A COMPARISON OF SCATTERING RESm.., TS
OBTAINED WITH THE PERIODIC SURFACE
MOMENT METHOD AND SEVERAL
APPROXIMATE SCATTERING THEORIES
USING WA VETANK DATA
By
BRENT O'LEARY
Bachelor of Science
Oklahoma State University
Stillwater, Oklahoma
1993
Submitted to the Faculty of the
Graduate College of
Oklahoma State University
In Partial Fulfillment of
the Requirements for
the Degree of
MASTER OF SCIENCE
July, 1996
A COMP ARlSON OF SCATTERING RESULTS
OBTAINED WITH THE PERIODIC SURFACE
MOMffiNTMffiTHODANDSEVERAL
APPROXIMATE SCATTERING TIffiORIES
USING WA VETANK DATA
Thesis Approved:
e rJt/~
Dean of the Graduate College
11
ACKNOWLEDGMENTS
I would like to extend thanks to the people who have helped me throughout
graduate school. To my advisor Dr. James West for his longdistance help and his
willingness to answer my email at any time regardless of how misguided the inquiry. I
would like to profusely thank my officemate Mike Stunn for the invaluable conversations
and especially for allowing me the use of his master1s thesis as a template for mine. I
would also like to thank my committee members, Dr. Ramakumar, and Dr. Scheets for
taking time to review this work.
This work was supported, in part, through funding by the Office of Naval Research
under contract NOOO 149251206.
This thesis is dedicated to my wife Tina, and son, Chandler. To my wife for her
understanding as I labored on this after coming home from work, and to my son for giving
me inspiration to complete this paper.
III
<
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION ..................................................... 1
II. ELECTROMAGNETIC ANALYSIS .......................... . . . . . . . . 6
Introduction ................................................... . ... 6
The General Scattering Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Moment Method Scattering Calculations ........................... 7
The Electric Field Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
The Magnetic Field Integral Equation ........................... 9
The Moment Method ............... . ..................... . . . . .. 9
Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11
Weighting Functions. .. ..... .. .. .. . . .. . . .. .. . .. ... ... . . . . .. . . 13
Traditional Moment Method Scattering 13
The Periodic Surface Moment Method . .. ................... . ... 15
Horizontal Polarization . . ............... .. . ... . . . . . . . . . . . . . . . 16
Vertical Polarization ..... . . . . . . . . . . . . .. . . . . . . .. . . . . . . . . . . . . . . 19
Scattering Calculations. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 20
Parameter Constraints. . . . .. . . . . . . . . . . .. . . . . . . . . . . .... . .. . . . . . . .. 20
Universal Series Evaluation . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .. 21
Approximate Scattering Theories 22
The Kirchoff Approximation . . .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . 22
The SmaUPerturbation Model ........... . . . ..... . ........... 23
III THE SURFACES Ar'ID THEIR PREPARATION ... . .......... . . . .... 25
Introduction . ......... . .................... . . . .............. . ..... . 25
Data Collection. . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
The Samples " . ..... . ........ .. .. . ........ . . . . , . . . . . . . . . . . . . . . .. 27
Data Processing. . . ... . . . ..... . . .. .. . ..... . . . . . . . . . . . . . . . . . . . .. . . . . . 28
IV
Surface Displacement . . ......... . .... . ... .. . . . . . . . .. . . . . . . .... . . 30
Independent Profiles ... . .. . .. ........ . . . . . .... . .. . . . . .. .... . . ... . .. 32
Spectral Estimation .. . ............. . ... . . . . . .. . . .. . . .... . . .... . . 34
IV THE RESULTS OBTAINED WIlli EACH SCATTERING
11ETHOD . . . . . . .. . . ...... . . . .. ...... . .. . .. . _ . _ . . . . . . . . . . . . . . . . . . . . . . 36
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Moment Method Parameters . . .. .. ..... . . _. _ .. . . . . . . . . . . . ... . . ... . . 36
Results . .. . .... . . ... ....... .. __ ... .. . . .. . . _ . . . . . . . . . . . . . . . . . . . . . . . . . 37
Discussion . . .. . ..... __ . . . . . . . . .. . ... . ... . _ . . . . . . . . . . . . . . . . . . . 43
V CONCLUSIONS ... . ..... .. .. . ............... . ......... .. ... . . . ... , . ' . 45
References .... . .. . .. . ... . . . ... . . .. . ..... . . . . . . .. .. .... .. ...... . .. . .. .. 47
v
*
Table
4.1
4.2
LIST OF TABLES
Page
Parameters Used for Moment Method Analysis. ... . . . . . . . . . . . . . . . . . 38
RMS Surface Height in Wavelengths... .... .. . ... . ... .. . . . .. ... . . .. 39
Vi
LIST OF FIGURES
Figure Title Page
2.1 Geometry for the General Scattering Problem . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 A Pulse Basis Function.... . ........... .. ....... ....... .. . . ........ . 12
2.3 A Stair Step Current Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12
2.4 A Surface Made Periodic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 16
2.5 Mechanism for Bragg Scattering. . .. ... ...... . . . . . .. . . . . . . . . . . . . . . .. 23
3.1 The Apparatus Used to Measure Slope Data . ....... .. ...... .. . . . .. 26
3.2 The Circular Wave Tank ... ... .... .. ...... . .... . .............. . .. .. 27
3.3 Surface Patch Dimensions .. __ .. _ .. . .. " ....... '" .. . . . . . . . . .. . . . . .. 28
3.4 A Slice of PreProcessed Slope Data .......... . .... . ..... . . . . . . .... 29
3.5 Zero Mean Slope Data .... . ........... . .. . ...... . ..... . .. . . ... . . . . . 29
3.6 A Period Boundary Discontinuity. .. . .. . .. .. . .. .. .. . .. . .. . . . . . . . . . . 30
3.7 The Raised Cosine Window .......................... . . . ....... . . . . 31
3.8 Removing the Period Boundary Edge ...... . , . . . . .. . . .. . . . ... . . .. .. 31
3.9 Slices Used for the Surface Generation and Autocorrelation . . . . . . . .. 32
3.1 The Cross Wind Autocorrelation Function .... . . . . . . . . . . . . . . . . . . . . .. 33
3.10 Roughness Spectrum Approximation ............... . . .. . . .. . .. . .... 34
3.11 dB Plot of Peri ado gram . ....... . . . . .. . ..... . . . . . . . .. . . . . .. . .. . .. . . . .. 35
4.1 Results for 8GHz . .. . .. .. . .. .. . . .. . . . . .. . .. . .. . . . . . . .. . .. . .. . . .. .. .. 40
4.2 Results for 18GHz ......................... ... ... . . .. . . ...... .. . . .. 40
4.3 Results for 28GHz .. . .. . ...................... . . .. ................. 41
4.4 Results for 38GHz .............................. . .................. 41
4.5 Results for 48GHz ...................................... . . . .. ...... 42
4.6 Results for 58GHz ........................... ........ .............. 42
Vll
SPM
KA
PSMM
EFIE
MFIE
HH
W
NOMENCLATURE
Small Perturbation Method
Kirchoff Approximation
Periodic Surface Moment Method
Electric Field Integral Equation
Magnetic Field Integral Equation
Horizontal Polarization
Vertical Polarization
V11l
CHAPTERl
INTRODUCTION
The most limiting factor in the perfonnance of oversea radar systems is usually
the surfacescatter clutter. This clutter can give high energy return signals which can
overwhelm the signal returned from actual targets, such as ships and low flying aircraft, or
cause false alanns when no target is present. The random roughness of the sea surface is
responsible for this clutter. Full understanding ofthe surface scattering mechanism that
leads to clutter signals will aid in the development of detection algoritluns that can extract
true signals from the clutter, reducing both the number of missed targets and the fa lse
alarm rates of such radars.
There are several analytically derived models to predict the radar backscatter from
rough surfaces, each of which are valid under certain conditions. The most popular of
these theories are the small perturbation method (SPM) [Rice 1958], Kirchoff (or physical
optics) approximation (KA) [Beckman and Spizzichino 1963], and the two scale model
[Wright 1968]. Because of the approximations made in the derivations of the models,
each is rigorously valid only under certain conditions. For example, the Kirchoff
approximation assumes electromagnetically largescale roughness, gently varying surfaces
(long surface correlation surface) and small to moderate incidence angles. It predicts the
scattering due to the physical optics current induced on the surface of the scatterer. SPM
1
on the other hand, was derived assuming short correlation lengths and moderate incidence
angles. First order SPM predicts the Braggresonant scattering, which is due to surface
components resonant with the illuminating field's wavelength. The two scale model
incorporates both of these models by applying KA to the electromagnetically large scale
surface roughness and SPM to the small scale roughness. None ofthese models directly
include the effects of surface selfshadowing, and as such, are expected to fail at the
largest incidence angles (smallest grazing angles).
Despite the approximations made in their derivations, the theoretical models have
often been shown to accurately predict roughsurface scattering outside of their known
regions of validity. For, example, Guinard and Daley (1970) showed experimentally that
the two scale model gives accurate seasurface scattering at angles of incidence to 85° at
vertical polarization. On the other hand, Chen and West (1995) showed that both SPM
and KA can give accurate scattering from numerically generated surfaces at horizontal
polarization and extremely large incidence angles under some surface roughness
conditions. For the models to be used to their fullest potential, the true ranges of validity
must be determined.
The moment method is a popular numerical technique that is often used to check
the accuracy of approximate models in scattering problems [Broschat, 1993; Chen and
Fung 1988; Kim et. a!. 1992; Chen and West 1995]. In this approach, the moment method
is used to solve electromagnetic integral equations, yielding the surface current. The
surface current is then numerically reradiated, giving the scattered field. This technique
has been used to confirm the validity of the scattering theories under the conditions for
2
,.
which they were rigorously derived [Durden and Vesecky 1990; Chen and Fung 1988;
Thorsos 1988].
Unfortunately, the standard moment method is not well suited to application at the
largest incidence angles. The surfaces modeled must be truncated, due to the limitations
of computer speed and physical storage, leading to nonphysical diffraction from the edges
in the numerical scattering that can mask the real scattering, especially at small grazing
angJes. One way to circumvent this limitation is to apply a tapered weighting window to
the incident electromagnetic field[Thorsos 1988]. This tapered window forces the
excitation to zero at the edges and reduces the diffraction. This method has the limitation
of not using the exact illuminating field. Also, electromagnetically valid weighting
functions require longer numerically modeled surfaces with increasing incidence
angles[Thorsos 1988]. At the largest incidence angles, the modeled surface must be so
long that application of the moment method is cost prohibitive. A second approach is to
force the surface to be periodic and include an infinite number of periods of the surface,
thereby eliminating the edges in the modeled surface [Rodriguez 1990] and allowing the
application of the technique at small grazing angles. The primary disadvantage of this
approach is that an infinite series must be evaluated for each element of the moment
interaction matrix, leading to computational inefficiency. A more efficient implementation
ofthe periodic surface moment method was developed by Chen and West ( 1995), and
used to investigate the validity of the scattering models from a limited class of surfaces
down to grazing incidence [West et. aI., 1995].
3
The most severe limitation in numerical studies of surface scattering is the method
used to represent the scattering surface. Typically. sample surfaces are generated from an
approximate power spectral density. Several roughness spectrum approximations have
been used to model the ocean surface, including the power law or PhiUips spectra
[Phillips, 1958] and the PiersonMoskewitz spectra [Broschat, 1993]. The
PiersonMoskewitz spectra is an approximation of the entire wave spectrum for the ocean
surface. parameterized by the speed of the wind generating the waves. The power law
spectra represents the saturated (large wave number) range of the PiersonMoskewitz
spectrum, and is not (to first order) a function of the wind speed. The saturated region
includes Braggresonant energy at most frequencies. These are approximate spectra only,
and as such their validity is not well established.
Only a few direct measurements of the wavenumber spectra of short ocean waves
exist. These measurements are usually taken with a scanning laser slope gauge and can
only resolve wave numbers from approximately 31 to 990 radim, which is not sufficient to
resolve small capillary waves. Laboratory data from wave tanks is the only reliable source
of such short wave data. There is some question of how well the results obtained with this
laboratory data can be extrapolated to the field conditions found in the open sea [Jahne
and Klinke, 1994].
The goal of this work is to examine the ranges of validity of the theoretical models
in describing the scattering from actual water surfaces. Experimentally measured slope
images taken in a closed wave tank, with wind generating waves, are integrated to obtain
height profiles.
4
These surfaces should have roughness similar to that of open water surfaces. The
scattering from upwind/downwind cuts of the surfaces is calculated using the
periodicsurface moment method of Chen and West (1995). This scattering is then used
to evaluate the ranges of validity ofthe scattering models when applied to actual water
surfaces. A detailed review of the periodic surface moment method used is given in
chapter two, as is a brief description of the SPM and KA scattering models. The
processing of the raw surface data to allow application of the PS!v1M is given in chapter
three, and the validity of the scattering models is examined in chapter four. Finally chapter
five provides conclusions to be drawn from this effort.
5
K
CHAPTER 2
ELECTROMAGNETIC ANALYSIS
Introduction
This chapter gives an overview of the periodicsurface moment method used to
predict scattering from perfectly conducting rough surfaces. The moment method is a
general numerical technique used to solve linear integrodifferential equations[Harrington,
1968]. When applied to rough surface scattering problems, the moment method is first
applied to integral equations that force the surface boundary condition to be met, yielding
the unknown surface currents. These currents can be reradiated to give the backscattered
field. Also included in this chapter is an overview of two approximate scattering theories,
the Kirchoff approximation (KA) and the small perturbation model (SPM). The Kirchoff
approximation reradiates the physical optics current to get the backscattered field. The
small perturbation model uses the roughness spectrum of the surface to predict the
scattering due to small resonant components of the surface.
The General Scattering Problem
Figure 2.1 shows the general rough surface scattering geometry to be considered
here.
6
y
1D ro~ unlConn in:r.,
per1e~conducttng observation p Jnt
r
r •
r = vector from origin to observa1ion point
l' = vector from origin to source
Figure 2.1 Geometry for the General Scattering Problem
Moment Method Scattering Calculations
The Electric Field Integral Equation
The scattering from a onedimensionally rough surface is best described by the
electric field integral equation when the illumination is horizontally polarized. The EFIE
insures that the boundary condition
E~ = an x(ES +E') = 0, (21)
is met. Where E'tan is the total tangential field at the surface 1In is a unit vector normal to
the surface, E" is the scattered electric field, and Ei is the incident electric field. For a
general, twodimensionally rough surface the EFIE is given by [Balanis 1989]
/fan x [k2 ff J,(r')G(r, r')ds'  If v' . J(r')V'G(r, r')cW] = an x ELm (22)
on S, where ~o = 41t X 107 , Eo = 3~1t X 109 is , T] = ~ , is the intrinsic impedance of free
space, k = til J~E = ~ is the electromagnetic wave number, A is the wavelength of the
incident field, CJJ is the radial frequency of the incident field, r is a vector from the origin to
an observation point on the surface, r' is a vector from the origin to a point on the source,
7
S is the scattering surface and G(r,r') is the three dimensional form of Green's function
given by
r' eJIdI
G(r, ) = 41tR ' (23)
where R is the distance from the source point to the observation point. R is expressed in
Cartesian coordinates as
(24)
The surface current density on the surface of the scatterer is found by solving equation
(22) for I.(r') using the moment method. The scattered field is then found by reradiating
the surface current using
ES(r) = j)lffi Is J,,(r'')G(r, r')ds' +~ V L VI . J,,(r')G(r, r')ds' (25)
This reradiation equation can be simplified for a one dimensionally rough
scattering surface with a horizontally polarized incident field. In this case the scattering
surface is described as y = f(x). The incident electric field E' has only a z component,
and the scattered electric fields are uniform in z.
Thus using,
I eJajii+t2 d _ '_u(2)( ) _ ~ t  j/1.l1o ax,
yX2+11
(26)
the EFIE reduces to
; IL Jz(p/)H~2) (kR)dl = E~(p). (27)
Here, L is the surface profile in the xy plane, and H~2) (kR) is the zero order Hankel
function of the second type. This equation is a scalar integral equation and is directly
solvable by the moment method. Similarly, the surface current reradiation reduces to
Eo! = Gz ; IL Jz(xl)H~2)(klp  p'I)dx'. (28)
8
The Magnetic Field Integral Equation
The magnetic field integral equation is used to describe scattering from a rough
surface at vertical polarization. The MFIE insures that the boundary condition
J!=anx(HJHi ) (29)
is met, where HS is the scattered magnetic field, and H is the incident magnetic field. For
a general 2D scattering problem the MFIE is given by [Balanis 1989]
J.~r) an xfJsASJZ(r')G(r, r')ds' = an xHi(r). (210)
The integration domain SilS indicates the principalvalue evaluation of the integral
around the singularity at r = rl. The MFIE is also a vector integral equation for the
surface current. Likewise, it can also be simplified for the two dimensional case. Using
the same arguments as in the last section and assuming a TM"' polarized incident wave,
equation 210 reduces to
h~r) +~ fL_6JJz(r')coso/H~2)(kR)dl' =H~(r) (211)
This scalar integral equation is also solved by direct application of the moment method.
The scattered field is given by
HS(r) = jmp. Is JsCr')G(r, r')dS + ~E V Is V' . Js(r)G(r, r')ds',
which for the twodimensional case reduces to
(212)
k (2) H; =~ fr Js(p')cos'¥H1 (kR)dl'. (213)
where 'I' is the angle between the distance vector and the normal vector at the observation
point.
The Moment Method
The moment method is used to approximate solutions to equations with the
general form [Harrington 1968]
L[ftR)] = g(R),
9
(214)
L is an arbitrary linear integradifferential equation, f is an unknown function to be
determined, and g is a known excitation function. In scattering problems, equation (214)
corresponds to the :MFIE or EFIE with g(R) as the illuminating field (or source) and f{R)
is the unknown surface current.
The first step in applying the moment method is to approximate the unknown
function as a weighted sum ofN known basis functions:
N
ftR) = L <x;Ni(R),
;=)
(215)
where NI(R) are the basis functions and <XI are unknown coefficients to be determined by
the moment method. Substituting equation 215 into 214 and recalling the properties of a
linear operator gives
N
L <x;L[Ni(R)] = g(R).
i=1
The residual of this approximate solution is
N
Res(R) = l: a;L[M(R)]  g(R)
1=1
The values of the coefficients are chosen to minimize this residual.
(216)
(217)
The moment method uses the method of weighted residuals to find the optimal
weighting coefficients. The weighted residuals are obtained by taking the inner product of
the residual and N weighting functions wj(R). The inner product is defined by
<wj(R),Res(R) >= fn wj(R)Res(R)dQ (218)
Setting these weighted residuals to zero and again taking advantage of the linearity of the
L operator gives the general moment equation:
L adn Wj (R)L [N; (R)]dQ = In Wj(R)g(R)dQ (219)
This equation has N linear algebraic equations and N unknowns, and can be readily solved
for a l using general linear algebra methods.
10
Examining equation (219), it is seen that the moment method is a two step
process. The first step is to "fill" the moment interaction matrix. This step includes a
numerical integration for each matrix element and increases processing time by ~ as more
basis functions are used to describe the surface. The second step is to solve the system of
equations generated for the unknown coefficients, a/so The direct linear algebra methods
usually used to solve for the uj coefficients are order N3. Because of this the solve time is
usually the limiting factor in the standard moment method. However, the fill time is
actually greater in the periodic surface implementation used here.
Basis Functions
The choices for basis functions are limitless. They can include either entire domain
functions valid over the entire surface or subdomain basis functions valid over only a
portion of the surface [Harrington 1968]. Subdomain basis functions are typically used
for electromagnetic scattering problems. Traditional choices for subdomain basis
functions in electromagnetic scattering problems include pulse functions, piecewise
sinusoid and piecewise linear functions [Balanis 1989]. The basis functions should be
chosen, if possible, to closely approximate the unknown function while striving to
minimize the computational effort expended. The basis functions used in this work
are subdomain pulse functions, as shown in Figure 2.2. With this method the surface is
divided into a series of small segments and the current density along the segment is
considered constant.
11
y
Pulse func.tlon, height = 1
1
rl+l
Figure 2.2 A Pulse Basis Function
Using this basis function produces a stairstep approximation to the surface current as
shown in Figure 2.3.
The pulse basis functions were chosen for their computational simplicity. The evaluation
of the linear operator in the EFIE and/or MFIE can be accurately evaluated without the
use of numerical integration [Harrington, 1968]. While fewer basis functions could
theoretically be used with "better" basis functions which more accurately approximate the
actual current density, in practice it has been shown that the actual reduction is small, and
any advantages are more than outweighed by the increased matrix fill time [Axline and
Fung 1978].
Jz
Figure 2.3 A Stairstep Current Approximation
12
y
Pulse fun.cll.on, height = 1
1
x
x1+1
Figure 2.2 A Pulse Basis Function
Using this basis function produces a stairstep approximation to the surface current as
shown in Figure 2.3 .
The pulse basis functions were chosen for their computational simplicity. The evaluation
of the linear operator in the EFIE and/or MFIE can be accurately evaluated without the
use of numerical integration [Harrington, 1968]. While fewer basis functions could
theoretically be used with "better" basis functions which more accurately approx.imate the
actual current density, in practice it has been shown that the actual reduction is small, and
any advantages are more than outweighed by the increased matrix fill time [Axline and
Fung 1978].
Jz
x
Figure 2.3 A Stairstep Current Approximation
12
Weighting Functions
As with the basis functions, many choices are available for weighting functions. In
this work the weighting functions are chosen to be Dirac delta functions, ( or impulse
functions) centered on the basis functions. This choice forces the surface boundary
conditions to be matched exactly at the point of the impulse. The primary advantage of
this approach is that the inner product in equation (218) reduces to the evaluation of the
operand at discrete points, thus eliminating the integration entirely. Again, this has been
shown to yield good results when applied to rough surface scattering [Chen and Fung].
Traditional MM Scattering
The moment method is now applied to the EFIE to yield the currents on a
onedimensional rough surface at horizontally polarized illumination. The EFIE in
equation (27) is first rewritten as
E~(x) = ~ J JI +h;(x') Jj(x')H~2)(kR)dxl, (220)
where hex') is the surface displacement and ~(x') is the first derivative, with respect to x',
of the displacement. The moment method is applied by expanding the unknown current as
a weighted sum of pulse basis functions:
Jj(x') = L~=] JnP[x' xn], (221)
where In are the unknown weighting coefficients,
P(X/) = { Xn  "'2 < x < Xn + 2 ,
1 I!.r., I!.r. }
o elsewhere
(222)
and"" , and .1xn are the center and length of the nth segment respectively. Substituting
(221) into (220) gives
E~(x)= ~ L~lJnJl!.rn Jl +h;(x')H~2)(kR)dx' (223)
13
the impulse weighting functions are now applied:
f:E~(x)8(xxm)dx = ~ L~] JnJllx" JI +h~(xm) f:Hi2\kR)o(x xm)dxdx' (224)
so
E~(xm) = ~ L!] J"fllx Jl +h;(xm) H~2)(kRm)dx'
r~" ' where Rm = J(xm _x')2 + [hl(xm) h(x')J2 .
(225)
Evaluating at the N segments, (225) can be rewritten as the matrix equation
(226)
where V m = EizCxJ and
Zmn = ; Jllxn J 1 + h;(xm) m2)(kRm)cix' (227)
Solving (226) for the In completes the moment method solution.
There is no closed form expression for the integral in equation (227), but if certain
conditions are met there are good approximations[Harrington 1968]. If the integration
length is electrically small and the observation point (xJ is not on the nth segment, the
integrand is approximately constant and (227) can be evaluated by
kr) rrl2'
Zm" = 711[,,11 (, (kRm,,) (228)
where
(229)
and
(230)
If the observation point is on the source segment, the integral is dominated by the behavior
of the integrand at the singularity at ~ = O. In this case the equation (227) is accurately
represented by
Zmm = ~ 11 [iH12) (1  j~ In i'k~,.,) (231 )
where 'Y = 1.781, is the Euler constant.
Use of the magnetic field integral equation with the moment method and point
matching is similar to this development with the EFIE [Axline and Fung, 1978].
14
The Periodic Surface Moment Method
Finite computer resources limit the size of the scattering surface that can be treated
with the standard moment method. The number of segments used to model the surface
increases linearly with the surface size, and the memory needed to store the interaction
matrix increases by~. Also the computational time needed to solve the system of linear
equations depends on N3. Thus both the CPU time needed to solve the equations, and the
memory needed to store the complex interaction matrix elements limit the size of the
surface that can be solved with this method, so the numerically modeled surface must be
artificially truncated. This truncation leads to nonphysical edge diffraction effects that
mask the physical scattering from the surface. The standard moment method avoids the
diffraction by applying a weighting function that smoothly reduces the incident field to
zero at the edges. However, Thorsos (1988) showed the electromagnetically valid
weighting windows become quite narrow beams at small grazing angles, leading to
unrealistic illumination of the surface features that gives incorrect scattering.
Many of the disadvantages of the standard moment method at small grazing angles
can be overcome by assuming that the scattering surface is periodic and infinitely
extending, as shown in Figure 24. Although only a finite length of surface is numerically
modeled, the assumption of periodicity eliminates the edges. Thus, no illumination
weighting function is needed to avoid the diffraction effects, so the technique can be
applied at arbitrarily small grazing angles [Kim et. al., 1992]. The primary disadvantage of
this approach is that a slowly converging infinite series must be evaluated during the fill
15
y
lpert~d~
Figure 2.4 A Surface Made Periodic
stage of the moment solution. Direct evaluation of the series is computationally
prohibitive at small grazing angles.
In this work the efficient implementation of the periodic surface moment method
developed by Chen and West (1995) is used. This approach is summarized here.
Horizontal Polarization
As mentioned earlier, the electric field integral equation is used for horizontal
po I arizatio n(IllI). The current on the periodic surface is given by
l(x' + pL) = eJkpLsin 9'l(x') . (232)
The form of the EFIE for periodic surfaces is obtained by substituting (232) into (28),
yielding
p=oo
Eo(x) = ~ J~~2 J 1 + h~(x') l(x') L eJkpLsin 9'H~2)(kRp)dxf (233)
p=>
J(x') is the unknown current density on the center period (p = 0), Sj is the incident angle,
L is the surface period, and ~ is the distance from the current source to the observation
point given by
Rp = J[x (x' +pL)P + [hex) h(x')J2 (234)
16
The moment method is applied as before, yielding
{
~f!JcpLSin9/JI(})(kRmnp) ~ }
Zh = kTJ6 ln r m n
mn 4 [1 _ j~ In(~) + 1eiKpLsin 9/1102) (kRmnp) m = n .
(235)
where Rmnp = J[xm  (Xn +pL)P + [h(xm) h(xn)P and 6In is defined earlier.
The matrix elements include an infinite series that has no closed fonn evaluation.
At lower incident angles the series converges quickly and only a few tenns are needed to
obtain accurate results. However, as the angle of incidence increases towards grazing
angles the series converges more and more slowly, and as the incidence angle approaches
90° direct evaluation of the series becomes quite time consuming. Thus, the matrix fill
time becomes prohibitive at large incidence angles if direct evaluation is used.
In Chen and West's approach, the matrix element equation is rewritten as
Zhm n  4b"jL A.l l 11 [S mh+ll + Smhn + Shmon] ,
where
Sh+ = ~ ejkpLsin 9; m2) (kR ) mn.L. 0 mnp
p=p<r+L ,
m~n }
nl =n '
(236)
(237)
(238)
(239)
Thus, the infinite series has been divided into an upper (h+), lower (h"), and center (hO)
summation. Proper choice of the cutoff period (pJ insures that all effects of the surface
displacements are included in the evaluation of the S~~. This subseries must therefore be
evaluated exactly for each matrix element. However, the lower and upper summations can
be calculated much more efficiently. The upper series is examined first.
17
When the source point is a great distance from the observation point the distance
between them can be approximated by
Rmnp =pL+&, p> ° (240)
where, ox = xn  x",. Substituting (237) into (235), replacing the Hankel function with
its large argument approximation and performing a Q order Taylor expansion gives
where
and
A  (2q1)!!
q  (2q)!!
(241)
(242)
(243)
These same arguments can be used to reduce equation 240, the lower summation, to
SZ;;, = ff ei(k&:+~) f A q Uq &q
q~
(244)
where
Uq = L eikpL{1+sin8t)_J_l
p=po+J (PL)q+ 2
(245)
The evaluation of both the upper and lower summations have been reduced to
evaluating a linear combination of the upper and lower "universal series" U~ and Uq. AJI
dependencies on m and n are contained solely in ox. For this reason the universal series
for each matrix element are identical and need only be evaluated a single time. This
approach reduces the calculation of the moment interaction matrix to evaluating the
universal series once and combining with it a few direct calculations for each elements'
center summation. This greatly improves the efficiency of evaluating the matrix terms.
18
Vertical Polarization
The vertical polarization development of the universal summation approach to the
periodicsurface moment method is similar to that taken for the horizontal polarization,
except the magnetic field integral equation is now used. The MFIE for uniform
illumination and a periodic surface reduces to [Kim et. al 1992].
Hi (x) = J~) +~ lJt J 1 +h;(x') J(x') i elkpLsin 9, COS '¥'Hi2) (kRp)cix' ,
Ll2 p=
(246)
where 'V' is the angle between the vector from the source to observation point and the
surface normal vector at the source point. Following a similar procedure as that for the
EFIE yields,
Z~n = tOmn +~ln~(S~ +Sv,;" +S~n)
where
U + and U  , are the universal series defined earlier and q q
I _ [x,,xm+pL]h .. (x,,)+[h(x.,}h(x,,)]
cos'¥mnp  J '
I h .. (x,,)
cos'¥n = ~
,,1+h;
R""", l+h~(x,,)
(247)
(248)
(249)
(250)
(251)
(252)
(253)
The evaluation of matrix elements has again been reduced to evaluating each
universal series once and the direct evaluation of a few center terms.
19
Scattering Calculations
The moment method yields the current induced on the scattering surface by the
incident field. The reradiation of the surface current is then used to find the scattered
field from which the surface radar crosssection is determined. The radar scattering
coefficient of a surface is defined as the radar cross section ofa surface divided by its
physical cross section. This work uses one dimensionally rough surfaces, so the scattering
coefficient calculations are therefore referenced to the surface length rather than an area.
In order to reduce the phase interference fading encountered when calculating the
backscattering from a single surface, the scattering from N. surfaces is averaged. The
onedimensional surface scattering coefficient is estimated by [Axline and Fung, 1982J
cr(B) = ~~[ Lfl IA; 12  ~.I Lfl A; 12] (254)
where ~. is the scattered field from the jib surface, R is the distance from the far field
observation point to the to the source point, and L is the length of the scattering surface.
At horizontal polarization, A: is the electric field scattered from a single surface period,
given by [Axline and Fung, 1982; Chen and West, 1995].
(255)
where e. is the scattering angle.
At vertical polarization A~ is the singleperiod scattered magnetic field given by J k e:r(kr+~)LN 6././i(x )cos'P eJk[xnsin9/+h(xn)cos9tldt (256) 8nr n=l n n n
Parameter Constraints
Chen and West derived several constraints on the parameters of the periodic
surface required for the validity of the moment method solution. These are now
summarized.
20
The scattering from a periodic surface is zero everywhere except on a grating
reradiation lobe, defined when
. e i). . e sm 3 = Tsm j. (257)
If a grating lobe exists at ±90o (horizontal), the infinite series in the periodic surface EFIE
and MFIE do not converge. This can be avoided by insuring
L fA.
:i: sin9,±! (258)
where t is any integer.
Approximations made in deriving equation (241), (244), (249), and (251)
require the following inequalities to be met:
P 10 0> kL 1
P 8(hmoxhminl2
0> V. '
P 22.4(hmaxhminl
0> L
where ~ = the maximum displacement of the surface, and hrnin is the minimum
displacement.
Universal Series Evaluation
(259)
(260)
(261)
When the incidence angle nears 90° the universal series converge very slowly and
direct evaluation becomes computationally prohibitive. The epsilon algorithm for
acceleration of series convergence was therefore applied to the universal series with
excellent results [Thatcher 1963].
21
Approximate Scattering Theories
The two most popular approximate rough surface scattering theories are the
Kirchoff approximation (KA) and the small perturbation model (SPM). A brief summary
of these theories is given here.
The Kirchoff Approximation
The Kirchoff approximation assumes that the current induced on the scatterer
surface can be approximated by treating the local region of the surface as an infinite,
perfectly conducting inclined plane [Beckman and Spizzichino, 1963]. Using this, the
surface current is then determined from the physical optics approximation:
Js = 2anxHi (262)
The KA is valid with electromagnetically long correlationlength surfaces or
largescale displacement surfaces at moderate incidence angles. The scattering
coefficients predicted by the Kirchoff approximation were determined using the approach
of Chen and Fung (1988). In this, the scattering coefficient is again calculated using
equation (254). However, the scattered fields are calculated from the physical optics
currents numerically determined from equation (262) rather than the surface currents
obtained via the moment method. Use of this approach insures that any differences in the
calculated lv1M and KA scattering coefficients will be due to fundamental limitations of the
Kirchoff approximation itself, rather than the additional approximations required to yield a
closed Conn KA expression as in Beckman and Spizzichino (1963).
22
The Small Perturbation Model
The small perturbation method finds the total field in the presence of a smooth
scatterer, and then perturbs these fields to account for the smallscale roughness. First
order SPM predicts the scattering to be entirely due to the "Braggresonant" surface wave
energy, whose wave number is given by
K = 2ksin 8; (263)
where K is the surface wave number. When this condition is met, the additional round trip
electrical path length between identical points on the surface wave but within different
periods is an integer multiple of the radar wavelength i.e. 2M sin 8; = n'A., as shown in
Figure 2.5. This yields constructive interference which overwhelms all other scattering
contributions. The scattering coefficients predicted by first order SPM (n=1) are
0'"" = 4K3(l +sin2(8)W(2Ksin(8))
O'hh =4K3cos4(8)(W(2Ksin(8)))
(264)
(265)
where W(K) is the surface roughness power spectral density. W(k) will be estimated from
the sample surface displacements to allow the calculation of the scattering coefficients
using SPM.
Figure 2.5 Mechanism for BraggResonance Scattering
23

Chen and Fung (1988) have shown the small perturbation model to be accurate
when the surface roughness standard deviation is small compared to the electromagnetic
wavelength and angle of incidence is between about 20° and 70°.
24
CHAPTER 3
THE SURFACES AND THEIR PREPARATION
Introduction
This chapter discusses the measurement and processing of the water surface
profiles that were used in the electromagnetic scattering calculations. Surface slope
profiles measured in a wave tank were provided by B. lahne and J. Klinke of Scripps
Institute of Oceanography. The measurement facilities and measurement procedure are
first discussed in this chapter. Then the procedure used to derive the surface displacement
profile from the slope is described, and the adjustments to the surface required to allow
the application of the periodic surface moment method are then examined. Finally, the
procedure used to estimate the wave height spectrum from the surface profiles is
described.
The Data Collection
The wave tank data used was collected from a circular wave tank facility at the
Institute for Environmental Physics at the University of He idle burg, Germany [Jahne and
Klinke, 1994]. The apparatus used for this data collection is depicted in Figure 3. 1.
25
I
I
,..1
I
.I
.•.
.:.: ~ )
: vertical rays to camera
....... ______ Flat Water Surface
, lUys are paraDel once they pUi through the ImJ
&ttom. GrTank
<:::: • :;;> Frelnel Len. <;: :;;;
, '
C=:::::::::==:;;;;=::J Ab5Grption WedEe
L_____ ...J DlfTu.sor
<a) (b)
Figure 3.1 The Apparatus Used to Measure Slope Data
This particular setup shines light from the under the bottom of the wave tank up
through the combination of an optical diffusor, an absortpion wedge, and a Fresnel lens at
one focal length distance from the wedge. A ccdcamera is placed at a large distance,
therefore all rays reaching the camera are vertical. The optical diffusor is meant to
simulate an isotropic light source by diffusing the light from the halogen lamps below.
The absorptive wedge provides a known intensity gradient. The light then passes to the
Fresnel lens, all rays emitted from a certain point on the diffusor are parallel once they
pass through the lens. If the water is flat the rays going to the camera will all come from
the center of the diffusor as shown in Figure 31a. If the water is sloped the light comes
26
I
I
l
I . ."_.<4 •4 )
4m
Figure 3.2 The Circular Wave Tank
from another point on the diffusor as shown in Figure 31b. Ifa linear absorption wedge
is used, the intensity at the camera is approximately linearly related to the slope. Iahne
and Schultz (1992) showed that the nonlinearities for the system used here are quite low.
The SampJes
The wavetank data provided was captured from a circular wave tank as pictured
in Figure 3.2. The wind was generated by a rotating paddle wheel mounted near the
ceiling of the water channel. The speed of the wind driving the waves was 10 meters per
second and the fetch of the waves produced is theoretically infinite [J ahne and Klinke
1994], mimicking the conditions in the open sea.
An image of a patch of the surface 18cm long in the alongwind direction and
14cm in the across wind direction was provided. The along wind dimension was sampled
27
A,
,
14cm cross wind
240 samples
<     ,                18cm along wind           >
: 496 samples
Wind Direction
Figure 3.3 Surface Patch Dimension
496 times, while the cross wind dimension was sampled 240 times, giving along wind and
across wind sampling intervals of 0.363 and 0.583 millimeters respectively. Since only
one dimensional surfaces can be treated with the moment method implementation of
chapter two, each of the 240 along wind slices was processed separately and scattering
from each was used for the backscattering coefficient calculations. Due to correlation
between adjacent alongwind slices, the number of independent surfaces is much less than
240, as discussed later.
Data Processing
Figure 3.4 shows a single alongwind slice of the surface slope profile. The
discontinuity of2.198355 when the slopes exceeded 1.0991775 is most likely due to an
28
I
~I it "I
1.5
0.5
8. 0 i
0.5
1
1.5
0
Kr 1\ ~ ~
1\ '"
0.05
~ ~
\ r..
0.1
x(cm)
~~

0.15
Figure 3.4 A Slice of preprocessed slope data
0.2
unsigned integer being treated as a signed integer in the data writing or reading scheme.
The discontinuities and mean offset were removed, as shown in Figure 3.5.
v.o
0.4
0.2
0 a
i
0.2
.Q.4
0.6
0.8
0 0.05 0.1
x(cm)
0.15
Figure 3.5 Zero Mean Slope Data
29
0.2
Surface Displacement
The moment method requires the surface displacement as well as the surface slope.
This was obtained by numerically integrating the slope profile. The displacement at the nih
sample was given by
Yn+l =Yll8x+Yll (31)
where yin is the is the surface slope at the nth sample and (5x is the alongwind sampling
interval. The integration was initialized by setting Yo=O.
The numerical scattering routine requires the rough surface to be periodic. Simply
assuming the integrated profile is periodic would lead to discontinuities in the surface
slope as shown in Figure 3.6. Note that there is no discontinuity in the height since the
average slope was forced to be zero, giving a zero displacement at both ends of the
surface profile. However, the slope discontinuity gives a sharp edge in the surface that
could lead to unrealistic scattering particularly at the higher frequencies examined.
displacement
Figure 3.6 A Period Boundary DisContinuity
30
.. ,I
it, .::.
i=10 i=NIO
Figure 3.7 The Raised Cosine Window
This was avoided by multiplying the height profile with the windowing function shown in
Figure 3.7. Each edge of the window represents one half cycle of a raised cosine function,
The weighting function is written mathematically as
W(X)={ f~:=~::~Xlt) ~,;x,;°x~=:}.
1 elsewhere
(32)
where Llx = °4~~m , the along wind sampling distance.
6x was chosen to be IOcx, so that 10 samples were modified on each side of the profile
(20 of the 496 total). Since the height data was changed by the window, the slope
changed also.
1st derivative discontinuity is removed by window
displacement
one period
Figure 3.8 Removing the Period Boundary Edges
31
.. ,I
it
EI
The slope at the boundaries was recalculated using the chain rule. The windowed height
data is given by
hex) = W(x)h(x) ,
so the windowed slope data is
h'(x) = ~[W(x)h(x)] = W(x)h'(x) + W'(x)h(x).
h'(x) is the slope profile.
Independent Profiles
(33)
(34)
As mentioned earlier, since the alongwind profiles were taken from the
same image, adjacent profiles are not independent. To estimate the number of
independent profiles available, the surface autocorrelation in the crosswind direction was
estimated.
Cross
Wind
Direction
I I I 1 I I
I I I I I I
Slope Image
248 I 496 points
 I . . ~ • •                       •         •  lag N
L '. ' •• _ I ., _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _          ... lag3
~ ... _ •.... ___  _ .  . . . _ .  . . . .       .  .  .       .Iag 2
~ :_ :_ : _: .,: _ _ _ __ _ _ _ _ _ _ _________ .. _ _ _ _ _ _ _ _ _ _ _ __ . lag 1
sample # 1 2 3 . .. N
AloDgwind direction
Figure 3.9 Slices used for Generating Surface Statistics
32
wi
This was accomplished by calculating the autocorrelation function for a single crosswind
cut, shown in Figure 3.9, using
RCn) = N~71 L':~ h k hk71 , (37)
where hk is the kth crosswind displacement sample. The autocorrelation functions
calculated for all 240 acrosswind cuts were then averaged to give the estimated
acrosswind autocorrelation for the entire surface. The results are shown plotted in Figure
3.10.
The surface autocorrelation reduces to one half at approximately at n = 7 in Figure
3.10. The correlation of 0.5 is used in conjunction with the widely accepted 3dB antenna
beamwidth to indicate uncorrelation between samples [U1aby, et. al. 1982]. Thus, there
are approximately 2;8 or 35 independent surfaces in the image. The nonnalized standard
deviation of the calculated scattering coefficients are therefore ),.. = 0.17 [UJaby, et.
0/ 35
a1.1982] giving an RMS error in the scattering coefficients of±0.7dB.
1.2r~
C O.B
.2
N (J)
t: 0.6 i ~ 0.4
0.2
oL~~~
o 20 40 60
lags (crosswind)
Figure 3.10 The Crosswind Autocorrelation Function
33
..'..... ". ")
..:1
.f .....
'l
.
:~ .....
oil
2.5
2
~ E 1.5 wi
:s ~
~~ 1
O.S
0
0
Spectral Estimation
139.626 279.253
k(radlm)
418.879
Figure 3.11 Roughness Spectrum Estimate
The surface power spectral density used in the smallperturbation scattering
calculations was estimated using the periodogram calculated from the independent upwind
surface profiles
[Bendat and Piersol, 1984]. The 496 point upwind slices of the surface profile were
extended to 1024 points by zero padding and converted to the frequency domain using an
FFT. The individual spectral Jines were then squared and normalized to the number of
points in the FFT (1024). The spectral lines at a given wave number were then averaged
across the independent surfaces to yield the final spectral estimate .. Again because
approximately 35 independent surfaces were used the RMS error in the spectral estimate
is about ± 17% (±0.7dB). Figure 3.11 shows the calculated periodogram. And Figure
3.12 shows the dB plot of the periodogram, along with the plots of several powerlaw
spectra from k 3 to k4 dependencies.
34
I
" !l
:a
'....
'.." ,. I
)
.~
As mentioned before, the power law spectrum is an estimate of the saturated range
of the power spectral density for the ocean surface. It has the form
W(k) = Wok<1.
values ofa ranging from 3 to 4 have been proposed. Expressing (38) in dB yields
(38)
(39)
.Figure 3.12 shows that a = 3.5 gives a good prediction of the measured power spectral
density in the saturated (high wave number) range.
0
15
iii' :E. 30
'[45 01
~ ~ .aJ !
75
1lO
10 100 1000 100: 1 Estimated  k"3  k"3.S  k~4 I
Figure 3.12 dB Plot ofPeriodograrn Estimate
35
.,.
'"
CHAPTER 4
THE RESULTS OBTAINED WITH
EACH SCATfERINGMETHOD
INTRODUCTION
The scattering from the processed surfaces was calculated at frequencies ranging
from 8GHz to 58GHz and incidence angles ranging from 5° to 89°. This frequency range
was selected to test the validity of the small perturbation and Kirchoff approximation
scattering theories for different levels of surface roughness. As the frequency increases
the illumination wavelength decreases, and the surface displacements become electrically
larger. The results from the periodic surface moment method, small perturbation, and
Kirchoff theoretical scattering models, are compared at both horizontal and vertical
polarizations in this section.
Moment Method Parameters
Several physical parameters of the surface had to be varied with the frequency and
incidence angle in order to meet the conditions summarized in chapter two. In particular,
the length of the scattering surface was truncated from the full 18 cm to meet equations
36
,I
I 'I
.......
...
(257) and (258). The number ofterrns in the infinite series of equations (235) and
(246) that were exactly evaluated was automatically determined from equations (259) ,
(261) and (262).
The number of basis functions used in the moment method description of the
surface was changed with frequency. Axline and Fung, (I978). showed that
approximately 10 basis functions are required per wavelength along the modeled surface
to yield an accurate MM: prediction of the scattering. Use of more basis functions would
result in unneeded computational expense, while use of fewer would yield to inaccurate
results. Once the length and corresponding number of basis functions were calculated, the
height profile was fesampled from a cubic spline fit of the surface. This resampled data
was used in the periodic surface calculations. The actual lengths of the modeled surface
used at each frequency and the associated numbers of basis functions are shown in table
4.1. Note that fewer basis functions are used as frequency decreases due to the longer
wavelength. Chen and West showed that a surface length of 5 wavelenbrths (50 basis
functions) is sufficient for accurate results at up to 89 degrees.
Results
Figures 4.1 through 4.6 show the calculated surface backscattering coefficient with
both horizontally and vertically polarized illumination at frequencies ranging from 8GHz
to 58GHz. The scattering coefficients calculated using the periodic surface moment
method, small perturbation model, and the Kirchoff approximation are shown. The RMS
37
,
.....
surface heights, expressed in wavelengths, corresponding to each frequency used are
summarized in table 4.2.
Table 4.1 Parameters Used for Moment Method Analysis
Frequency 8GHz 18GHz 28GHz 38GHz 48GHz 58GHz
9, L(m) andN L(m) andN L(m)andN L(m) andN L(m) andN L(m) and N
5 O.22m O.19m O.19m O.18m O.2m O.19m
70 segments 13 9 segments 208 segments 277 segments 381 segments 450 segments
10 O.22m 0.19m O.19m a.18m O.18m a.18m
71 segments 141 segments 211 segments 281 segments 351 segments 422 segments I
t
15 O.19m O.18m O.18m O.18m 0.18m a.18m
; 60 segments 132 segments 204 segments 276 segments 348 segments 423 segments I
20 D.2m O.18m 0.18m 0.18m 0.18m O.18m
66 segments 131 segments 206 segments 281 segments 355 segments 421 segments
25 0.2m O.18m 0.18m 0.18m 0.18m 0.18m
63 segments 134 segments 204 segments 275 segments 353 segments 422 segments
30 0.19m 0.18m O.19m I 0.18m O.ISm a.18m
63 segments 132 segments 208 segments 278 segments 347 segm.ents 423 segments
35 O.2m 0.19m 0.18m O.ISm O.ISm 0.18m
: 64 segments 135 segments 205 segments 275 segments 352 segments 422 segments
40 a.19m 0.19m O.18m 0.I8m 0.18m 0.18m
61 segments 135 segments 208 segments 275 segments 34S segments 421 segments
50 0.19m O.l9m 0.18m 0.18m 0.1801 0.1801
61 segments 135 segments 208 segments 275 segments 348 segments . 423 segments
60 0.19m 0.18m a.19m O.I8m a .18m a.18m
63 segments 132 segments 208 segments 278 segments 347 segments 423 segments
65 a.2m a.18m O.18m a.18m a.18m O.18m
63 segments 134 segments 2a4 segments 275 segments 353 segments 423 segments
70 0.2m a.lSm a.18m O.18m O.18m a.18m
66 segments 131 segments 206 segments 281 segments 355 segments 421 segments
75 O.19m a.I8m O.ISm · 0.18m a .18m a.lSm
60 segments 132 segments 204 segments 277 segments 34S segments 420 segments
78 O.18m a.18m O.18m O.ISm 0.18m 0.18m
6a segments 133 segments 207 segments 281 segments 355 segments 426 segments
80 0.22m O.19m a.19m 0.18m a.18m a.18m
71 segments . 141 segments 211 segments 281 segments 351 segments 422 segments
82 D.2m O.l8~1 a.19m O.19m a.18m a.19m
66 segments 131 segments 218 segments 283 segments 349 segments 436 segments
38
85
87
88
89
8GHz
18GHZ
28GHz
38GHz
48GHz
58GHz
Table 4.1 (cont)
0.22m O.19m O.19m O.18m 0.2m O.19m
70 segments 139 segments 208 segments 277 segments 381 segments 450 segments
0.36m O.24m O.21m O.19m O.21m O.2m
115 segments 173 segments 230 segments 288 segments 402 segments 460 segments
O.27m O.24m O.23m O.23m O.22m O.19m
87 segments 173 segments 259 segments 345 segments 431 segments 431 segments
O.54m O.24m O.31m O.23m O.27m O.22m
172 segments 172 segments 344 segments 344 segments 516 segments 516 segments
Table 4.2 RMS Surface Height in Wavelengths
Frequency RMS Surface Height (in wavelengths)
0.0857744
0.1929924
0.3002104
0.4074284
0.5146464
0.6218644
39
I
t • • ..
'4
)
.......
...
20
• • ~
m 0 ;; ..
~
J!. ... c:
Q.) 20 :0
~
Q.)
8 ~ SPM HH
0)
.I. .i.:. SPMW
Q.)
== 60 ~ Moment Method W i L~ Moment Method HH
co
.D 80 OKA
100~~~L~~
o 20 40 60 80 100 incident anale (dearees) ....
Figure 4.1 Backscattering coefficients predicted at 8GHz by the small perturbation ~
)
l
method (SPM), periodic surface moment method (pSMM) and Kirchoff
approximation (KA) at vertical (VV) and horizontal (DB) polarization. Note that ...
KA yields identical coefficients at HH and VV polarizations. polarizations.
20
CD 0
~ c:
Q.) 20
:~
:s:::
Q)
8 40   SPM HH 0)
c:
.~ SPMW
t:: 60 ~ ~ Moment Method VV
uco !::.~ Moment Method HH
.D 80 oKA
100
0 20 40 60 80 100
incident angle (degrees)
Figure 4.2 Same as 4.1 but at 18GHz
40
....
co
~ c:
Q)
:~ Q)
8
C'l
c:
";;:
Q)
t:: j
u ro
.0
80
incident angle (degrees)
Figure 4.3 same as 4.1 but at 28GHz
20
~ ""
0
20
40   SPM HH
SPMW
~o • Moment Method W
C, Moment Method HH
80 OKA
_100L~L~L~
o 20 40 60 80 100
incident angle (degrees)
Figure 4.4 same as 4.1 but at 38GHz
41
• )
l .. 4
.4.
.. )
l
20
• fi) 0
"0 c:::
:QQ) ·20
:t::
Q)
8 40
OJ c:::
';:=
Q)
.. , "" ~~£ ~ ':,,: ~oooM~
'  t: i_~ .
~ _. SPM HH
::~ .... , A
~Nt~.
SPMW
:,,"\
:".\:
t GO ~ (,) ro
.D 80
r '''\
A Moment Method W r~;.
~J Moment Method HH
A: ~ ... ",
I
KA
,J .,
0 ;:. ...:
·100 I I I
o 20 40 60 80 10C
incident angle (degrees)
Figure 4.5 same as 4.1 but at 48GHz
20
S
as 0
~ c
Q) ·20 '0
!i:
Q)
8 40   SPM HH OJ c:::
';:= SPMW Q)
:t:: GO !S A Moment Method W
: .. ~
~ ':') Moment Method H H (,) ro
.D 80 .
0 KA ;: .. :
100
0 20 40 60 80 10:
incident angle (degrees)
Figure 4.6 same as 4.1 but at 58GHz
42
Discussion
At 8GHz, the surface standard deviation is 0.086", meeting the smallness criteria
for SPM to be valid at moderate incidence angles given in chapter two[Ulaby et.a!' 1982].
This is confirmed in Figure 4.1 where SPM and PSMM agree to within 3 dB at all angles
examined above 20° for horizontal polarization and from 20° to 87° incidence at vertical
polarization. Above 87° the M:MVV scattering drops rapidly, and is 12dB below the
SPM predictions at 89°. The Kirchoff approximation is accurate to within 2dB at all
incidence angles below 20°. These results are similar to those found by Chen and West
(1995) in their investigation of scattering from smallscale rough surfaces that had
Gaussianweighted roughness spectra ..
The operating frequency was increased to 18GHz in Figure 4.2, giving a surface
standard deviation ofO.193ft... The roll off of the SPMVV scattering now occurs at a
smaller incidence angle of 82° most likely due to the increased self shadowing resulting
from the greater electromagnetic roughness. The horizontal results still proved accurate
to 89° incidence. KA is accurate to 25° at both polarizations at this frequency, slightly
higher than at 8GHz, and again, due to the increased surface roughness.
The general trend of the PS1'vfMVV scattering rolloff beginning at lower
incidence and KA scattering being accurate to higher incidence with increasing frequency
is continued in Figures 4.3 through 4.6. At 58GHz in Figure 4.6, the surface standard
deviation is 0.629". Here the strong rolloffin the PSMMVV occurs at about 75°
incidence, reaching a maximum error of more than 25dB at 89°. SPMIDI gives excellent
43
... .•..
a 4
)
i
agreement with the corresponding MM results through 88° and overpredicts the scattering
by only 7dB at 89° incidence. This result is not in agreement with the largescale
roughness results found by Chen and West (1995). This disagreement arises from the fact
that Chen and West used a Gaussian power spectral density to describe the surface
roughness, which includes no Braggresonant energy at high frequencies/large roughness.
KA is still valid at this frequency up to about 40°. Note that PSMM yields similar
scattering coefficients at the two polarizations at the smallest incidence angles at aU
frequencies, and the maximum angle at which they agree increases with increasing
frequency. This allows KA, which includes no polarization dependence, to accurately
predict the scattering for both polarizations up to these angles.
As discussed earlier, the power spectral density for the experimentally measured
surfaces is in agreement with the powerlaw spectra sometimes used to describe the ocean
surface. Thus these results are quite different from the scattering from the
Gaussianweighted spectrum surfaces presented by West and Chen(1995), but similar to
that obtained by West et. aI. (1995) when a power law surface was used.
44
....
oil
• )
t
l
CHAPTER FIVE
CONCLUSIONS
The validity of the small perturbation and Kirchoff approximation models in
predicting electromagnetic scattering from rough water surfaces has been examined. The
scattering predicted by the models was directly compared with the numerically calculated
"exact" scattering from sample water surfaces. Use ofa periodicsurface moment method
for scattering calculations allowed the comparison at incidence angles up to 89°,
considerably higher than that possible using the standard windowedillumination moment
method.
Often the greatest limitation of numerical studies such as this, is the method used
to represent the scattering surface. The statistics of the roughness of open water surfaces
are not well known and accurate direct measurements of the roughness with resolution
fine enough to resolve the small Braggresonant ripple waves do not exist. Thus surfaces
have typically been generated from idealized roughness spectra that are at best only rough
approximations of the actual surface spectra.
In this work, the scattering surfaces were derived from direct measurements of the
upwind/downwind slopes of windgenerated water surfaces in a circular wave tank. The
slopes were processed to yield several independent, one dimensionally rough scattering
surface to which the numerical scattering algorithm was directly applied. While the
45
I
I •• «
,•• • , . ..
I
I·i .I
surfaces are of course also not truly representative of the open sea surface, this approach
does allow the Braggresonant ripples to be resolved allowing an accurate representation
of a very important scattering mechanism with windgenerated water surfaces.
When the frequency was chosen so that the scattering surface roughness was
electromagnetically small(8GHz) the small perturbation theory was found to be accurate at
incidence angles up to at least 89° for horizontal polarization. At vertical polarization,
SPM was accurate to 87° and rolled off sharply at higher incidence. The incidence angle
at which this rolloff occurred reduced with increasing frequency down to about 75° at
58GHz, indicating that SPM is valid over a wider range of surface roughness and
incidence angles at horizontal polarization. The Kirchoff approximation was found to be
accurate at small and moderate grazing angles, with the highest angle of validity
increasing with frequency. A surprising result is that KA seems to accurately predict the
scattering for vertical polarization up to 85° at 58GHz, with no shadowing correction.
These results indicate that the approximate scattering models may be valid over a wider
range than previously thought.
46
I
REFERENCES
Axline R.M. and Fung AK., "Numerical Computation of Scattering from perfectly
Conducting Random Surface," IEEE Transactions on Antennas and Propagation Vol.
AP26 pp. 482488. 1978.
Balanis, C.A, Advanced Engineering Electromagnetics, New York, John Wiley and Sons
1989.
Balanis, C. A, Antenna Theory Analysis and Design, New York, John Wiley and Sons
1982.
•
Surfaces. Macmillan, New York, 1963.
•
~;
Beckman, P., and Spizzichino, A The Scattering of Electromagnetic Waves From Rough
Breipohl A.M., and Shanmugan K.S., Random Signals Detection, Estimation and Data
Analysis, New York, John Wiley and Sons 1988.
Broschat, S.L., "The phase perturbation approximation for rough surface scattering from a
PiersonMoskowitz sea surface," IEEE Transactions on Geoscience and Remote
Sensing., vol. 31, no 1 pp. 278283, January 1993.
Brown, Gary S./ "Backscattering from a Gaussiandistributed perfectly conducting rough
surface," IEEE Transactions on Antennas and Propogation., vol. AP26, no. 3, pp. 472 
481, May 1978.
47
Chen KS. and Fung AK "A Comparison ofBackscattering Models for Rough
Surfaces," IEEE Transactions on Geoscience and Remote Sensing. Vol. 33 no. 1 pp.
195200
Chen M.F., and Fung AK, "A Numerical Study of the Regions of Validity of the Kirchoff
and Small Perturbation Rough Surface Scattering Models," Radio Science, Vol. 23 No.
2 pp. 163  170. 1988.
Chen, Ruimin., "Numerical Investigation Of Electromagnetic Scattering From The Ocean
Surface At Extreme Grazing Angles," Diss., Oklahoma State University 1993.
Durden, S.L., and J. Vesecky., "A Numerical Study of the Separation Wavenumber in the
TwoScale Scattering Approximation, II IEEE Transactions on Geoscience and Remote
Sensing., voL 28, no. 2, pp. 271272, March 1990.
Fung, AK, Moore, RK., and Ulaby, P.T., Microwave Remote Sensing Volume 11,
reading, MA: Addison Wesley, 1982.
Guinard, N.W., "An Experimental Study of a Sea Clutter Model", Proceedings of the
IEEE, 58(4), 543550, 1970.
Hanington R, Field Computation by Moment of Methods, MacmiUan, New York, 1968.
Jackson n .R and Thorsos E.L., "The Validity of the Perturbation Approximation for
Rough Surface Scattering Using a Gaussian Roughness Spectrum," Journal of the
Acoustical Society of America. Vol. 86 No.3. pp. 261277. 1988.
Jahne, Bernd, and Klinke, Jochen, "Wave Number Spectra of Short Wind WavesLaboratory
Results and Extrapolation to the Ocean", Draft copy, June 22, 1994.
lahne, B. and Shultz H. "Calibration and Accuracy of Optical Slope and Height
48
Measurements for Short Wind Waves", SPIE Proceedings 1749, in press 1992.
Kim Y., Rodriguez E. and Durden S.L., "A Numerical Assessment of Rough Surface
Scattering Theories: Vertical Polarization," Radio Science Vol. 27 pp. 515527. 1992.
Kim Y., Rodriguez E. and Durden S.L., "A Numerical Assessment of Rough Surface
Scattering Theories: Horizontal Polarization," Radio Science Vol. 27 pp. 495513.
1992.
Rice S.O., "Reflection of Electromagnetic Waves from Slightly Rough Surfaces,"
Communications on pure and applied mathematics Vol. 4 pp. 351378. 1951 .
Sturm, J.M., "Iterative Methods For Solving Large Linear Systems In The Moment
Method Analysis Of Electromagnetic Scattering," Thesis, Oklahoma State University
1993.
Thacher, HT. "Algorithm 215 : SHANKS," Communications of the ACM, vol. 6, no. 11 ,
pp. 662, Nov. 1963
West. James c., and Ruimin Chen, "Analysis of scattering from rough surfaces at small
grazing angles using a periodicsurface moment method," IEEE Transactions on
Geoscience and Remote Sensing., vol 31, no 1 pp. 278283, September 1995.
West, James c., "Numerical Prediction of Shadowing in EM Scattering From a Rough
Ocean Wave at Grazing Incidence," currently under review
West, James C. et.al., "The SlightlyRough Facet Model in Ra.dar Imaging of the ocean
surface," International Journal of Remote sensing, vol. ] 1, no. 4, pp. 617637, April
1990.
West, James c., Ruimin Chen, and Brent O'leary., "Numerical Calculation of Scattering
49
From Rough Surfaces with PowerLaw Spectra Using a Periodic Surface Moment
Method", 1995 Progress in Electromagnetics Research Symposiu.in (PIERS 1995),
Seattle Washington, July 2428, 1995.
50
l
~
VITA
Brent Sean O'Leary
Candidate for the Degree of
Master of Science
Thesis: A COMPARISON OF SCA TfERING RESULTS OBTAINED WITH
THE PERIODIC SURFACE MOMENf METHOD AND SEVERAL
APPROXIMATE SCATTERING THEORIES USING WA VET ANK
DATA
Major Field: Electrical Engineering
Biographical:
Personal Data: Born in Wichita Kansas, May 26, 1970, the son of Mr. and Mrs.
James Edward O'Leary. Wife Tina, and one son, Chandler.
Education: Graduated from Hooker High School, Hooker Oklahoma, in May 1988;
Received Bachelor of Science Degree in Electrical Engineering from
Oklahoma State University in December, 1993; Completed the requirements
for the Master of Science Degree at Oklahoma State University in July, 1996.
Professional Experience: Electronic Systems Engineer, Lockheed Martin
VoughtSystems, June 1995 to present. Research Assistant, Department of
Electrical Engineering, Oklahoma State University, January 1993 to June 1995;
Professional Memberships: IEEE, NSPE, Eta Kappa Nu and Phi Kappa Phi