 dB V-pol for the 45 degree azimuth and -17.3
dB H-pol and -16.2 dB V-pol for the 90 degree azimuth. The velocity of the HEBW was
such that the wave left the radar footprint early in its evolution (t = 1.1 sec). Therefore,
the radar data can only partially characterize the HEBW and likely underestimates the
peak RCS.
A plot of the peak RCS per unit area for each azimuth run is plotted for the
LEBW (Fig. 3.9a) and HEBW (Fig. 3.9b).
The plotted values indicate the peak
backscatter averaged over three samples (6 ms). The short averaging allowed for fine
temporal isolation o f the peak backscatter at each azimuth and the determination of the
corresponding wave surface. The values for azimuths other than upwave are most likely
“spike” related and may not be as representative of the surface scattering properties as the
54
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Low Energy Breaking W ave
-1 0
HH
vv
50
100
150
A zlm jfi An g it R elafv e 10 W ay# Travel fdegl
200
(a)
High Energy Breaking W ave
-1 0
HH
VV
-20
50
100
ISO
Aztautfi A ngli f le h f v e ® W ave Travel (dag)
200
(b)
Figure 3.9 Plot of the peak backscatter as a function of azimuth angle for the (a) LEBW and (b)
HEBW measurements at 40 degrees angle of incidence. Values were averaged over 3 samples (6ms).
55
Reproduced with permission of the copyright owner. Further reproduction prohibited w ithout permission.
averaged RCS previously quoted. With the exception of the LEBW 90 degree azimuth
measurements, the peak V-pol returns were greater than or equal to the peak H-pol
returns at each azimuth.
3.2.2 Spectral Data
The temporal data from all the runs were Fast Fourier Transformed (FFT) over
one eighth of a second intervals, yielding Doppler resolutions of 8 Hz. Plots of the
Doppler spectra for 6-7 consecutive time intervals are displayed for the LEBW (Figs.
3.10a,b) and HEBW (Figs. 3.10c,d). In the upwave measurements (Figs. 3.10a,c), the
spectra were dominated by a single narrow peak located at 122.0 Hz for the LEBW and
208.0 Hz, for the HEBW. These Doppler shifts corresponded to the radial components
(in the direction of the radar) of the scatterer velocities of 1.18 m/s and 2.01 m/s
respectively. These values are approximately equal to the phase speeds of the dominant
waves generated by the wave maker, and are similar to the observations of Kwoh and
Lake (1984a).
The narrow Doppler peak is consistent with linear wave theory and recent particle
tracking velocimetry (PTV) studies by Perlin (1995) which showed that particles along
the steepened wave face have orbital velocities with approximately the same radial
component. Therefore, scattering from the particles on the steepened wave face would
produce the observed narrow Doppler spectra. This supports the assertion that the peak
upwave backscatter was caused by specular or near-specular scattering from the
steepening wave face.
The largest Doppler shift in the upwave measurements corresponded temporally
to the time of maximum radar backscatter, as was observed by Jessup, Melville and
Keller (1990, 1991a,b). Although the peak upwave spectra were dominated by a narrow
spectral peak, some additional returns can be seen in the upwave spectra corresponding to
56
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Low Energy BfaaMng Wava (Dowmraval
100
t
•200
too
Oopplar Frequency
Fine (tec)
D octor Frequency (Kz|
(a)
(b)
High Energy Breaking Wave (Upwave]
Hfeh Energy BreaUig Wave (Downwavel
(
Powar <v*2|
SO
10
10
8
o
200
too
too
00
at
Desptea Frequency (Hz)
Tfene ( m c |
Oojvler Frequency (Hz)
(c)
(d)
Figure 3.10 Plot of the Doppler spectra as a function of time, for the (a) LEBW upwave, (b) LEBW
downwave, (c) HEBW upwave and (d) HEBW downwave measurements at 40 degrees angle of
incidence. Each spectrum was based upon 64 points (12S ms) of radar data.
57
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
slower scatterer velocities.
This indicates that as the wave evolved, other scattering
mechanisms were present in the upwave direction, but were small when compared to the
specular or near-specular backscatter components. As with the temporal data, the spectral
content of the radar return from the HEBW can not be determined after it exited the radar
footprint.
Therefore, the null spectra beyond time = 1.0 seconds in the upwave
measurements and time = 1.1 in the downwave measurements should not be
misinterpreted.
Like the upwave measurements, the largest Doppler shifts in the downwave
measurements (Figs. 3.10b,d) corresponded temporally to the time of maximum radar
backscatter in that direction. The spectra in the downwave measurements were broader
than those in the upwave direction, and were split into two distinct groups of peaks. This
implies that the primary scatterers had a distribution o f velocities. With the amplitude of
the wave decreasing after breaking, it can be assumed that the orbital velocities of the
surface particles will decrease. Further, it may be inferred that the decrease in velocity
would occur at different rates, and therefore, broad Doppler spectra could be expected.
The split Doppler spectra most likely corresponds to the free and bound scatterers also
observed by Kwoh and Lake (1984b) and Lee et al. (1995).
3.2.3 Waveprobe Data
Dissipation energies were calculated for the LEBW and HEBW by analyzing the
spectra derived from the pre- and post-break waveprobe temporal data. Fig. 3.10a shows
a representative waveprobe measurement 55 cm prior to breaking for the HEBW. As can
be seen, the breaking wave is highly peaked and has achieved the highest amplitude in the
wave group (Fig. 3.11a, time = 33.6 sec). Fig. 3.11b shows a representative waveprobe
measurement 37 cm beyond breaking for the HEBW. At this point, the wave crest has
collapsed to approximately one fourth of its peak amplitude (Fig. lib , time = 35.0 sec).
58
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Break -55cm
Break + 37cm
!! !
j
i !,
-"V'N |
1}
a a ,
^ • A i! ; ’.i\ VV
U
f*
■
III' M
i\i !\:\
.
a tw
n m
33 i««
ri«« B«>*
37 h i
*t m
*« m
« in
l |t« l
n m
*l«e
(a)
s«et
)i
u
‘
‘i
r
m
t<* !qgor *
<• h i
(«•«)
(b)
Prt- and Pott-Btaafc Spacta
LEBW
05
Frequency(M
x|
(c)
Figure 3.11 Plot of wave height versus time for the (a) pre-break (break-55 cm) and (b) post-break
(break+37 cm) waveprobes, along with the corresponding low frequency wave height spectra.
59
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The temporal data from the pre- and post-break waveprobes were broken into 64
segments covering the passage of the entire wave group. Each segment consisted of 64
data points, which were weighted by a Hanning window and fast Fourier transformed to
determine the spectrum for each segment. Individual spectrum were ensemble averaged
over the wave group, to yield the final spectra, [Meadows et al., 1995].
The amount of energy dissipated by wave breaking was calculated by integrating
the final spectra, derived from the pre-break (break - 55 cm) and post-break (break + 37
cm) waveprobes, over the range of wavemaker frequencies (Fig. 3.11c). Both the LEBW
and HEBW dissipated a significant amount of energy within the first 37 cm of the wave
breaking process. Dissipation rates were found to be 11.6% for the LEBW, and 21.8%
for the HEBW, which are consistent with the findings of Rapp and Melville (1990).
Considering that the peak radar backscatter for the HEBW was obtained when the wave
was at the far edge of the antenna pattern, it may be inferred that the HEBW would have
produced a larger backscatter RCS had it been boresighted at that point. Therefore, the
radar backscatter appears to increase with the rate of dissipation of surface wave energy
as observed by Melville et al. (1988). However, it should be noted that the backscatter
values for these two studies are not directly comparable. Melville et al. (1988) used
spatially and temporally averaged RCS values, whereas this study used peak RCS values.
3.3 Experimental Results (Grazing Angles of Incidence)
For the X-band FMCW radar, the power at a given range (or frequency) for a
desired time interval was found by taking the FFT of the temporal data in that interval.
An image of the backscattered power could then be created by stacking the results of
several sequential FFTs. This technique was used to generate the backscatter images for
the H-pol upwave (Fig. 3.12a), V-pol upwave (Fig. 3.12b) and V-pol downwave (Fig.
3.12c) HEBW measurements. Figs 3.12a-c represent the results of 150 sequential 512
60
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
point (16 ms) FFTs. As can be seen, the V-pole return tracks in with the wave as
observed by Lewis and Olin (1980). The X-band FMCW radar had a noise floor of -15
dB, which precluded the ability to detect returns from the LEBW and also the roughness
returns at H-pol for the HEBW. The peak radar return in the two upwave measurements
(Fig 3.12a,b, time = 0.9 sec) corresponded to the point in the wave evolution where the
wave face was visually observed to be near vertical steepness. The wave exhibited a
prominent ‘toe’ with incipient breaking occurring immediately below it. The peak RCS
observed were 4.3 dB for H-pol and 4.5 dB for V-pol.
Such a large polarization
independent return at grazing incidence, suggests that the radar return was caused by
specular scattering from tangential facets on the ‘toe’ or vertical face of the wave. This
observation is consistent with the findings of Sletten and Wu (1996).
The peak upwave returns were followed by a period of significant radar returns
for both H-pol and V-pol, (time = 1.0 to 1.1 sec). This corresponded temporally to when
the wave crest elongated and the splash-up feature became prominent. The average RCS
for this period were -11.6 dB for H-pol and -4.3 dB for V-pol. Thus, the splash-up
appears to be a significant source of radar backscatter, even at grazing angles of
incidence. However, unlike at moderate angles of incidence, the V-pol to H-pol ratio is
approximately 9 dB. The assertion that scattering was from the ‘toe’ of the wave and
subsequent splash-up, is supported by the absence of any significant grazing backscatter
from the LEBW, which did not possess these features.
From time = 1.1 to 1.8 seconds, the splash-up feature elongated and wave profile
was characterized by broad patches of small scale roughness which covered the
underlying long wave. The downwave V-pol measurements lagged the upwave V-pol
measurements by approximately 0.1 seconds. This is most likely due to the time it takes
for the small scale roughness to present itself on the backside of the underlying long
wave. Average RCS over this interval were -8.54 dB for the V-pol upwave and -10.86
dB for the V-pol downwave HEBW measurements. The H-pol returns due to roughness
61
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
can not be determined, however, they are upper bounded by the radar’s noise floor of -15
dB.
The grazing angle power plots were converted to RCS per unit area by tracking
the wave in range and determining the RCS from the appropriate range gate, for each
FFT. These plots were then averaged over several runs with results shown for the H-pol
upwave (Fig. 3.13a), V-pol upwave (Fig. 3.13b) and V-pol downwave (Fig. 3.13c)
HEBW measurements. The differences in the shapes of the H-pol and V-pol returns from
time = 0.9 to 1.0 may be due to multipath effects as discussed by Sletten and Wu (1996).
To examine this possibility, the complex dielectric constant was calculated to be
8 = 60.14 + J32.85 using the Debye relation and an experimental fresh water temperature
of 21.1°C, [Ulaby et al., 1982].
Applying this value to the Fresnel equations for a
specular air-water interface at 85 degrees angle of incidence, yielded reflection coefficient
magnitudes of 0.96 for H-pol and 0.04 for V-pol. Thus, the vertically polarized fields are
significantly reduced upon reflection from the water surface due to the incidence angles
extreme proximity to the Brewster angle (83.1 deg). Therefore, strong multipath fading
would only be expected in the H-pol returns.
3.4 Numerical Modeling
Modeling of the theoretical backscatter, for the scattering geometry shown in Fig.
2.3, will be divided into pre- and post-break surface profiles. For the deterministic pre­
break surfaces, theoretical scattering calculations will use iterative techniques derived
from the magnetic field integral equation (see Chap II, section 2.4.1). The post-break
surface profiles will be analyzed using a modified SPM which takes into account the local
angle of incidence over the surface (see Chap II, section 2.2).
62
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0
4.2 0
Range(m)
6.6
Range(m)
(a)
0
4.2
Range (m)
(b)
(C)
Figure 3.12 Plot of the backscattered power as a function of range and time for the (a) HEBW V-pol
upwave, (b) HEBW H-poi upwave measurement and (c) HEBW V-pol downwave measurement, at 85
degrees angle of incidence. Each image consists of 150 consecutive 512 points (16 ms) FFTs of radar
data.
63
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
RCS (OSdag AOt. H(gh Hrmgy Brtakrig Wtvt)
0
s
<* -s
oa
-1 0
-IS
0.7
0.0
0.0
13
1.4
1.5
1.6
1.7
1.3
1.4
IS
1.6
1.7
(a)
-s
-10
- I S 1------ L-10.7
0 .0
o
0.0
(b)
■i— VV Qownwaiva
-1 0
- I S 1-------- >0.7
0.0
0.0
1.1
1.9
Ebpsad Tina fue)
1.4
IS
1.6
1.7
(c)
Figure 3.13 Plot of the RCS as a function of time for the (a) HEBW H-pol upwave, (b) HEBW V-pol
and (c) HEBW V-pol downwave measurements averaged over several runs, at 85 degrees angle of
incidence.
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3.4.1 Pre-Break (Moderate Angles of Incidence)
To derive theoretical radar cross section estimates comparable to experimental
results, the incident plane wave must be weighted in a manner commensurate with the
actual radar. This was accomplished by setting the weighting functions equal to the
dimension of the one-way 3 dB antenna pattern on the water surface. From calibration,
the K-band Radar had one-way 3 dB beamwidths of 12.8 degrees (E-plane) and 14.2
degrees (H-plane). This yielded weighting functions of 37.9 cm by 25.8 cm for H-pol
and 34.1 cm by 28.7 cm for V-pol (alongtank by crosstank) for a height above water of
88.3 cm and 40 degrees angle of incidence. Similarly, the X-band radar’s one-way 3dB
beamwidths were found to be 11.4 degrees (E-plane) and 14.3 degrees (H-plane). This
yielded crosstank weighting functions of 63.6 cm for V-pol (height above water 22.1 cm)
and 81.4 cm for H-pol (height above water 35.6 cm) at 85 degrees angle of incidence.
The surface profiles used for analysis, were obtained by digitizing the video data
with a commercial frame grabber utility. The surface profiles were extracted from the
frames of interest and then scaled as determined by calibration. The surface profiles were
then smoothly transitioned onto a one-dimensional (along x axis) 3 m surface mesh. A
1000 point grid, equally spaced over the x axis, was applied to the mesh. At each grid
point, the surface elevation, x-directed slope and surface unit normal were determined.
This information, along with the incident field at each grid point, allowed for the
numerical solution of the one-dimensional surface current integral equations (Eq. 2.43).
The LEBW profile which corresponded to the maximum upwave backscatter (Fig.
3.14a) was input into a 1000 point surface mesh and the backscatter was solved for
iteratively as previously described. The bistatic RCS is plotted for the zeroth order iterate
(PO solution), along with the backscattered values determined experimentally at 40
degrees angle of incidence (Fig. 3.15a). As can be seen the agreement is quite good
between the physical optics solution and the experiment values. Experimental values
65
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averaged approximately 1 dB lower than the PO solution, which may be due to the pec
assumption inherent in the theoretical calculations.
As was discussed in the experimental results section, the LEBW continued to
produce significant backscatter even after wave breaking, in the upwave direction. To
gain insight into this time period, the mean surface of Fig. 3.14b (time = 1.29 sec) was
analyzed, yielding the bistatic RCS displayed in Fig. 3.15b. As can be seen, the radar
backscatter is still strong (3.4 dB V-pol, 1.0 dB H-pol), and the bistatic scattering pattern
is similar to that of Fig. 3.15a, with differences attributed to a slight decrease in slope
near the wave crest. From video data, the roughness generated from breaking was
observed to occur below the steepest portion of the wave crest. Therefore, the roughness
did not perturb the surface identified as the dominant source o f radar backscatter, and
significant returns could be expected until the wave crest collapsed. This is consistent
with the experimental results where the upwave radar returns remained strong until the
wave crest collapsed at time =1.31 seconds.
The location of the scatterers, dominating backscatter at 40 degrees angle of
incidence, may be inferred from the PO (or Kirchhoff) currents induced on the wave
surface. Figs. 3.16a-d show the surface height, surface slope, surface current phase slope
and surface current magnitude corresponding to Fig. 3.14a. The surface current phase
slope is defined as the change in phase per horizontal unit length over the wave surface.
The point on the wave corresponding to the maximum slope (37 deg) is indicated by a
broken vertical line in all of the plots. As can be seen, large current densities exist at the
steepest point of the wave face just below the wave crest. The corresponding phase slope
approaches zero in this region, indicating a near-specular presentation of the surface to
the radar. The reradiation pattern expected from steepest point o f the wave face would be
a broad pattern, centered about the local angle of coherent backscatter. The proximity of
the local coherent scattering angle (34 deg) at this point on the wave crest, with the
backscattering angle (40 deg), allows us to infer that the peak radar return was caused by
66
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Figure 3.14 Laser sheet enhanced video image corresponding to (a) the peak upwave return (time =
1.25 seconds) )and (b) the period of significant radar returns (time = 1.29 seconds), for the LEBW at
40 degrees angle of incidence.
.....
4....
(b)
Figure 3.15. Theoretical bistatic physical optics scattering solution corresponding to (a) the surface
profile in Fig. 14a and (b) the surface profile in Fig. 14b. V-pol results are represented by a solid line
and H-pol by a dotted line. The experimental results, corresponding to Fig. 14a, are also indicated
for V-pol (x) and for H-pol (o).
67
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Local Wave Hgt
S
Local Stopo
p ; ....................
i |
i :
0.05
0.05
: |
-
0 .1
-
0.1
-0.05
0
(a)
4
(b)
Currant Density (phase slope)
Z
E
Current Density (magnitude)
2 15
■
atoa
q.2
o
«
£«
0
-
0.1
-0.05
0
0
-0.5
D istFrom Radar B right (m)
0
DistFrom Radar Brsght (m)
Figure 3.16 Plot of the (a) surface profile, (b) surface slope, (c) physical optics surface current phase
slope and (d) physical optics surface current magnitude corresponding to Fig 14a, the peak radar
return at 40 degrees of incidence, in the upwave direction.
68
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near-specular scattering from the steepened wave crest. This supports the assertion of
Jessup et al. (1991a,b) that the backscatter was caused by a decrease in the local
incidence angle with the steepened wave face.
Although one breaking wave was considered here, one must remember that there
exists a spectrum of ocean waves with various wave heights and wavelengths. It is
reasonable to assume that the ocean contains many waves which approach breaking, but
do not possess the energy to actually break. From the above analysis, it is reasonable to
assume that if these waves steepen enough, they may cause a rise in radar backscatter
without any visual whitecapping. This “near breaking wave” may explain the findings of
several authors such as Lee et al. (1995), and Lewis and Olin (1980) who detected
significant radar backscatter without any visual whitecapping.
3.4.2 Pre-Break (Grazing Angles of Incidence)
The detail of the ‘toe’ of the HEBW, corresponding to the maximum radar return
at grazing angles of incidence, was obscured by blossoming of the laser light on the wave
surface. To determine if the peak scatter could come from the ‘toe’ of a steepened wave,
surfaces were extracted from the evolution of a similar spilling-plunging breaking wave
(to be described in Chapter IV).
The breaking wave was generated by the same
techniques and over a common range of chirps. Although the characteristics of the wave
were similar to the HEBW, I do not assert that this wave was the same as the HEBW, and
will therefore, limit my analysis to the PO scattering from the ‘toe’ of the steepened
breaking wave. The wave sequences used for analysis are shown in Figs 3.17a-c and
roughly correspond to time = 0.9 to 1.1 seconds in the HEBW evolution.
To avoid undersampling at the ‘toe’ of the wave, the iterative procedure was
modified by defining an along-surface coordinate system instead of the x-axis. Fig 3.16d
shows the theoretical backscatter RCS determined for the mean wave profiles in Fig
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3.16a-c, at 85 degrees angle o f incidence. As can be seen, large RCS were obtained when
the ‘toe’ of the wave was prominent and the splash-up was minimal (Figs. 3.16a,b). The
backscatter significantly decreased after the wave crest collapsed (Fig 3.16c) and the
splash-up became prominent. This is consistent with experimental results, and supports
the assertion that the peak radar return was due to specular returns from the ‘toe’ of the
HEBW, at grazing angles of incidence.
3.4.3 Post-Break
The validity of the SPM, described in Chapter II, was shown by Brown (1978) to
be valid when a roughness parameter, ^ ,was much less than unity (see Eq. 2.27). For the
K-band moderate angle of incidence measurements, this criterion is not met. However,
measurements with the X-band radar at grazing measurements are well within this limit
(4 = 0.087) and will be the subject of the following analysis.
A direct application of Eqs. 2.28a-c was attempted, by letting 0 equal the angle of
incidence and assuming that the HEBW achieved the saturation wave spectrum defined
by Walker et al. (1996). A theoretical backscatter value of -16.1 dB for V-pol and -49.5
dB for H-pol were obtained for an incidence angle of 85 degrees. While the H-pol return
was unverifiable due to the radar’s noise floor, the V-pol return is significantly less than
the experimentally determined average V-pol RCS of -8.5 dB for the upwave and -10.8
dB for the downwave measurements. To better approximate the experimental conditions,
the wave evolution was divided into 0.1 second time intervals, from 1.1 to 1.7 seconds.
The local angle of incidence was determined for each time period based upon the local tilt
of the roughness extracted from the video data. Roughness tilt angles ranged from 0 to
30 degrees for the upwave measurements and from 0 to 25 degrees for the downwave
slopes. Eqs. 2.28a-c were then solved for each time interval, using the local angle of
70
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(C)
(d)
Figure 3.17 Image of the surface profile used to model scattering from (a) the incipient ‘toe’ feature,
(b) the prominent ‘toe’ feature and (c) the diminished ‘toe’ feature. Corresponding polarization
averaged physical optics backscatter RCS for the mean surface profiles in Figs 3.17a-c at 85 degrees
angle of incidence in the upwave direction are shown in (d).
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incidence and the local Bragg wavenumber. The backscatter computed for each time
interval was integrated over the radar footprint (bounded by the range resolution cell of
the radar), and then averaged over the entire time interval.
This yielded averaged
backscatter RCS of -7.5 dB V-pol and -29.2 dB H-pol upwave and -10.7 dB V-pol and
-36.8 dB H-pol downwave for the HEBW measurements. The V-pol results fall within
1.0 dB of the experimental values and confirm that Bragg scattering dominated the post­
break radar returns for X-band, at grazing angles of incidence.
3.5 Conclusions
Temporal and spectral radar data have been presented for two transient breaking
waves. For both waves, a strong azimuthal dependence of the radar backscatter was
observed, and attributed to a combination of the surface tilt effects and a possible
anisotropy of the small scale roughness.
The moderate angle of incidence upwave
measurements were presented with steeply sloped wave faces and were dominated by
specular (HEBW) or near-specular (LEBW) scattering, just prior to wave breaking. Peak
radar returns were polarization independent and had Doppler spectra, whose dominant
peaks corresponded to the phase speed of the wave. A physical optics scattering solution
was found to model the experimental backscatter quite accurately.
Analysis of the
physical optics currents, over the surface which produced the peak backscatter, confirmed
that the maximum backscatter was due to specular or near-specular scattering from the
steepened wave face near the crest.
The peak radar returns in the 0, 45, 90 and 135 degree azimuth measurements
(LEBW), and 0, 45 and 90 degree azimuth measurements (HEBW) occurred after wave
breaking, as the steepened wave crest collapsed. Radar returns at these azimuths were
“spikey” in appearance and had polarization ratios near unity. Downwave measurements
produced spectra that were broad and split into two distinct groups of peaks. The split is
72
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most likely caused by the existence of both free and bound scatterers, while the broad
spectra are a result of the distribution in orbital velocities of the post-break small scale
roughness.
The HEBW upwave grazing measurements were presented with steeply sloped
wave faces with a prominent ‘toe’. The peak radar return was found to be caused by
specular scattering from the ‘toe’ of the wave prior to the wave crest collapsing. After
breaking the radar return was dominated by the post-break small scale roughness. The Vpol radar backscatter was accurately modeled using the small perturbation method.
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CHAPTER IV
THE HIGH-RESOLUTION TRANSIENT BREAKING WAVE EXPERIMENT
4.0 Experimental Design and Equipment
The high-resolution transient breaking wave experiment was the second of two
comprehensive laboratory experiments designed to isolate and identify the scattering
mechanisms in deep-water breaking waves. The experiment was conducted in the gravity
wave facility at the University of Michigan’s Marine Hydrodynamics Laboratory. The
gravity wave tank was 35 m long, 0.7 m wide and had an experimental water level of 1.14
m (see Fig. 4.1, side view). The wave tank was above ground and allowed for more
precise wave characterization than could be obtained in the earlier transient breaking
wave experiment.
Further, the radar was more readily positioned to obtain radar
backscatter data throughout the breaking wave evolution. To take advantage of these
features, the radar backscatter frequency and angle of incidence dependence studies were
conducted in the gravity wave facility.
A point gauge, mounted to the side of the wave tank, was used to maintain the
water level throughout the experiment. A “master” PC initiated wave generation by
sending a command signal to the servo controlled, wedge shaped wave maker, and then
monitored its output via a feedback loop and auxiliary electronics. To initiate data
acquisition, the “master” PC sent TTL trigger signals to the “slave” PC based radar data
acquisition system and the high speed imager’s CPU. Runs were commenced twenty
minutes apart to ensure all measurements started from a still water condition, and the
water surface was cleaned by a surface skimming system between each series of runs.
74
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Radar Sled
Wavemaker &
Electronics
Laser Sheet
Optics
/
0-\
*
«
►
£
Point Gauge
Laser
/
Command
Signal
“Master”
Control
PC
TTL Trigger #1
TTL Trigger #2
Imager
Cntrl. &
CPU
“Slave”
Radar DAQ
PC
8 Channels
Side View
Direction
of Wave
Propagation
Top View
Figure 4.1 Diagram of the high-resolution transient breaking wave experimental setup, with the
radar in the upwave configuration.
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4.0.1. Wave Generation
The breaking wave generation techniques used in this experiment were described
in great detail in Perlin et al. (1996), but will be summarized here for completeness. The
wave maker was programmed to produce a sinusoidal output whose frequency was
linearly decreased, at each zero crossing of the wavemaker with the mean surface, over a
small bandpass of frequencies. The frequency components were chosen, based upon
linear wave theory, to create waves that would add constructively at a given location in
the wave tank, as described by Davis and Zamick (1964).
The Davis and Zamick
technique was modified by applying the wave maker transfer function obtained from
dynamically calibrating the wave maker over the frequencies used to generate breaking
waves. The wave generation technique was further modified by adjusting the amplitude
of the generated waves between each zero crossing, such that the local wave steepness,
ka, was constant for all the waves in the wave group. This acted to increase or decrease
ka evenly over the wave group as the wave maker gain was changed, and to spread non­
linear effects approximately equally over the wave group.
For this experiment, several types of breaking waves were chosen for analysis.
The first wave analyzed was a spilling breaking wave. This wave was similar to those
studied by Duncan et al. (1994) and was characterized by minimal post-break small scale
roughness on the forward wave face. The spilling breaking wave sequence is shown in
Fig. 4.2. A spilling-plunging breaking wave, similar to those analyzed by Rapp and
Melville (1990), was also examined (Fig. 4.3). The spilling-plunging breaking wave was
characterized by a prominent wave crest which collapsed into the wave face (Fig. 4.3,
time = 0.6 sec) causing a large splash-up (Fig. 4.3, time = 0.725 sec) and significant post­
break small scale roughness.
The most energetic wave analyzed was the plunging
breaking wave (Fig. 4.4), identical to the one studied in Perlin et al. (1996).
76
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The
Spilling Breaking Wave (Gain = 36001
time = 0.45s
time = 0.55s
time = 0.475s
time = 0.575s
time = 0.5s
time = 0.525s
time = 0.6s
time = 0.625s
time = 0.65s
time = 0.675s
time = 0.7s
time = 0.725s
time = 0.75s
time = 0.775s
time = 0.8s
time = 0.825s
time = 0.85s
time = 0.875s
time = 0.9s
Figure 4.2 The spilling breaking wave image sequence at 25 ms intervals. The spatial reference is
indicated in black, with distances in centimeters to an arbitrary reference point. The imager was
repositioned at times 0.6 and 0.775 sec.
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time = 0.85
time = 0.875s
time = 0.9s
time = 0.925s
Figure 43 The spilling-plunging breaking wave image sequence at 25 ms intervals. The spatial
reference is indicated in black, with distances in centimeters to an arbitrary reference point. The
imager was repositioned at times 0.5,0.65 and 0.8 sec.
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Plunging Breaking Wave (Gain = 4200)
time = 0.3s
time = 0.4s
time = 0.5s
time = 0.6s
time = 0.625s
time = 0.65s
time = 0.675s
time = 0.7s
time = 0.725s
time = 0.75s
time = 0.775s
time = 0.8s
time = 0.825s
time = 0.85s
time = 0.875s
time = 0.9s
time = 0.925s
time = 0.95s
time = 0.975s
time = 1.0s
time = 1.025s
time = 1.125s
time = 1.225s
time = 1.325s
Figure 4.4 The plunging breaking wave image sequence at 25 and 100 ms intervals. The spatial
reference is indicated in black, with distances in centimeters to an arbitrary reference point. The
imager was repositioned at times 0.6,0.7,0.85 and 1.125 sec.
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plunging breaking wave was characterized by a prominent jet of water which protruded
from the wave crest (Fig. 4.4, time = 0.65 sec), impacted forward o f the wave face (Fig.
4.4, time = 0.75 sec), and generated a large splash-up (Fig. 4.4, time = 0.8 sec) with
significant post-break small scale roughness.
The spatial coordinates in the images are indicated by a black horizontal line with
vertical marks every 5 cm. The distance, in cm, from an arbitrary reference point is listed
below each vertical reference mark. As can be seen in Figs. 4.2-4, the breaking waves
evolve quite rapidly both temporally and spatially, making breaking wave analysis
extremely difficult.
The various breaking waves were generated by adjusting the gain to the wave
maker. The spilling, spilling-plunging and plunging breaking waves used gains of 3600,
4000 and 4200 respectively. All breaking waves were generated using frequencies from
0.8 Hz to 2.0 Hz, and may be considered deep water gravity waves. Using linear wave
theory and the deep water approximation, the phase speeds can be found to range from
0.78 m/s to 1.95 m/s [Crapper, 1984].
4.0.2. Radar Measurements
A X/K-band CW radar (Fig. 4.5a,b) was used to obtain the radar backscatter
measurements in this experiment [Dano and Kletzli, 1995]. The X/K-band radar was a
transceiver based system, capable of making multi-polarization, multi-frequency radar
backscatter measurements. The radar utilized X and K-band transceivers (M/A-Com
#MA86735, #MA86843) and mated pyramidal hom antennas (M/A-Com #MA86552,
#MA86554). For each frequency, one transceiver/antenna pair was positioned to obtain
horizontal polarization (H-pol), and the other for vertical polarization (V-pol). With its
array of four transceiver/antenna pairs, the radar was capable of true simultaneous
frequency and polarization measurements. The outputs from each transceiver unit were
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the basebanded In-Phase (I) and Quadrature (Q) voltage signals. Each of these signals
were passed through DC removal circuits, a selectable gain amplifier and a lowpass filter
prior to sampling. The radar data was sampled at rate of 1 KHz per channel by a PC
based data acquisition system utilizing a National Instruments MIO-16X, 16 bit A/D
board. Data acquisition was triggered via a TTL high-low signal from the “master” PC.
The X/K-band CW radar was mounted on a movable radar sled which traversed
the top of the wave tank, and could be positioned to obtain either upwave or downwave
radar measurements (Fig. 4.5b). Backscatter measurements were taken at azimuth angles
of 30, 45 and 60 degrees for both the upwave and downwave directions. The radar was
positioned to take measurements in 10 cm increments, so that its boresight on the water
surface ranged from approximately 40 cm prior to wave breaking to 60 cm beyond
breaking. Due to a vertical offset between the K- and X-band antenna apertures, the Xband boresight on the water surface lagged/led the K-band boresight by approximately
10.8 cm for the upwave/downwave runs, at all angles of incidence. One measurement
was conducted at each boresight position, for each incidence angle and wave type, in both
the upwave and downwave directions. Radar absorbent material was positioned on the
wavetank (Fig. 4.1, top view) to prevent contamination from secondary reflections and
limit the radars footprint on the water surface. Radar absorbent material was also placed
below the radar apertures to prevent any contamination from nadir reflections.
Runs were commenced by determining the exact point and time past commandsignal for each wave type to break. The radar was then positioned, relative to an arbitrary
reference point, to probe the breaking wave evolution in 10 cm increments. For each run,
the “master” PC would initiate wave generation and then drive the TTL trigger signal to
its low state at a predetermined time. The TTL signal would trigger the “slave” PC to
commence radar data acquisition. The radar was set to trigger one second prior to the
determined wave breaking time. Two seconds of X/K-band CW radar were collected for
each respective measurement and stored on the “slave” PC.
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Signal
Conditioning
HH
Vin
W
Signal
Conditioning
(a)
(b)
Figure 4.5 Schematic (a) and photograph (b) of the X/K-Band CW radar, radar sled and laser sheet
optics (far right side of photo).
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X/K-Band CW radar: K-Band NocmaRzad Anttma Patlarn
0 .9
0.B
0 .7
0.6
0.3
60
30
20
20
30
40
(a)
X /K -B and CW ra d a r. X -B an d N o tm a fe a d A n ta m a P afiain
0.B
0.7
0.6
0.9
40
90
(b)
Figure 4.6 Normalized antenna gain patterns for the X/K-band radar’s (a) K-band pyramidal horn
antennas and (b) X-band pyramidal horn antennas.
83
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The radar was internally and externally calibrated as described in Ulaby et al.
(1982).
Internal calibrations determined gain errors, phase errors and DC biasing
inherent to the radars components and circuitry. External calibrations determined the
returned power from targets of known radar cross section (RCS) and the antenna’s
normalized radiation pattern.
These calibration parameters were used to determined
target RCS within +/- 1 dB. The 2-way normalized antenna gain patterns, determined
from calibration, are shown for the K-band (Fig. 4.6a) and X-band (Fig. 4.6b) pyramidal
horn antennas. The 2-way 3 dB beam widths are 28.4 deg (E-plane) and 19.6 deg (Hplane) for the X-band antennas and 16.8 deg (E-plane) and 16.1 deg (H-plane) for the Kband antennas.
The dynamic noise floor of the radar system was found to be
approximately -20 dB.
The K-band antennas were ideal for the laboratory measurements. Both V-pol
and H-pol radiation patterns were nearly symmetric, which minimized antenna pattern
effects on the radar return. The “fan” pattern of the X-band antennas was by no means
ideal for radar backscatter measurements. However, by limiting the radiation pattern and
carefully analyzing the illumination on the water surface, The X-band also yielded
meaningful results.
4.0.3 Wave Characterization
A non-intrusive laser sheet and imager technique was used to capture the high
resolution morphology of the breaking waves, [Perlin et al., 1993]. The laser sheet was
generated by a Spectra-Physics 164, 5W, Argon-Ion laser and model 265 exciter. The
laser light was redirected by 4 dielectric mirrors, and formed into an alongtank laser sheet
through the use of two cylindrical lenses with focal lengths of 34 mm and 25 mm
respectively (Newport #04, #CKX025). The laser sheet was directed 14 cm in from the
side of the wave tank to prevent contamination from boundary layer effects.
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Approximately 30 cm of the water surface was illuminated in the along-tank direction
using this technique. To enhance imaging, a florourescein dye was added to the water
which fluoresced the 488.0 nm wavelength line of the argon-ion laser.
The illuminated area on the water surface was imaged by a Kodak Ektapro
intensified imaging system. The imaging system consisted of a high speed intensified
imager (model #1012), using a standard 50 mm camera lens, an intensified imager
controller and a dedicated CPU. For each measurement, the imager and laser sheet were
positioned so that their field of views coincided with the radar footprint on the water
surface. The TTL trigger signal to the imager was then adjusted so that the breaking
wave was fully imaged as it transited the radar footprint. The imager continuously
acquired 400 images, at a 1 KHz rate, until a TTL trigger signal was received from the
“master” PC. Once triggered, the imager saved the 200 images immediately preceding
and following the trigger signal. The 400 video images (239 horizontal pixels by 192
vertical pixels) were downloaded to a separate PC for processing via a standard GPEB
interface. Points on the wave surface extracted from the images, were accurate within 1
pixel or about 0.8 mm.
Image data was calibrated by imaging a calibration target etched with 2 cm
squares at the intersection of the laser sheet and the water surface. This provided a means
of focusing the video system as well as determining horizontal and vertical scaling
factors. Wave consistency was monitored by imaging the wave as it passed several
reference marks on the wave tank. Individual wave runs were found to be repeatable
within +/- 5 ms throughout the experiment. This variance is most likely due to slight
changes in the water level and can serve as an upper bound for temporal repeatability of
the generated breaking waves.
4.0.4 Specular Facet Detection
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All measurements were repeated with the radar replaced by the high speed imager.
The imager was positioned so that its focal point coincided with the radar boresight for a
given measurement location. Two high power halogen lamps were placed on either side
of the imager and were set to the same angle of incidence as the imager. Using this
procedure, optically specular facets showed up as bright patches in the images. Since the
imager was triggered using the same command signal that was used for wave
characterization, the observed specular returns could be synchronized in time with a
specific wave profile and RCS measurement. When interpreting the specular detection
results, one should remember that optically specular does not necessarily mean that the
surface is specular at microwave frequencies. Kwoh and Lake (1984a) point out that if
the radius of curvature of the surface is small when compared to the microwave
wavelength it will still be optically specular, but the backscatter will be described by
Rayleigh vice geometric scattering.
4.1 Experimental Results
4.1.1 Temporal Data
The peak radar backscatter (averaged over 6 ms) was determined at each 10 cm
sampling position along the wavetank, in the up and downwave directions. The RCS
versus time plots are shown for the spilling breaking wave at K-band (Fig 4.7a) and at Xband (Fig. 4.7b). Similarly, peak backscatter plots are included for the spilling-plunging
breaking wave (Figs. 4.8a,b) and the plunging breaking wave (Figs. 4.9a,b). Taking radar
measurements every 10 cm allowed for analysis of the full wave progression without
antenna pattern weighting becoming a factor, as it is in single position measurements.
Thus, the radar plots included in Figs 4.7-9a,b can be thought of as the radar backscatter
that would be obtained if the radar was moving with the breaking wave
86
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
K-band peak RCS (Spilling Breaking Wave)
Upwavs
30 deg.
Downwav*
a2a
(0
8
(0
ff
CP
-2 0
-20
0
O.S
0
20
45 deg.
0 .5
20
5 o
8as
S 0
8
tt
-2 0
-20
0.5
0.5
20
« 0
ee
60 deg.
<0
s
CP
-2 0
-20
0
O.S
Tima (sac)
0.5
Tima
(m c ]
(a)
X-band peak RCS (Spilling Breaking Wave)
(Jpwava
Downwava
20
30 deg.
to
V
a
-20
-20
o
45 deg.
o.s
0.5
20
20
2 0
2 0
cg
tr
g
-20
-20
0.S
0.5
20
60 deg.
20
at
S o
* 0
ucr
CO
CO
V
a
-2 0
-20
O.S
Tima itac)
1
O.S
Tima (*ac)
I
(b)
Figure 4.7 Plot of peak RCS as a function of time at (a) K-band and (b) X-band for the spilling
breaking wave at 30,45 and 60 degrees angle of incidence (V-pol solid line, H-pol dashed line).
87
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
K-band peak RCS (Spilling-Plunging Breaking Wave)
Upwava
Downwava
20
0
30 deg.
-2 0
-2 0
o
o.s
0
O.S
20
45 deg.
m
*a
8EE
S
0
3
EC
-20
-20
0
0.S
O.S
20
20
aj
■a
60 deg.
■oS 0
cO
co
O
EC
EE
-20
-20
0
O.S
Tima (me)
O.S
Tima 1*ac)
1
(a)
X-band peak RCS (Spilling-Plunging Breaking Wave)
Upwavt
Downwav«
20
30 deg.
aTS
5 0
a
g
-20
-20
0.5
45 deg.
0
O
TO
3
m
Tl
3EE
3IE
-20
0.5
-20
0
0
20
O.S
m
T3
O.S
c«*o 0
60 deg.
CE
-20
-20
0
O.S
Tima (sac)
O.S
Tima (sac)
(b)
Figure 4.8 Plot of peak RCS as a function of time at (a) K-band and (b) X-band for the spillingplunging breaking wave at 30,45 and 60 degrees angle of incidence (V-pol solid line, H-pol dashed
line).
88
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
K-band peak RCS (Plunging Breaking Wave)
llpwava
D o w rw a v a
20
30 deg.
20
m
2
0
20
45 deg.
8
1.5
0
c8c
20
0
20
to
1
I5
20
e
60 deg.
to 0
8
20
0
1.5
0
<0
u
«
tr
0
20
0
20
I5
0
e
20
•20
0
05
0
1.5
05
Tima (aac)
Tima (aac)
I
15
(a)
X-band peak RCS (Plunging Breaking Wave)
Upwava
Dowriwava
20
20
0
0
30 deg.
20
•20
0
05
0
1.5
20
45 deg.
0
0
E
20
20
0
0
1 .5
20
60 deg.
a3>
15
8
8
ec
05
20
20
0
10
(0
«s
0
«E
20
0
20
05
1 .5
Tima (aac)
0
Tima iaae)
(b)
Figure 4.9 Plot of peak RCS as a function of time at (a) K-band and (b) X-band for the plunging
breaking wave at 30,45 and 60 degrees angle of incidence (V-pol solid line, H-pol dashed line).
89
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
and took a measurement every 10 cm o f horizontal travel.
For all breaking waves considered, the peak radar backscatter was polarization
independent and occurred in the upwave direction prior to wave breaking. These peak
radar returns were all coincident with the detection of a specular patch by the specular
facet detector, indicating that the dominant scattering mechanism was in fact specular
scattering from the steepened wave face in the upwave direction.
This result lends
experimental proof to the hypothesis o f Jessup et al. (1991a,b) and Loewen and Melville
(1991) who hypothesized that specular scattering was the cause of the large polarization
independent radar returns at moderate angles of incidence.
At 30 degrees angle of incidence, the peak K-band RCS, over all waves, varied
between 19.0 to 21.4 dB for V-pol and 17.3 to 18.4 dB for H-pol. X-band values varied
between 7.8 to 10.1 dB for V-pol and 5.3 to 8.3 dB for H-pol. Specular returns at 45 and
60 degrees angle of incidence were only observed for the spilling-plunging and plunging
breaking waves. For those two waves, at 45 degrees angle of incidence, the K-band
values varied between 11.5 to 12.1 dB for V-pol and 9.7 to 11.5 dB for H-pol. X-band
values varied between 9.1 to 12.6 dB for V-pol and 6.8 to 11.2 dB for H-pol. Similarly,
at 60 degrees angle of incidence, the K-band values varied between 1.3 to 1.4 dB for Vpol and -0.3 to 2.3 dB for H-pol. X-band values varied between -2.4 to 2.4 dB for V-pol
and -4.1 to 0.3 dB for H-pol. The difference between V-pol and H-pol returns can be
explained by antenna pattern differences for both frequencies.
Fig. 4.10a shows the upwave specular facet detector images which corresponded
temporally to the peak radar backscatter for the three breaking waves considered, at 30,
45 and 60 degrees angle of incidence. As can be seen, significant specular facets were
detected for the spilling breaking wave only at 30 degrees angle of incidence. This is
consistent with the radar returns in Figs. 4.7a,b, where a large radar return was only
observed at 30 degrees angle of incidence. Significant specular facets were detected at all
angles of incidence, in the upwave direction, for the spilling-plunging and plunging
90
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
breaking waves. Specular facets were initially detected at 30 degrees angle of incidence
and then followed by 45 and 60 degree angles of incidence as the wave face steepened
through those respective angles. This is consistent with Figs. 4.8a,b and 4.9a,b, where
large radar returns were observed at all angles of incidence, in the order described.
The 30 degree angle of incidence measurements produced the largest peak RCS,
followed by the 45 and 60 degree angle of incidence measurements respectively. This
difference in radar backscatter may be explained by examining the size o f the specular
patches in Fig. 4.10a. As can be seen, the largest specular patches were observed at 30
degrees angle of incidence and were roughly the same size for all three breaking waves
considered. This explains how three waves of drastically different energies produced
approximately the same radar backscatter at 30 degrees angle of incidence. The specular
patches observed at 45 and 60 degree angles of incidence were smaller than those
observed at 30 degrees and most likely explains the reduction in the peak radar
backscatter as the angle of incidence increased.
The dominance of the K-band radar return over the X-band return can be
explained by the relative size of the specular scattering patch at each incidence angle.
Due to the frequency differences, the specular patch will have twice the dimensions, in
wavelengths, for K-band than it will for X-band.
Therefore, the K-band coherent
reradiation pattern, from a fixed size specular patch, could be expected to be stronger and
more directive than the X-band coherent reradiation pattern from the same patch.
Because of this, the K-band radar return should dominate the X-band radar return at
specular or near-specular scattering angles.
At non-specular scattering angles, the
broader X-band reradiation pattern may actually dominate the overall radar return. This
conjecture is supported by experimental observations.
The actual location of the specular facets was resolved by analyzing the
deterministic surfaces corresponding to the peak radar returns. As expected, the peak
radar backscatter corresponded to the wave face obtaining steepnesses equal to the radar
91
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
angle of incidence, and thus producing strong specular backscatter in the upwave
direction. At 30 degrees angle of incidence the backscatter came from the lower part of
the wave face which maintained 30 degree specular facets throughout the pre-breaking
sequence. The 45 degree and 60 degree specular facets were located near the middle and
top of the wave face respectively.
As the wave face steepened, the specular scattering patches identified by the
specular facet detector were observed to split into two regions, one near the bottom of the
wave face and the other near the rounded wave crest (Fig. 4.10b). Radar backscatter
returns during these periods showed a strong interference pattern due to the coherent
addition of scattering from the two specular patches. An example of this type of radar
return is included for the spilling-plunging breaking wave, at 45 degrees angle of
incidence (Fig. 4.11).
The radar backscatter after the peak scattering event remained significant for all of
the breaking waves considered. From analysis of the images obtained by the specular
facet detector, and the corresponding deterministic wave profiles, it was found that the
continued large backscatter was due to returns from the steepened top of the wave face
prior to the wave crest collapsing.
collapsed quite rapidly (Fig. 4.2,
For the spilling breaking wave, the wave crest
time = 0.65 sec), and therefore, the large radar
backscatter was eliminated quickly.
The top of the spilling-plunging breaking wave’s wave face remained steep until
the wave crest collapsed and the splash up feature was generated (Fig. 4.2, time = 0.65 0.775 sec). Fig. 4.10c shows a sequence of corresponding side view and specular facet
detector images for the spilling-plunging breaking wave at 45 degrees angle o f incidence.
As can be seen, there is a noticeable specular return (indicated in the top of the images by
large light patches) located above the post break roughness (indicated in the middle of the
images by small dots of light). In a similar manner, returns from the protruding jet were
found to be the source of the continued large backscatter from the plunging breaking
92
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Wave Type
30 degrees
45 degrees_______ 60 degrees
time = 0.460s
time = 0.564s
time = 0.324s
time = 0.432s
Spilling
Brk. Wave
time = 0.572s
SpillingPlunging
Brk. Wave
time = 0.508s
Plunging
Brk. Wave
time = 0.412s
time = 0.530s
time = 0.626s
30 degrees
45 degrees
60 degrees
tune = .5s
tune = .5s
tune = .525s
(a)
SpillingPlunging
Brk. Wav
tune = 0 .6 s
tune = 0.65s
time = 0.7s
time = 0.75s
«
(c)
Figure 4.10 Images from the specular facet detector showing facet detection at (a) the peak radar
returns, (b) the interference dominated returns and (c) the large post-peak radar returns.
93
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
K-band. 45 dag AOI
.
0.2
breaking
.-.■■M.
,Jk_.
0 .9
0 .4
0 .5
0.6
0 .7
0.8
0 .9
Tima (sac)
Figure 4.11 Plot of the radar return after wave face slope exceeds the radar angle of incidence for
the spilling-plunging breaking wave at 45 degrees angle on incidence.
94
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
wave.
The downwave radar backscatter was negligible for the time periods discussed
thus far.
However, once the wave crests collapsed the downwave radar backscatter
became discernible with peak RCS approximately equal to the upwave direction, during
the same time interval, for all of the breaking waves considered. While it is impossible to
classify the distribution of the post-break radar backscatter from a series of point
measurements, the average properties of the post-break backscatter may be used to yield
interesting results.
The average post-break V-pol RCS for K-band, over all waves, were 1.67, -2.93
and -8.73 dB in the upwave direction, and -2.12, -3.8 and -10.28 dB in the downwave
direction for 30, 45 and 60 degree angles of incidence respectively. Similarly, X-band Vpol returns, over all breaking waves, were -0.73, -3.92 and -9.55 dB in the upwave
direction, and -5.40, -3.73 and -7.49 dB in the downwave direction, for 30, 45 and 60
degree angles of incidence respectively. The H-pol returns averaged 3.11 dB and 3.84 dB
lower than the V-pol returns for K- and X-band measurements respectively.
While
antenna pattern effects account for some of the polarization difference, the V-pol return
can still be said to have dominated the H-pol return in the post-break regime. For both
frequencies, over all breaking waves, the downwave RCS averaged approximately 2 dB
lower than the upwave RCS.
4.1.2 Spectral Data
The temporal data from all the runs were fast Fourier transformed (FFTed) over
0.128 second intervals (128 points), yielding Doppler resolutions of 7.8 Hz. Plots of the
K- and X-band Doppler spectra, corresponding temporally to the peak radar return at each
10 cm measurement location, are displayed for the spilling breaking wave (Fig. 4.12, Kband; Fig 4.13 X-band), spilling-plunging breaking wave (Fig. 4.14, K-band; Fig 4.15 X-
95
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
band), and plunging breaking wave (Fig. 4.16, K-band; Fig 4.17 X-band). The maximum
Doppler shifts were located at frequencies which corresponded to the radial components
(in the direction of the radar) of a scatterer velocity of approximately 1.0 m/s. This
velocity corresponds roughly to the linear phase speed of the center frequency of the
generated wave packet, as was also observed by Loewen and Melville (1991).
The largest Doppler peak in the upwave measurements corresponded temporally
to the time of maximum radar backscatter. In fact, the progression o f the specular returns
from the steepening wave face may be observed in the delay between the initial Doppler
peaks in the 30, 45 and 60 degree measurements in the spilling-plunging and plunging
breaking waves. Although the peak upwave spectra were initially dominated by a single
narrow spectral peak, some additional returns can be seen in the upwave spectra
corresponding to various scatterer velocities.
This indicates that other scattering
mechanisms were present in the upwave direction, but were small when compared to the
specular or near-specular backscatter components.
As the wave crest collapsed, the
upwave Doppler spectra broadened and decreased in magnitude as the amount of post­
break small scale roughness increased. This is consistent with Jessup et al. (1991a,b).
Like the upwave measurements, the largest Doppler peaks in downwave
measurements corresponded temporally to the time of maximum radar backscatter in that
direction. The largest Doppler shifts also corresponded to the phase speed of the center
frequency of the generated wave packet. The downwave spectra became prominent only
after the wave crest, which shadowed the wave face from the radar footprint, collapsed.
The spectra in the downwave measurements were broader than those in the upwave
direction, and were split into several peaks. This implies that the primary scatterers had a
distribution of velocities. With the amplitude of the wave decreasing after breaking, it
can be assumed that the orbital velocities of the surface particles will decrease. Further, it
may be inferred that the decrease in velocity would occur at different rates, and therefore,
broad Doppler spectra could be expected.
96
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
K-band Doppler Spectra (Spilling Breaking Wave)
Upwave
Downwave
K - lia i tts«fl AOL (*•••«•. 0« * - MOO
<C-6wd.«ed*0 AOI.OawnwtM. OaH • * 0 0
too
>300
DappteFtaqiM
wy0*1
IC-*M. 4Cd«4 AOt. Upwav*. <!•* • MOO
;
I
J
f
Z
•
....
.•*
-
•*.
.;■■■
"s..
!:
ot
Figure 4.12 Series plots of the K-band Doppler spectra for the upwave (left side) and downwave
(right side) measurements, at 30 (top), 45 (middle) and 60 degree (bottom) angles of incidence, for the
spilling breaking wave.
97
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
X-band Doppler Spectra (Spilling Breaking Wave)
Upwave
Downwave
:
i"
.
:
\Z
100 04
K-fcaaO.4(d«8A<X.Upvav«.0*l•IIOB
X-Oaad.4CdagAOI.Oovctwava.Gefc• 9CO
O
100
(00 04
Ttaah
JC-taaa.fOdagAOI.Upwav*.data•9400
20
tC
0I
OopplarFtaqaaneynot
Figure 4.13 Series plots of the X-band Doppler spectra for the upwave (left side) and downwave
(right side) measurements, at 30 (top), 45 (middle) and 60 degree (bottom) angles of incidence, for the
spilling breaking wave.
98
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
K-band Doppler Spectra (Spilling-Plunging Breaking Wave)
Upwave
Downwave
(-tad.Maasaoi.Upwava.Oato•M
OO
C-bwd.ttdas AOI. Oomwava. Qafc . 4000
000
0;
I
tapptat FM«Mer OteJ
Ootptar Fraoeaacr
( • i a d .K t i s A0I.Upwav«.0*i • 4000
X-0aad.4fdag AOI. D am an*. Qato • 4000
CO
O
Oopplar FMmaicy n d |
K-ft«d.Mdae AOt.Upwava.Oafti. 4000
X-bdM.SOdao AOt.Ooanaova.Gato • 4000
; «o
Depplor iNatM qr OO]
Figure 4.14 Series plots of the K-band Doppler spectra for the upwave (left side) and downwave
(right side) measurements, at 30 (top), 45 (middle) and 60 degree (bottom) angles of incidence, for the
spilling-plunging breaking wave.
99
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
X-band Doppler Spectra (Spilling-Plunging Breaking Wave)
Upwave
Downwave
X -tlB l IQIIg AOt. UgardV*. a a k • 4090
K-band.SOdag AOt. B aw w na. Oak - 4009
X-baad. 4Sd«g AOt. Ugnava. Oak • 4000
K-baad.4fdag AOl.Oowmava.Oak* 4000
US'.
400 •
{ - .
too
0I
;
:f
t
><
;
r
o
-100
•<o
Oopptar FreauaRcy
K-baad. €0d«g AOt. Upaava. Oak • 4000
K-baad. Udag AOI. De«n«ava. Oak « 4000
10
10
0
0-7
T k a Im c |
e
■too
Oopglar Ftaaaaacy ota)
Figure 4.15 Series plots of the X-band Doppler spectra for the upwave (left side) and downwave
(right side) measurements, at 30 (top), 45 (middle) and 60 degree (bottom) angles of incidence, for the
spilling-plunging breaking wave.
100
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
K-band Doppler Spectra (Plunging Breaking Wave)
Upwave
Downwave
K-llM. Mttafl AOI. Upvava. 0 *1 • 4200
K-baad.SOdas AOi.OownwaM. Gait. 4200
Datptar Fnqaaaqr IKM
QAOl. Oovnwava. Galt • <200
K-tand.COdafl AOI. Oownwava. Oalt • 4200
Oipotar Fmomoct iKzi
200
Figure 4.16 Series plots of the K-band Doppler spectra for the upwave (left side) and downwave
(right side) measurements, at 30 (top), 45 (middle) and 60 degree (bottom) angles of incidence, for the
plunging breaking wave.
101
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
X-band Doppler Spectra (Plunging Breaking Wave)
Upwave
Downwave
C-tM. 900*8AOt.Upw«v«.0 * • 4200
tee
JUtwtf.SMap AOt. Downweve. Ga» • 4200
Ooppkr fteaweaqr (M
il
K -tM .ilM g AOI. Upwave. Gate • 4200
X-bead.4Sdeo AOt. Oewaweve. Gefc . 4200
0
100
too
Hr fteaaener (MU
JUbaM. 100*8 AOI. Upwave. Gate • 4200
1.<00*8 AOl. Downwave. Oato ■ 4200
<00
'400
0
DeppHr Frequency (Mi|
Dept*Frequency(M
U
100 04
Figure 4.17 Series plots of the X-band Doppler spectra for the upwave (left side) and downwave
(right side) measurements, at 30 (top), 45 (middle) and 60 degree (bottom) angles of incidence, for the
plunging breaking wave.
102
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4.2 Numerical Modeling
Modeling of the theoretical backscatter was achieved using the same techniques
described in Chapter ID.
The wave surfaces were divided into pre- and post-break
profiles. For the deterministic pre-break surfaces, theoretical scattering calculations used
the iterative techniques derived from the magnetic field integral equation (see Chap II,
section 2.4.1). The post-break surface profiles were analyzed using a modified small
perturbation method which took into account the local angle of incidence over the surface
(see Chap II, section 2.2).
4.2.1 Pre-break
Calculations will be presented for the upwave measurements, using the geometry
shown in Fig. 2.3. To derive theoretical radar cross section comparable to experimental
results, the incident plane wave must be weighted in a manner commensurate with the
actual radar. This was accomplished by setting the weighting functions equal to the
dimension o f the 1-way 3 dB antenna pattern on the still water surface. From calibration,
the K-band Radar had 1-way 3 dB beamwidths of 23.7 degrees (E-plane) and 26.0
degrees (H-plane). This yielded alongtank by crosstank weighting functions of 22.1 cm
by 20.8 cm for V-pol and 24.4 cm by 18.9 cm for H-pol (30 degrees AOI), 34.2 cm by
25.5 cm for V-pol and 38.0 cm by 23.1 cm for H-pol (45 degrees AOI) and 75.4 cm by
36.0 cm for V-pol and 85.7 cm by 32.7 cm for H-pol (60 degrees AOI). Similarly, the Xband radar’s 1-way 3 dB beamwidths were found to be 40.6 degrees (E-plane) and 28.0
degrees (H-plane). This yielded alongtank by crosstank weighting functions of 35.1 cm
by 19.6 cm for V-pol and 23.1 cm by 29.0 cm for H-pol (30 degrees AOI), 58.3 cm by
103
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
24.0 cm for V-pol and 36.2 cm by 35.6 cm for H-pol (45 degrees AOI) and 170.7 cm by
33.9 cm for V-pol and 83.4 cm by 50.3 cm for H-pol (60 degrees AOI).
Weighted plane wave approximations allow for computational speed by avoiding
an integration over the radar aperture during the initial and final field calculations.
However, care should be taken when using weighted plane wave approximations, since
the weighted plane wave simulates the amplitude characteristics of the radiated fields, but
not the phase. To ensure the accuracy of the weighted plane wave approach, the physical
optics scattering solutions were determined for several surfaces using both weighted
plane wave and actual TE10 illuminations [Balanis, 1987].
Results for all antenna
configurations, except the X-band H-pol, were within +/-1 dB and validated the weighted
plane wave approach for those configurations. Due to the large E-plane pattern and
horizontal presentation of the dominant scatterer, the phase taper across the water surface
was nontrivial for the X-band H-pol configuration. To account for the effects of phase
taper, a first order correction factor was established for each angle of incidence, and then
applied to the X-band H-pol calculations.
The deterministic surfaces profiles used for numerical scattering calculations were
obtained from the files of 400 images captured by the high speed imager. The surface
profiles were extracted from the images of concern, and then scaled based upon
calibration parameters. All surface profiles were centered, and smoothly transitioned
onto a one-dimensional (along x axis) 3 m surface mesh. This was done to ensure that the
surface was large enough that edge effects, caused by illuminating the end of the mesh,
were avoided. A 1000 point grid, equally spaced over the x axis was applied to the mesh.
Grid points in between extracted surface points were determined using clamped cubic
splines. At each grid point, the surface elevation, x directed slope, and surface unit
normal were determined. This information, along with the incident field at each grid
point,
allowed
for
the
numerical
solution
of the
one-dimensional
104
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
surface
current integral equation (Eq. 2.43).
Examples of the theoretical bistatic RCS, for the wave profiles, which caused the
peak backscatter at 30, 45 and 60 degree angles of incidence for the spilling-plunging
breaking wave, are shown in Figs. 4.18a-20a for K-band and Figs. 4.18b-20b for X-band.
The peak measured values are also plotted for comparison at the appropriate angle of
incidence. As can be seen, the numerical and measured values agree quite well. An
interesting note is that the main lobe of the coherent scattering pattern, at all incidence
angles, is stronger and more directive for the K-band radiation than it is for the X-band.
This is due to the relative size of the specular scattering area as previously discussed.
The interference pattern at larger scattering angles in Figs. 4.18a,b-20a,b is due to the
coherent addition of energy scattered from various parts of the wave, and may be
analyzed using a geometric optics approach.
The theoretical radar backscatter was obtained in a similar manner for the surface
profiles which corresponded to the peak upwave radar backscatter and detection of
specular facets, at all three experimental angles of incidence. After one iteration of the
surface current integral equation (Eq. 2.43), it was found that the physical optics solution
predicted the experimental RCS over a wide range of values within approximately 3 dB.
In general, the theoretical RCS tended to over predict the experimental values. This
overprediction might be expected since the iterative procedure was based on the
assumption that the water surface was a perfect electric conductor, which unfortunately is
not true.
To correct for finite conductivity, a first order correction as developed by Kwoh
and Lake (1984a) was applied to the numerical calculations. The correction is based
upon the small perturbation method (SPM), which assumes the backscattered power is
proportional to the reflection coefficients squared ( gw2 , ghh2 ).
To solve for the
reflection coefficients required the complex dielectric constant at each frequency (see
Eqs. 2.28b,c). The complex dielectric constants were found for K-band (31.07 - j35.66)
105
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Bistatfc RC8 (K -band. 3Qd«a AOI. d a In • 4000)
90
VV
20
vv
RCS (dB)
to
to
20
90
60
60
90
-2 0
20
O e a tta rln o A ngla (d a g )
60
(a)
a la.ta tic RC S ( X - b a n d . 9 0 d a g AOI. Q a ln _ 4 0 0 0 )
40
20
RCS (dB)
IO
IO
20
90
VV
HH
VV
HH
60
60
90
60
-2 0
20
S c a tta tln g A ngla (d a g )
40
90
(b)
Figure 4.18 Plot of the numerically calculated bistatic and measured RCS at 30 degrees angle of
incidence, at (a) K-band and (b) X-band, for the spilling-plunging breaking wave.
106
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Blfttadc RCS (K -band. 4 6 d ag AOI. O ain —4000)
90
20
to
13
oe
IO
20
VV
VV
HH
50
90
60
-2 0
O
20
8 c a t t « r l n a A n a l* ( d a g )
60
90
(a)
B latad c RC S ( X - b a n d . 4 6 d a g AOI. d a In - 4 0 0 0 )
90
20
IO
nee
to
20
90
VV
HH
VV
HH
60
60
-2 0
6er
8 c * tt* rln a
20
60
A n a l* ( d a a )
(b)
Figure 4.19 Plot of the numerically calculated bistatic and measured RCS at 45 degrees angle of
incidence, at (a) K-band and (b) X-band, for the spilling-plunging breaking wave.
107
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Bl«tatlc RC8 (K-band. SOdag AOI. Q aln —4000)
ao
20
to
20
30
VV
vv
60
60
• O
20a
l)
-20
•O
S c a tte rin g A n g le (d « a )
(a)
B ia ta d c R C S ( X -b a n d . SO dag AOI. G a in - 4 0 0 0 )
40
20
IO
8QC
IO
20
30
VV
HH
40
VV
HH
60
60
•O
60
-20
88 ce ra n a r l n g
20
A n g le (d a g )
60
60
(b)
Figure 4.20 Plot of the numerically calculated bistatic and measured RCS at 60 degrees angle of
incidence, at (a) K-band and (b) X-band, for the spilling-plunging breaking wave.
108
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and for X-band (60.14 - j32.85), using the Debye relation [Ulaby et al., 1982] for a water
temperature of 21.1°C. The correction factors were obtained by determining Igw-2) and
|ghh2| from Eqs. 2.28b,c for a perfect electric conductor and the actual dielectric constant,
and then subtracting the difference. This technique yielded K-band V-pol corrections of
-2.64 dB, -3.18 dB and -4.32 dB, and H-pol corrections of -1.99 dB, -1.63 dB and -1.15
dB for 30, 45 and 60 degree angles of incidence respectively. Similarly, X-band V-pol
corrections of -2.34 dB, -2.81 dB and -3.82 dB, and H-pol corrections of -1.77 dB, -1.45
dB and -1.02 dB were obtained.
Fig. 4.21a,b shows the dielectric corrected physical optics numerical calculations
plotted against the measured experimental values for K- and X-band respectively. As can
be seen the agreement is strikingly good between the dielectric corrected numerical
solutions and the experiment values.
The spread in the data is most likely due to
interference of returns from different locations on the wave face. The K-band data has a
mean error between numerical predictions and experimental values of -0.23 dB with a
standard deviation of 2.30 dB. The X-band data has a mean error of 1.24 dB with a
standard deviation of 3.63 dB.
The location of the scatterers, dominating the peak scattering events, may also be
isolated by analyzing the PO (or Kirchhoff) currents induced on the wave surface. Figs.
4.22a-d show the surface height, surface slope, surface current phase slope and surface
current magnitude corresponding to the surface which produced the peak radar
backscatter at 45 degrees angle of incidence, for the spilling-plunging breaking wave.
The surface current phase slope is defined as the change in phase per horizontal unit
length over the wave surface. The point on the wave corresponding to the maximum
slope (44.1 deg.) is indicated by a broken vertical line in all of the plots.
As can be seen, large current densities exist at the steepest point of the wave face
just below the wave crest. The corresponding phase slope approaches zero in this region,
indicating a near-specular presentation of the surface to the radar.
The reradiation
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2£
K-Band Thaoratfcal va. U aaaurad RCS
8 10
20
2S
U a a a u ra d RCS fdB)
(a)
X -B and Tlvaoratieal va. U aaa u rad RC S
20
c8
-6
20
U a a a u ra d RCS (dB)
(b)
Figure 4.21 Comparison of the numerically calculated theoretical RCS and experimentally measured
RCS over all wave profiles and angles of incidence for (a) K-band and (b) X-band.
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Local Wava H gt
Local Slopa
0.1
0.0S
0
-0.5
0.05
-
0 .1
----------------------
-0.1
-0.05
0.05
-0.5
0
(a)
4
Currant Danaity (phaaa alopa)
2
Currant Danaity (magnitude)
h-a
a
2.2
o
■5
a
01-------- —
-0.1 -0.05
0
0.05
01-
-0.5
0.1
DiatFrom Radar Braght (m)
0
DiatFrom Radar Braght (m)
(C)
Figure 4.22 Plot of the (a) surface profile, (b) surface slope, (c) physical optics surface current phase
slope and (d) physical optics surface current magnitude corresponding to the profile determined to
cause the peak backscatter at 45 degrees angle of incidence, for the spilling-plunging breaking wave.
I ll
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
pattern, expected from this portion of the wave face, would be a broad pattern centered
about the local angle of coherent backscatter.
The proximity of the local coherent
scattering angle (44.1 deg) at this point on the wave face, with the backscattering angle
(45 deg), allows us again to assert that the peak radar return was caused by specular
scattering from the steepened wave face.
4.2.2 Post-break
As previously mentioned, the SPM was shown by Brown (1978) to be valid when
the roughness parameter,
is much less than unity (see Eq. 2.27). It appears quite
unlikely that this criterion will be met at either K or X-band at moderate angles of
incidence, for the violent initial wave breaking. However, as the breaking wave evolves,
the height variance of the post break roughness may subside sufficiently that the
roughness criterion is eventually met.
To see if the SPM could predict the post-break radar backscatter required
knowledge of the small-scale roughness as well as the large-scale slopes. Since spectral
measurements were not available, an attempt was made to “bound” the amount of smallscale roughness.
The upper limit was set by using the Walker saturation spectrum
[Walker et al., 1996], which was derived from an actively breaking stationary wave. The
lower limit was set by using the Phillips spectrum [Phillips, 1985], which was derived for
an ocean in equilibrium. Because the wave numbers considered were beyond the model
limits of the Philips spectrum, results were only used to define an approximate lower
limit of SPM validity. The large-scale slopes which were used to obtain the local surface
tilt and corresponding Bragg wavenumbers, were obtained from the smoothed wave
profiles extracted from image data throughout the breaking wave evolution.
The spilling breaking wave was chosen for analysis. The spilling breaking wave
is arguably the most common type of breaking wave on the ocean and avoids the large
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splash-up features which are infrequent occurrences in the open ocean. The surface
profiles corresponding to the peak radar backscatter from the spilling breaking wave at
each 10 cm measurement location were extracted from the image data. For each surface,
Eqs. 2.28a-c were solved at each point on the water surface using the local angle of
incidence.
With the SPM based RCS determined at each surface point, a weighted
average was then taken based upon the antenna pattern over the water surface. The
resulting SPM derived RCS for the both the Walker based (upper dot-dashed line) and
Phillips based (lower dot-dashed line) spectrum are plotted in Fig. 4.23a for K-band and
4.23b for X-band at 30 degrees angle of incidence. Experimental values in Figs. 4.23a,b
are shown as solid lines.
For comparison, the physical optics solutions were calculated from the same
smoothed wave surfaces used in the previous calculations. The physical optics solutions
were then corrected for the effects of small scale roughness in a manner similar to that
described in Thompson (1988). Basically, the specular returns described by the physical
optics solution are attenuated by a roughness factor of,
(4.1)
2
where <r| > is the height variance associated with wavenumbers greater than one third of
the Bragg wave number, t„ is the time of peak radar backscatter and t is the characteristic
transition time derived experimentally.
The small-scale height variance was allowed to transition from zero prior to
breaking, to a maximum value given by
00
(4.2)
C
where k,. is the cutoff wavenumber equal to one third of the nominal Bragg wave number
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and S(k) is the wavenumber spectrum in an actively breaking region given by Walker et
al. (1996),
S(k) = 0.003k-3-5.
(4.3)
The height variance was given by,
C
(
(n2 )(t)= (n2)
(ll2)(t)= (r)2)
max
w .)
1 -e *
for 0 < t- to < T
for
max
(4.4)
t-to > t.
From the above analysis, ^ was found to equal 2.917 for X-band (<ri2>max = 2.00* 10'5
m2) and 4.410 for K-band Ocrj^max = 5.73* IO"6 m2) at 30 degrees angle of incidence. The
characteristic transition time (x) was estimated to be 0.2 seconds and the time of peak
radar backscatter (to ) was 0.46 seconds. The resulting roughness corrected physical optics
solutions are plotted as dotted lines in Figs. 4.23a,b.
As can be seen, then pre-breaking (time < 0.55 sec) and transient-breaking (time =
0.55-0.65 sec), backscatter was explained quite well by the roughness corrected physical
optics solutions. As the post break roughness spread and the wave crest continued to
collapse (time = 0.65-0.875 sec) the roughness corrected physical optics solution fell well
below the observed experimental values. However, the V-pol radar backscatter was
predicted quite well in this time interval by the SPM using the Walker saturation
spectrum. The H-pol return tended to be underpredicted by both the roughness corrected
physical optics and SPM solutions. Beyond this time (time > 0.875 sec) the surface could
not be imaged due to an obstruction in the wavetank structure. However, it appears that
as the post-break roughness continued to evolve, the radar backscatter for both
polarizations became consistent with the SPM.
The same analysis was performed for the 45 degrees angle of incidence
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K -Band. SOdag AOI. G ain-3600. Upwava
30
20
W
-20
0.6
0 .7
0.S
0.8
HH
-20
- 3 0 L0 .3
0£
0.7
0.6
0.3
0.8
Ttana <«ac]
(a)
X -B a n d . S O dtg AOI. G ain> 9600. Upwava
W
sTl
o» 0
“ -IB
-20
- 3 0 L0.3
0.4
OS
0.6
0.7
0.8
0.9
HH
at
■a
m
w 0
* -10
-20
0.4
0.6
0.7
Tima (sac]
0.8
0.9
(b)
Figure 4.23 Plot of the measured RCS (solid line), roughness corrected PO solution (dotted line),
SPM with Walker spectrum (top dashed line) and SPM with Phillips spectrum (bottom dashed line)
for (a) K-band and (b) X-band at 30 degrees angle of incidence.
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K-Band. 45dag AOI. GaiitM9600. Upwava
*-10
to
0-20
w
-9 0
os
0.6
0 .7
0.8
0 .9
S-io
to
0 -2 0
HH
0.9
0£
0.6
0 .7
0 .9
0 .9
Thna (sac)
(a)
X-Band. 45dag AOI. Gain>3600. Upwava
“ -I0
-90
W
0.4
0.6
0.7
0.9
0.9
XI“
O -20
-90 -40 >0.9
HH
0.4
0.7
0.6
0.9
0.9
T im a (sac]
(b)
Figure 4.24 Plot of the measured RCS (solid line), PO solution (dotted line), SPM with Walker
spectrum (top dashed line) and SPM with Phillips spectrum (bottom dashed line) for (a) K-band and
(b) X-band at 45 degrees angle of incidence.
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measurements, with results shown in Fig. 4.24a for K-band and 4.24b for X-band. The
wave face did not reach specular slopes with respect to the radar at 45 degrees and
therefore, the measured backscatter is underpredicted by the physical optics solution,
plotted in Figs. 4.24a,b as dotted lines. As the wave broke and the crest began to collapse
(time = 0.55-0.65 sec) the measured RCS remained slightly larger than predicted by the
SPM. However, as the wave crest collapsed and the post break roughness began to
spread (time = 0.65-0.875 sec), the V-pol return was again predicted quite accurately by
the SPM using the Walker saturation spectrum. The measured H-pol radar return was
consistently underpredicted during this time interval. As the wave continued to evolve
(time > 0.875 sec), the radar backscatter for both polarizations appeared to be consistent
with the SPM.
4.3 Conclusions
Temporal and spectral data have been presented for breaking waves of various
energies, at several angles of incidence, and in both the upwave and downwave
directions. The largest radar returns were polarization independent and detected in the
upwave direction. Peak radar backscatter and Doppler spectra were observed to occur
earlier, relative to the passage of the wave crest, in the 30 degree angle of incidence
measurements, than in the 45 and 60 degree measurements. The deterministic surfaces
and specular detection images, corresponding to these times of peak radar backscatter,
were examined for each angle of incidence angle. Specular scattering from the forward
wave face, as it steepened through specular angles in relation to the radar, was found to
produce the largest instantaneous radar backscatter for all wave profiles considered at 30
degrees incidence and also at 45 and 60 degrees incidence for the spilling-p lunging and
plunging breaking waves. The importance of specular scattering was further confirmed
by analyzing the physical optics (or Kirchhoff) surface currents over the experimental
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wave profiles. Backscatter showing an interference like pattern was observed for periods
when the specularly sloped wave face steepened to slopes greater than the radar angle of
incidence, creating multiple specular patches.
The upwave radar returns, and corresponding Doppler spectra, remained strong
until the wave crest collapsed, for all the waves considered.
After the wave crest
collapsed, the downwave backscatter measurements became prominent, and both the
upwave and downwave radar returns became dominated by the post-break small scale
roughness. Radar returns during this period were significantly smaller than the pre-break
specular scattering observed in the upwave direction. V-pol returns dominated the H-pol
returns by an average of 3.11 dB for K-band and 3.83 dB for X-band. The upwave
returns dominated the downwave measurements by approximately 2 dB. Doppler spectra
during this period were “spikey” in appearance due to the distribution in orbital velocities
of the post-break small scale roughness.
For upwave look directions at angles of incidence smaller than the maximum
observed surface slope, the radar backscatter transitioned from a specular scattering
dominated pre-break region, predicted by physical optics analysis, to a roughness
dominated post-break region. In the transition region, the backscatter was predicted by a
physical optics model with corrections for the small-scale roughness effects on the
specular facets. In the post-break region, the V-pol radar returns were consistent with
calculations using the small perturbation method.
The radar returns for angles of
incidence larger than the maximum surface slope were dominated by the backscatter from
the post-break small-scale roughness.
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CHAPTER V
SUMMARY AND CONCLUSIONS
5.0 Summary
This dissertation has described the results from two comprehensive laboratory
investigations into the radar backscatter from mechanically generated, transient deepwater breaking waves of various energy states.
The history of field and laboratory
experiments leading up to this thesis work were described in Chapter I. In general, it may
be stated that the exact nature of the radar backscatter from breaking waves was
uncertain, based upon the numerous scattering mechanisms proposed by various authors.
In Chapter II the most prominent scattering models were either derived or discussed. The
dominant scattering models (PO, SPM, TSM) were shown to accurately predict the radar
backscatter only at certain angles of incidence or sea states.
A derivation is then
presented for the iterative method, which was the primary numerical analysis method
used in this thesis.
The transient breaking wave experiment was presented in Chapter HI.
The
experiment was conducted in the 110 m x 7.6 m x 4 m deep model basin at the University
of Michigan’s Marine Hydrodynamics Laboratory. The wide experimental basin allowed
for azimuthal and grazing angle radar measurements that are not possible in smaller wave
facilities. Spilling (LEBW) and spilling-plunging (HEBW) breaking waves were probed
at azimuth angles of 0, 45, 90, 135 and 180 degrees, relative to the direction of wave
travel, at 40 degrees angle of incidence, with a K-band CW radar. Pseudo-simultaneous
H-pol and V-pol data were obtained and analyzed, both temporally and spectrally, to
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determine the dominant scattering mechanisms in transient breaking waves, at moderate
angles of incidence. The HEBW was also analyzed at azimuth angles of 0 and 180
degrees, at 85 degrees angle of incidence, with a X-band FMCW radar. Independent Hpol and V-pol backscatter data were obtained and analyzed, both temporally and spatially,
to determine the dominant scattering mechanisms at grazing angles of incidence.
Radar measurements were complemented by wave characterization measurements
using a laser sheet and video camera technique, along with capacitance wave probes
located throughout the model basin. The pre-break deterministic wave profiles were
extracted from the video data and analyzed using the iterative method, derived in Chapter
II.
The post-break wave profiles were analyzed using a modified SPM technique,
described in Chapter n.
The high-resolution transient breaking wave experiment was presented in Chapter
IV. The experiment was conducted in a 35 m long by 0.7 m wide gravity wave facility
with an experimental water level of 1.14 m. The wave tank was above ground and
allowed for more precise wave characterization and spatial positioning of the radar
throughout the breaking wave evolution.
Spilling, spilling-plunging and plunging
breaking waves were created from gravity waves with wavelengths from 0.4 to 2.4 m and
had breaking crest heights o f up to 6 cm. The breaking waves were probed at 30, 45 and
60 degree angles of incidence, with a X/K-band CW radar, in the upwave and downwave
directions.
Simultaneous H-pol and V-pol data were obtained and analyzed, both
temporally and spectrally, to isolate the scattering mechanisms in transient breaking
waves of various energies. Radar measurements were taken in 10 cm increments along
the progression of the breaking wave in both the upwave and downwave directions.
Wave characterization was carried out through the use of a high speed digital
imager and laser sheet system. Imaging rates matched that of the radar data acquisition (1
KHz) and were accurate to 0.8 mm.
The breaking waves were also analyzed after
replacing the radar with the imager and two halogen lamps. The imager and lights were
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positioned such that their field of view matched that of the radar at each measurement
location, and served as a means to detect specular facets on the evolving breaking waves.
The pre-break deterministic wave profiles were extracted from the image data and
analyzed using the iterative method. The post-break wave profiles were analyzed using a
modified SPM technique.
5.1 Conclusions
Specular scattering from the forward wave face was found to produce the largest
instantaneous radar backscatter in the upwave direction at 30 degrees angle of incidence
for all the breaking waves considered and also at 45 and 60 degrees angles of incidence
for the higher energy breaking waves considered.
In the transient breaking wave
experiment (Chapter EH), the largest RCS, for both breaking waves, was observed in the
upwave direction just prior to wave breaking. This corresponded temporally to when the
wave face reached specular (HEBW) or near-specular (LEBW) slopes in relation to the
radar. Peak radar returns were polarization independent and had Doppler spectra whose
dominant peaks corresponded to the phase velocity of the wave.
This result lends
experimental evidence in support of Jessup et al. (1991a,b) and Loewen and Melville
(1991) who hypothesized that specular scattering was the cause of the large polarization
independent radar returns at moderate angles of incidence
Both the LEBW and HEBW were found to dissipate a significant amount of
energy early in the wave breaking process. Dissipation rates were found to be 11.6% for
the LEBW, and 21.8% for the HEBW, which are consistent with the findings of Rapp and
Melville (1990). The radar backscatter appeared to increase with the rate of dissipation of
surface wave energy as observed by Melville et al. (1988). However, it should be noted
that the backscatter values for these two studies are not directly comparable. Melville et
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al. (1988) used spatially and temporally averaged RCS values, whereas this study
considered peak RCS values.
The importance of specular scattering in transient breaking waves was confirmed
in greater detail in the high resolution breaking wave experiment (Chapter IV). In this
experiment, the peak radar backscatter and Doppler spectra were observed at earlier times
in the 30 degree angle of incidence measurements, relative to the passage of the wave
crest, than in the 45 and 60 degree measurements. The peak returns at 30 degrees angle
of incidence temporally corresponded to the wave face steepening through specular
slopes for all waves considered. Similarly, the peak radar returns at 45 and 60 degree
angles of incidence temporally corresponded to the wave face steepening through
specular slopes for the spilling-plunging and plunging breaking waves. Like the earlier
experiment, the largest radar returns were polarization independent and detected in the
upwave direction prior to wave breaking.
Backscatter showing an interference-like
pattern was also observed for periods when the specularly sloped wave face steepened to
slopes greater than the radar angle of incidence, creating multiple specular patches.
Although infrequently observed on the open ocean, the splash-up feature proved
to be a significant source o f radar backscatter in both experiments.
In the transient
breaking wave experiment, significant radar backscatter from the splash-up probably
resulted in increased radar returns for the 135 degree azimuth measurements as well as
for the upwave grazing measurements of the HEBW. In the high-resolution transient
breaking wave experiment, the splash-up features, observed in the spilling-plunging and
plunging breaking waves, were found to be a significant source of radar backscatter in the
upwave direction.
Another feature which was found to strongly influence the radar backscatter was
the ‘toe’ of a steepened breaking wave. In the transient breaking wave experiment the
‘toe’ was found to be a locally strong source of radar backscatter, at grazing angles of
incidence, prior to the wave crest collapsing. This observation was supported by the
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physical optics scattering solution from waves with a similar ‘toe-like’ feature and is
consistent with the findings of Sletten and Wu (1996) who also found the ‘toe-like’
feature to be a strong source of radar backscatter.
The azimuthal dependence of the radar backscatter was determined in the transient
breaking wave experiment. During the time of the peak backscatter detection in the
upwave direction, radar returns in the 0, 45, 90 and 135 degree azimuth directions were
negligible. After the wave crest collapsed, prominent radar returns were obtained at all
azimuth angles. The backscatter during this period was dominated by the post-break
small scale roughness and appeared to be due to a combination of the surface tilt effects
and a possible anisotropy of the roughness. The largest post-break radar returns were
found in the up and downwave look directions when the underlying long waves tilted the
roughness in their respective directions. The smallest radar returns were observed in the
crosstank direction, where the roughness was never tilted toward the radar. All post­
break radar returns were “spikey” in appearance and independent of polarization. The
downwave measurements produced spectra that were broad and split into several groups
of peaks. The split was attributed to the existence of both free and bound scatterers,
while the broad spectra were attributed to the distribution in orbital velocities of the post­
break small scale roughness.
These results were confirmed in the high-resolution transient breaking wave
experiment in the upwave and downwave directions. After the wave crest collapsed both
the upwave and downwave radar returns became dominated by the post-break small scale
roughness. V-pol returns dominated the H-pol returns by an average of 3.83 dB, and the
upwave returns dominated the downwave measurements by approximately 2 dB.
Doppler spectra during this period were “spikey” and broad banded in appearance, which
again, is attributed to the distribution in orbital velocities of the post-break small scale
roughness. These findings are consistent with observations by Kwoh and Lake (1984b)
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and Lee et al. (1995) who observed broad spectra split into groups of free and bound
scatterers.
The frequency dependence of radar backscatter was primarily detected in the radar
returns from the specular patches, where the peak RCS was observed to increase with
increasing frequency. This effect was attributed to the relative size, in wavelengths, of
the specular scattering patch, which increases with frequency. Much like an antenna
aperture, as the relative dimensions of the radiating area increase, the radiation pattern
becomes stronger and more directive. Frequency effects were weak after wave breaking.
This is consistent with the SPM, where it can be seen that if the wave height spectrum has
a k -4 dependence then the RCS should be mostly frequency independent.
The deterministic wave profiles, corresponding to the detection of the peak radar
backscatter in both experiments, were analyzed using the iterative method. Again, the
peak radar returns were found to be caused by specular scattering and were modeled quite
effectively using the physical optics scattering solution. In the high-resolution transient
breaking wave experiment, a first order correction for the dielectric properties of water
were applied to the numerical PO, solutions which yielded even better agreement
between theoretical and experimental results. The spread in the observed radar data was
attributed to interference between returns from different parts of the wave face.
The evolution of the radar backscatter from transient breaking waves was
observed in the high-resolution breaking wave experiment. For upwave look directions at
angles of incidence smaller than the maximum observed surface slope, the radar
backscatter transitioned from a specular scattering dominated pre-break region, predicted
by physical optics analysis, to a roughness dominated post-break region. In the transition
region, the backscatter was predicted by a physical optics model with corrections for the
small-scale roughness effects on the specular facets. In the post-break region, the V-pol
radar returns were consistent with calculations using the small perturbation method. The
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radar returns for angles of incidence larger than the maximum surface slope were
dominated by the backscatter from the post-break small-scale roughness.
An important point to note is that remote sensing systems such as scatterometers
determine temporally and spatially averaged radar returns. Therefore, care should be
taken when applying the peak RCS values quoted in this thesis directly to such remote
sensing applications. For example, if a scatterometer probed a region of the ocean where
active wave breaking was occurring, the peak RCS observed would be much less than the
experimental values described in this thesis. This is because the scatterometer spatially
averages over the entire radar footprint, which includes non-breaking as well as breaking
regions.
In the laboratory experiments presented in this thesis, the breaking waves
“beam-filled” the entire antenna pattern and therefore, produced larger peak RCS values.
Similarly, if a radar system has a large integration time the temporally long radar returns
from the post-break roughness may be more important to the overall radar backscatter
than the temporally short (but much stronger) specular scattering returns.
5.3 Future Work
Although this thesis has isolated and identified several scattering mechanisms in
transient breaking waves, there still remains a considerable amount of information yet to
be determined about the radar backscatter from breaking waves. To further study the
radar backscatter from breaking waves, it would be useful to
•
Repeat the high resolution experiment with wind generated breaking waves using
the wind-wave facility currently under construction at the University of Michigan
Marine Hydrodynamics Laboratory.
•
Determine the two-dimensional surface statistics of the pre-break surface profiles
using the High Resolution Surface Slope Imager (HRSSI) [Meadows et al., 1995].
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•
Determine the statistics of the one- or two-dimensional post-break roughness
spectra throughout the evolution of wave breaking.
To apply the results of radar backscatter from breaking waves to remote sensing
applications, it will be useful to:
•
Incorporate the strong radar backscatter effects of wave breaking into two-scale
ocean scattering models using the statistics of breaking waves on the ocean [see
e.g. Glazman, 1986].
•
Determine how the spatially and temporally averaged radar backscatter from
breaking waves will bias scatterometry derived wind vectors over the ocean.
•
Determine how the spatially and temporally averaged radar backscatter from
breaking waves will bias SAR derived ocean wave spectra.
126
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APPENDICES
127
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APPENDIX A
CW AND FMCW RADAR THEORY
A.0 CW Radar Theory
The continuous wave (CW) radars used in this thesis generated their radio
frequency (RF) energy using either a Gunn diode (transceiver based radars) or a voltage
controlled oscillator (VCO). For either type, the transmitted field ( E ^ ) can be written in
complex phasor form as,
(A.l)
where E0 is the amplitude o f the radiated field, lq is the wave number, x is the distance
from the transmitter and © is the radial frequency. For a target at a distance x from the
transmitter, the received field ( E re) can be written as,
j(2krx - k x im-m t+codt d )
(A.2)
where cod is the Doppler shift imparted on the transmitted energy, td is the time from the
scattering event to reception and xint is the distance from the transmitting element to the
receiving element. T is the reflection coefficient of the scatterer and will be proportional
to the square root of the radar cross section.
In the transceiver based radars, xint is the distance from the Gunn diode to the
Schottky diode used as the receiving element. The Schottky diode will mix both the
transmitted field (Eq. A.1, x = xim ) and the received field (Eq. A.2) to yield the
intermediate frequency (IF) signal. Most transceiver units utilize two Schottky diodes
spaced an eighth of a wavelength apart. This will offset the signal received by the diodes
ninety degrees in phase and create an in-phase (I) and quadrature (Q) signal.
128
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In VCO based scatterometers, xjnt will be the path length from the VCO to the
transmitted field input into the mixer. The quadrature signal in a VCO based radar is
created by running the received signal through a ninety degree hybrid, which outputs the
input signal and the input signal with a ninety degree phase shift. Mixing of the in-phase
and quadrature signals are accomplished using discrete microwave mixers.
For transceiver or VCO based radars, the IF outputs of the mixers can be shown
[Saunders, 1990] to be,
E j = E J k e j( 2 k r x - 0)dt d )
(A 3)
T E q j(2krx-a>dt d + | )
e Q = — 2— e
(A"4)
where E{ is the in-phase Field and E q is the quadrature field. These signals will be
used to form the real and imaginary parts of a complex time series of radar data. The IF
will be equal to the Doppler shift imparted by the scatterer (27tad). To ensure the signal
is sampled without aliasing, the sampling rate must be twice that of the largest expected
Doppler frequency [Proakis and Manolakis, 1992].
The total radar return will be due to the simultaneous superposition of all the
scatterers in the radar footprint. Since each scatterer will have a different reflection
coefficient and range, the magnitude and phase of the return will vary with time based
upon where the scatterers are in the footprint and how strongly they scatter. CW radars
do not have any ranging capability and will not receive the scattered fields from
stationary objects. Therefore, caution should be used when comparing CW radar derived
RCS with pulse or FMCW radar derived RCS.
A.1 FMCW Radar Theory
129
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Frequency
PRI
Time
Fig. A.1 FMCW Radar RF output from the VCO
Frequency modulated continuous wave (FMCW) radars vary their RF output over
a range of frequencies. This RF sweep is accomplished by exciting the radar’s VCO with
a varying voltage created by an internal function generator. Various excitation schemes
may be used to sweep the frequency in an FMCW radar. In this thesis, the FMCW
radar’s VCO was excited by a triangle wave voltage input. This input produced a linearly
swept frequency output from the VCO as shown in Fig A.I. The pulse repetition interval
(PRI) represents the temporal length of a single cycle of thetriangular wave input or
sweep output.During one PRIthe frequency is swept from
higher frequency (co0 +
Aco)
the initialfrequency (oo0) to a
and then back again. The bandwidth of the sweep
(A co)
will
ultimately determine the range resolution of the FMCW radar.
The radial frequency for a region of increasing frequency may be written as,
G)(t)=G)0 +
0
~
PRI 2
t
(A.5)
where t represents the time from the start of the cycle. Integrating Equation A.5 with
respect to time yields,
0(t) = coot+ - ^ - t 2.
0
PRI; 2
130
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(A.6)
Thus, the transmitted field can be written in complex phasor notation as,
Etx = p E0 e
Afl) 2v
j(C0° t+ PRI
PRI
.
(A.7)
The transmitted energy will get backscattered and received after a time delay
equal to twice the scatterer range (R J divided by the speed of light (c). Therefore, the
received field will be,
2o)0R 0
E „ = r - p - E 0 -eJQ)0t
c
Aco
2
+ PR lt
4 R 0 Aco
c-PRI t+
4A coR 0 2
c2PRI
(A 8 )
The transmitted field (Eq. A.7) and received field (Eq. A.8) are mixed to yield the IF
signal,
r
t;
2
if—
E if = — ^ - e
2(0° R° + 4Ao) R ° 2 _
c
c2p r i
4 R ° Aco n
c-p r i
(A 9 )
with a radial EF frequency of,
—
4Ag>•RQ
- ^ f -
(A. 10)
In the sweep region of decreasing frequency, the negative of the radial frequency in Eq.
A. 10 will be obtained.
The IF signal will have a spectrum of frequencies spanning from DC to one half
of the sampling rate, as dictated by the Nyquest criterion [Proakis and Manolakis, 1992].
To ensure that returns from the maximum experimental range are not aliased, requires a
minimum sampling rate (MSR) of,
4Aoo • R 0
MSR = ------ —r .
71•c •PRI
131
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(A. 11)
v
’
From Eq. A .ll, one might think to select a PRI as large as possible to minimize the
required sampling rate and to increase the unaliased observation range. However, the
power transmitted will vary based upon the instantaneous frequency being transmitted. If
a data set is temporally short compared to the PRI, it will be biased depending on which
portion of the frequency sweep it corresponded to. To avoid this biasing, the PRI should
be chosen to be much shorter than the required data length.
The required data length will be determined by the type of results desired. For
remote sensing purposes, the magnitude and spectrum of the radar return for a given
range interval is desired. From Eq. A. 10, the range of a scatterer uniquely determines its’
IF. Therefore, the data may be Fourier transformed to obtain the scatterered power versus
range. If successive subsets o f a large data set are transformed and “stacked” in time, a
scatterer power versus range and time plot may also be obtained. Taking a second
Fourier transform of the complex power along a given range interval, will yield the
spectra for that interval. The sampling rate for the second Fourier transform will equal
the inverse of the temporal length of the subset of data used in the first Fourier transform.
Therefore, the length of each data subset must be short enough to ensure that the
sampling rate for the second Fourier transform is twice the highest Doppler frequency to
be considered.
From signal theory, the narrowest bandpass of radial frequency that can be
detected is 2^/PRI [Ulaby et al., 1982]. Substituting co = 27r/PRI into Eq. A. 10, and
rearranging for R, yields the theoretical range resolution of the radar (R,.es),
Rres=
T~T
47 tA cd
•
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(A. 12)
As can be seen, the range resolution is directly related to the inverse o f the sweep
bandwidth. Looking at Eq. A. 12, one may say to increase the sweep bandwidth (Aco) to
get the best possible range resolution. However, as can be seen in Eq. A. 10, this would
drive the IF frequencies so high that they may not be sampled without aliasing. Further,
instabilities in the radar system usually limit the range resolution to a value greater than
that predicted by theory. In practice, the optimal PRI and bandwidth settings will be
determined based on available sampling rate, required observation range, and required
range resolution.
For a FMCW radar, the total radar return for a given range interval will be due to
the simultaneous superposition of all the scatterers in the radar footprint. The magnitude
and phase of the return will vary with time based upon which range intervals the
scatterers are in and how strongly they scatter. The ability of FMCW radars to separate
returns into several range intervals, allows them to be used at grazing angles, where CW
radar footprints become unmanageable. Unlike the CW radar, The FMCW radar will
receive the scattered fields from stationary objects.
133
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APPENDIX B
RADAR CALIBRATION
B.O General
The CW and FMCW radars were calibrated at the experimental range facility in
the Environmental Research Institute of Michigan (ERIM). Several internal and external
calibrations were performed to ensure the ability to obtain accurate absolute radar cross
section measurements within +/-1 dB.
B.l Calibration Setup
Calibration of the CW Doppler radars used in this thesis, required a moving
calibration target of known radar cross section. The FMCW radar, although calibrated in
a similar manner, required a stationary calibration target. For both radar types, a metallic
sphere with a radius (r) of 7.5 cm was chosen. The calibrating sphere presented a target
in the geometric (or optic) region for X-band (A. = 2.85 cm, x = 16.5) and K-band (A. =
1.24, x = 38.0), where % is the size parameter defined by 2itr/X. Using simplified Mie
scatter theory [Ulaby et al., 1982] the theoretical cross section of the sphere can be found
to be 176.6 cm2 for both X- and K-band.
The radar to be calibrated was placed on a computer controlled turn table (Model
538 Pedestal Control Unit, Interface Engineering Inc., Model PNO AL-360-1P Antenna
Positioner, Orbit Advanced Technologies,) and directed towards the calibrating target as
illustrated in Fig. B.l. The sphere was linearly oscillated through a distance of 2.86 cm,
at a rate of approximately 2 Hz. The oscillation distance ensured enough phase change in
the radar returns to properly determine the calibration parameters.
134
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PC based
data acq.
Power
supply
Radar
Calibration
Sphere
\
\
Calibration target
oscillator
I
o
/
Pedestal
control unit
t
Antenna
positioner
Radar absorbent
material
Fig. B.1 Radar calibration setup
The antenna gain pattern and calibration parameters were determined by taking
backscatter measurements in one degree increments through the antenna’s E-, H- and 45
degree diagonal planes. At each point, four seconds o f the complex time series (CW
radars) or real time series (FMCW radar) data were sampled at 512 Hz. Data acquisition
was performed by a 486 based PC utilizing a 16 bit, National Instruments MIO-16X A/D
board.
B.2 Internal Calibrations
The Internal calibrations determined the amplitude errors, phase errors and DC
biasing inherent to the radars components and circuitry. The derivation of these errors
and corresponding corrections will be presented in this section.
To illustrate the
calibration process, a plot of uncorrected I/Q voltage data, taken against a sphere of
constant RCS, was included in Fig. B.2a. The data represented in this plot will be
displayed after each calibration correction.
135
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B.2.1 DC Bias
In the CW radars, the DC components of the received signals came from the
reflected energy of objects that were not moving in the radar footprint. The initial DC
components (approximately 1.5 V) were removed by highpass filters in the CW radar’s
signal conditioning circuitry, prior to amplification. Even after highpass filtering the
data, a residual DC component (approximately 0.5 mV) still existed. When amplified,
this residual DC component became a significant bias in the recorded signal, as seen in
the offset of the center of the ellipse from the reference axis in Fig. B.2a. The DC bias
varied depending on the experimental setup and was calculated on an individual basis for
each measurement.
To find the DC biases, the statistical maximum and minimum voltages were
determined for each channel. The DC bias was merely the average of the maximum and
minimum voltages. The DC bias was removed, for a given channel, by subtracting the
bias voltage from the data. The DC bias corrected I/Q data plot is shown in Fig. B.2b.
As can be seen, the plot is now centered at the origin. If the primary axis of the ellipse
was at an angle of 45 or 135 degrees, relative to the positive real axis, then a pure phase
error would exist. If the primary axis of the ellipse was at an angle of 0 or 90 degrees,
relative to the positive real axis, than a pure amplitude error would exist. All data must
be DC corrected prior to being phase or amplitude corrected.
In most data sets against distributed targets, the DC bias will equal the mean of
the data. In these cases, the mean for each channel may be subtracted off directly from
the data to remove the DC bias. However, this technique will not work for low level
signals, where random noise may bias the mean voltage of the signal, or for radar
calibration, where the calibrator oscillation length will cause the I/Q data plot to be
unevenly distributed about the origin.
136
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In the FMCW radar, the DC component of the received signal came from internal
reflections from within the radar. When the FMCW data was Fourier transformed, as
described in Appendix A, the DC component was effectively removed from the
experimental ranges of concern. Therefore, the DC bias correction used on the CW radar
data was not necessary for the FMCW radar data. Since the FMCW produced a real time
series of data, no amplitude or phase corrections were applicable.
B.2.2 Phase Error
Ideally, the quadrature channel should be phase shifted 90 degrees from the InPhase channel.
The phase shift is accomplished by spacing the receiving diodes
(transceiver based radars) so that the radiation passing the first diode goes through an
additional 90 degrees of phase before passing the second diode.
This condition is
achieved when the spacing between diodes (Ax) is,
n -X
for n = 0,1,2,3....
(B.l)
A phase error will occur if the diodes are not spaced properly, or if each channel’s signal
is processed by components which have different phase responses. The phase error can
be found by examining the real parts o f the actual in-phase and quadrature voltages
received.
Taking the real part of the received voltages, Eq. A.3 and A.4, yields,
Vj = Aicos(Q )
(B.2)
and
Vq = Aq-cos(Q+ <|>ap),
137
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(B.3)
where Q is equal to 2kx+(»dtd and Aj and A,, are the DC bias corrected maximum
amplitudes of the in-phase and quadrature voltages respectively. (j>ap is the actual phase
difference between the in-phase and quadrature voltages (ideally 90 degrees). Taking a
cosine expansion of Vq and substituting in B.2 yields,
q -
-7 L-cos(<t>ap) = -
-sin(4»ap) .
(B.4)
Squaring both sides and simplifying using trigonometric properties yields,
2
v q V:
cos^(<t>ap) - 2 ——
•cos(<j)ap) +
Aq Aj
(y
vq
V A q y
= 0.
(B.5)
x r i - 1
Equation B.5 can be solved using the quadratic equation. Disregarding the extraneous
root, the actual phase difference (<j)ap) is found by taking the arccosine of the root from the
quadratic equation. The phase error (A<|>) can then be defined as,
A(l> =
‘Pap "
n
2
(B.6)
With the phase error determined, the data may be phase corrected to obtain the proper 90
degree offset between the in-phase and quadrature voltages.
The desired quadrature voltage (Vqd) will have a phase shift of n/2 and can be
written as,
it.
Vqd = A q •C0S(Q + - ) = A q ' sin(Q)
(B.7)
Taking a cosine expansion of Vq and substituting in equations B.2 and B.7 yields,
Aq
Vq = ^7-Vi.COS((f>ap )+ Vqd -sinC^ap).
138
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(B.8)
Rearranging for Vqd yields the corrected quadrature voltage equation,
V q-^-V i-C O sO K p )
(B.9)
As seen in Eq. B.9, the phase correction can be determined from the uncorrected voltages
(V;, Vq), their DC corrected maximum amplitudes (A;, Aq) and the actual phase (<j>ap).
Each I/Q data pair is phase corrected by replacing the received quadrature voltage (Vq)
with the corrected quadrature voltage (Vqd). The DC bias and phase corrected data plot is
shown in Fig. B.2c.
B.2.3 Amplitude Error
A mismatch in amplitude occurs when the in-phase and quadrature channels are
processed by circuitry with slightly different amplitude responses. When scaling the
channels, one must remember two boundary conditions: 1) The power before scaling
must equal the power after scaling,
Af+A^=
C{ -A} +Cq •Aq,
(B.10)
and 2) The power of the scaled in-phase and quadrature channels must be equal,
(B .ll)
where A( and Aq represent the amplitudes of the voltages before scaling and Q and Cq
equal the in-phase and quadrature scaling factors.
139
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Raw CaBrafeo O ta
X Conactad C tftm fe n Oat*
DC a Phaaa Contend Catiraton O tft
DC. P te u a Anpfltuda Contend C ritn fen Da*
•flnjOA mwipmo
-OS
0
In-Ptiau Vdvga
OS
OS
(c)
0
iiKPtoau Vohifia
OS
(d)
Fig. B.2 Calibration correction sequence for data gathered against a constant RCS spherical
calibration target. Raw data (a), DC corrected data (b) DC and phase corrected data (c) and DC,
phase and amplitude corrected data (d) are shown.
140
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Eqs. B. 10 an B.l 1 can be solved simultaneously to yield the values of the scaling
coefficients,
C. =
(B.12)
(B. 13)
The amplitude error is corrected by scaling all o f the DC and phase corrected in-phase
and quadrature data by Q and Cq respectively. With the DC bias, phase and amplitude
errors corrected, the I/Q data plot will be error free and appear as a circle with a radius
equal to the power received from the calibration target, as seen in Fig. B2d.
The order of the phase and amplitude corrections did not matter. However, if the
DC bias was not removed prior to phase and amplitude correcting the data, the values of
Aj and Aq would have been incorrect, and have lead to inaccurate calibration corrections.
B.3 External Calibrations
External calibrations determined the returned power from targets of known radar
cross section and the antenna’s normalized radiation patterns. Results from the external
calibrations were ultimately used to determine experimental target RCS (See Appendix
C).
B.3.1 Normalized Antenna Gain Pattern
The antenna gain pattern was the most important of all the calibration
measurements. This calibration determined how the radar weighted what it “saw” and
how its footprint was defined across a distributed target. The normalized gain pattern
141
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was found by placing the radar on a computer controlled turn table (See Fig. B.l) at a
specified distance (Rcal) fr°m the calibration sphere. The radar was leveled so that its
boresight axis corresponded to the sphere’s axis of oscillation. Data was taken along the
E-, H- and 45 degree diagonal planes o f the radar, at 1 degree increments from boresight
to +/- 45 degrees. This range allowed for characterization of the both main lobe and
major side lobes of the antenna pattern.
The DC bias, phase and amplitude corrected power was determined for each angle
measurement.
The power at each point was then normalized with respect to the
maximum power obtained (PCAl)> yielding the normalized antenna gain pattern (gn(0,(j>)).
The normalized E- and H-plane antenna gain patterns were shown previously the for the
K-band CW radar (Fig. 3.2a), X-band FMCW radar (Fig. 3.2b) and X/K-band CW radar
(Figs. 4.6a,b) used in this thesis.
All experimental data presented in this thesis were taken at far field ranges. The
far field range (Rff) of an aperture radar can be approximated as [Ulaby et al. 1982],
where d is the largest aperture dimension of the radar and X is the radar wavelength. For
ranges greater than the far field range, the normalized antenna pattern does not change
and PCALwill vary inversely with RcAL to the fourth power, times a calibration constant.
In the near field, PCAL, as well as the normalized antenna gain pattern, can not be
extrapolated from calibration results at a different range. Therefore, when conducting
experiments in the near field, the calibration must be conducted at the planned
experimental range. A more detailed near field calibration procedure against distributed
targets is described in Sarabandi et al. (1992).
142
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APPENDIX C
RADAR CROSS SECTION DETERMINATION
C.O General
This appendix will show the derivation of the average radar cross section per unit
area for a distributed target. The standard radar equation for an distributed target can be
written as [Moore, 1990],
P rxt(rt
f
I
=
?tx •G(r,0,(j))2 -Gsys -A.2 •<r°(r,0,()))
------------------------------------
ra d *
footpr int
(4 *
-dA (C.l)
•r j t ( 0 ,«
where,
PK =
G(r, 0,<f>) =
GsyS =
X =
CT°(r, 0,(j)) =
rtgt (9><t») =
Radar power transmitted
Radar antenna gain
Radar system gain (due to radar circuitry)
Radar wavelength
Target radar cross-section
Radar aperture to target distance
The antenna gain can be broken into the more useful parts as the maximum gain
(G0) times the normalized antenna gain pattern (g„) yielding,
Ptx-G0(r)2 -gn(r,0,<j>)2 Gsys X2 -cyo(r,0,c
p “
<8‘ =
8
radar
1 a. 3
(4 ;t) -rtgt( 6 .W
( C 2 )
footpr int
Since the experiments described in this thesis were conducted in the far field at
moderate angles of incidence, to good approximation the maximum gain and the
normalized antenna gain pattern will not vary with range over the footprint. This will
allow the maximum gain to be pulled out of the integral and the normalized antenna gain
pattern to be simplified. Eq. C.2 can be further simplified by noting that CW Doppler
143
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radars have no ranging capability, and therefore, can only detect the average radar cross
section per unit area (which will be weighted by the normalized antenna gain pattern).
This allows the target radar cross section to be removed from the integral and designated
with an overbar to represent the average RCS per unit area,
Pr - g J - x2
rx tgt
J
(4 *7C)'
radar
g n (e .* r
dA.
(C.3)
rtgt (©» 4*)
footpr int
To get an absolute RCS reference, the radars were calibrated against a metal
sphere, as described in Appendix B. The power received from the calibration sphere at a
fixed range (rcal) and at radar boresight (gn(0,<J>) = 1) can be written as,
P t x - G o ’^
’ Gsys'CTcal
(C.4)
rxCal
(4 •7t)3 •rc4ai
The target radar equation (Eq. C.3) can be divided by the calibration radar
equation (Eq. C.4) and rearranged to solve for the average RCS per unit area,
-l
(P
r r x tgt
c t° =
P
V rxcai y
^cal
r
4
v rcal '
J
gn M )
dA
(C.5)
radar
rtgt(® »^ )
V footpr int
It is important to realize that the spatially averaged RCS will be most accurate
when the scatterers illuminated by the radar are uniformly distributed over the radar
footprint. In the case of a dominant point or line scatterer, the RCS per unit area will
underpredict that of the actual line or point scatterer.
This underprediction can be
minimized by restricting your radar footprint with radar absorbent material as previously
described in the experimental sections.
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The integral in Eq. C.5 is not the weighted area commonly referred to in the
literature. The target range term is left in the integral for increased accuracy. To convert
the last term to weighted area, you would have to multiply by rtgt4.
C.l The RCS Integral
The Fourier transform of the fields on an radar aperture are well known to
produce the antenna patterns [Balanis, 1989]. For a rectangular aperture in a TEi0 mode ,
the H-plane will be cosine weighted. This leads to a broadened beam with reduced
sidelobes that is well approximated by a Gaussian shape function.
The E-plane is
uniformly distributed, which leads to a sine shaped pattern. Although sine shaped, the
main lobe of the E-plane pattern is also well approximated by a Gaussian shape function.
Thus, the normalized antenna pattern can be written as,
(C.6)
where p0 and P^, are the 2-way, half power beam widths, (determined during radar
calibration) for the E-plane and H-plane respectively. \\f is the radar look angle as defined
from the downward vertical, and 0 and ® are the pseudo-spherical coordinates referenced
to the radar boresight as shown in Fig C.l (0 is the standard 0 of spherical coordinates
when <(>= jr/2 (E-plane), and O is the standard 0 of spherical coordinates when (j) = 0 (Hplane)). The range term can also be found in terms of the pseudo-spherical coordinates.
From basic geometric relations,
rtgt(0,cl>) = h / (cos0-cos<l> ),
where h is the height of the radar aperture above the water surface.
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(C-7)
The integral can be solved by either transferring Eqs. C.6 and C .l to the targets
Cartesian coordinates and integrating over the footprint, or by transferring the target
limits to 0 and O and integrating over the pseudo-spherical coordinates. The target range
and normalized antenna gain pattern are already in 0 and O coordinates, and therefore,
the later technique will be utilized.
To transfer from the target coordinates to the pseudo spherical coordinates,
requires a Jacobian type correction to find dA in terms of 0 and <t>. From the scattering
geometry,
x = h •tan(0)
(C.8)
therefore,
dx = h / cos^ (0)
and,
y = ~ ^ q ' tan(<f>)
(C.10)
dy = h / cos2 (<)>)• cos(0) dd>.
(C. 11)
therefore,
d0
(C.9)
Combining Eqs. C.9 and C.l 1 yields,
h2
dA = — r--------------- dO •d 0 .
cos (0)-cos (<j>)
(C.12)
With the integral converted to pseudo-spherical coordinates, Eq. C.5 can now be
readily and accurately solved using various numerical integration procedures [see e.g.
Burden and Fraries, 1993]. The value of the integral will not change as long as the
experimental parameters (h, vj/, p0, p0 and area illuminated) are held constant. Therefore,
for constant experimental parameters, the average RCS per unit area can be written as a
the power received from a target multiplied by a series of constants,
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______ CTcal______
a 0 = Prx tgt
(C.13)
v Pr xcal •R-cai ■AaJ
where,
-2.772^
Aa =
JJ
cos(0) •cos2 (<j)) •e
-2.772^
e
^
•d0d(D
(C.14)
Antenna
Pattern
The integral limits will be determined by the portion of the antenna pattern that is
illuminating the target area.
+©
,k
.
<D «
* + d>
r
-
3
As viewed from lie radar aperture
Fig. C.l Pseudo-spherical coordinates used for RCS calculations
147
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APPENDIX D
LINEAR WAVE THEORY
D.O General
This appendix is intended to provide basic insight into the linearized
hydrodynamic equations that can be used to determine the phase speeds of waves on the
ocean. For an in depth description of the hydrodynamics of the open ocean, one is
directed to Phillips (1966), or for the hydrodynamics of waves Crapper (1984), Kinsman
(1965).
D.l Equations of Motion
Before the equations of motion can be analyzed, a distinction must be made
between the material or Eulerian derivative and Lagrangian derivative. The Eulerian
derivative shows the change of a fluid parcel at a fixed point with respect to time. This is
the type measurement that may be obtained from a fixed wave staff or capacitive wave
probes. The Lagrangian derivative shows the change in a single fluid parcel as it flows
with respect to time.
These type of measurements could be obtained from surface
drifters. The conversion from the Lagrangian to the Eulerian derivative, for a fuction F in
Cartesian coordinates (x, y, z) and time (t) is,
(D.l)
with gradient,
(D.2)
148
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In Eq. D.l, v is the fluid velocity vector L^i + U2j + V k , where Ui an U2 represent the
surface currents in the x and y directions respectively. The left side of Eq. D.l represents
the Eulerian derivative. The second term on right side of Eq. D1 is the Lagrangian or
local time derivative.
D. 1.1 Conservation of Mass
From the conservation of mass, it follows that themass density in a volume will
decrease at the rateof mass flow out of the volume.This can be written in the form of a
continuity equation,
^ = -V opv,
at
(D.3)
where p is fluid density. Assuming that the fluid is incompressable yields,
V o v = 0.
(D-4)
Assuming that the the fluid velocity is irrotational, allows the velocity vector to be
written in terms of a scalar potential <j)s,
v = -V(j)s.
(D.5)
Equations D.4 and D.5 can be combined to yield Laplace’s equation,
V2<t>s = 0 .
D.l.2 Conservation of Momentum
149
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(D.6)
The momemtum equation can be written in various forms. The form including
pressure and gravitational forces but ignoring Coriolis and centrifugal forces can be
written as [Philips, 1966],
p ^ + VP-pg = f.
dt
(D.l)
The first term in Eq. D.7 is the mass acceleration, P is the pressure, g is the gravational
force vector and and f represents the frictional force per unit volume. Since our analysis
is for long gravity waves, the effects of viscosity may be ignored (Crapper, 1984).
Applying the inviscid assumption simplifies the momentum equation to,
p - p + VP - pg = 0.
dt
(D.8)
Eq. D.8 can be converted to a more useful form by applying Eq. D .l, Eq. D.5 and
a vector identity to yield,
—a<t>s - i
, vp
_
V ——+ v - q + ---- + gVz = 0
dt
2
p
(D.9)
where q is themagnitude of the velocity vector (v ) and g was written as gVz and
assumed to act verticallydownwards. This eqaution may be integrated bycollecting the
gradients and integrating both sides. This leads to the unsteady form of Bernoulli’s
equation,
d<t>s
1, P
^T - + r q 2 + - + gz = F(t)
dt
2
p
where F(t) is an arbitrary function of time only.
D.2 Boundary Conditions
150
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(D.10)
The boundary conditions will be categorized as either kinematic or dynamic.
D.2.1 Kinematic Boundary Conditions
The kinematic conditions state that for any surface (S) bounding and moving with
a fluid, the same particles will always comprise S. This can be written as,
f-0.
(D.U)
or from Eq. D.l,
20
_
— 4- v o VS =
dt
0.
(D.12)
The free surface can be defined as z = ri(x,y,t). Therefore, the free surface of the
fluid is defined as,
S = T i(x ,y ,t) -z = 0
(D.13)
Substituting this result into Eq. D.12 yields the free surface kinematic boundary
condition,
^ + u 3 + U2 f 1- V = 0
at
ax
ay
on
z = T |.
(D.U)
Similarly, the kinematic bottom boundary condition can be found by letting z = -h(x,y),
where h is the distance from mean sea level to the ocean floor. Assuming the ocean floor
is constant with respect to time yields,
U i — + U2 — - V = 0
dx.
dy
on
z = -h.
(D.l 5)
Assuming a flat bottom and writing V in terms of the velocity potential yields,
3<f>s
= 0.
dz
151
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(D.l 6)
D.2.2 Dynamic Boundary Conditions
The dynamic boundary condition applies only to the free surface, and implies that
the pressure exerted upon the water surface is only due to surface tension and the
atmospheric pressure.
It can be shown that by analyzing a small two-dimensional
displacement on the water surface [Crapper, 1984], the pressure may be written as,
P=
(D.l 7)
tk
where x is the surface tension coefficient and k is the curvature of the surface defined as,
d2q / d x 2
K = ---------TUI'
(1 + ( 5n /dx ) 2 ) 3/2
(D-18)
Eq. D .l7 and D .l 8 may be substituted into Bernoulli’s equation (Eq. D.10) to yield the
dynamic surface boundary condition,
P 5<|>s 1 2
x —^
- + — + - q +gz = F ( t ) - - ( V o n )
on
p
dt 2
p
z = r|
(D.l 9)
D.2.3 Linerized Boundary Conditions
As can be seen, the boundary conditions (Eq. D. 14, D.l6 and D.l9) are nonlinear.
To yield the simplest analytic solutions, the boundary conditions are linearized using
small perturbation and Taylor expansion methods on q and <j), and then collecting all first
order terms. For this treatment to be valid in deep water, the wave steepness (ka) must be
much less than 1, where a is the wave amplitude [Crapper, 1984]. Thus it can be stated
that the steepness must be small for the nonlinear terms to be insignificant in deep water
breaking waves. Acasual observation of breaking waves on the open ocean will prove
this assumption to be incorrect for capillary waves. However, long gravity waves on the
152
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ocean are highly linear and are approximated well by the equations derived from the
small amplitude assumption.
D.3 LINEAR WAVE PHASE SPEEDS
By solving Laplace’s equation using linearized boundary conditions yields the
dispersion relation for deep water ocean waves.
For deep water gravity waves the
dispersion relation, which arises from the dynamic boundary condition, is,
C
p
=
( 0 . 20)
1/ f
and for capillary waves,
Cp=P
(D.2I)
VP
where CP is the phase speed of the wave.
These relations provide a means of helping to isolate the scattering mechanisms
dominating the radar backscatter measurements of breaking waves.
When the radar
returns are Fourier transformed, the phase speed of the scatterers may be found from the
Doppler relation for an on axis target,
2C
fd = — ^-sin(0)
Kt
(D.22)
where fd is the doppler shift observed in the spectra of the radar return, 7^ is the radar
wavelength and 0 is the lookangle of the radar referenced tonadir.
Byobserving the
changes in theDoppler spectra over short periods, thetemporalcharacteristics
of the
scattering mechanisms may be inferred as well as the location of the scatterer on the
wave.
153
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BIBLIOGRAPHY
154
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